# Conformal Data for the O(3) Wilson-Fisher CFT from Fuzzy Sphere Realization of Quantum Rotor Model

Arjun Dey, Loïc Herviou, Christopher Mudry, Andreas M. Läuchli

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Arjun Dey, Loïc Herviou, Christopher Mudry, Andreas M. Läuchli. Conformal Data for the O(3) Wilson-Fisher CFT from Fuzzy Sphere Realization of Quantum Rotor Model. 2025. hal-05346784

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# Conformal Data for the O(3) Wilson-Fisher CFT from Fuzzy Sphere Realization of Quantum Rotor Model

Arjun Dey,¹,² Loic Herviou,³ Christopher Mudry,¹,² and Andreas Martin Läuchli¹,²

¹Laboratory for Theoretical and Computational Physics,
PSI Center for Scientific Computing, Theory and Data, 5232 Villigen PSI, Switzerland

²Institute of Physics, École Polytechnique Fédérale de Lausane (EPFL), 1015 Lausane, Switzerland

³Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France

We present a model for strongly interacting fermions with internal O(3) symmetry on the fuzzy sphere that (i) preserves the rotational symmetry of the fuzzy sphere and (ii) undergoes a quantum phase transition in the (2+1)-dimensional O(3) Wilson-Fisher universality class. Using exact diagonalization (ED) and density matrix renormalization group (DMRG), we locate the quantum critical point via conformal perturbation theory and obtain scaling dimensions from finite-size spectra. We identify 24 primary operators and determine some of their operator product expansion coefficients through first-order conformal perturbation theory. The results are benchmarked against conformal bootstrap and large quantum-number expansions and reveal a weakly irrelevant operator that plays a role in dimerized antiferromagnets. Our work establishes the fuzzy sphere as a general framework for quantitatively accessing conformal data in non-Abelian conformal field theories (CFTs).

**Introduction.**— CFTs in (d + 1)-dimensional space-time [(d + 1)D] are fully specified by their conformal data, scaling dimensions, and operator product expansion (OPE) coefficients, which govern universal critical behavior, and some measurable responses [1–7]. While (1+1)D CFTs [2, 8] enjoy remarkable theoretical completeness through Virasoro algebra [9] and integrability [10], the landscape of (2+1)D CFTs remains largely unexplored, presenting rich opportunities for discovering new universal physics and testing fundamental principles of quantum field theory.

Standard approaches have limitations when it comes to accessing detailed information about operator spectra and OPE data of (2+1)D CFTs. (i) Large-scale Monte Carlo simulations on lattices are effective for extracting critical exponents [11], but they face significant challenges when trying to resolve additional low-lying operators or obtain OPE coefficients. (ii) The conformal bootstrap has achieved remarkable precision for the Ising universality class [12, 13], providing tight bounds and detailed information for the lowest-lying operators. However, for theories with non-Abelian symmetries such as O(3), it typically yields constraints for only a limited set of low-lying operators, especially in higher-spin sectors, and becomes increasingly computationally intensive. Moreover, it does not offer a direct Hamiltonian realization. (iii) Analytic approaches often produce asymptotic series that require resummation and extrapolation to the physical dimension of space or the desired symmetry group, with growing uncertainties for spinning/higher-rank operators and specific OPE coefficients [14, 15]. (iv) Fuzzy sphere regularization [16–27] provides a direct Hamiltonian approach to studying (2+1)D CFTs through diagonalization of quantum systems on a sphere, offering access to both spectra and correlation functions while preserving full symmetries. A prior fuzzy sphere study of the (2+1)D CFT with internal

O(3) symmetry confirmed feasibility but was limited by Hilbert-space growth and identified only a limited number of operators [28].

Here, we introduce a fuzzy sphere realization (FZR) of a truncated quantum rotor model (TQRM) [7, 16] that retains the full internal O(3) and spatial SO(3) symmetry of the (2+1)-dimensional O(3) Wilson-Fisher universality class and is tractable to ED and DMRG. Building on the well-known correspondence between O(3) quantum rotors and O(3) quantum criticality [7], the idea is to use a conformal map to identify a compactification of two-dimensional space with the unit 2-sphere $S^2$ and to leverage the state-operator correspondence to extract conformal data. This is done by relating the finite-size gaps of the FZR of the TQRM to the scaling dimensions of the CFT operators. We identify 24 low-lying primary operators, with their scaling dimensions summarized in Tables I and II (Appendix A). Selected OPE coefficients are also determined. The full set of scaling dimensions and OPE values can be found in the Supplementary Material [29]. We benchmark our results with bootstrap results [30] and large-charge predictions [31, 32], wherever they are available.

**Quantum rotor model.**— A rigid rotor, a particle of mass $\mu$ constrained to move on the surface of a sphere of radius $R$, serves as the prototypical quantum model of rotational motion. Its energy is purely kinetic, arising from quantized angular momenta with moment of inertia $I = \mu R^2 > 0$. On a square lattice, an array of such rotors with nearest-neighbor ferromagnetic Heisenberg coupling $J > 0$ realizes an O(3) magnetic transition when the dimensionless coupling $IJ$ is tuned to quantum criticality [7].

The full two-dimensional interacting lattice Hamiltonian for O(3) quantum rotors reads $\hat{H}_{\infty} := \hat{H}_{\text{kin}} + \hat{H}_{\text{int}}$,
$$ \hat{H}_{\text{kin}} := \frac{1}{2I} \sum_{i=1}^{N} \hat{\mathbf{L}}_{i}^{2}, \qquad \hat{H}_{\text{int}} := -J \sum_{\langle ij \rangle} \hat{\mathbf{n}}_{i} \cdot \hat{\mathbf{n}}_{j}, \qquad (1a) $$

where the pair of operator-valued 3-component vectors ($\hat{\mathbf{L}}_i, \hat{\mathbf{n}}_i$) obey the equal-time O(3) rotor algebra

$$ \begin{aligned} [\hat{\mathbf{L}}_i^a, \hat{\mathbf{L}}_j^b] &:= i\hbar \epsilon^{abc} \hat{\mathbf{L}}_i^c \delta_{ij}, \\ [\hat{\mathbf{L}}_i^a, \hat{\mathbf{n}}_j^b] &:= i\hbar \epsilon^{abc} \hat{\mathbf{n}}_i^c \delta_{ij}, \quad a, b = 1, 2, 3 \equiv x, y, z, \quad (1b) \\ [\hat{\mathbf{n}}_i^a, \hat{\mathbf{n}}_j^b] &:= 0, \end{aligned} $$

with the local constraints

$$ \hat{\mathbf{L}}_i^2 = \hbar^2 l(l+1)\hat{\mathbb{I}}, \qquad \hat{\mathbf{n}}_i^2 := \hat{\mathbb{I}}. \qquad (1c) $$

Here, $l \in \mathbb{Z}_{\ge 0}$ fixes a global representation of each quantum rotor, $\langle ij \rangle$ are nearest-neighbors sites on the square lattice with $N$ the number of sites.

In the limit $IJ = \infty$, $\hat{H}_{\infty}$ simplifies to the classical O(3) ferromagnetic Heisenberg model

$$ \hat{H}_{\text{int}} = -J \sum_{\langle ij \rangle} \mathbf{n}_i \cdot \mathbf{n}_j, \qquad \mathbf{n}_i^2 = 1, \qquad (2) $$

with the ferromagnetic ground state $|\mathbf{n}_1, \dots, \mathbf{n}_N\rangle = |\mathbf{n}, \dots, \mathbf{n}\rangle$ whereby $\hat{\mathbf{n}}_i |\mathbf{n}_1, \dots, \mathbf{n}_N\rangle = \mathbf{n}_i |\mathbf{n}_1, \dots, \mathbf{n}_N\rangle$. In the limit $IJ = 0$, $\hat{H}_{\infty}$ simplifies to

$$ \hat{H}_{\text{kin}} = \sum_{i=1}^{N} \sum_{l=0}^{\infty} \sum_{m_i=-l}^{+l} \frac{\hbar^2 l(l+1)}{2I} |l, m_i\rangle \langle l, m_i| \quad (3) $$

with $|l, m_i\rangle$ the eigenstate of $\hat{\mathbf{L}}_i^2$ and $\hat{\mathbf{L}}_i^z$ with eigenvalues $\hbar^2 l(l+1)$ and $\hbar m_i$, respectively. The ground state of $\hat{H}_{\text{kin}}$ is the singlet state with $l=0$. It is believed that, in the thermodynamic limit $N \uparrow \infty$, there exists an unstable quantum critical point when $IJ \approx 1$ that separates the long-range ferromagnetically ordered phase controlled by the fixed point Hamiltonian $\hat{H}_{\text{int}}$ from the paramagnetic phase controlled by the fixed point Hamiltonian $\hat{H}_{\text{kin}}$. This quantum critical point is believed to belong to the $(2+1)$D O(3) Wilson-Fisher universality class.

Numerical simulations of a lattice Hamiltonian requires a Hilbert space of finite dimensionality. We therefore truncate each quantum rotor on the right-hand side of Eq. (3) at a maximum angular momenta $l_{\text{max}} = 1$, retaining all states $|l, m\rangle$ with $0 \le l \le l_{\text{max}}$ and $-l \le m \le l$. This procedure, by which $\hat{H}_{\infty}$ becomes $\hat{H}_{l_{\text{max}}}$, preserves the full O(3) symmetry, since each $l$ multiplet is kept intact [29]. Moreover, this procedure is not expected to change the topology of the phase diagram at vanishing temperature.

**Fuzzy sphere many-body electronic Hamiltonian.** — To access conformal data, we map the quantum rotor model

to a system of $N$ fermions with internal degrees of freedom, moving on a sphere [33] and projected to the lowest Landau level (LLL). In this limit, the noncommuting coordinate operators generate the fuzzy sphere algebra [34], thereby providing a natural short-distance cutoff (the magnetic length) that preserves rotational symmetry.

We focus on the case where each fermion has $F = 4$ flavors and the system resides at quarter filling. At quarter filling, strong interactions enforce (on average) single occupancy per LLL orbital, analogous to the large-$U$ limit of the Hubbard model.

The microscopic fuzzy sphere Hamiltonian that we shall study numerically is defined to be

$$ \hat{H}_{\text{fzs}} := \hat{P}_{\text{LLL}} \left( u \hat{H}_{\text{Hub}} - v \hat{H}_{\text{Heis}} + h \hat{H}_{\text{trans}} \right) \hat{P}_{\text{LLL}}. \quad (4) $$

Here, $\hat{P}_{\text{LLL}}$ denotes the projector to the LLL. The dimensionless couplings $\delta u \equiv u - u_c$, $\delta v \equiv v - v_c$, $\delta h \equiv h - h_c \in \mathbb{R}$ measure the deviations away from a putative quantum critical point $u_c, v_c, h_c$ in the $(2+1)$D O(3) Wilson-Fisher universality class. Hamiltonian $\hat{H}_{\text{Hub}}$ is a short-range two-body repulsive U(4)-flavor-symmetric density-density interaction between the fermions that implements the constraint that every LLL are singly occupied in the limit $u \uparrow \infty$, $v, h$ fixed. Hamiltonian $\hat{H}_{\text{Heis}}$ is a short-range two-body Heisenberg-like interaction that selects a ground-state that breaks spontaneously the internal SO(3) symmetry down to the subgroup SO(2) in the limit $v \uparrow \infty$, $u, h = 0$ fixed. Hamiltonian $\hat{H}_{\text{trans}}$ is a one-body term that breaks the U(4)-flavor-symmetry down to the subgroup U(1)×U(3) that selects a non-degenerate gapped singlet ground state in the limit $|h| \uparrow \infty$, $u, v$ fixed. The information about the quantum rotor is encoded inside $\hat{H}_{\text{Heis}}$ and $\hat{H}_{\text{trans}}$. The explicit representation of $\hat{H}_{\text{fzs}}$ can be found in the Supplementary Material [29].

**Symmetries.** — Hamiltonian (4) exhibits three symmetries: (a) spatial rotational symmetry SO(3), arising from conservation of angular momenta with the quantum number $L$; (b) internal SO(3) spin symmetry with the quantum number $S$; (c) and a $\mathbb{Z}_2$ symmetry corresponding to the conservation of the parity of the number of electrons occupying $p$ orbitals in internal spin space. The quantum number for the SO(3) × $\mathbb{Z}_2 \cong$ O(3) is denoted by $S^{\pm}$, where the superscript denotes the $\mathbb{Z}_2$ parity.

**Accessing quantum criticality through the state-operator correspondence.** — We first assume the existence of a quantum critical point $\delta u = \delta v = \delta h = 0$ at which $\hat{H}_{\text{fzs}}(R)$ exhibits the conformal invariance of the $(2+1)$D O(3) Wilson-Fisher universality class in the thermodynamic limit $R \propto \sqrt{N} \uparrow \infty$. If we denote eigenstates of $\hat{H}_{\text{fzs}}(R)$ by the label $o$ (see Table I), the state-operator correspondence [3] implies that the eigenergy $E_o(R)$ measured relative to that of the ground state $E_o(R)$
<table><caption>Table I. Summary of selected primary operators. S labels the internal SO(3) spin representation, L the SO(3) angular momenta on the fuzzy sphere, and ± the $\mathbb{Z}_2$ parity (improper part) of the internal O(3) symmetry, so that states are labeled by $S^±$ and L. The index I orders energy eigenstates within each symmetry sector. Scaling dimensions are extracted via ED and DMRG, and are compared to conformal bootstrap (CB); asterisks denote exact values, and parentheses indicate system size. The two bold-faced entries in the $\Delta$ (CB) column are used as input.</caption><thead><tr><th>S<sup>±</sup></th><th>L</th><th>I</th><th>o</th><th>Δ(ED)</th><th>Δ(DMRG)</th><th>Δ(CB)</th></tr></thead><tbody><tr><td>1<sup>-</sup></td><td>0</td><td>1</td><td>σ</td><td>0.51893 (12)</td><td>0.51893 (28)</td><td><strong>0.518936</strong> [30]</td></tr><tr><td>0<sup>+</sup></td><td>0</td><td>2</td><td>ε</td><td>1.56189 (12)</td><td>1.61781 (26)</td><td><strong>1.59488</strong> [30]</td></tr><tr><td>0<sup>+</sup></td><td>0</td><td>4</td><td>ε'</td><td>3.77279 (12)</td><td>3.80177 (14)</td><td>3.76680 [35]</td></tr><tr><td>0<sup>+</sup></td><td>2</td><td>1</td><td>T<sub>μν</sub></td><td>2.97749 (12)</td><td>3.01062 (20)</td><td>3*</td></tr><tr><td>1<sup>+</sup></td><td>1</td><td>1</td><td>j<sub>μ</sub></td><td>1.88253 (12)</td><td>1.97204 (26)</td><td>2*</td></tr><tr><td>2<sup>+</sup></td><td>0</td><td>1</td><td>t<sub>(2)</sub></td><td>1.25407 (12)</td><td>1.23767 (26)</td><td>1.20954 [30]</td></tr><tr><td>4<sup>+</sup></td><td>0</td><td>1</td><td>t<sub>(4)</sub></td><td>3.26858 (11)</td><td>3.16459 (26)</td><td>2.99056 [30]</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Figure 1. Panel (a) shows the dependence on $h$ of the relevant coupling $g_\epsilon(h)$ for a given $R \propto \sqrt{N}$. The critical point $h_c$ is the root to $g_\epsilon(h) = 0$ after the thermodynamic limit $N \uparrow \infty$ has been taken. Panel (b) shows the dependence of the root to $g_\epsilon(h) = 0$ as a function of $1/\sqrt{N} \propto 1/R$. Fitting this dependence with an appropriate power law and using the first irrelevant anomalous dimension 0.7688 [35], we extract the critical value $h_c = 14.992 \pm 0.018$ and the critical exponent $1/\nu = 1.3908 \pm 0.0852$, which is in close agreement with the known value 1.406 [36].</figcaption></figure>

scales like

$$ \delta E_o(R) \equiv E_o(R) - E_0(R) \sim \frac{c}{R} \Delta_o \quad (\hbar = 1), \quad (5) $$

in the limit $R \propto \sqrt{N} \uparrow \infty$. Here, the radius $R$ of the fuzzy sphere is interpreted as the radius of compactification in the radial quantization of a CFT, $\Delta_o$ the scaling dimension of the CFT operator labeled by o and c the system dependent speed of “light” in the CFT.

If we switch on a small perturbation $\delta h = h - h_c \neq 0$ in $\hat{H}_{fzs}(R)$ holding $\delta u = \delta v = 0$ fixed, the energy shift to first-order in perturbation theory is captured by adding the single relevant scalar operator $\epsilon$ (see Table I) with the coupling $g_\epsilon(h)$, together with irrelevant operators to the (2+1)D O(3) Wilson-Fisher critical point. Motivated by

first-order conformal perturbation theory (CPT) [37, 38], we make the scaling Ansatz

$$ \delta E_o(R, h) = \frac{c}{R} \Delta_o(R) + g_\epsilon(h) f_{o\epsilon o}(R). \quad (6) $$

Here $\Delta_o(R) = \Delta_o + \mathcal{O}(R^{-\omega})$ deviate from its value at quantum criticality due to the presence of the leading irrelevant operator perturbing the CFT with the scaling exponent $\omega > 0$. The value of $h_c$ at the quantum critical point is identified by extrapolating the value of the root to $g_\epsilon(h) = 0$ to its limiting value when $R \uparrow \infty$. The interpretation of $f_{o\epsilon o}(R) = f_{o\epsilon o} + \mathcal{O}(R^{-\omega})$ in the limit $R \uparrow \infty$ is the following. (i) It is the (universal) operator product expansion coefficient corresponding to fusing operators o and $\epsilon$ into o, if o labels a scalar primary field. (ii) It is proportional to $f_{o_p, \epsilon o_p}(R)$ with the propotionality constant a known function of $\Delta_{o_p}$ and $\Delta_\epsilon$ if o labels the descendant of a primary field labeled by $o_p$ [37, 38]. (iii) Finally, it is a combination of operator-product expansion coefficients [38] if o labels a field with non-vanishing conformal spin.

Our strategy to establish the existence of the (2+1)D O(3) Wilson-Fisher critical point from exact diagonalization of $\hat{H}_{fzs}(R)$ proceeds in two steps. First, we choose two eigenstates (CFT operators) of $\hat{H}_{fzs}(R)$ labeled by $o_1$ and $o_2$, for which estimates of $\Delta_{o_i}$ and $f_{o_i \epsilon o_i}$ with $i = 1, 2$ at the (2+1)D O(3) Wilson-Fisher critical point are known, to solve for $c$ and $g_\epsilon(h)$ from the pair of equations (6) they generate. Second, for any eigenstate (CFT operator) of $\hat{H}_{fzs}(R)$ labeled by o, we extract $\Delta_o(R)$ by using the extracted value of $c$ in Eq. (6) with $g_\epsilon(h) = 0$ and then extract $f_{o\epsilon o}$ from the dependence on $R$ of Eq. (6) with $g_\epsilon(h) \neq 0$.

Results for $c$ and $h_c$. — To determine $c$ and $g_\epsilon(h)$, we choose the pair of eigenstates labeled by $o_1 \equiv \sigma$ and $o_2 \equiv \partial_\mu \sigma$, where $\sigma$ labels the lowest eigenenergy of $\hat{H}_{fzs}(R)$ with the quantum numbers $S = 1^-$ and $L = 0$, while $\partial_\mu \sigma$ labels the lowest eigenenergy of $\hat{H}_{fzs}(R)$ with the quantum numbers $S = 1^-$, $L = 1$. In CFT jargon, $\sigma$ labels the most relevant primary field and $\partial_\mu \sigma$ is its first descendent. We insert in Eq. (6) $o_1 \equiv \sigma$ and $o_2 \equiv \partial_\mu \sigma$, the conformal bootstrap (CB) estimates $\Delta_\sigma = 0.518936$ from Table I and $\Delta_{\partial_\mu \sigma} = 1 + \Delta_\sigma = 1.518936$ together with the CB estimates for the OPE coefficient $f_{\sigma\epsilon\sigma} = 0.525$ and $f_{\partial_\mu\sigma\epsilon\partial_\mu\sigma} = f_{\sigma\epsilon\sigma} A_{\sigma,\epsilon}$ with $A_{\sigma,\epsilon} = 1 + [\Delta_\epsilon(\Delta_\epsilon - 3)/(6\Delta_\sigma)]$ and $\Delta_\epsilon$ given by its CB value quoted in Table I. The factor $A_{\sigma,\epsilon}$ accounts for the fact that the perturbation affects the first descendent differently from the primary [37, 38]. Solving these equations, we obtain the speed of light $c \sim 0.0122$ and the coupling $g_\epsilon(h)$. We then solve for the root of $g_\epsilon(h) = 0$ for each value of $N$, to obtain the finite-size critical point as shown in Fig. 1(a). By extrapolating the root of $g_\epsilon(h) = 0$ to $N \uparrow \infty$ as shown in Fig. 1(b), we obtain the critical point of the system as $h_c = 14.992$.
<figure><img src="image_4.png" /></figure>

Figure 2. Dependence on $R^{-\omega}$ of the dimensionless number $\Delta_o(R)$ defined by the scaling Ansatz (6) with $h$ chosen to be the root of $g_{\varepsilon}(h) = 0$ for each value of $R$. Here, the value $\omega \equiv \Delta_{\varepsilon'} - 3 \approx 0.7668$ is deduced from the CB estimate for the scaling dimension $\Delta_{\varepsilon'}$ of the leading irrelevant CFT perturbation (see Table I). The limiting value $\Delta_o$ of $\Delta_o(R)$ as $R \uparrow \infty$ is interpreted as the scaling dimension of the CFT operator labeled by $o$. The choice of the eigenstates labeled by $o$ is organized by $S^{\pm}$ sectors across panels; and by angular momenta $L$ within each panel. Dashed lines indicate conformal bootstrap predictions with the color code green for primary operators and red for descendants. Solid lines represent exact CFT values as applicable to the stress-energy tensor and the conserved Noether current (using the same color code). Operator names, such as $\varepsilon$ and $\varepsilon_{\mu}$, are indicated in red boxes. Only selected sectors are shown here; the complete spectra are available in the Supplementary Material [29]. Panel (a) displays the $S = 0^{+}$ sector, panel (b) the $S = 1^{+}$ sector, panel (c) the $S = 1^{-}$ sector, and panel (d) the $S = 2^{+}$ sector. Note that, except for the $\sigma$ and $\partial_{\mu}\sigma$ levels used to determine $c$ and $h_c$, all the plotted values of $\Delta_o(R)$ as $R \uparrow \infty$ are parameter-free outputs.

**Results for scaling dimensions.**— The dependence on $R^{3-\Delta_{\varepsilon'}}$ of the dimensionless number $\Delta_o(R)$ defined by the scaling Ansatz (6) with $h$ chosen to be the root of $g_{\varepsilon}(h) = 0$ for each value of $R$ is shown in Fig. 2 for a selection of eigenstates of $\hat{H}_{fs}(R)$ obtained from ED and DMRG. Here, $\Delta_{\varepsilon'}$ is the scaling dimension of the leading irrelevant perturbation, which is estimated to be 3.7668 by CB. Our estimates for the scaling dimensions $\Delta_o$, obtained from the largest available ED and DMRG system sizes rather than from an explicit extrapolation to $R \uparrow \infty$, are reported in Table I when $o$ labels a primary operator whose scaling dimension is known either from the conformal bootstrap or exactly, and in Table II (Appendix A)

otherwise. A systematic list for the values of the scaling dimensions for all symmetry sectors can be found in the supplementary material [29].

Figure 2 reveals signatures of emergent conformal symmetry that includes the expected descendant structure, the stress energy tensor, and the conserved SO(3) Noether currents [16].

The CB value of the scaling dimension $\Delta_{\varepsilon}$ for the primary operator $\varepsilon$ corresponding to the quantum numbers $S = 0^{+}$ and $L = 0$ is consistent with its ED lower bound and DMRG upper bound from Table I, respectively. The exact value of the scaling dimension $\Delta_{T_{\mu\nu}} = 3$ for the stress-energy tensor $T_{\mu\nu}$ corresponding to the quantum
numbers $S = 0^+$ and $L = 2$ is consistent with the ED lower bound and the DMRG upper bound from Table I. The exact value of the scaling dimension $\Delta_{j_\mu} = 2$ for the conserved SO(3) Noether current $j_\mu$ corresponding to the quantum numbers $S = 1^+$, $L = 1$, that reflects the internal SO(3) symmetry is also consistent with the ED and DMRG estimates from Table I. In addition, the CB predictions for the scaling dimensions of the rank-2 $t_{(2)}$ and rank-4 $t_{(4)}$ O(3) primaries are consistent with the ED and DMRG estimates from Table I. Finally, according to Fig. 2 and Table II (Appendix A), there is no CB prediction for the scaling dimension $\Delta_{\sigma_{(3)}} \approx 2.124$ where $\sigma_{(3)}$ corresponds to the quantum numbers $S = 3^-$ [rank 3 O(3) pseudotensor] and $L = 0$ [SO(3) scalar].

In addition, a variety of irrelevant primaries appear, including dangerously irrelevant [39] ones. Among these, the most significant are $\epsilon'$, a scalar under both O(3) and SO(3) [see Fig. 2(a)], and $\epsilon_\mu$ a pseudoscalar under O(3) and a pseudovector under SO(3) [see Fig. 2(b)]. These operators govern leading corrections to scaling in lattice simulations [40, 41]. Notably, $\epsilon_\mu$ provides insight into a long-standing discrepancy in the critical exponents of columnar versus staggered dimerized antiferromagnets [40–43], once misattributed to different universality classes [44]. Our finite-size extrapolation using a single power law with exponent 0.7668 yields the scaling dimension $\Delta \approx 3.931$, in reasonable agreement with the five-loop $\epsilon$-expansion result of 3.78796 [45]. The identification of this operator as a three-body term with two derivatives and dimension 3.5 in the free O(3) theory further corroborates predictions in the literature [40].

At last, the spectrum of scaling exponent that we found numerically is compared with those obtained from the large-$S$ expansion [see Refs. [29, 31, 32] and Appendix B], i.e., a Laurent expansion in powers of $1/S$ of the scaling dimensions of some low-lying primaries in the internal SO(3) symmetry sector labeled by the quantum number $S \gg 1$. Our data agree with this large $S$ expansion, in particular for the low-$L$ primaries. They are consistent with the existence of both a gapless, linearly dispersing phonon mode and of a dispersive gapped Goldstone mode.

**Results for OPE coefficients.**— We extract $f_{o\epsilon\epsilon}(R)$ in the scaling Ansatz (6) with the help of

$$f_{o\epsilon\epsilon}(R) = \left. \frac{\partial \delta E_o(R)}{\partial g_{\epsilon}(R)} \right|_{g_{\epsilon}(R)=0} . \qquad (7)$$

In Fig. 3, we plot the dependence on $R^{-\omega}$ of $f_{o\epsilon\epsilon}(R)$ for some of these OPE coefficients and we compare their limiting values as $R \uparrow \infty$ to the available CB data. Other OPE coefficients are plotted and tabulated in the Supplementary Material [29].

**Discussion and outlook.**— We have mapped the square-lattice quantum rotor model, whose Hamiltonian exhibits a quantum critical point in the (2+1)-dimensional O(3)

<figure><img src="image_5.png" /><figcaption>Figure 3. Dependence on $R^{-\omega}$ with $\omega \equiv \Delta_{\epsilon'} - 3 \approx 0.7668$ of the coefficient $f_{o\epsilon\epsilon}(R) = f_{o\epsilon\epsilon} + \mathcal{O}(R^{-\omega})$ defined by Eq. (7) with $o$ chosen in the $S = 0^\pm$ symmetry sectors. Solid lines denote CB estimates. Dashed red lines include CPT-derived descendant factors applied to the bootstrap data [37]. The states (CFT operators) selected by $o$ include $\epsilon, \epsilon_\mu, T_{\mu\nu}$.</figcaption></figure>

Wilson-Fisher universality class, onto the fuzzy sphere Hamiltonian (4) by placing interacting fermions with four internal flavors on the sphere. Our method remains computationally tractable using ED and DMRG even for a maximum of $N = 26$ interacting fermions and provides evidences for an emergent conformal symmetry consistent with the (2+1)-dimensional O(3) Wilson-Fisher universality class. We identify 24 primary operators labeled by the index $o$ in Table I and Table II (Appendix A) and compute their scaling dimensions $\Delta_o$ as well as their diagonal OPE coefficients $f_{o\epsilon\epsilon}$. The values that we find agrees with published ones and confirm predictions from the large-$S$ expansion [31, 32, 46] (see Appendix B).

While analyzing the (2+1)-dimensional O(3) Wilson-Fisher CFT, we identified a weakly irrelevant operator $\epsilon_\mu$ relevant to the criticality of staggered dimerized antiferromagnets. This operator transforms as a pseudoscalar under O(3) and a pseudovector under Lorentz [SO(3)] symmetry, originating from a cubic two-derivative term in the free theory, reminiscent of the topological $\theta$-term. Its behavior in the presence of a boundary remains an intriguing question.

Our framework naturally generalizes to the study of O(N) Wilson-Fisher quantum criticality, with explicit Hamiltonian constructions provided in the supplementary material [29]. Furthermore, it enables systematic exploration of models exhibiting richer internal symmetries, such as $O(n_1) \times O(n_2)$ [47]. This versatility grants access to both Wilson-Fisher and biconal fixed points within a unified approach. By introducing anisotropic terms that break $O(n)$ symmetry down to a discrete hypercubic subgroup, our method also allows investigation
of the stable cubic universality class for $n \ge 3$.

*Acknowledgments.*— The authors thank F Mila, Slava Rychkov and Gabriele Cuomo for useful discussions. LH acknowledges the Tremplin funding from CNRS Physique. AD acknowledges useful discussions with João Penedones, Johan Henriksson and Giuseppe Mussardo. AD thanks IHES, Paris for hospitality, where these results were first presented by AD at the workshop "Fuzzy Sphere Meets Conformal Bootstrap", June 2-6, 2025.

*Note added.*— While finishing this article, we became aware of an independent related work [48].

## Appendix A: Primary operators

Table II presents the primary operators whose scaling dimensions we have computed, for which conformal bootstrap results are not currently available.

<table><caption>Table II. Selected primary operators with the same conventions as in Table I.</caption><thead><tr><th>S</th><th>L</th><th>I</th><th>o</th><th>Δ(ED)</th><th>Δ(DMRG)</th></tr></thead><tbody><tr><td>0<sup>-</sup></td><td>1</td><td>1</td><td>ε<sub>μ</sub></td><td>2.96067 (12)</td><td>3.2396 (18)</td></tr><tr><td>1<sup>+</sup></td><td>1</td><td>3</td><td>φ<sub>μ</sub></td><td>3.70828 (12)</td><td>3.75703 (18)</td></tr><tr><td>1<sup>-</sup></td><td>0</td><td>3</td><td>σ'</td><td>3.87329 (12)</td><td>-</td></tr><tr><td>1<sup>-</sup></td><td>1</td><td>2</td><td>σ<sub>μ</sub></td><td>2.87055 (12)</td><td>3.04373 (26)</td></tr><tr><td>2<sup>+</sup></td><td>0</td><td>3</td><td>t'<sub>(2)</sub></td><td>3.67128 (12)</td><td>3.68121 (16)</td></tr><tr><td>2<sup>+</sup></td><td>0</td><td>4</td><td>t''<sub>(2)</sub></td><td>4.31261 (12)</td><td>-</td></tr><tr><td>2<sup>+</sup></td><td>2</td><td>1</td><td>t'<sub>(2)μν</sub></td><td>3.02538 (12)</td><td>3.02825 (26)</td></tr><tr><td>2<sup>-</sup></td><td>1</td><td>1</td><td>t<sub>(2)μ</sub></td><td>2.68970 (12)</td><td>2.78187 (26)</td></tr><tr><td>2<sup>-</sup></td><td>2</td><td>1</td><td>t'<sub>(2)μν</sub></td><td>3.29489 (12)</td><td>3.5341 (24)</td></tr><tr><td>3<sup>+</sup></td><td>1</td><td>1</td><td>χ<sub>(3)μ</sub></td><td>3.68769 (12)</td><td>3.75082 (26)</td></tr><tr><td>3<sup>+</sup></td><td>2</td><td>1</td><td>χ<sub>(3)μν</sub></td><td>4.22326 (12)</td><td>-</td></tr><tr><td>3<sup>-</sup></td><td>0</td><td>1</td><td>σ<sub>(3)</sub></td><td>2.17310 (12)</td><td>2.12486 (26)</td></tr><tr><td>3<sup>-</sup></td><td>0</td><td>3</td><td></td><td>4.88895 (11)</td><td>-</td></tr><tr><td>3<sup>-</sup></td><td>2</td><td>1</td><td>σ<sub>(3)μν</sub></td><td>4.02221 (12)</td><td>3.9825 (24)</td></tr><tr><td>4<sup>+</sup></td><td>0</td><td>3</td><td></td><td>6.25369 (11)</td><td>-</td></tr><tr><td>4<sup>+</sup></td><td>2</td><td>1</td><td>t<sub>(4)μν</sub></td><td>5.17757 (11)</td><td>-</td></tr><tr><td>4<sup>-</sup></td><td>1</td><td>1</td><td></td><td>4.85139 (11)</td><td>4.87182 (26)</td></tr><tr><td>4<sup>-</sup></td><td>2</td><td>1</td><td></td><td>5.29724 (11)</td><td>-</td></tr></tbody></table>

## Appendix B: Large-$S$ expansion

The large-$S$ expansion provides a controlled semiclassical framework for computing the scaling dimensions of local operators in a CFT with internal O(3) symmetry when these local operators transform according to the $2S + 1$-dimensional irreducible representation of O(3). In the sector of the CFT with $S \gg 1$, some scaling dimensions can be associated with the dispersions of two weakly interacting fields associated to the explicit symmetry breaking of the internal symmetry SO(3) down to

<figure><img src="image_6.png" /><figcaption>Figure 4. (a) For each quantum number $S = 1, 2, \dots$, two scaling dimensions are reported. The lowest one corresponds to the CFT state in the symmetry sector with the spin quantum number $S^{\kappa(S)} = 1^{-}, 2^{+}, \dots$ with $\kappa(S) = (-1)^S$ for the internal symmetry group O(3) = $\mathbb{Z}_2 \times \text{SO}(3)$ and the angular momenta $L = 0$. The larger one corresponds to the CFT state with the quantum numbers $S^{\kappa(S+1)} = 1^{+}, 2^{-}, \dots$ and $L = 1$. The scaling dimensions obtained from ED and DMRG of $\hat{H}_{\text{fzs}}(R)$ and reported as $+(\times)$ for $\kappa(S) = +(-)$. For fuzzy sphere results, number of electrons is 26 (DMRG) for $S = 1, 2, 3, 4$ and 12 (ED) for $S = 5, 6, 7$. The scaling dimensions obtained from fitting the large-$S$ expansions (B1) and (B2) with bootstrap data are reported as $\bigcirc(\square)$ for $\kappa(S) = +(-)$. (b) Scaling dimensions as a function of $R^{-\omega}$ with $\omega \approx 0.7668$ for phonon primaries and their descendants obtained from ED of $\hat{H}_{\text{fzs}}(R)$ in the symmetry sector $S = 2^{+}$. The green lines report the scaling dimensions obtained from fitting the large-$S$ expansion (B3) with bootstrap data.</figcaption></figure>

SO(2). One is massless and called a phonon mode. The other is gapful and called a gapped Goldstone mode.

Three key predictions from the large-$S$ expansion are the following when spacetime is three dimensional.

**1. Lowest scalar primaries.** The scaling dimension of the lowest scalar ($L = 0$) operator in each $S \gg 1$ sector obeys the semiclassical expansion

$$ \Delta_0(S^{\kappa(S)}, L=0) = \alpha S^{3/2} + \beta S^{1/2} - 0.0937256 + \gamma S^{-1/2} + \mathcal{O}(1/S), \quad (B1) $$

where $\kappa(S) = (-1)^S$ determines the parity. The Wilson coefficients $\alpha = 0.31076$, $\beta = 0.29818$, and $\gamma = 0.00370$ are obtained by fitting the lowest level in the sectors $1^{-}, 2^{+}, 4^{+}$ to bootstrap data for $\sigma, t_{(2)}$, and $t_{(4)}$.

**2. Gapped Goldstone modes.** The scaling dimensions of the state with spin $S \gg 1$, parity $\kappa(S+1) =$
$-\kappa(S)$, and non-vanishing angular momenta $L \ge 1$ obeys the semiclassical expansion

$$ \Delta(S^{\kappa(S+1)}, L) = \Delta_0(S^{\kappa(S)}, L=0) + \mu(S) + \chi \frac{L(L+1)}{2\mu(S)} + \mathcal{O}\left(\frac{L^4}{\mu^3}\right), \qquad (\text{B2a}) $$

where

$$ \mu(S) := \frac{\partial \Delta_0(S, L=0)}{\partial S} \qquad (\text{B2b}) $$

is the chemical potential that breaks explicitly the internal continuous symmetry from SO(3) to SO(2). The term $L(L+1)/(2\mu)$ encodes the dispersion of the gapped Goldstone boson. The value of the Wilson coefficient $\chi$ is obtained by fitting the lowest scaling dimension in the sector $1^+$ to the value 2 of the scaling dimension of the Noether current $j_\mu$.

**3. Gapless phonon modes.** The scaling dimension of the state with the quantum numbers $S \gg 1$ for the spin, parity $\kappa(S) = (-1)^S$, and the angular momenta $L = 0, 1, \dots$ obeys the semiclassical expansion

$$ \Delta_0(S^{\kappa(S)}, L) = \Delta_0(S^{\kappa(S)}, L=0) + \sum_L n_L \omega_L, \qquad (\text{B3a}) $$

where

$$ \omega_L = \sqrt{L(L+1)/2}, \qquad (\text{B3b}) $$

is the energy of the phonon with angular momenta $L$ and the integer $n_L$ is its occupancy. The value $L = 1$ correspond to a phonon descendant, while the values $L \ge 2$ correspond to phonon primaries.

The agreement between these theoretical predictions and fuzzy sphere numerical results (shown in Fig. 4) provides strong evidence for the presence of both gapped Goldstone modes and gapless phonon modes in the $(2+1)$-dimensional O(3) Wilson-Fisher CFT, validating the large-$S$ expansion framework.

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Supplemental Material for:

Conformal Data for the O(3) Wilson-Fisher CFT from Fuzzy Sphere Realization of Quantum Rotor Model

Arjun Dey,¹,² Loic Herviou,³ Christopher Mudry,¹,² and Andreas Martin Läuchli¹,²

¹Laboratory for Theoretical and Computational Physics,
PSI Center for Scientific Computing, Theory and Data, 5232 Villigen PSI, Switzerland

²Institute of Physics, École Polytechnique Fédérale de Lausane (EPFL), 1015 Lausane, Switzerland

³Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France

CONTENTS

<table><tbody><tr><td>I. Hamiltonian</td><td>5</td></tr><tr><td>II. Truncated quantum rotor matrix elements</td><td>6</td></tr><tr><td>III. Relation of truncated quantum rotor with bilayer heisenberg model</td><td>7</td></tr><tr><td>IV. O(N) generalization: Hamiltonian and matrix elements</td><td>7</td></tr><tr><td>V. Conformal primaries and descendants</td><td>8</td></tr><tr><td>A. Angular momenta structure of descendants</td><td>8</td></tr><tr><td>B. Examples</td><td>9</td></tr><tr><td>C. Conserved quantities and their descendants</td><td>9</td></tr><tr><td>VI. Conformal perturbation theory</td><td>10</td></tr><tr><td>VII. Large-S expansion</td><td>10</td></tr><tr><td>VIII. Details on the simulations</td><td>12</td></tr><tr><td>IX. Scaling dimension data for all symmetry sectors</td><td>13</td></tr><tr><td>A. S = 0:</td><td>13</td></tr><tr><td>S = 0<sup>+</sup>, L = 0</td><td>13</td></tr><tr><td>S = 0<sup>+</sup>, L = 1</td><td>14</td></tr><tr><td>S = 0<sup>+</sup>, L = 2</td><td>14</td></tr><tr><td>S = 0<sup>+</sup>, L = 3</td><td>15</td></tr><tr><td>S = 0<sup>+</sup>, L = 4</td><td>15</td></tr><tr><td>S = 0<sup>+</sup>, L = 5</td><td>15</td></tr><tr><td>S = 0<sup>+</sup>, L = 6</td><td>16</td></tr><tr><td>S = 0<sup>+</sup>, L = 7</td><td>16</td></tr><tr><td>S = 0<sup>-</sup>, L = 0</td><td>16</td></tr><tr><td>S = 0<sup>-</sup>, L = 1</td><td>17</td></tr><tr><td>S = 0<sup>-</sup>, L = 2</td><td>17</td></tr><tr><td>S = 0<sup>-</sup>, L = 3</td><td>17</td></tr><tr><td>S = 0<sup>-</sup>, L = 4</td><td>18</td></tr><tr><td>S = 0<sup>-</sup>, L = 5</td><td>18</td></tr><tr><td>S = 0<sup>-</sup>, L = 6</td><td>18</td></tr><tr><td>S = 0<sup>-</sup>, L = 7</td><td>19</td></tr><tr><td>B. S = 1:</td><td>20</td></tr><tr><td>S = 1<sup>+</sup>, L = 0</td><td>20</td></tr><tr><td>S = 1<sup>+</sup>, L = 1</td><td>20</td></tr><tr><td>S = 1<sup>+</sup>, L = 2</td><td>21</td></tr><tr><td>S = 1<sup>+</sup>, L = 3</td><td>21</td></tr></tbody></table>
S = 1+, L = 4 22

S = 1+, L = 5 22

S = 1+, L = 6 23

S = 1+, L = 7 23

S = 1-, L = 0 23

S = 1-, L = 1 24

S = 1-, L = 2 24

S = 1-, L = 3 25

S = 1-, L = 4 25

S = 1-, L = 5 26

S = 1-, L = 6 26

S = 1-, L = 7 26

C. S = 2: 27

S = 2+, L = 0 27

S = 2+, L = 1 28

S = 2+, L = 2 28

S = 2+, L = 3 29

S = 2+, L = 4 29

S = 2+, L = 5 30

S = 2+, L = 6 30

S = 2+, L = 7 30

S = 2-, L = 0 31

S = 2-, L = 1 31

S = 2-, L = 2 32

S = 2-, L = 3 32

S = 2-, L = 4 33

S = 2-, L = 5 33

S = 2-, L = 6 33

S = 2-, L = 7 34

D. S = 3: 35

S = 3+, L = 0 35

S = 3+, L = 1 35

S = 3+, L = 2 36

S = 3+, L = 3 36

S = 3+, L = 4 37

S = 3+, L = 5 37

S = 3+, L = 6 37

S = 3+, L = 7 38

S = 3-, L = 0 38

S = 3-, L = 1 38

S = 3-, L = 2 39

S = 3-, L = 3 39

S = 3-, L = 4 40

S = 3-, L = 5 40

S = 3-, L = 6 40

S = 3-, L = 7 41

E. S = 4: 42

S = 4+, L = 0 42

S = 4+, L = 1 42

S = 4+, L = 2 43

S = 4+, L = 3 43

S = 4+, L = 4 44

S = 4+, L = 5 44

S = 4+, L = 6 44

S = 4+, L = 7 45

S = 4-, L = 0 45
$S = 4^{-}, L = 1$

$S = 4^{-}, L = 2$

$S = 4^{-}, L = 3$

$S = 4^{-}, L = 4$

$S = 4^{-}, L = 5$

$S = 4^{-}, L = 6$

$S = 4^{-}, L = 7$

F. $S = 5:$

$S = 5^{+}, L = 0$

$S = 5^{+}, L = 1$

$S = 5^{+}, L = 2$

$S = 5^{+}, L = 3$

$S = 5^{+}, L = 4$

$S = 5^{+}, L = 5$

$S = 5^{+}, L = 6$

$S = 5^{+}, L = 7$

$S = 5^{-}, L = 0$

$S = 5^{-}, L = 1$

$S = 5^{-}, L = 2$

$S = 5^{-}, L = 3$

$S = 5^{-}, L = 4$

$S = 5^{-}, L = 5$

$S = 5^{-}, L = 6$

$S = 5^{-}, L = 7$

G. $S = 6:$

$S = 6^{+}, L = 0$

$S = 6^{+}, L = 1$

$S = 6^{+}, L = 2$

$S = 6^{+}, L = 3$

$S = 6^{+}, L = 4$

$S = 6^{+}, L = 5$

$S = 6^{+}, L = 6$

$S = 6^{+}, L = 7$

$S = 6^{-}, L = 0$

$S = 6^{-}, L = 1$

$S = 6^{-}, L = 2$

$S = 6^{-}, L = 3$

$S = 6^{-}, L = 4$

$S = 6^{-}, L = 5$

$S = 6^{-}, L = 6$

$S = 6^{-}, L = 7$

H. $S = 7:$

$S = 7^{+}, L = 0$

$S = 7^{+}, L = 1$

$S = 7^{+}, L = 2$

$S = 7^{+}, L = 3$

$S = 7^{+}, L = 4$

$S = 7^{+}, L = 5$

$S = 7^{+}, L = 6$

$S = 7^{+}, L = 7$

$S = 7^{-}, L = 0$

$S = 7^{-}, L = 1$

$S = 7^{-}, L = 2$

$S = 7^{-}, L = 3$

$S = 7^{-}, L = 4$

$S = 7^{-}, L = 5$

45

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51

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61

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66

66

67
$S = 7^-, L = 6$ 67

$S = 7^-, L = 7$ 67

X. OPE Data for all sectors 68

A. $S = 0$: 68

$S = 0^+$ 68

$S = 0^-$ 68

B. $S = 1$: 69

$S = 1^+$ 69

$S = 1^-$ 69

C. $S = 2$: 72

$S = 2^+$ 72

$S = 2^-$ 72

D. $S = 3$: 73

$S = 3^+$ 73

$S = 3^-$ 73

E. $S = 4$: 74

$S = 4^+$ 74

$S = 4^-$ 74

F. $S = 5$: 75

$S = 5^-$ 75

G. $S = 6$: 75

$S = 6^+$ 75

References 75
# I. HAMILTONIAN

We begin by recalling the Hamiltonian as defined in the main text:

$$ \hat{H}_{\text{fzs}} := \hat{P}_{\text{LLL}} (u \hat{H}_{\text{Hub}} - v \hat{H}_{\text{Heis}} + h \hat{H}_{\text{trans}}) \hat{P}_{\text{LLL}}. \quad (1) $$

Here, $\hat{P}_{\text{LLL}}$ is the projector onto the lowest Landau level (LLL), and the terms $\hat{H}_{\text{Hub}}$, $\hat{H}_{\text{Heis}}$, and $\hat{H}_{\text{trans}}$ are defined as follows.

The unprojected Hamiltonian in real space, in terms of local densities built from the four-flavor rotor spinor, reads:

$$ \hat{H} = u \hat{H}_{\text{Hub}} - v \hat{H}_{\text{Heis}} + h \hat{H}_{\text{trans}}, \quad (2a) $$

$$ \hat{H}_{\text{Hub}} = \sum_{a,b=1}^{N} \int_{S^2} d\Omega_a d\Omega_b U_{ab} [n^0(\Omega_a) n^0(\Omega_b)], \quad (2b) $$

$$ \hat{H}_{\text{Heis}} = \sum_{a,b=1}^{N} \int_{S^2} d\Omega_a d\Omega_b U_{ab} [n^{\mathcal{R}}(\Omega_a) n^{\mathcal{R}}(\Omega_b)], \quad (2c) $$

$$ \hat{H}_{\text{trans}} = \sum_{a=1}^{N} \int_{S^2} d\Omega_a n^{\mathcal{L}}(\Omega_a). \quad (2d) $$

The local densities are defined by

$$ n^{\alpha}(\Omega) = \begin{pmatrix} \hat{\psi}_{0,0}^{\dagger} & \hat{\psi}_{1,-1}^{\dagger} & \hat{\psi}_{1,0}^{\dagger} & \hat{\psi}_{1,1}^{\dagger} \end{pmatrix} \mathcal{M}^{\alpha} \begin{pmatrix} \hat{\psi}_{0,0} \\ \hat{\psi}_{1,-1} \\ \hat{\psi}_{1,0} \\ \hat{\psi}_{1,1} \end{pmatrix}, \quad (3) $$

with $\mathcal{M}^0 = I$, $\mathcal{M}^L = \text{diag}(0, 2, 2, 2)$, and $\mathcal{M}^R$ the two-body rotor matrix elements (defined below via Eq. (6)).

The LLL projector acts on the field operator as

$$ \hat{P}_{\text{LLL}} \hat{\psi}(\Omega) \hat{P}_{\text{LLL}} = \sum_{m=-Q}^{Q} Y_{Q,Q,m}(\Omega) \hat{c}_{m}, \quad (4a) $$

$$ \mathbf{c}_{m}^{\dagger} \equiv (c_{m,(0,0)}^{\dagger}, c_{m,(1,-1)}^{\dagger}, c_{m,(1,0)}^{\dagger}, c_{m,(1,1)}^{\dagger}), \quad (4b) $$

where $Y_{Q,Q,m}$ are the LLL monopole harmonics and the four components of $\mathbf{c}_m^\dagger$ correspond to the truncated rotor flavors $\{(l, m) = (0, 0), (1, -1), (1, 0), (1, 1)\}$. Here $m$ denotes the z-component of the angular momenta and it can take $2Q+1$ values from $-Q$ to $Q$.

Projecting the Hamiltonian explicitly to the LLL yields

$$ \hat{H}_{\text{fzs}} = u \hat{H}_{\text{Hub}}^{\text{LLL}} - v \hat{H}_{\text{Heis}}^{\text{LLL}} + h \hat{H}_{\text{trans}}^{\text{LLL}}, \quad (5a) $$

$$ \hat{H}_{\text{Hub}}^{\text{LLL}} = \sum_{m_1, m_2, m = -Q}^{Q} V_{m_1, m_2, m_2-m, m_1+m} \mathbf{c}_{m_1}^{\dagger} \mathbf{c}_{m_2}^{\dagger} \mathbf{c}_{m_1+m} \mathbf{c}_{m_2-m}, \quad (5b) $$

$$ \hat{H}_{\text{Heis}}^{\text{LLL}} = \sum_{m_1, m_2, m = -Q}^{Q} V_{m_1, m_2, m_2-m, m_1+m} \mathbf{c}_{m_1}^{\dagger} \mathbf{c}_{m_2}^{\dagger} \mathcal{M}^{\mathcal{R}} \mathbf{c}_{m_1+m} \mathbf{c}_{m_2-m}, \quad (5c) $$

$$ \hat{H}_{\text{trans}}^{\text{LLL}} = \sum_{m=-Q}^{Q} \mathbf{c}_{m}^{\dagger} \mathcal{M}^{\mathcal{L}} \mathbf{c}_{m}. \quad (5d) $$
Here $V_{m_1,m_2,m_3,m_4}$ are the usual LLL two-body matrix elements on the sphere. The rotor matrix elements $\mathcal{M}^R$ acting on the four-flavor space is the truncated-basis matrix built from spherical-harmonic couplings; in components it reads

$$ \mathcal{M}^{\mathcal{R}}_{(l_1,m_1),(l_2,m_2);(l_3,m_3),(l_4,m_4)} = \sum_{\alpha=-1,0,1} (-1)^{-m_1-m_2-\alpha} \frac{\sqrt{9(2l_1+1)(2l_2+1)(2l_3+1)(2l_4+1)}}{4\pi} \\ \times \begin{pmatrix} l_1 & 1 & l_4 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_2 & 1 & l_3 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & 1 & l_4 \\ -m_1 & -\alpha & m_4 \end{pmatrix} \begin{pmatrix} l_2 & 1 & l_3 \\ -m_2 & \alpha & m_3 \end{pmatrix}, \quad (6) $$

which evaluates the interaction terms $\sum_{\alpha=-1,0,1} \hat{n}_i^\alpha \cdot \hat{n}_j^\alpha$ in the truncated $\{|l, m\rangle\}$ rotor basis with $l \in \{0, 1\}$.

Eq. (5) provides the LLL-projected, second-quantized form used in our ED/DMRG calculations. The matrix $\mathcal{M}^L$ encodes the single-site flavor weights, while $\mathcal{M}^R$ captures the two-body rotor matrix elements within the truncated $\{(0, 0), (1, -1), (1, 0), (1, 1)\}$ manifold. The coefficients $V_{m_1,m_2,m_3,m_4}$ contain interaction dependence through the standard monopole harmonic integrals on $\mathbb{S}^2$.

The interaction $U(\Omega_{ab})$ is encoded through matrix elements $V_{m_1,m_2,m_3,m_4}$ using Haldane pseudopotentials [1], following the convention established in Ref. [2]. These pseudopotentials provide a systematic decomposition of rotationally invariant interactions on the sphere into components of definite relative angular momenta. These are expressed using Wigner-3j symbols as

$$ V_{m_1,m_2,m_3,m_4} = \sum_{l=0}^{L} V_l(4s - 2l + 1) \begin{pmatrix} s & s & 2s-l \\ m_1 & m_2 & -m_1 - m_2 \end{pmatrix} \begin{pmatrix} s & s & 2s-l \\ m_3 & m_4 & -m_3 - m_4 \end{pmatrix}. \quad (7) $$

We restrict to nonzero $V_0$ and $V_1$. Pseudopotentials represent the projection of central interactions onto the lowest Landau level, decomposed by total angular momenta. Specifically, $V_0$ corresponds to the projection of the contact interaction $\delta(\bar{r} - \bar{r}')$, while $V_1$ captures that of $\nabla^2\delta(\bar{r} - \bar{r}')$. We determine the optimal values for the Hamiltonian parameters to be $V_0 = 6.5$, $V_1 = 1.0$, $u = 1.0$, and $v = 1.4$, and all the simulations for the conformal data are done with these parameters.

## II. TRUNCATED QUANTUM ROTOR MATRIX ELEMENTS

We briefly recall the quantum rotor model: each site hosts a quantum rotor, whose configuration is described by a unit vector $\mathbf{n}$ on the sphere ($\mathbf{n}^2 = 1$). The components of $\mathbf{n}$ are the fundamental degrees of freedom, and the Hilbert space is spanned by spherical harmonics $|l, m\rangle$, as in the previous section.

The position operator $\hat{\mathbf{n}}$ acts as a vector operator in this space, with spherical components given by

$$ n_q = R \sqrt{\frac{4\pi}{3}} Y_{1,q}(\Omega), \quad (8) $$

where $q = -1, 0, 1$ labels the spherical basis, following the notation of the previous section.

To compute the matrix elements of the interaction term $\hat{\mathbf{n}}_i \cdot \hat{\mathbf{n}}_j$ in the truncated rotor basis, we evaluate

$$ \begin{align} & \sum_{\alpha=-1,0,1} \langle l_1, m_1 | \langle l_2, m_2 | \hat{n}_i^\alpha \cdot \hat{n}_j^\alpha | l_3, m_3 \rangle | l_4, m_4 \rangle \nonumber \\ &= \sum_{\alpha=-1,0,1} \int d\Omega_i d\Omega_j Y_{l_1,m_1}^*(\Omega_i) Y_{l_2,m_2}^*(\Omega_j) \nonumber \\ & \qquad \times Y_{1,\alpha}^*(\Omega_i) Y_{1,\alpha}(\Omega_j) Y_{l_3,m_3}(\Omega_j) Y_{l_4,m_4}(\Omega_i) \nonumber \\ &= \sum_{\alpha=-1,0,1} (-1)^{-m_1-m_2-\alpha} \frac{\sqrt{9(2l_1+1)(2l_2+1)(2l_3+1)(2l_4+1)}}{4\pi} \nonumber \\ & \qquad \times \begin{pmatrix} l_1 & 1 & l_4 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_2 & 1 & l_3 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & 1 & l_4 \\ -m_1 & -\alpha & m_4 \end{pmatrix} \begin{pmatrix} l_2 & 1 & l_3 \\ -m_2 & \alpha & m_3 \end{pmatrix}. \tag{9} \end{align} $$

This expression gives the explicit matrix elements of the two-site interaction in the truncated rotor basis, and corresponds to the matrix elements of $\mathcal{M}^R$ as shown in Eq. (6) in the previous section.
### III. RELATION OF TRUNCATED QUANTUM ROTOR WITH BILAYER HEISENBERG MODEL

Interestingly, the $O(3)$ rotor model has a correspondence with the bilayer Heisenberg antiferromagnet (AFM); in particular, the rotor can capture the low-energy physics of the bilayer AFM. In the strong inter-layer coupling limit, a bilayer AFM has, on every lattice site, a singlet ground state and a gapped triplet excitation formed by the two spins on that site. These levels coincide with the $l=0$ and $l=1$ states of an $O(3)$ quantum rotor once one identifies [3]

$$ \hat{\mathbf{L}}_i \leftrightarrow \hat{\mathbf{S}}_{1,i} + \hat{\mathbf{S}}_{2,i}, \quad \hat{\mathbf{n}}_i \leftrightarrow \hat{\mathbf{S}}_{1,i} - \hat{\mathbf{S}}_{2,i}. \qquad (10) $$

so that the angular momenta of the rotor is the total spin of the rung and its orientation vector is the staggered (Néel) moment. Intra-layer coupling of the antiferromagnet then hops the triplet between sites exactly as the rotor coupling $-J\hat{\mathbf{n}}_i \cdot \hat{\mathbf{n}}_j$ propagates the $l=1$ excitation, making the quantum rotor a compact, symmetry-preserving low-energy description of the bilayer magnet [3]. Motivated by this correspondence, we keep only the singlet and triplet, i.e. set $l_{max}=1$, thereby retaining the full low-energy manifold while discarding higher-spin states that lie far above the gap.

### IV. $O(N)$ GENERALIZATION: HAMILTONIAN AND MATRIX ELEMENTS

We generalize to an $O(N)$ rotor with unit vector $\hat{\mathbf{n}} = (n^1, \dots, n^N)$ on $S^{N-1}$ and a truncated on-site Hilbert space spanned by hyperspherical harmonics $\{|l, \mu\}$, corresponding to rank-$l$ irreducible representations of $SO(N)$ (with $\mu$ denoting the degeneracy index within a given representation). In the $\{l=0 \oplus 1\}$ truncation, the local “flavor” space is $(1 \oplus N)$, i.e. one singlet plus $N$ vector components. We denote the $(N+1)$-component field spinor as

$$ \hat{\Psi}^{\dagger} = (\hat{\psi}_{0}^{\dagger}, \hat{\psi}_{1,1}^{\dagger}, \dots, \hat{\psi}_{1,N}^{\dagger}), $$

where the first entry is the $l=0$ singlet and the rest are the $l=1$ vector components.

The $O(N)$ analogue of Eq. (1) reads

$$ \hat{H}_{\text{fzs}}^{(N)} = \hat{P}_{\text{LLL}} (u \hat{H}_{\text{Hub}} - v \hat{H}_{\text{Heis}}^{(N)} + h \hat{H}_{\text{trans}}) \hat{P}_{\text{LLL}}, \qquad (11) $$

with the unprojected real-space form

$$ \hat{H} = u \hat{H}_{\text{Hub}} - v \hat{H}_{\text{Heis}}^{(N)} + h \hat{H}_{\text{trans}}, \qquad (12a) $$

$$ \hat{H}_{\text{Hub}} = \sum_{a,b=1}^{N_s} \int_{S^2} d\Omega_a d\Omega_b U_{ab} n^0(\Omega_a) n^0(\Omega_b), \qquad (12b) $$

$$ \hat{H}_{\text{Heis}}^{(N)} = \sum_{a,b=1}^{N_s} \int_{S^2} d\Omega_a d\Omega_b U_{ab} \sum_{A=1}^{N} n^A(\Omega_a) n^A(\Omega_b), \qquad (12c) $$

$$ \hat{H}_{\text{trans}} = \sum_{a=1}^{N_s} \int_{S^2} d\Omega_a n^L(\Omega_a), \qquad (12d) $$

where $N_s$ is the number of electrons. The local densities are bilinears $n^\alpha(\Omega) = \hat{\Psi}^\dagger(\Omega) M^\alpha \hat{\Psi}(\Omega)$ with

$$ M^0 = \text{diag}(1, 1, \dots, 1), \quad M^L = \text{diag}(0, w, \dots, w), $$

where $w$ fixes the relative weight of the $l=1$ sector (e.g. $w=2$ for $N=3$).

Projecting to the LLL yields

$$ \hat{H}_{\text{fzs}}^{(N)} = u \hat{H}_{\text{Hub}}^{\text{LLL}} - v \hat{H}_{\text{Heis}}^{(N)\text{LLL}} + h \hat{H}_{\text{trans}}^{\text{LLL}}, \qquad (13a) $$

$$ \hat{H}_{\text{Hub}}^{\text{LLL}} = \sum_{m_1, m_2, m=-s}^{s} V_{m_1, m_2, m_2-m, m_1+m} \mathbf{c}_{m_1}^{\dagger} \mathbf{c}_{m_2}^{\dagger} \mathbf{c}_{m_1+m} \mathbf{c}_{m_2-m}, \qquad (13b) $$

$$ \hat{H}_{\text{Heis}}^{(N)\text{LLL}} = \sum_{m_1, m_2, m=-s}^{s} V_{m_1, m_2, m_2-m, m_1+m} \mathbf{c}_{m_1}^{\dagger} \mathbf{c}_{m_2}^{\dagger} \mathcal{M}^{\mathcal{R},(N)} \mathbf{c}_{m_1+m} \mathbf{c}_{m_2-m}, \qquad (13c) $$

$$ \hat{H}_{\text{trans}}^{\text{LLL}} = \sum_{m=-s}^{s} \mathbf{c}_{m}^{\dagger} \mathcal{M}^{\mathcal{L}} \mathbf{c}_{m}, \qquad (13d) $$
where $c_m^\dagger$ creates the $(1 \oplus N)$ flavor at orbital $m$, and $V_{m_1,m_2,m_3,m_4}$ are the standard LLL two-body matrix elements (constructed via pseudopotentials). The $N$-dependence enters only through $M^{\mathcal{R},(N)}$.

a. **Quantum numbers and degeneracies.** For an $O(N)$ rotor, the wavefunctions are hyperspherical harmonics $Y_{l\mu}^{(N)}(\Omega)$ on $S^{N-1}$, which transform as the symmetric traceless rank-$l$ representations of $SO(N)$. They are labeled by a single angular momenta quantum number $l = 0, 1, 2, \dots$ and a multi-index $\mu$ that distinguishes the degenerate components within a given $l$. The degeneracy of the level $l$ is

$$d_l = \frac{(2l + N - 2)(l + N - 3)!}{l!(N - 2)!}, \qquad (14)$$

so that $d_0 = 1$ (singlet) and $d_1 = N$ (vector representation). Thus, the $\{l = 0, 1\}$ truncation yields a local Hilbert space of dimension $1 + N$.

b. **Two-body rotor matrix elements.** The two-body matrix elements that enter the $O(N)$ rotor Hamiltonian, denoted $M^{\mathcal{R},(N)}$, encode the internal structure of the model and depend on the overlaps between single-particle states and the components of the unit vector $\hat{n}$ on the sphere. While their explicit analytic form involves group-theoretic quantities such as Clebsch-Gordan coefficients and integrals over hyperspherical harmonics, in practice these matrix elements can be efficiently computed numerically for any $N$ and for the truncated $(1 \oplus N)$ Hilbert space used in our simulations.

With this approach, Eq. (13) defines the $O(N)$ extension of the truncated quantum rotor Hamiltonian: the monopole harmonic coefficients $V_{m_1,m_2,m_3,m_4}$ remain unchanged, while the internal structure enters only via $M^L$ and $M^{\mathcal{R},(N)}$, enabling straightforward ED/DMRG simulations with an $O(N)$ rotor flavor space.

## V. CONFORMAL PRIMARIES AND DESCENDANTS

In conformal field theory, operators are organized into families. Each family consists of a primary operator and its descendants, which are generated by acting with momentum operators (derivatives).

A primary operator $O$ is defined by its transformation properties under conformal symmetry. At the origin (in radial quantization), it satisfies:

$$[K_i, O(0)] = 0, \qquad D O(0) = \Delta_O O(0), \qquad (15)$$

where $K_i$ are special conformal generators, $D$ is the dilatation operator, and $\Delta_O$ is the scaling dimension.

Descendants are generated by applying momentum operators (derivatives) to the primary operator:

$$\text{Level 1: } \partial_{\mu} O \qquad (16a)$$

$$\text{Level 2: } \partial_{\mu} \partial_{\nu} O \qquad (16b)$$

$$\text{Level n: } \partial_{\mu_1} \partial_{\mu_2} \cdots \partial_{\mu_n} O \qquad (16c)$$

On $\mathbb{R} \times S^2$, this translates to simple energy rules: the primary creates a state with energy $\Delta_O$, and each application of a momentum operator $P_i$ increases the energy by exactly 1. Therefore, all level-$n$ descendants have energies $\Delta_O + n$. Throughout the following discussion, we use the notation $L_P$ to denote the angular momenta of the primary operator and $L_D$ to denote the angular momenta of its descendants.

### A. Angular momenta structure of descendants

For scalar primaries i.e. primaries with angular momenta $L_P = 0$, at descendant level $n$, the allowed angular momenta are:

$$L_D \in \{n, n-2, n-4, \dots\} \quad \text{down to} \quad \begin{cases} 0, & \text{if } n \text{ is even} \\ 1, & \text{if } n \text{ is odd} \end{cases} \qquad (17)$$

For spinning primaries i.e. primaries with $L_P > 0$, at descendant level $n$, the allowed angular momenta are:

$$L_D \in \{L_P + n, L_P + n - 2, L_P + n - 4, \dots, |L_P - n|\} \qquad (18)$$

Each angular momenta multiplet $L$ contains $2L+1$ states with magnetic quantum numbers $m = -L, -L+1, \dots, +L-1, +L$.
## B. Examples

### Scalar primary ($L_P = 0$):

* Level $n = 1$: Energy $\Delta + 1$, angular momenta $L_D = 1$ (corresponds to $\partial_\mu O$)

* Level $n = 2$: Energy $\Delta + 2$, angular momenta $L_D = 2, 0$ (corresponds to $\partial_\mu \partial_\nu O$)

* Level $n = 3$: Energy $\Delta + 3$, angular momenta $L_D = 3, 1$ (corresponds to $\partial_\mu \partial_\nu \partial_\rho O$)

### Vector primary ($L_P = 1$):

* Level $n = 1$: Energy $\Delta + 1$, angular momenta $L = 2, 1, 0$

- $L = 2$: corresponds to traceless symmetric part of $\partial_\mu O_\nu$

- $L = 1$: corresponds to antisymmetric part $\partial_{[\mu}O_{\nu]}$

- $L = 0$: corresponds to trace part $\partial_\mu O^\mu$

* Level $n = 2$: Energy $\Delta + 2$, angular momenta $L = 3, 2, 1, 0$ (corresponds to $\partial_\mu \partial_\rho O_\nu$)

## C. Conserved quantities and their descendants

Conserved quantities are somewhat special since they are generators of symmetries, as a consequence they have a very special descendant structure. The most important examples are:

**Stress-energy tensor ($T_{\mu\nu}$):** The stress-energy tensor is a symmetric, traceless, conserved tensor that generates spacetime symmetries (translations, rotations, scale transformations, and special conformal transformations). On $\mathbb{R} \times \mathbb{S}^2$:

* Primary: $T_{\mu\nu}$ with scaling dimension $\Delta = 3$ and angular momenta $L_P = 2$

* Level $n = 1$ descendants: Energy $\Delta + 1 = 4$, angular momenta $L_D = 3, 2, 1$

- $L = 3$: traceless symmetric part of $\partial_\rho T_{\mu\nu}$

- $L = 2$: mixed symmetry combinations

- $L = 1$: trace and divergence parts $\partial_\rho T^{\rho\nu} = 0$ (vanishes due to conservation)

**Noether current ($J_\mu$):** For internal symmetries like $O(3)$, the Noether current is a conserved vector that generates internal rotations:

* Primary: $J_\mu$ with scaling dimension $\Delta = 2$ and angular momenta $L_P = 1$

* Level $n = 1$ descendants: Energy $\Delta + 1 = 3$, angular momenta $L = 2, 1, 0$

- $L = 2$: traceless symmetric part of $\partial_\rho J_\mu$

- $L = 1$: antisymmetric part $\partial_{[\rho}J_{\mu]}$

- $L = 0$: divergence $\partial_\mu J^\mu = 0$ (vanishes due to conservation)

The conservation laws $\partial_\mu T^{\mu\nu} = 0$ and $\partial_\mu J^\mu = 0$ eliminate certain descendant states, leading to shortened multiplets compared to generic operators.
## VI. CONFORMAL PERTURBATION THEORY

For any D-dimensional CFT on $\mathbb{R} \times \mathbb{S}^{D-1}$, the state-operator correspondence allows to reinterpret the dilatation operator as a “quantum Hamiltonian” $\hat{H}_{CFT}$ whose eigenstate $|o\rangle$ has as eigenvalue the scaling dimension $\Delta_o$ of a CFT operator labeled by o,

$$ \hat{H}_{CFT} |o\rangle = \Delta_o |o\rangle. \qquad (19) $$

Conformal perturbation theory provides a systematic framework to describe deviations from criticality originating either from a CFT-breaking perturbation or from finite-size effects. Following Ref. [4, 5], the effective Hamiltonian in Eq. (19) is replaced by $\hat{H} := \hat{H}_{CFT} + \delta\hat{H}$, where the perturbation is expressed as

$$ \delta\hat{H} = \sum_n \int_{\mathbb{S}^{D-1}} d\Omega g_n(\Omega) \hat{\chi}_n(0, \Omega), \qquad (20) $$

with $\hat{\chi}_n$ local CFT operators inserted at $\tau = 0$ on $\mathbb{R} \times \mathbb{S}^{D-1}$, and $g_n(\Omega)$ the corresponding coupling functions. The perturbing operators must respect the symmetries of the microscopic theory. For the O(3) rotor model, $\hat{\chi}_n$ must preserve both SO(3) rotational symmetry and O(3) internal spin symmetry.

The energy eigenvalues of $H$ are then given by

$$ \begin{aligned} E_{o_m} = E_{o_m, \text{CFT}} &+ \langle o_m | \delta \hat{H} | o_m \rangle \\ &+ \sum_{i \neq m} \frac{\langle o_m | \delta \hat{H} | o_i \rangle \langle o_i | \delta \hat{H} | o_m \rangle}{E_{o_m, \text{CFT}} - E_{o_i, \text{CFT}}} \\ &+ \mathcal{O}((\delta \hat{H})^3), \end{aligned} \qquad (21) $$

where $E_{o_m, \text{CFT}} \equiv \Delta_{o_m}$. To first order, only the diagonal matrix elements $\langle o_m | \delta \hat{H} | o_m \rangle$ contribute. These are determined by OPE coefficients between $\hat{\chi}_n$ and $\hat{o}_m$. For $o_m$ labeling a scalar primary and $\partial o_m$ labeling its first descendant, the first-order shifts are

$$ \delta E_{o_m} = g_{\chi_n} f_{o_m, \chi_n, o_m}, \qquad (22a) $$

$$ \delta E_{\partial o_m} = g_{\chi_n} f_{o_m, \chi_n, o_m} A_{o_m, \chi_n}. \qquad (22b) $$

respectively. Here, $g_{\chi_n} = \int_{\mathbb{S}^{D-1}} g_n(\Omega)$ is the integrated coupling, while the multiplicative factor $A_{o_m, \chi_n}$ is fixed by the conformal symmetry to be

$$ A_{o_m, \chi_n} = 1 + \frac{\Delta_{\chi_n}(\Delta_{\chi_n} - 3)}{6\Delta_{o_m}}. \qquad (23) $$

## VII. LARGE-$S$ EXPANSION

If the CFT has a continuous internal symmetry group $G$, there exists a conserved Noether current and a corresponding conserved charge operator $\hat{Q}$, with eigenvalues $Q$ that scale with the area $R^{D-1}$ in the thermodynamic limit. We define the charge and energy densities as

$$ \rho := \frac{Q}{R^{D-1}}, \qquad (24a) $$

$$ \delta\varepsilon := \frac{\delta E}{R^{D-1}}. \qquad (24b) $$

To study states at fixed $\rho$ and $\delta\varepsilon$, we introduce a chemical potential $\mu$ and consider the Hamiltonian $\hat{H}_{CFT} + \mu\hat{Q}$. Dimensional analysis then gives

$$ \rho = c_{\rho} \mu^{D-1}, \qquad (25a) $$

$$ \delta\varepsilon = c_{\varepsilon} \mu^D, \qquad (25b) $$
where $c_ρ$ and $c_ε$ are dimensionless constants of order one. Inverting, we have

$$ \mu = \left( \frac{\rho}{c_\rho} \right)^{1/(D-1)}, \qquad (26a) $$

$$ \delta\epsilon = c \rho^{D/(D-1)}, \quad c = c_{\epsilon} c_{\rho}^{-D/(D-1)}. \qquad (26b) $$

Combining state-operator correspondence with Eqs. (24b), and (26b), we obtain the leading scaling of the scaling dimension at large charge:

$$ \Delta_o = c Q^{D/(D-1)}. \qquad (27) $$

This leading behavior is the first term in a systematic large-charge (semiclassical) expansion [6–9]:

$$ \Delta_{\mathcal{O}}^{\text{sc}} = \sum_{n=0}^{\infty} c_n^{\text{sc}} Q^{\frac{D}{D-1}-n}, \qquad (28) $$

where the coefficients $c_n^{\text{sc}}$ can be computed in effective field theory. The expansion in Eq. (28) does not include a $Q^0$ term at leading order, but such a term appears at one-loop order [7]:

$$ \Delta_o^{\text{sc}+1-\text{loop}} = \Delta_o^{\text{sc}} + c^{1-\text{loop}}. \qquad (29) $$

Explicit methods for computing these coefficients are available in the literature [6–16]. In what follows, we focus on the case $D=3$ and $G=O(3)$.

In the thermodynamic limit $R \propto \sqrt{N} \to \infty$ at zero temperature, the phase diagram of the fuzzy sphere Hamiltonian $\hat{H}_{\text{fzs}}(R)$ consists of: (i) a long-range ordered phase that spontaneously breaks SO(3) down to SO(2); (ii) a quantum disordered phase with a gapped, non-degenerate, symmetric ground state; and (iii) a quantum critical point separating the two, described by the $(2+1)$-dimensional O(3) Wilson-Fisher CFT. In the ordered phase, Goldstone’s theorem predicts two gapless modes (transverse to the order parameter), while the longitudinal mode (associated with the unbroken SO(2)) is not sharp. Approaching the critical point from the ordered side, the longitudinal gap closes continuously. In the disordered phase, all modes are gapped and strongly interacting.

If we instead consider the Hamiltonian $\hat{H}_{\text{fzs}}(R) + \mu \hat{Q}^z$, where $\hat{Q}^z$ is the conserved charge associated with the Cartan subalgebra of $\mathfrak{so}(3)$, the situation changes: the chemical potential $\mu$ explicitly breaks time-reversal symmetry and precludes emergent Lorentz invariance [17–20]. Of the two Goldstone modes present at $\mu=0$, only one remains gapless (the phonon), while the other becomes a sharp but gapped (pseudo-)Goldstone mode, with a gap set by $\mu$ [21, 22]. The scale $\mu$ introduces a UV length scale $1/\mu \propto R/\sqrt{S}$, where $S \gg 1$ is the eigenvalue of $\hat{Q}^z$ for the states of interest. In this regime, the CFT at large charge is described by a weakly interacting effective theory for the gapless and gapped Goldstone modes, and the expansion (29) applies.

Let us now make the large-$S$ expansion explicit for the lowest-lying states in each symmetry sector. For the scalar primary with $S^{\kappa(S)} = 1^-, 2^+, \dots$ (with $\kappa(S) = (-1)^S$) and $L=0$, the scaling dimension admits the expansion

$$ \begin{aligned} \Delta_0(S^{\kappa(S)}, L=0) = & \alpha (S^{\kappa(S)})^{3/2} + \beta (S^{\kappa(S)})^{1/2} \\ & - 0.0937256 + \gamma (S^{\kappa(S)})^{-1/2} \\ & + \mathcal{O}\left(\frac{1}{S^{\kappa(S)}}\right), \end{aligned} \qquad (30) $$

where $\alpha, \beta$, and $\gamma$ are Wilson coefficients. These can be fixed by matching to the scaling dimensions of the lowest scalar primaries, as estimated from the conformal bootstrap (CB):

$$ \begin{aligned} \Delta_{\sigma}(1^{-}, 0) & \approx 0.518936, \\ \Delta_{t_2}(2^{+}, 0) & \approx 1.20954, \\ \Delta_{t_4}(4^{+}, 0) & \approx 2.99056. \end{aligned} \qquad (31) $$

Solving, we find

$$ \alpha \approx 0.31076, \quad \beta \approx 0.29818, \quad \gamma \approx 0.00370. \qquad (32) $$

This expansion captures the scaling of the lowest-energy state in each $S^{\kappa(S)}$ sector, and the alternation of parity $\kappa(S) = (-1)^S$ is confirmed numerically.
For the lowest-lying state in the $S^{\kappa(S+1)} = 1^+, 2^-, \dots$ sector (with $\kappa(S) = (-1)^S$) and $L = 0, 1, \dots$, corresponding to the gapped Goldstone mode, the scaling dimension is predicted [8, 9] to be

$$ \Delta(S^{\kappa(S+1)}, L) = \Delta_0(S^{\kappa(S)}, 0) + \mu(S^{\kappa(S)}) \\ + c \frac{L(L+1)}{2\mu(S^{\kappa(S)})} + \mathcal{O}\left(\frac{L^4}{\mu^3(S^{\kappa(S)})}\right), \qquad (33a) $$

where

$$ \mu(S) := \frac{\partial \Delta_0(S, 0)}{\partial S} \qquad (33b) $$

is the chemical potential, and $c$ is a Wilson coefficient. The term proportional to $L(L+1)$ describes the gapped dispersion of the pseudo-Goldstone mode. The value of $c$ can be fixed by matching to the scaling dimension of the conserved current operator, $\Delta_{j\mu}(1^+, 1) = 2$.

The gapless phonon mode is also captured in the large-$S$ expansion. For the $S^{\kappa(S)} = 1^-, 2^+, \dots$ sector with $L = 0, 1, 2, \dots$, the scaling dimension is given by [8, 9]

$$ \Delta_0(S^{\kappa(S)}, L) = \Delta_0(S^{\kappa(S)}, 0) + \sum_L n_L \omega_L, \qquad (34) \\ \omega_L = \sqrt{L(L+1)/2}, $$

where $n_L$ is the occupation number of the phonon with angular momenta $L$ and single-particle energy $\omega_L$. Phonons with $L = 1$ generate descendants, while $L \ge 2$ correspond to new primary operators.

VIII. DETAILS ON THE SIMULATIONS

We use exact diagonalization (ED) and density matrix renormalization group (DMRG) to determine the critical point and extract the conformal data. In ED, we exploit all available symmetries of the Hamiltonian to reduce computational cost and access larger system sizes.

ED simulations.— We first implement the $U(1)$ symmetry associated with orbital $SO(3)$ by conserving $L_z$, which partitions the Hilbert space into sectors labeled by $L_z$. Although ED does not resolve the total angular momenta $L$, each $L$ multiplet spans eigenstates with $L_z = -L, \dots, L$. This structure can be inverted to recover $L$ by comparing eigenvalues across different $L_z$ sectors. For example, if an eigenvalue is present at $L_z = 3$ but absent at $L_z = 4$, it must belong to an $L = 3$ multiplet. By recursively eliminating contributions from higher-$L$ multiplets in lower-$L_z$ sectors, the remaining unmatched eigenvalues at $L_z = \ell$ can be uniquely assigned to $L = \ell$. This method enables multiplet resolution even when only $L_z$ is conserved.

Further refinement is achieved by exploiting the $L_z \to -L_z$ symmetry in the $L_z = 0$ sector. This symmetry allows the Hilbert space at $L_z = 0$ to be decomposed into even and odd parity sectors, depending on the behavior of the eigenstate under $L_z$ sign reversal. Since even-$L$ multiplets have even parity under this transformation and odd-$L$ multiplets have odd parity, the $L_z = 0$ even and odd sectors effectively separate even-$L$ and odd-$L$ contributions. For instance, an eigenvalue identified as part of an $L = 3$ multiplet from the $L_z = 3$ sector must have a corresponding $L_z = 0$ odd sector, any remaining eigenvalues must arise from the $L = 1$ multiplet. Consequently, the $L_z = 1$ sector becomes redundant for multiplet reconstruction, as all information it contains can be extracted from symmetry-resolved $L_z = 0$ data.

Similarly, we use the $U(1)$ symmetry generated by $S_z$ to resolve spin sectors, along with the symmetry $S_z \to -S_z$ in the $S_z = 0$ sector. This splits the $S_z = 0$ space into even and odd parity sectors and allows us to reconstruct spin multiplets in the same manner as for angular momenta.

We also utilize the discrete $Z_2$ subgroup of the internal $O(3)$ symmetry. This symmetry classifies states according to their parity under internal inversion and further reduces the Hilbert space by separating sectors that do not mix.

Combining all these symmetries, the largest Hilbert space sector becomes $(L_z, S_z) = (2, 2)$ instead of the typical $(0, 0)$. Using this fully symmetry-resolved basis, we simulate systems with up to 12 electrons in 48 orbitals. The full Hilbert space of dimension around 70 billion is reduced to around 300 million in the largest sector. ED is parallelized across all symmetry sectors.
DMRG simulations.— DMRG simulations use the same base code as Ref. 5. We use ITensors.jl [23, 24] as the underlining tensor network library. The four degrees of freedom representing the O(3) rotor on the fuzzy sphere are represented as a chain of 4 × N spinless electrons, in ascending order of orbital momenta. We implemented an Abelian subgroup of O(3) (conservation of $l_z$ and of the parity of $l$), as well as the U(1) conservation of the electronic charge and the orbital momenta. The MPO were computed as standard, discarding contributions of amplitude $V_{m_1,m_2,m_3,m_4}$ below $10^{-12}$ and compressed using the default compression of ITensors.jl. We use noisy two-site DMRG [25, 26] to be ergodic in phase space due to the complexity of the model. The amplitude of the noise was generally kept at $10^{-5}$. Our convergence criterion at a given $\chi$ are variations of the energy and the mid-chain entropy below $10^{-7}$. To compute excited states, we successively incorporate into the effective Hamiltonian the weighted projectors onto previously-computed low-energy states.

## IX. SCALING DIMENSION DATA FOR ALL SYMMETRY SECTORS

### A. $S=0$:

<figure><img src="image_7.png" /><figcaption>FIG. 1: Spectrum in the $S=0^+$ sector.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 2: Spectrum in the $S=0^-$ sector.</figcaption></figure>
<table><caption>TABLE I: Scaling dimension for the sector $S = 0^+$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (Ω)</td><td>ED</td><td>0.0000 (12), 0.0000 (11), 0.0000 (10)</td><td>4 (ε')</td><td>ε</td><td>6.0000 (0), 7.6364 (1), 3.7720 (2)</td></tr><tr><td>1 (Ω)</td><td>DMRG</td><td>0.0000 (26), 0.0000 (24), 0.0000 (22)</td><td>4 (ε')</td><td>1/N</td><td>6.0000 (0), 4.9192 (1), 6.1362 (2)</td></tr><tr><td>1 (Ω)</td><td>ε</td><td>0.0000 (0)</td><td>5</td><td>ED</td><td>4.4389 (12), 4.3820 (11), 4.3200 (10)</td></tr><tr><td>1 (Ω)</td><td>1/N</td><td>0.0000 (0)</td><td>5</td><td>ε</td><td>8.0000 (0), 12.3636 (1), 0.9512 (2)</td></tr><tr><td>2 (ε)</td><td>ED</td><td>1.5619 (12), 1.5533 (11), 1.5434 (10)</td><td>5</td><td>1/N</td><td>8.0000 (0), 6.5590 (1), 11.0212 (2)</td></tr><tr><td>2 (ε)</td><td>DMRG</td><td>1.6178 (26), 1.6114 (24), 1.6065 (22)</td><td>6</td><td>ED</td><td>5.5679 (11), 5.5399 (10), 5.5163 (9)</td></tr><tr><td>2 (ε)</td><td>ε</td><td>2.0000 (0), 1.4545 (1), 1.6104 (2), 1.4954 (3), 1.7890 (4)</td><td>6</td><td>ε</td><td>10.0000 (0), 18.1818 (1)</td></tr><tr><td>2 (ε)</td><td>1/N</td><td>2.0000 (0), 1.6397 (1), 1.3011 (2)</td><td>6</td><td>1/N</td><td>10.0000 (0), 8.1987 (1), 18.4517 (2)</td></tr><tr><td>3 (∂∂ε<sup>(0)</sup>)</td><td>ED</td><td>3.5492 (12), 3.5512 (11), 3.5526 (10)</td><td>7</td><td>ED</td><td>5.6467 (11), 5.6298 (10), 5.6091 (9)</td></tr><tr><td>3 (∂∂ε<sup>(0)</sup>)</td><td>DMRG</td><td>3.5582 (14)</td><td>8</td><td>ED</td><td>5.8026 (11), 5.8577 (10), 5.8002 (9)</td></tr><tr><td>3 (∂∂ε<sup>(0)</sup>)</td><td>ε</td><td>4.0000 (0), 4.0000 (1), 3.4298 (2), 4.7127 (3), 0.9315 (4)</td><td>9</td><td>ED</td><td>6.1103 (10), 6.0896 (9), 6.0600 (8)</td></tr><tr><td>3 (∂∂ε<sup>(0)</sup>)</td><td>1/N</td><td>4.0000 (0), 3.2795 (1), 3.1214 (2)</td><td>10</td><td>ED</td><td>6.2424 (10), 6.3343 (9), 6.1421 (8)</td></tr><tr><td>4 (ε')</td><td>ED</td><td>3.7728 (12), 3.7623 (11), 3.7534 (10)</td><td>11</td><td>ED</td><td>6.2569 (10), 6.4425 (9), 6.1518 (8)</td></tr><tr><td>4 (ε')</td><td>DMRG</td><td>3.8018 (14)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE II: Scaling dimension for the sector $S = 0^+$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂ε)</td><td>ED</td><td>2.5214 (12), 2.5092 (11), 2.4949 (10)</td><td>4</td><td>ED</td><td>5.1604 (11), 5.0957 (10), 5.0289 (9)</td></tr><tr><td>1 (∂ε)</td><td>DMRG</td><td>2.6058 (24), 2.5907 (20), 2.5777 (18)</td><td>5</td><td>ED</td><td>5.3386 (11), 5.3058 (10), 5.2748 (9)</td></tr><tr><td>2 (∂∂∂ε<sup>(1)</sup>)</td><td>ED</td><td>4.5680 (12), 4.5518 (11), 4.5289 (10)</td><td>6</td><td>ED</td><td>5.5565 (11), 5.5140 (10), 5.4739 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.7811 (12), 4.7817 (11), 4.7842 (10)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE III: Scaling dimension for the sector $S = 0^+$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (T<sub>μν</sub>)</td><td>ED</td><td>2.9775 (12), 2.9731 (11), 2.9678 (10)</td><td>5</td><td>ED</td><td>4.6677 (11), 4.5933 (10), 4.5132 (9)</td></tr><tr><td>1 (T<sub>μν</sub>)</td><td>DMRG</td><td>3.0090 (24), 3.0106 (20), 3.0042 (18)</td><td>6</td><td>ED</td><td>5.0871 (11), 5.0647 (10), 5.0336 (9)</td></tr><tr><td>2 (∂∂ε<sup>(2)</sup>)</td><td>ED</td><td>3.4715 (12), 3.4485 (11), 3.4209 (10)</td><td>7</td><td>ED</td><td>5.1936 (11), 5.1512 (10), 5.0958 (9)</td></tr><tr><td>2 (∂∂ε<sup>(2)</sup>)</td><td>DMRG</td><td>3.6154 (24), 3.5736 (18), 3.5456 (16)</td><td>8</td><td>ED</td><td>5.3751 (11), 5.3196 (10), 5.2538 (9)</td></tr><tr><td>3 (ε∂T<sub>μν</sub>)</td><td>ED</td><td>4.0733 (11), 4.0694 (10), 4.0605 (9)</td><td>9</td><td>ED</td><td>5.3942 (11), 5.3376 (10), 5.2860 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.3268 (11), 4.2745 (10), 4.2185 (9)</td><td>10</td><td>ED</td><td>5.5768 (11), 5.5570 (10), 5.5292 (9)</td></tr></tbody></table>

<table><caption>TABLE IV: Scaling dimension for the sector $S = 0^+$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂T<sub>μν</sub>)</td><td>ED</td><td>3.8970 (11), 3.8663 (10), 3.8265 (9)</td><td>5</td><td>ED</td><td>5.3751 (11), 5.2893 (10), 5.1948 (9)</td></tr><tr><td>1 (∂T<sub>μν</sub>)</td><td>DMRG</td><td>4.0529 (24)</td><td>6</td><td>ED</td><td>5.7486 (11), 5.6716 (10), 5.5783 (9)</td></tr><tr><td>2 (∂∂ε<sup>(3)</sup>)</td><td>ED</td><td>4.2604 (11), 4.2005 (10), 4.1296 (9)</td><td>7</td><td>ED</td><td>5.8559 (11), 5.7642 (10), 5.6604 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.9108 (11), 4.8447 (10), 4.7661 (9)</td><td>8</td><td>ED</td><td>5.9263 (11), 5.8897 (10), 5.8065 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.0701 (11), 5.0077 (10), 4.9297 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE V: Scaling dimension for the sector $S = 0^+$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.5766 (11), 4.4834 (10), 4.3665 (9)</td><td>6</td><td>ED</td><td>5.6082 (11), 5.4957 (10), 5.3586 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.8152 (11), 4.7233 (10), 4.6159 (9)</td><td>7</td><td>ED</td><td>5.7541 (11), 5.6541 (10), 5.5432 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.9547 (11), 4.8895 (10), 4.8091 (9)</td><td>8</td><td>ED</td><td>5.9585 (11), 5.8266 (10), 5.6734 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.4425 (11), 5.3297 (10), 5.1975 (9)</td><td>9</td><td>ED</td><td>6.0619 (11), 5.9813 (10), 5.8504 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.5136 (11), 5.4339 (10), 5.3445 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE VI: Scaling dimension for the sector $S = 0^+$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.9507 (11), 4.7893 (10), 4.6050 (9)</td><td>6</td><td>ED</td><td>6.0837 (11), 5.9411 (10), 5.7783 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.1981 (11), 5.0605 (10), 4.9019 (9)</td><td>7</td><td>ED</td><td>6.2203 (11), 6.0680 (10), 5.8828 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.5209 (11), 5.3891 (10), 5.2320 (9)</td><td>8</td><td>ED</td><td>6.3105 (11), 6.1723 (10), 5.9956 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.7672 (11), 5.5989 (10), 5.4018 (9)</td><td>9</td><td>ED</td><td>6.3631 (11), 6.1897 (10), 6.0337 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.9554 (11), 5.7946 (10), 5.6073 (9)</td><td>10</td><td>ED</td><td>6.3613 (10), 6.1933 (9), 6.0181 (8)</td></tr></tbody></table>

<table><caption>TABLE VII: Scaling dimension for the sector $S = 0^+$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.1130 (11), 4.9142 (10), 4.6978 (9)</td><td>6</td><td>ED</td><td>6.1250 (10), 5.9081 (9), 5.6583 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.2177 (10), 5.0150 (9), 4.7895 (8)</td><td>7</td><td>ED</td><td>6.2000 (10), 5.9805 (9), 5.7048 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.5627 (10), 5.3421 (9), 5.0903 (8)</td><td>8</td><td>ED</td><td>6.2363 (10), 6.0608 (9), 5.8563 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.7306 (10), 5.4865 (9), 5.2085 (8)</td><td>9</td><td>ED</td><td>6.2791 (10), 6.0859 (9), 6.0601 (8)</td></tr><tr><td>5</td><td>ED</td><td>5.9562 (10), 5.7315 (9), 5.4784 (8)</td><td>10</td><td>ED</td><td>6.4977 (10), 6.3031 (9), 6.1167 (8)</td></tr></tbody></table>

<table><caption>TABLE VIII: Scaling dimension for the sector $S = 0^+$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.9669 (10), 4.7393 (9)</td><td>6</td><td>ED</td><td>6.1595 (10), 5.9030 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.2696 (10), 5.0470 (9)</td><td>7</td><td>ED</td><td>6.2812 (10), 6.0202 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.5836 (10), 5.3387 (9)</td><td>8</td><td>ED</td><td>6.3428 (10), 6.1994 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.7507 (10), 5.4652 (9)</td><td>9</td><td>ED</td><td>6.4283 (10), 6.3599 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.0309 (10), 5.7842 (9)</td><td>10</td><td>ED</td><td>6.5796 (10), 6.3630 (9)</td></tr></tbody></table>
<table><caption>TABLE IX: Scaling dimension for the sector $S = 0^-$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂ε<sub>μ</sub><sup>(0)</sup>)</td><td>ED</td><td>3.8465 (11), 3.8200 (10), 3.7934 (9)</td><td>2</td><td>ED</td><td>5.8726 (11), 5.8546 (10), 5.8325 (9)</td></tr><tr><td>1 (∂ε<sub>μ</sub><sup>(0)</sup>)</td><td>ε</td><td>6.0000 (0), 4.5000 (1)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE X: Scaling dimension for the sector $S = 0^-$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (ε<sub>μ</sub>)</td><td>ED</td><td>2.9607 (12), 2.9231 (11), 2.8816 (10)</td><td>5</td><td>ED</td><td>5.2288 (11), 5.1885 (10), 5.1464 (9)</td></tr><tr><td>1 (ε<sub>μ</sub>)</td><td>DMRG</td><td>3.2396 (24), 3.1372 (18), 3.0859 (16)</td><td>6</td><td>ED</td><td>5.4907 (11), 5.4758 (10), 5.4480 (9)</td></tr><tr><td>2 (ε∂ε<sub>μ</sub>)</td><td>ED</td><td>3.8797 (12), 3.8594 (11), 3.8394 (10)</td><td>7</td><td>ED</td><td>5.6812 (11), 5.6260 (10), 5.5851 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.7683 (12), 4.7462 (11), 4.7231 (10)</td><td>8</td><td>ED</td><td>5.8775 (11), 5.8512 (10), 5.8119 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.9282 (12), 4.9136 (11), 4.8970 (10)</td><td>9</td><td>ED</td><td>6.0208 (11), 5.9360 (10), 5.8569 (9)</td></tr></tbody></table>

<table><caption>TABLE XI: Scaling dimension for the sector $S = 0^-$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂ε<sub>μ</sub><sup>(2)</sup>)</td><td>ED</td><td>3.6813 (12), 3.6319 (11), 3.5776 (10)</td><td>5</td><td>ED</td><td>5.5374 (11), 5.4948 (10), 5.4406 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.5118 (11), 4.4649 (10), 4.4160 (9)</td><td>6</td><td>ED</td><td>5.6334 (11), 5.5610 (10), 5.4874 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.7853 (11), 4.7559 (10), 4.7235 (9)</td><td>7</td><td>ED</td><td>5.7835 (11), 5.7242 (10), 5.6494 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.4114 (11), 5.3630 (10), 5.3090 (9)</td><td>8</td><td>ED</td><td>5.8271 (10), 5.7405 (9), 5.6441 (8)</td></tr></tbody></table>
<table><caption>TABLE XII: Scaling dimension for the sector $S = 0^-, L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>3.9558 (11), 3.8915 (10), 3.8167 (9)</td><td>5</td><td>ED</td><td>5.2900 (11), 5.2361 (10), 5.1915 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.2468 (11), 4.1646 (10), 4.0711 (9)</td><td>6</td><td>ED</td><td>5.4046 (11), 5.3422 (10), 5.2740 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.6344 (11), 4.5495 (10), 4.4567 (9)</td><td>7</td><td>ED</td><td>5.6168 (11), 5.5420 (10), 5.4541 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.1711 (11), 5.1314 (10), 5.0674 (9)</td><td>8</td><td>ED</td><td>5.7798 (11), 5.6935 (10), 5.5992 (9)</td></tr></tbody></table>

<table><caption>TABLE XIII: Scaling dimension for the sector $S = 0^-, L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.5031 (11), 4.4034 (10), 4.2886 (9)</td><td>5</td><td>ED</td><td>5.8664 (11), 5.8002 (10), 5.7140 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.7047 (11), 4.5797 (10), 4.4369 (9)</td><td>6</td><td>ED</td><td>5.9833 (11), 5.8903 (10), 5.7832 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.0960 (11), 4.9784 (10), 4.8487 (9)</td><td>7</td><td>ED</td><td>6.0862 (11), 6.0018 (10), 5.8591 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6366 (11), 5.5198 (10), 5.3910 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XIV: Scaling dimension for the sector $S = 0^-, L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.8462 (11), 4.7038 (10), 4.5425 (9)</td><td>6</td><td>ED</td><td>6.2600 (11), 6.1198 (10), 5.9537 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.9976 (11), 4.8289 (10), 4.6400 (9)</td><td>7</td><td>ED</td><td>6.3262 (11), 6.1749 (10), 5.9972 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.3805 (11), 5.2243 (10), 5.0495 (9)</td><td>8</td><td>ED</td><td>6.3790 (11), 6.2222 (10), 6.0399 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6873 (11), 5.5854 (10), 5.4647 (9)</td><td>9</td><td>ED</td><td>6.5134 (11), 6.3601 (10), 6.1892 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.8466 (11), 5.6970 (10), 5.5396 (9)</td><td>10</td><td>ED</td><td>6.4389 (10), 6.2714 (9), 6.0832 (8)</td></tr></tbody></table>
<table><caption>TABLE XV: Scaling dimension for the sector $S = 0^-$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.0167 (11), 4.8396 (10), 4.6451 (9)</td><td>6</td><td>ED</td><td>6.2784 (10), 6.0591 (9), 5.8148 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.0923 (11), 4.9449 (10), 4.7204 (9)</td><td>7</td><td>ED</td><td>6.3025 (10), 6.0964 (9), 5.8442 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.3207 (10), 5.1020 (9), 4.8583 (8)</td><td>8</td><td>ED</td><td>6.3445 (10), 6.1172 (9), 6.0792 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.7780 (10), 5.6165 (9), 5.4761 (8)</td><td>9</td><td>ED</td><td>6.5120 (10), 6.3054 (9), 6.1742 (8)</td></tr><tr><td>5</td><td>ED</td><td>5.9033 (10), 5.7251 (9), 5.5152 (8)</td><td>10</td><td>ED</td><td>6.5837 (10), 6.4015 (9), 6.2423 (8)</td></tr></tbody></table>

<table><caption>TABLE XVI: Scaling dimension for the sector $S = 0^-$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.8979 (10), 4.6900 (9), 4.4578 (8)</td><td>6</td><td>ED</td><td>6.3169 (10), 6.0637 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.9697 (10), 4.7244 (9)</td><td>7</td><td>ED</td><td>6.3439 (10), 6.0966 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.3183 (10), 5.0716 (9)</td><td>8</td><td>ED</td><td>6.3827 (10), 6.3456 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.8361 (10), 5.6977 (9)</td><td>9</td><td>ED</td><td>6.5678 (10), 6.4360 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.0313 (10), 5.8044 (9)</td><td>10</td><td>ED</td><td>6.6739 (10), 6.4762 (9)</td></tr></tbody></table>
B. $S = 1$:

<figure><img src="image_9.png" /><figcaption>FIG. 3: Spectrum in the $S = 1^+$ sector.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 4: Spectrum in the $S = 1^-$ sector.</figcaption></figure>

<table><caption>TABLE XVII: Scaling dimension for the sector $S = 1^+$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.9527 (11), 4.9255 (10), 4.8969 (9)</td><td>3</td><td>ED</td><td>5.2989 (11), 5.2711 (10), 5.2480 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.9828 (11), 4.9628 (10), 4.9430 (9)</td><td>4</td><td>ED</td><td>6.3998 (10), 6.3982 (9), 6.1968 (7)</td></tr></tbody></table>
<table><caption>TABLE XVIII: Scaling dimension for the sector $S = 1^+$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (j<sub>μ</sub>)</td><td>ED</td><td>1.8825 (12), 1.8669 (11), 1.8483 (10)</td><td>3 (φ<sub>μ</sub>)</td><td>ε</td><td>7.0000 (0), 7.4545 (1)</td></tr><tr><td>1 (j<sub>μ</sub>)</td><td>DMRG</td><td>1.9720 (26), 1.9619 (24), 1.9564 (22)</td><td>3 (φ<sub>μ</sub>)</td><td>1/N</td><td>3.0000 (0), 6.0000 (1)</td></tr><tr><td>1 (j<sub>μ</sub>)</td><td>ε</td><td>3.0000 (0), 2.0000 (1)</td><td>4</td><td>ED</td><td>3.9435 (12), 3.9180 (11), 3.8904 (10)</td></tr><tr><td>1 (j<sub>μ</sub>)</td><td>1/N</td><td>2.0000 (0)</td><td>5</td><td>ED</td><td>4.0142 (12), 3.9775 (11), 3.9373 (10)</td></tr><tr><td>2 (ε∂j<sub>μ</sub>)</td><td>ED</td><td>2.6825 (12), 2.6644 (11), 2.6461 (10)</td><td>6</td><td>ED</td><td>4.4245 (12), 4.3675 (11), 4.3052 (10)</td></tr><tr><td>2 (ε∂j<sub>μ</sub>)</td><td>DMRG</td><td>2.8546 (24), 2.8262 (20), 2.7937 (18)</td><td>7</td><td>ED</td><td>4.7726 (12), 4.7663 (11), 4.7533 (10)</td></tr><tr><td>2 (ε∂j<sub>μ</sub>)</td><td>ε</td><td>5.0000 (0), 4.1818 (1)</td><td>8</td><td>ED</td><td>4.9129 (11), 4.9080 (10), 4.8877 (9)</td></tr><tr><td>2 (ε∂j<sub>μ</sub>)</td><td>1/N</td><td>1.0000 (0), 4.0000 (1)</td><td>9</td><td>ED</td><td>4.9721 (11), 4.9544 (10), 4.9542 (9)</td></tr><tr><td>3 (φ<sub>μ</sub>)</td><td>ED</td><td>3.7083 (12), 3.7108 (11), 3.7147 (10)</td><td>10</td><td>ED</td><td>5.1663 (11), 5.1027 (10), 5.0381 (9)</td></tr><tr><td>3 (φ<sub>μ</sub>)</td><td>DMRG</td><td>3.7570 (18), 3.7364 (16), 3.7190 (14)</td><td>11</td><td>ED</td><td>5.3278 (11), 5.2990 (10), 5.2723 (9)</td></tr></tbody></table>

<table><caption>TABLE XIX: Scaling dimension for the sector $S = 1^+$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂j<sub>μ</sub>)</td><td>ED</td><td>2.7159 (12), 2.6806 (11), 2.6403 (10)</td><td>6</td><td>ED</td><td>4.7258 (11), 4.6670 (10), 4.6017 (9)</td></tr><tr><td>1 (∂j<sub>μ</sub>)</td><td>DMRG</td><td>2.9292 (24), 2.8984 (20), 2.8678 (18)</td><td>7</td><td>ED</td><td>4.8183 (11), 4.8033 (10), 4.7848 (9)</td></tr><tr><td>2 (ε∂j<sub>μ</sub>)</td><td>ED</td><td>3.5974 (11), 3.5795 (10), 3.5586 (9)</td><td>8</td><td>ED</td><td>5.0833 (11), 5.0148 (10), 4.9404 (9)</td></tr><tr><td>2 (ε∂j<sub>μ</sub>)</td><td>DMRG</td><td>3.7893 (24), 3.7215 (18), 3.6818 (16)</td><td>9</td><td>ED</td><td>5.2251 (11), 5.1750 (10), 5.1144 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.3147 (11), 4.2546 (10), 4.1896 (9)</td><td>10</td><td>ED</td><td>5.2641 (11), 5.2082 (10), 5.1558 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.4810 (11), 4.4158 (10), 4.3425 (9)</td><td>11</td><td>ED</td><td>5.5686 (11), 5.5326 (10), 5.4877 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.6611 (11), 4.6149 (10), 4.5571 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE XX: Scaling dimension for the sector $S = 1^+$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂∂j<sub>μ</sub>)</td><td>ED</td><td>3.3328 (11), 3.2650 (10), 3.1878 (9)</td><td>7</td><td>ED</td><td>5.0284 (11), 4.9468 (10), 4.8496 (9)</td></tr><tr><td>1 (∂∂j<sub>μ</sub>)</td><td>DMRG</td><td>3.8055 (24), 3.6704 (18), 3.5861 (16)</td><td>8</td><td>ED</td><td>5.1000 (11), 5.0496 (10), 4.9593 (9)</td></tr><tr><td>1 (∂∂j<sub>μ</sub>)</td><td>ε</td><td>5.0000 (0), 4.0000 (1), 4.0172 (2), 4.0331 (3)</td><td>9</td><td>ED</td><td>5.1602 (11), 5.0800 (10), 5.0230 (9)</td></tr><tr><td>1 (∂∂j<sub>μ</sub>)</td><td>1/N</td><td>1.0000 (0), 4.0000 (1), 4.0823 (2), 4.0658 (3)</td><td>10</td><td>ED</td><td>5.3170 (11), 5.2260 (10), 5.1211 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.0654 (11), 4.0529 (10), 4.0353 (9)</td><td>11</td><td>ED</td><td>5.3305 (11), 5.2363 (10), 5.1289 (9)</td></tr><tr><td>2</td><td>DMRG</td><td>4.1178 (24), 4.0236 (16)</td><td>12</td><td>ED</td><td>5.4695 (11), 5.3999 (10), 5.3212 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.4157 (11), 4.3589 (10), 4.2886 (9)</td><td>13</td><td>ED</td><td>5.5570 (11), 5.4743 (10), 5.3818 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.6994 (11), 4.6160 (10), 4.5212 (9)</td><td>14</td><td>ED</td><td>5.5950 (11), 5.5147 (10), 5.4143 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.8350 (11), 4.7625 (10), 4.6803 (9)</td><td>15</td><td>ED</td><td>5.6795 (11), 5.6103 (10), 5.5195 (9)</td></tr><tr><td>6</td><td>ED</td><td>4.9493 (11), 4.8672 (10), 4.7749 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XXI: Scaling dimension for the sector $S = 1^+$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂∂∂j<sub>μ</sub>)</td><td>ED</td><td>3.8063 (11), 3.6998 (10), 3.5774 (9)</td><td>8</td><td>ED</td><td>5.5513 (11), 5.4337 (10), 5.3014 (9)</td></tr><tr><td>1 (∂∂∂j<sub>μ</sub>)</td><td>DMRG</td><td>4.5565 (24), 4.3335 (18), 4.2069 (16)</td><td>9</td><td>ED</td><td>5.6744 (11), 5.5476 (10), 5.4004 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.8267 (11), 4.7381 (10), 4.6230 (9)</td><td>10</td><td>ED</td><td>5.6997 (11), 5.5834 (10), 5.4378 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.0163 (11), 4.9240 (10), 4.8208 (9)</td><td>11</td><td>ED</td><td>5.7598 (11), 5.6191 (10), 5.4610 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.2333 (11), 5.1163 (10), 4.9851 (9)</td><td>12</td><td>ED</td><td>5.7819 (11), 5.6632 (10), 5.5372 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.3302 (11), 5.2305 (10), 5.1147 (9)</td><td>13</td><td>ED</td><td>5.9617 (11), 5.8451 (10), 5.7096 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.4415 (11), 5.3199 (10), 5.1813 (9)</td><td>14</td><td>ED</td><td>6.0013 (11), 5.8841 (10), 5.7464 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.4721 (11), 5.3655 (10), 5.2432 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE XXII: Scaling dimension for the sector $S = 1^+$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.0906 (11), 3.9400 (10), 3.7700 (9)</td><td>5</td><td>ED</td><td>5.7034 (11), 5.5446 (10), 5.3641 (9)</td></tr><tr><td>1</td><td>ε</td><td>7.0000 (0), 6.0000 (1), 6.0193 (2), 6.0366 (3)</td><td>6</td><td>ED</td><td>5.7747 (11), 5.6223 (10), 5.4463 (9)</td></tr><tr><td>1</td><td>1/N</td><td>3.0000 (0), 6.0000 (1), 6.0873 (2), 6.0693 (3)</td><td>7</td><td>ED</td><td>5.8508 (11), 5.7136 (10), 5.5356 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.1501 (11), 4.9933 (10), 4.8144 (9)</td><td>8</td><td>ED</td><td>5.8852 (11), 5.7482 (10), 5.5929 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.4371 (11), 5.2931 (10), 5.1283 (9)</td><td>9</td><td>ED</td><td>5.9324 (11), 5.7912 (10), 5.6148 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6073 (11), 5.4532 (10), 5.2808 (9)</td><td>10</td><td>ED</td><td>5.9640 (11), 5.7983 (10), 5.6271 (9)</td></tr></tbody></table>

<table><caption>TABLE XXIII: Scaling dimension for the sector $S = 1^+$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.2206 (11), 4.0355 (10), 3.8328 (9)</td><td>6</td><td>ED</td><td>5.7691 (10), 5.5466 (9), 5.2931 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.0911 (10), 4.8848 (9), 4.6596 (8)</td><td>7</td><td>ED</td><td>5.8714 (10), 5.6539 (9), 5.3912 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.4320 (10), 5.2176 (9), 4.9785 (8)</td><td>8</td><td>ED</td><td>5.9094 (10), 5.6848 (9), 5.5047 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.6161 (10), 5.4024 (9), 5.1603 (8)</td><td>9</td><td>ED</td><td>5.9319 (10), 5.7133 (9), 5.5605 (8)</td></tr><tr><td>5</td><td>ED</td><td>5.6727 (10), 5.4537 (9), 5.2162 (8)</td><td>10</td><td>ED</td><td>6.0003 (10), 5.7711 (9), 5.6286 (8)</td></tr></tbody></table>

<table><caption>TABLE XXIV: Scaling dimension for the sector $S = 1^+$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.0579 (10), 3.8407 (9), 3.6083 (8)</td><td>6</td><td>ED</td><td>5.8131 (10), 5.5611 (9), 5.5738 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.1327 (10), 4.9190 (9), 4.9450 (8)</td><td>7</td><td>ED</td><td>5.9292 (10), 5.6690 (9), 5.6538 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.4485 (10), 5.2069 (9), 5.1384 (8)</td><td>8</td><td>ED</td><td>5.9590 (10), 5.8089 (9), 5.7928 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.6578 (10), 5.4111 (9), 5.2726 (8)</td><td>9</td><td>ED</td><td>5.9930 (10), 5.8581 (9), 5.9262 (8)</td></tr><tr><td>5</td><td>ED</td><td>5.7096 (10), 5.4793 (9), 5.3811 (8)</td><td>10</td><td>ED</td><td>6.0665 (10), 5.8947 (9), 5.9607 (8)</td></tr></tbody></table>
<table><caption>TABLE XXV: Scaling dimension for the sector $S = 1^-$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (σ)</td><td>ED</td><td>0.5207 (12), 0.5201 (11), 0.5196 (10)</td><td>3 (σ')</td><td>1/N</td><td>6.5000 (0), 6.5450 (1)</td></tr><tr><td>1 (σ)</td><td>DMRG</td><td>0.5197 (26), 0.5205 (24), 0.5196 (22)</td><td>4 (∂∂∂∂σ<sup>(0)</sup>)</td><td>ED</td><td>4.5544 (11), 4.5514 (10), 4.5482 (9)</td></tr><tr><td>1 (σ)</td><td>ε</td><td>1.0000 (0), 0.5000 (1), 0.5103 (2), 0.5195 (3), 0.5158 (4), 0.5260 (5), 0.4975 (6), 0.5946 (7), 0.2212 (8)</td><td>5</td><td>ED</td><td>5.0420 (11), 5.0255 (10), 5.0095 (9)</td></tr><tr><td>1 (σ)</td><td>1/N</td><td>0.5000 (0), 0.5450 (1), 0.5342 (2), 0.4994 (3)</td><td>6</td><td>ED</td><td>5.1259 (11), 5.1344 (10), 5.1525 (9)</td></tr><tr><td>2 (∂∂σ<sup>(0)</sup>)</td><td>ED</td><td>2.5164 (12), 2.5053 (11), 2.4924 (10)</td><td>7</td><td>ED</td><td>5.5228 (11), 5.4632 (10), 5.4009 (9)</td></tr><tr><td>2 (∂∂σ<sup>(0)</sup>)</td><td>ε</td><td>5.0000 (0), 5.5909 (1)</td><td>8</td><td>ED</td><td>5.7429 (11), 5.6680 (10), 5.5901 (9)</td></tr><tr><td>2 (∂∂σ<sup>(0)</sup>)</td><td>1/N</td><td>4.5000 (0), 4.5450 (1)</td><td>9</td><td>ED</td><td>5.8939 (10), 5.8712 (9), 5.8398 (8)</td></tr><tr><td>3 (σ')</td><td>ED</td><td>3.8733 (12), 3.8595 (11), 3.8458 (10)</td><td>10</td><td>ED</td><td>6.2140 (10), 6.1874 (9), 6.1545 (8)</td></tr><tr><td>3 (σ')</td><td>ε</td><td>7.0000 (0), 9.7727 (1)</td><td>11</td><td>ED</td><td>6.3657 (10), 6.3581 (9), 6.1888 (8)</td></tr></tbody></table>

<table><caption>TABLE XXVI: Scaling dimension for the sector $S = 1^-$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂σ)</td><td>ED</td><td>1.5197 (12), 1.5195 (11), 1.5194 (10)</td><td>5</td><td>ED</td><td>4.1899 (12), 4.1611 (11), 4.1314 (10)</td></tr><tr><td>1 (∂σ)</td><td>DMRG</td><td>1.5194 (26), 1.5196 (24), 1.5194 (22)</td><td>6</td><td>ED</td><td>4.7569 (12), 4.7419 (11), 4.7251 (10)</td></tr><tr><td>1 (∂σ)</td><td>ε</td><td>4.0000 (0), 2.9545 (1)</td><td>7</td><td>ED</td><td>4.8193 (11), 4.8093 (10), 4.8004 (9)</td></tr><tr><td>1 (∂σ)</td><td>1/N</td><td>3.5000 (0), 3.0047 (1)</td><td>8</td><td>ED</td><td>5.1422 (11), 5.1163 (10), 5.0867 (9)</td></tr><tr><td>2 (σ<sub>μ</sub>)</td><td>ED</td><td>2.8706 (12), 2.8465 (11), 2.8185 (10)</td><td>9</td><td>ED</td><td>5.1869 (11), 5.1564 (10), 5.1268 (9)</td></tr><tr><td>2 (σ<sub>μ</sub>)</td><td>DMRG</td><td>3.0437 (26), 3.0190 (24), 3.0084 (22)</td><td>10</td><td>ED</td><td>5.3474 (11), 5.3760 (10), 5.3944 (9)</td></tr><tr><td>3 (∂∂∂σ<sup>(1)</sup>)</td><td>ED</td><td>3.4782 (12), 3.4619 (11), 3.4427 (10)</td><td>11</td><td>ED</td><td>5.4546 (11), 5.4507 (10), 5.4070 (9)</td></tr><tr><td>3 (∂∂∂σ<sup>(1)</sup>)</td><td>DMRG</td><td>3.5880 (24), 3.5631 (18), 3.5348 (16)</td><td>12</td><td>ED</td><td>5.5302 (11), 5.4959 (10), 5.5087 (9)</td></tr><tr><td>4</td><td>ED</td><td>3.7443 (12), 3.7236 (11), 3.7019 (10)</td><td>13</td><td>ED</td><td>5.5678 (11), 5.5579 (10), 5.5532 (9)</td></tr><tr><td>4</td><td>DMRG</td><td>3.8365 (16), 3.7982 (14)</td><td>14</td><td>ED</td><td>5.7838 (11), 5.7084 (10), 5.6324 (9)</td></tr></tbody></table>
<table><caption>TABLE XXVII: Scaling dimension for the sector $S = 1^{-}$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂∂σ<sup>(2)</sup>)</td><td>ED</td><td>2.5642 (12), 2.5588 (11), 2.5511 (10)</td><td>7</td><td>ED</td><td>4.3751 (11), 4.3464 (10), 4.3172 (9)</td></tr><tr><td>1 (∂∂σ<sup>(2)</sup>)</td><td>DMRG</td><td>2.5729 (26), 2.5722 (24), 2.5749 (22)</td><td>8</td><td>ED</td><td>4.6179 (11), 4.5854 (10), 4.5474 (9)</td></tr><tr><td>2</td><td>ED</td><td>3.3557 (12), 3.3155 (11), 3.2689 (10)</td><td>9</td><td>ED</td><td>4.9881 (11), 4.9512 (10), 4.9100 (9)</td></tr><tr><td>2</td><td>DMRG</td><td>3.5937 (24), 3.5700 (20), 3.5337 (18)</td><td>10</td><td>ED</td><td>5.0579 (11), 5.0276 (10), 4.9914 (9)</td></tr><tr><td>3</td><td>ED</td><td>3.6584 (11), 3.6098 (10), 3.5542 (9)</td><td>11</td><td>ED</td><td>5.2208 (11), 5.2027 (10), 5.1790 (9)</td></tr><tr><td>3</td><td>DMRG</td><td>3.9841 (24), 3.9061 (18), 3.8469 (16)</td><td>12</td><td>ED</td><td>5.3232 (11), 5.2994 (10), 5.2676 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.0371 (11), 4.0067 (10), 3.9723 (9)</td><td>13</td><td>ED</td><td>5.4556 (11), 5.4031 (10), 5.3426 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.2043 (11), 4.1791 (10), 4.1527 (9)</td><td>14</td><td>ED</td><td>5.4706 (11), 5.4210 (10), 5.3763 (9)</td></tr><tr><td>6</td><td>ED</td><td>4.3020 (11), 4.2790 (10), 4.2504 (9)</td><td>15</td><td>ED</td><td>5.5574 (11), 5.5195 (10), 5.4698 (9)</td></tr></tbody></table>

<table><caption>TABLE XXVIII: Scaling dimension for the sector $S = 1^{-}$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂∂∂σ<sup>(3)</sup>)</td><td>ED</td><td>3.4317 (11), 3.3837 (10), 3.3231 (9)</td><td>8</td><td>ED</td><td>5.1156 (11), 5.0675 (10), 5.0112 (9)</td></tr><tr><td>1 (∂∂∂σ<sup>(3)</sup>)</td><td>DMRG</td><td>3.6279 (24), 3.6043 (20), 3.6012 (18)</td><td>9</td><td>ED</td><td>5.2135 (11), 5.1644 (10), 5.1062 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.0296 (11), 3.9647 (10), 3.8907 (9)</td><td>10</td><td>ED</td><td>5.2957 (11), 5.2578 (10), 5.2014 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.3016 (11), 4.2209 (10), 4.1285 (9)</td><td>11</td><td>ED</td><td>5.3981 (11), 5.3411 (10), 5.2839 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.5139 (11), 4.4933 (10), 4.4681 (9)</td><td>12</td><td>ED</td><td>5.5137 (11), 5.4634 (10), 5.3969 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.7490 (11), 4.6948 (10), 4.6328 (9)</td><td>13</td><td>ED</td><td>5.6106 (11), 5.5674 (10), 5.5102 (9)</td></tr><tr><td>6</td><td>ED</td><td>4.8163 (11), 4.7667 (10), 4.7101 (9)</td><td>14</td><td>ED</td><td>5.6774 (11), 5.6066 (10), 5.5351 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.0115 (11), 4.9435 (10), 4.8563 (9)</td><td>15</td><td>ED</td><td>5.7799 (11), 5.7000 (10), 5.6110 (9)</td></tr></tbody></table>
<table><caption>TABLE XXIX: Scaling dimension for the sector $S = 1^{-}$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>3.9886 (11), 3.8814 (10), 3.7548 (9)</td><td>9</td><td>ED</td><td>5.6352 (11), 5.5768 (10), 5.4680 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.6339 (11), 4.5360 (10), 4.4198 (9)</td><td>10</td><td>ED</td><td>5.7030 (11), 5.6060 (10), 5.5311 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.7506 (11), 4.6286 (10), 4.4897 (9)</td><td>11</td><td>ED</td><td>5.7994 (11), 5.7021 (10), 5.5890 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.9948 (11), 4.9254 (10), 4.8450 (9)</td><td>12</td><td>ED</td><td>5.8259 (11), 5.7311 (10), 5.6127 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.2719 (11), 5.1846 (10), 5.0705 (9)</td><td>13</td><td>ED</td><td>5.8853 (11), 5.7760 (10), 5.6564 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.4009 (11), 5.3261 (10), 5.2470 (9)</td><td>14</td><td>ED</td><td>6.0084 (11), 5.8986 (10), 5.7679 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.4630 (11), 5.3824 (10), 5.2594 (9)</td><td>15</td><td>ED</td><td>6.0161 (11), 5.9300 (10), 5.8295 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.5468 (11), 5.4271 (10), 5.3047 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XXX: Scaling dimension for the sector $S = 1^{-}$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.2715 (11), 4.1112 (10), 3.9297 (9)</td><td>6</td><td>ED</td><td>5.8483 (11), 5.6807 (10), 5.4867 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.0149 (11), 4.8504 (10), 4.6661 (9)</td><td>7</td><td>ED</td><td>5.9059 (11), 5.7638 (10), 5.5979 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.0540 (11), 4.9069 (10), 4.7348 (9)</td><td>8</td><td>ED</td><td>5.9720 (11), 5.8480 (10), 5.6935 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.4655 (11), 5.3469 (10), 5.2086 (9)</td><td>9</td><td>ED</td><td>6.0310 (11), 5.8744 (10), 5.7066 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.6441 (11), 5.4897 (10), 5.3156 (9)</td><td>10</td><td>ED</td><td>6.1394 (11), 5.9737 (10), 5.7815 (9)</td></tr></tbody></table>

<table><caption>TABLE XXXI: Scaling dimension for the sector $S = 1^{-}$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.5577 (12), 4.3794 (11), 4.1845 (10)</td><td>6</td><td>ED</td><td>5.8243 (10), 5.6053 (9), 5.4436 (8)</td></tr><tr><td>2</td><td>ED</td><td>4.9479 (10), 4.7365 (9), 4.5061 (8)</td><td>7</td><td>ED</td><td>5.9356 (10), 5.7088 (9), 5.5733 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.0900 (10), 4.8689 (9), 4.6174 (8)</td><td>8</td><td>ED</td><td>6.0030 (10), 5.8023 (9), 5.6677 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.5525 (10), 5.3509 (9), 5.1261 (8)</td><td>9</td><td>ED</td><td>6.0568 (10), 5.8316 (9), 5.7456 (8)</td></tr><tr><td>5</td><td>ED</td><td>5.6338 (10), 5.4342 (9), 5.2105 (8)</td><td>10</td><td>ED</td><td>6.1042 (10), 5.8985 (9), 5.8086 (8)</td></tr></tbody></table>
<table><caption>TABLE XXXII: Scaling dimension for the sector $S = 1^{-}$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.1933 (10), 3.9703 (9), 3.7335 (8)</td><td>6</td><td>ED</td><td>5.9072 (10), 5.6909 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.9866 (10), 4.7661 (9), 4.6229 (8)</td><td>7</td><td>ED</td><td>5.9683 (10), 5.8249 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.1556 (10), 4.9041 (9)</td><td>8</td><td>ED</td><td>6.0639 (10), 5.9439 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6145 (10), 5.3940 (9)</td><td>9</td><td>ED</td><td>6.0949 (10), 6.0134 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.6938 (10), 5.4618 (9)</td><td>10</td><td>ED</td><td>6.1954 (10), 6.0756 (9)</td></tr></tbody></table>

### C. $S = 2$:

<figure><img src="image_11.png" /><figcaption>FIG. 5: Spectrum in the $S = 2^{+}$ sector.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIG. 6: Spectrum in the $S = 2^{-}$ sector.</figcaption></figure>
<table><caption>TABLE XXXIII: Scaling dimension for the sector $S = 2^+$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (t<sub>(2)</sub>)</td><td>ED</td><td>1.2541 (12), 1.2543 (11), 1.2548 (10)</td><td>3 (t'<sub>(2)</sub>)</td><td>ε</td><td>6.0000 (0), 7.3636 (1)</td></tr><tr><td>1 (t<sub>(2)</sub>)</td><td>DMRG</td><td>1.2377 (26), 1.2413 (24), 1.2413 (22)</td><td>3 (t'<sub>(2)</sub>)</td><td>1/N</td><td>5.0000 (0), 6.0808 (1)</td></tr><tr><td>1 (t<sub>(2)</sub>)</td><td>ε</td><td>2.0000 (0), 1.1818 (1), 1.2318 (2), 1.1741 (3), 1.2956 (4), 0.9603 (5), 2.0644 (6)</td><td>4 (t''<sub>(2)</sub>)</td><td>ED</td><td>4.3126 (12), 4.2668 (11), 4.2155 (10)</td></tr><tr><td>1 (t<sub>(2)</sub>)</td><td>1/N</td><td>1.0000 (0), 1.3603 (1), 1.3386 (2)</td><td>4 (t''<sub>(2)</sub>)</td><td>ε</td><td>8.0000 (0), 12.0909 (1)</td></tr><tr><td>2 (∂∂t<sup>(0)</sup><sub>(2)</sub>)</td><td>ED</td><td>3.4465 (12), 3.4572 (11), 3.4667 (10)</td><td>4 (t''<sub>(2)</sub>)</td><td>1/N</td><td>7.0000 (0), 8.4410 (1)</td></tr><tr><td>2 (∂∂t<sup>(0)</sup><sub>(2)</sub>)</td><td>DMRG</td><td>3.4130 (16), 3.4278 (14)</td><td>5 (∂∂∂∂t<sup>(0)</sup><sub>(2)</sub>)</td><td>ED</td><td>4.9647 (12), 4.9505 (11), 4.9360 (10)</td></tr><tr><td>2 (∂∂t<sup>(0)</sup><sub>(2)</sub>)</td><td>ε</td><td>4.0000 (0), 3.7273 (1), 3.2742 (2), 4.2128 (3), 1.5738 (4), 10.6276 (5)</td><td>6</td><td>ED</td><td>5.4530 (11), 5.4320 (10), 5.4173 (9)</td></tr><tr><td>2 (∂∂t<sup>(0)</sup><sub>(2)</sub>)</td><td>1/N</td><td>3.0000 (0), 3.7205 (1)</td><td>7</td><td>ED</td><td>5.6115 (11), 5.6039 (10), 5.5942 (9)</td></tr><tr><td>3 (t'<sub>(2)</sub>)</td><td>ED</td><td>3.6713 (12), 3.6712 (11), 3.6750 (10)</td><td>8</td><td>ED</td><td>5.9238 (11), 5.8730 (10), 5.8081 (9)</td></tr><tr><td>3 (t'<sub>(2)</sub>)</td><td>DMRG</td><td>3.6812 (16), 3.6745 (14)</td><td>9</td><td>ED</td><td>5.9710 (11), 6.0579 (10), 6.0661 (9)</td></tr></tbody></table>

<table><caption>TABLE XXXIV: Scaling dimension for the sector $S = 2^+$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂t<sub>(2)</sub>)</td><td>ED</td><td>2.2862 (12), 2.2870 (11), 2.2878 (10)</td><td>7</td><td>ED</td><td>5.2022 (11), 5.1801 (10), 5.1608 (9)</td></tr><tr><td>1 (∂t<sub>(2)</sub>)</td><td>DMRG</td><td>2.2727 (24), 2.2747 (20), 2.2775 (18)</td><td>8</td><td>ED</td><td>5.2522 (11), 5.2277 (10), 5.2053 (9)</td></tr><tr><td>2 (∂∂∂t<sup>(1)</sup><sub>(2)</sub>)</td><td>ED</td><td>3.9102 (12), 3.8845 (11), 3.8548 (10)</td><td>9</td><td>ED</td><td>5.4933 (11), 5.4503 (10), 5.4096 (9)</td></tr><tr><td>2 (∂∂∂t<sup>(1)</sup><sub>(2)</sub>)</td><td>DMRG</td><td>3.9620 (14)</td><td>10</td><td>ED</td><td>5.5950 (11), 5.5730 (10), 5.5545 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.4954 (12), 4.4856 (11), 4.4704 (10)</td><td>11</td><td>ED</td><td>5.8176 (11), 5.7920 (10), 5.7614 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.7440 (12), 4.7570 (11), 4.7704 (10)</td><td>12</td><td>ED</td><td>5.8261 (11), 5.8043 (10), 5.7870 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.8464 (12), 4.8254 (11), 4.8044 (10)</td><td>13</td><td>ED</td><td>5.9086 (11), 5.8907 (10), 5.8713 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.0648 (11), 5.0080 (10), 4.9469 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE XXXV: Scaling dimension for the sector $S = 2^+$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (t'<sub>(2)μν</sub>)</td><td>ED</td><td>3.0254 (12), 3.0239 (11), 3.0219 (10)</td><td>10</td><td>ED</td><td>5.1904 (11), 5.1596 (10), 5.1126 (9)</td></tr><tr><td>1 (t'<sub>(2)μν</sub>)</td><td>DMRG</td><td>3.0282 (26), 3.0272 (24), 3.0280 (22)</td><td>11</td><td>ED</td><td>5.3185 (11), 5.2744 (10), 5.2364 (9)</td></tr><tr><td>1 (t'<sub>(2)μν</sub>)</td><td>ε</td><td>4.0000 (0), 3.0000 (1), 3.0083 (2), 3.0163 (3), 3.0148 (4)</td><td>12</td><td>ED</td><td>5.3373 (11), 5.2921 (10), 5.2383 (9)</td></tr><tr><td>1 (t'<sub>(2)μν</sub>)</td><td>1/N</td><td>3.0000 (0), 3.0721 (1), 3.0269 (2)</td><td>13</td><td>ED</td><td>5.4474 (11), 5.4333 (10), 5.4194 (9)</td></tr><tr><td>2 (∂∂t<sup>(2)</sup><sub>(2)</sub>)</td><td>ED</td><td>3.3100 (12), 3.3020 (11), 3.2909 (10)</td><td>14</td><td>ED</td><td>5.4975 (11), 5.4800 (10), 5.4591 (9)</td></tr><tr><td>2 (∂∂t<sup>(2)</sup><sub>(2)</sub>)</td><td>DMRG</td><td>3.3297 (24), 3.3339 (18), 3.3283 (16)</td><td>15</td><td>ED</td><td>5.6353 (11), 5.6062 (10), 5.5451 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.0910 (11), 4.0701 (10), 4.0332 (9)</td><td>16</td><td>ED</td><td>5.6561 (11), 5.6241 (10), 5.5686 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.1735 (11), 4.1439 (10), 4.1133 (9)</td><td>17</td><td>ED</td><td>5.6882 (11), 5.6396 (10), 5.5892 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.2821 (11), 4.2371 (10), 4.1949 (9)</td><td>18</td><td>ED</td><td>5.7046 (11), 5.6508 (10), 5.6218 (9)</td></tr><tr><td>6</td><td>ED</td><td>4.5763 (11), 4.5059 (10), 4.4301 (9)</td><td>19</td><td>ED</td><td>5.8026 (11), 5.7601 (10), 5.7273 (9)</td></tr><tr><td>7</td><td>ED</td><td>4.7107 (11), 4.6580 (10), 4.5977 (9)</td><td>20</td><td>ED</td><td>5.9181 (11), 5.8666 (10), 5.8160 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.0252 (11), 4.9916 (10), 4.9528 (9)</td><td>21</td><td>ED</td><td>5.9301 (10), 5.8682 (9), 5.7834 (8)</td></tr><tr><td>9</td><td>ED</td><td>5.0983 (11), 5.0564 (10), 5.0080 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XXXVI: Scaling dimension for the sector $S = 2^+$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>3.9708 (11), 3.9413 (10), 3.9025 (9)</td><td>9</td><td>ED</td><td>5.5042 (11), 5.4447 (10), 5.3769 (9)</td></tr><tr><td>1</td><td>DMRG</td><td>4.0967 (24)</td><td>10</td><td>ED</td><td>5.7175 (11), 5.6548 (10), 5.5751 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.1517 (11), 4.1010 (10), 4.0374 (9)</td><td>11</td><td>ED</td><td>5.7370 (11), 5.6722 (10), 5.5779 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.7264 (11), 4.6516 (10), 4.5682 (9)</td><td>12</td><td>ED</td><td>5.7737 (11), 5.7160 (10), 5.6437 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.8445 (11), 4.7704 (10), 4.6862 (9)</td><td>13</td><td>ED</td><td>5.7909 (11), 5.7456 (10), 5.6854 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.9242 (11), 4.8476 (10), 4.7579 (9)</td><td>14</td><td>ED</td><td>5.8398 (11), 5.7725 (10), 5.7384 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.0665 (11), 5.0074 (10), 4.9315 (9)</td><td>15</td><td>ED</td><td>5.9263 (11), 5.8786 (10), 5.8105 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.2573 (11), 5.1794 (10), 5.0950 (9)</td><td>16</td><td>ED</td><td>5.9585 (11), 5.9035 (10), 5.8288 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.3604 (11), 5.2692 (10), 5.1649 (9)</td><td>17</td><td>ED</td><td>5.9983 (11), 5.9354 (10), 5.8449 (9)</td></tr></tbody></table>
<table><caption>TABLE XXXVII: Scaling dimension for the sector $S = 2^+$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.6463 (11), 4.5568 (10), 4.4474 (9)</td><td>9</td><td>ED</td><td>5.7895 (11), 5.6559 (10), 5.5022 (9)</td></tr><tr><td>1</td><td>ε</td><td>6.0000 (0), 5.0000 (1), 5.0169 (2), 5.0317 (3), 5.0259 (4)</td><td>10</td><td>ED</td><td>5.8133 (11), 5.6935 (10), 5.5530 (9)</td></tr><tr><td>1</td><td>1/N</td><td>5.0000 (0), 5.0858 (1), 5.0537 (2)</td><td>11</td><td>ED</td><td>5.8244 (11), 5.7119 (10), 5.5866 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.7173 (11), 4.6082 (10), 4.4795 (9)</td><td>12</td><td>ED</td><td>5.9668 (11), 5.8897 (10), 5.7981 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.0247 (11), 4.9758 (10), 4.9036 (9)</td><td>13</td><td>ED</td><td>6.0334 (11), 5.9569 (10), 5.8609 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.3045 (11), 5.2021 (10), 5.0756 (9)</td><td>14</td><td>ED</td><td>6.0871 (11), 6.0041 (10), 5.8732 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.3360 (11), 5.2384 (10), 5.1161 (9)</td><td>15</td><td>ED</td><td>6.1243 (11), 6.0356 (10), 5.9319 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.3687 (11), 5.2661 (10), 5.1684 (9)</td><td>16</td><td>ED</td><td>6.1601 (11), 6.0767 (10), 5.9551 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.4314 (11), 5.3177 (10), 5.1984 (9)</td><td>17</td><td>ED</td><td>6.1780 (11), 6.0836 (10), 5.9923 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.6153 (11), 5.5019 (10), 5.3670 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XXXVIII: Scaling dimension for the sector $S = 2^+$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.9661 (11), 4.8107 (10), 4.6349 (9)</td><td>6</td><td>ED</td><td>5.7953 (11), 5.6550 (10), 5.4869 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.0837 (11), 4.9256 (10), 4.7442 (9)</td><td>7</td><td>ED</td><td>5.9290 (11), 5.7996 (10), 5.6157 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.5992 (11), 5.4664 (10), 5.2865 (9)</td><td>8</td><td>ED</td><td>5.9609 (11), 5.8198 (10), 5.6875 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6488 (11), 5.4879 (10), 5.3188 (9)</td><td>9</td><td>ED</td><td>6.0623 (11), 5.8851 (10), 5.6928 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.6886 (11), 5.5216 (10), 5.3424 (9)</td><td>10</td><td>ED</td><td>6.1386 (11), 5.9693 (10), 5.7794 (9)</td></tr></tbody></table>

<table><caption>TABLE XXXIX: Scaling dimension for the sector $S = 2^+$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.0792 (11), 4.8899 (10), 4.6855 (9)</td><td>5</td><td>ED</td><td>5.7145 (10), 5.5093 (9), 5.2735 (8)</td></tr><tr><td>1</td><td>ε</td><td>8.0000 (0), 7.0000 (1), 7.0189 (2), 7.0353 (3), 7.0285 (4)</td><td>6</td><td>ED</td><td>5.8371 (10), 5.6207 (9), 5.3796 (8)</td></tr><tr><td>1</td><td>1/N</td><td>7.0000 (0), 7.0882 (1), 7.0601 (2)</td><td>7</td><td>ED</td><td>5.9575 (10), 5.7355 (9), 5.4749 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.0734 (10), 4.8483 (9), 4.5969 (8)</td><td>8</td><td>ED</td><td>6.0064 (10), 5.7781 (9), 5.6007 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.5728 (10), 5.3397 (9), 5.0797 (8)</td><td>9</td><td>ED</td><td>6.0615 (10), 5.8508 (9), 5.6749 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.6252 (10), 5.4117 (9), 5.1777 (8)</td><td>10</td><td>ED</td><td>6.1132 (10), 5.9074 (9), 5.7738 (8)</td></tr></tbody></table>
<table><caption>TABLE XL: Scaling dimension for the sector $S = 2^+$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.9063 (10), 4.6941 (9), 4.4733 (8)</td><td>6</td><td>ED</td><td>5.8998 (10), 5.6642 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.1235 (10), 4.8757 (9)</td><td>7</td><td>ED</td><td>6.0232 (10), 5.7760 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.5905 (10), 5.3426 (9)</td><td>8</td><td>ED</td><td>6.0629 (10), 5.8914 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6695 (10), 5.4418 (9)</td><td>9</td><td>ED</td><td>6.1309 (10), 5.9250 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.7845 (10), 5.5404 (9)</td><td>10</td><td>ED</td><td>6.1778 (10), 6.0142 (9)</td></tr></tbody></table>

<table><caption>TABLE XLI: Scaling dimension for the sector $S = 2^-$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 ($\partial t_{(2)\mu}^{(0)}$)</td><td>ED</td><td>3.8120 (12), 3.8059 (11), 3.8001 (10)</td><td>7</td><td>ED</td><td>6.4610 (10), 6.4522 (9), 6.2308 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.0590 (12), 5.0907 (11), 5.1042 (10)</td><td>8</td><td>ED</td><td>6.4713 (10), 6.4983 (9), 6.2690 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.4659 (11), 5.4155 (10), 5.3631 (9)</td><td>9</td><td>ED</td><td>6.4735 (10), 6.6107 (9), 6.2951 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.9003 (11), 5.8846 (10), 5.8643 (9)</td><td>10</td><td>ED</td><td>6.4745 (10), 6.7557 (9), 6.3046 (8)</td></tr><tr><td>5</td><td>ED</td><td>6.1463 (10), 6.1130 (9), 6.0714 (8)</td><td>11</td><td>ED</td><td>6.4877 (10), 6.8338 (9), 6.3302 (8)</td></tr><tr><td>6</td><td>ED</td><td>6.1822 (10), 6.1576 (9), 6.1280 (8)</td><td>12</td><td>ED</td><td>6.5358 (10), 6.8407 (9), 6.3508 (8)</td></tr></tbody></table>
<table><caption>TABLE XLII: Scaling dimension for the sector $S = 2^-$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (t<sub>(2)μ</sub>)</td><td>ED</td><td>2.6897 (12), 2.6731 (11), 2.6538 (10)</td><td>10</td><td>ED</td><td>5.4846 (11), 5.4863 (10), 5.5268 (9)</td></tr><tr><td>1 (t<sub>(2)μ</sub>)</td><td>DMRG</td><td>2.7819 (26), 2.7735 (24), 2.7663 (22)</td><td>11</td><td>ED</td><td>5.7578 (11), 5.7523 (10), 5.7226 (9)</td></tr><tr><td>1 (t<sub>(2)μ</sub>)</td><td>ε</td><td>6.0000 (0), 5.5000 (1)</td><td>12</td><td>ED</td><td>5.8919 (11), 5.8105 (10), 5.7372 (9)</td></tr><tr><td>1 (t<sub>(2)μ</sub>)</td><td>1/N</td><td>-2.0000 (0), 2.5000 (1), 2.9053 (2)</td><td>13</td><td>ED</td><td>5.9072 (11), 5.8289 (10), 5.7519 (9)</td></tr><tr><td>2 (ε∂t<sup>(1)</sup><sub>(2)μ</sub>)</td><td>ED</td><td>3.6051 (12), 3.5941 (11), 3.5834 (10)</td><td>14</td><td>ED</td><td>6.0794 (11), 6.0557 (10), 6.0243 (9)</td></tr><tr><td>2 (ε∂t<sup>(1)</sup><sub>(2)μ</sub>)</td><td>DMRG</td><td>3.7288 (24), 3.6879 (18), 3.6576 (16)</td><td>15</td><td>ED</td><td>6.1194 (11), 6.1024 (10), 6.0746 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.1472 (12), 4.1209 (11), 4.0941 (10)</td><td>16</td><td>ED</td><td>6.1393 (11), 6.1283 (10), 6.0905 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.6675 (12), 4.6654 (11), 4.6625 (10)</td><td>17</td><td>ED</td><td>6.1785 (11), 6.1449 (10), 6.1168 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.7917 (12), 4.7838 (11), 4.7776 (10)</td><td>18</td><td>ED</td><td>6.2524 (10), 6.2165 (9), 6.1584 (8)</td></tr><tr><td>6</td><td>ED</td><td>5.1143 (11), 5.0868 (10), 5.0561 (9)</td><td>19</td><td>ED</td><td>6.2761 (10), 6.2309 (9), 6.2258 (8)</td></tr><tr><td>7</td><td>ED</td><td>5.1339 (11), 5.1111 (10), 5.0847 (9)</td><td>20</td><td>ED</td><td>6.4161 (10), 6.3860 (9), 6.3477 (8)</td></tr><tr><td>8</td><td>ED</td><td>5.1796 (11), 5.1441 (10), 5.1111 (9)</td><td>21</td><td>ED</td><td>6.4929 (10), 6.4586 (9), 6.3882 (8)</td></tr><tr><td>9</td><td>ED</td><td>5.4442 (11), 5.4196 (10), 5.3615 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XLIII: Scaling dimension for the sector $S = 2^-$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (t'<sub>(2)μν</sub>)</td><td>ED</td><td>3.2949 (12), 3.2535 (11), 3.2059 (10)</td><td>11</td><td>ED</td><td>5.5375 (11), 5.5042 (10), 5.4430 (9)</td></tr><tr><td>1 (t'<sub>(2)μν</sub>)</td><td>DMRG</td><td>3.5341 (24), 3.5033 (20), 3.4668 (18)</td><td>12</td><td>ED</td><td>5.5681 (11), 5.5106 (10), 5.4694 (9)</td></tr><tr><td>2 (∂t<sup>(2)</sup><sub>(2)μ</sub>)</td><td>ED</td><td>3.5671 (12), 3.5353 (11), 3.4977 (10)</td><td>13</td><td>ED</td><td>5.6789 (11), 5.6467 (10), 5.5954 (9)</td></tr><tr><td>2 (∂t<sup>(2)</sup><sub>(2)μ</sub>)</td><td>DMRG</td><td>3.7565 (24), 3.7358 (20), 3.7040 (18)</td><td>14</td><td>ED</td><td>5.7740 (11), 5.6991 (10), 5.6204 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.0326 (11), 3.9948 (10), 3.9544 (9)</td><td>15</td><td>ED</td><td>5.8395 (11), 5.7882 (10), 5.7018 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.2648 (11), 4.2492 (10), 4.2357 (9)</td><td>16</td><td>ED</td><td>5.8699 (11), 5.8343 (10), 5.7878 (9)</td></tr><tr><td>5</td><td>ED</td><td>4.5352 (11), 4.5141 (10), 4.4883 (9)</td><td>17</td><td>ED</td><td>5.9220 (11), 5.8633 (10), 5.8172 (9)</td></tr><tr><td>6</td><td>ED</td><td>4.9692 (11), 4.9319 (10), 4.8917 (9)</td><td>18</td><td>ED</td><td>5.9415 (11), 5.8882 (10), 5.8391 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.0349 (11), 5.0068 (10), 4.9729 (9)</td><td>19</td><td>ED</td><td>5.9695 (11), 5.9253 (10), 5.8704 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.3154 (11), 5.2925 (10), 5.2492 (9)</td><td>20</td><td>ED</td><td>6.0124 (11), 5.9634 (10), 5.9078 (9)</td></tr><tr><td>9</td><td>ED</td><td>5.3817 (11), 5.3372 (10), 5.2952 (9)</td><td>21</td><td>ED</td><td>6.0339 (10), 5.9624 (9), 5.9329 (8)</td></tr><tr><td>10</td><td>ED</td><td>5.4566 (11), 5.4064 (10), 5.3608 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE XLIV: Scaling dimension for the sector $S = 2^-$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>3.8934 (11), 3.8198 (10), 3.7370 (9)</td><td>12</td><td>ED</td><td>5.8031 (11), 5.7250 (10), 5.6285 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.2347 (11), 4.1605 (10), 4.0742 (9)</td><td>13</td><td>ED</td><td>5.8817 (11), 5.8055 (10), 5.7214 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.6377 (11), 4.5875 (10), 4.5288 (9)</td><td>14</td><td>ED</td><td>5.9417 (11), 5.8503 (10), 5.7479 (9)</td></tr><tr><td>4</td><td>ED</td><td>4.7665 (11), 4.7195 (10), 4.6669 (9)</td><td>15</td><td>ED</td><td>5.9915 (11), 5.9000 (10), 5.7979 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.1017 (11), 5.0647 (10), 5.0169 (9)</td><td>16</td><td>ED</td><td>6.0374 (11), 5.9529 (10), 5.8559 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.2490 (11), 5.2027 (10), 5.1402 (9)</td><td>17</td><td>ED</td><td>6.0941 (11), 6.0181 (10), 5.9248 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.3563 (11), 5.3115 (10), 5.2649 (9)</td><td>18</td><td>ED</td><td>6.1176 (11), 6.0406 (10), 5.9503 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.5031 (11), 5.4430 (10), 5.3729 (9)</td><td>19</td><td>ED</td><td>6.1537 (11), 6.0670 (10), 5.9685 (9)</td></tr><tr><td>9</td><td>ED</td><td>5.6107 (11), 5.5394 (10), 5.4582 (9)</td><td>20</td><td>ED</td><td>6.2040 (11), 6.1375 (10), 6.0370 (9)</td></tr><tr><td>10</td><td>ED</td><td>5.6928 (11), 5.6254 (10), 5.5492 (9)</td><td>21</td><td>ED</td><td>6.2111 (11), 6.1649 (10), 6.0740 (9)</td></tr><tr><td>11</td><td>ED</td><td>5.7623 (11), 5.6813 (10), 5.5925 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XLV: Scaling dimension for the sector $S = 2^-$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.3690 (11), 4.2605 (10), 4.1372 (9)</td><td>12</td><td>ED</td><td>6.0851 (11), 5.9824 (10), 5.8652 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.7237 (11), 4.6023 (10), 4.4632 (9)</td><td>13</td><td>ED</td><td>6.1597 (11), 6.0453 (10), 5.9091 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.9659 (11), 4.8979 (10), 4.8199 (9)</td><td>14</td><td>ED</td><td>6.1793 (11), 6.0935 (10), 5.9813 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.2648 (11), 5.1667 (10), 5.0517 (9)</td><td>15</td><td>ED</td><td>6.2777 (11), 6.1555 (10), 6.0149 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.4722 (11), 5.3951 (10), 5.2978 (9)</td><td>16</td><td>ED</td><td>6.2997 (11), 6.1845 (10), 6.0540 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.5916 (11), 5.5483 (10), 5.4668 (9)</td><td>17</td><td>ED</td><td>6.3080 (11), 6.2045 (10), 6.0769 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.6939 (11), 5.6097 (10), 5.5241 (9)</td><td>18</td><td>ED</td><td>6.3409 (11), 6.2233 (10), 6.0984 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.7913 (11), 5.6751 (10), 5.5530 (9)</td><td>19</td><td>ED</td><td>6.4179 (11), 6.3121 (10), 6.1747 (9)</td></tr><tr><td>9</td><td>ED</td><td>5.8051 (11), 5.7139 (10), 5.6077 (9)</td><td>20</td><td>ED</td><td>6.4428 (11), 6.3407 (10), 6.2084 (9)</td></tr><tr><td>10</td><td>ED</td><td>5.9908 (11), 5.8764 (10), 5.7480 (9)</td><td>21</td><td>ED</td><td>6.4604 (11), 6.3447 (10), 6.2320 (9)</td></tr><tr><td>11</td><td>ED</td><td>6.0465 (11), 5.9504 (10), 5.8346 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE XLVI: Scaling dimension for the sector $S = 2^-$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.6657 (11), 4.5158 (10), 4.3475 (9)</td><td>6</td><td>ED</td><td>6.0269 (11), 5.8809 (10), 5.7095 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.0147 (11), 4.8480 (10), 4.6611 (9)</td><td>7</td><td>ED</td><td>6.0637 (11), 5.9066 (10), 5.7301 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.4069 (11), 5.2956 (10), 5.1661 (9)</td><td>8</td><td>ED</td><td>6.1725 (11), 6.0671 (10), 5.9179 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6262 (11), 5.4787 (10), 5.3109 (9)</td><td>9</td><td>ED</td><td>6.2141 (11), 6.0835 (10), 5.9281 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.9363 (11), 5.7905 (10), 5.6200 (9)</td><td>10</td><td>ED</td><td>6.2671 (11), 6.1193 (10), 5.9899 (9)</td></tr></tbody></table>
<table><caption>TABLE XLVII: Scaling dimension for the sector $S = 2^-$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.8087 (11), 4.6248 (10), 4.4237 (9)</td><td>6</td><td>ED</td><td>6.0070 (10), 5.7790 (9), 5.5200 (8)</td></tr><tr><td>2</td><td>ED</td><td>4.9611 (10), 4.7415 (9), 4.4995 (8)</td><td>7</td><td>ED</td><td>6.0377 (10), 5.8470 (9), 5.7957 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.4942 (10), 5.3117 (9), 5.1068 (8)</td><td>8</td><td>ED</td><td>6.2128 (10), 6.0361 (9), 5.8553 (8)</td></tr><tr><td>4</td><td>ED</td><td>5.6286 (10), 5.4270 (9), 5.2006 (8)</td><td>9</td><td>ED</td><td>6.2480 (10), 6.0404 (9), 5.9932 (8)</td></tr><tr><td>5</td><td>ED</td><td>5.9182 (10), 5.7052 (9), 5.4543 (8)</td><td>10</td><td>ED</td><td>6.3472 (10), 6.1410 (9), 6.0517 (8)</td></tr></tbody></table>

<table><caption>TABLE XLVIII: Scaling dimension for the sector $S = 2^-$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.6555 (10), 4.4389 (9), 4.2061 (8)</td><td>6</td><td>ED</td><td>6.0118 (10), 5.7602 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.0024 (10), 4.7652 (9)</td><td>7</td><td>ED</td><td>6.1303 (10), 6.0405 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.5641 (10), 5.3626 (9)</td><td>8</td><td>ED</td><td>6.2859 (10), 6.1068 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.6811 (10), 5.4504 (9)</td><td>9</td><td>ED</td><td>6.2938 (10), 6.2452 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.9494 (10), 5.6952 (9)</td><td>10</td><td>ED</td><td>6.4197 (10), 6.3157 (9)</td></tr></tbody></table>
D. $S = 3$:

<figure><img src="image_13.png" /><figcaption>FIG. 7: Spectrum in the $S = 3^+$ sector.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIG. 8: Spectrum in the $S = 3^-$ sector.</figcaption></figure>

<table><caption>TABLE XLIX: Scaling dimension for the sector $S = 3^+$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂χ<sup>(0)</sup><sub>(3)μ</sub>)</td><td>ED</td><td>4.8259 (11), 4.8234 (10), 4.8233 (9)</td><td>5</td><td>ED</td><td>6.7672 (11), 6.7497 (10), 6.7445 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.2651 (11), 6.2858 (10), 6.3136 (9)</td><td>6</td><td>ED</td><td>6.8450 (11), 6.9052 (10), 6.8764 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.6407 (11), 6.6205 (10), 6.5905 (9)</td><td>7</td><td>ED</td><td>6.9287 (11), 7.2564 (10), 7.3783 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.7135 (11), 6.6795 (10), 6.6448 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE L: Scaling dimension for the sector $S = 3^+$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (χ<sub>(3)μ</sub>)</td><td>ED</td><td>3.6877 (12), 3.6735 (11), 3.6572 (10)</td><td>10</td><td>ED</td><td>6.4997 (11), 6.4700 (10), 6.4372 (9)</td></tr><tr><td>1 (χ<sub>(3)μ</sub>)</td><td>DMRG</td><td>3.7508 (26), 3.7489 (24), 3.7430 (22)</td><td>11</td><td>ED</td><td>6.5641 (11), 6.5623 (10), 6.5440 (9)</td></tr><tr><td>1 (χ<sub>(3)μ</sub>)</td><td>ε</td><td>5.0000 (0), 3.7273 (1)</td><td>12</td><td>ED</td><td>6.6437 (11), 6.6248 (10), 6.6217 (9)</td></tr><tr><td>1 (χ<sub>(3)μ</sub>)</td><td>1/N</td><td>-3.0000 (0), 3.0000 (1), 4.0808 (2)</td><td>13</td><td>ED</td><td>6.6585 (11), 6.6610 (10), 6.6431 (9)</td></tr><tr><td>2 (ε∂χ<sup>(1)</sup><sub>(3)μ</sub>)</td><td>ED</td><td>4.6687 (11), 4.6611 (10), 4.6539 (9)</td><td>14</td><td>ED</td><td>6.7557 (11), 6.7239 (10), 6.6998 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.1889 (11), 5.1732 (10), 5.1604 (9)</td><td>15</td><td>ED</td><td>6.7801 (11), 6.7673 (10), 6.7606 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.4946 (11), 5.4776 (10), 5.4631 (9)</td><td>16</td><td>ED</td><td>6.8196 (11), 6.8078 (10), 6.8044 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.7342 (11), 5.7249 (10), 5.7133 (9)</td><td>17</td><td>ED</td><td>6.9944 (11), 6.9307 (10), 6.8657 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.7789 (11), 5.7652 (10), 5.7526 (9)</td><td>18</td><td>ED</td><td>7.0016 (11), 6.9762 (10), 6.9155 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.8330 (11), 5.8197 (10), 5.8080 (9)</td><td>19</td><td>ED</td><td>7.0236 (11), 7.0399 (10), 6.9756 (9)</td></tr><tr><td>8</td><td>ED</td><td>6.1831 (11), 6.1685 (10), 6.1543 (9)</td><td>20</td><td>ED</td><td>7.0566 (11), 7.0605 (10), 7.0601 (9)</td></tr><tr><td>9</td><td>ED</td><td>6.4545 (11), 6.4308 (10), 6.4058 (9)</td><td>21</td><td>ED</td><td>7.0663 (11), 7.1412 (10), 7.1115 (9)</td></tr></tbody></table>

<table><caption>TABLE LI: Scaling dimension for the sector $S = 3^+$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (χ<sub>(3)μν</sub>)</td><td>ED</td><td>4.2233 (12), 4.1830 (11), 4.1375 (10)</td><td>12</td><td>ED</td><td>6.5524 (11), 6.5103 (10), 6.4640 (9)</td></tr><tr><td>2 (∂χ<sup>(2)</sup><sub>(3)μ</sub>)</td><td>ED</td><td>4.5861 (12), 4.5550 (11), 4.5178 (10)</td><td>13</td><td>ED</td><td>6.5935 (11), 6.5519 (10), 6.5029 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.0499 (11), 5.0093 (10), 4.9662 (9)</td><td>14</td><td>ED</td><td>6.6433 (11), 6.5989 (10), 6.5400 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.4088 (11), 5.4066 (10), 5.4040 (9)</td><td>15</td><td>ED</td><td>6.6579 (11), 6.6138 (10), 6.5642 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.5717 (11), 5.5365 (10), 5.4952 (9)</td><td>16</td><td>ED</td><td>6.7080 (11), 6.6678 (10), 6.6195 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.6084 (11), 5.5700 (10), 5.5317 (9)</td><td>17</td><td>ED</td><td>6.7557 (11), 6.7068 (10), 6.6543 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.9022 (11), 5.8783 (10), 5.8547 (9)</td><td>18</td><td>ED</td><td>6.7897 (11), 6.7477 (10), 6.6956 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.9894 (11), 5.9469 (10), 5.9007 (9)</td><td>19</td><td>ED</td><td>6.7975 (10), 6.7592 (9), 6.7080 (8)</td></tr><tr><td>9</td><td>ED</td><td>6.0990 (11), 6.0674 (10), 6.0314 (9)</td><td>20</td><td>ED</td><td>6.8811 (10), 6.8201 (9), 6.7464 (8)</td></tr><tr><td>10</td><td>ED</td><td>6.3114 (11), 6.2785 (10), 6.2439 (9)</td><td>21</td><td>ED</td><td>6.9361 (10), 6.8652 (9), 6.7826 (8)</td></tr><tr><td>11</td><td>ED</td><td>6.4621 (11), 6.4466 (10), 6.4184 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE LII: Scaling dimension for the sector $S = 3^+$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.6308 (11), 4.5539 (10), 4.4680 (9)</td><td>12</td><td>ED</td><td>6.3866 (11), 6.3346 (10), 6.2744 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.9033 (11), 4.8288 (10), 4.7431 (9)</td><td>13</td><td>ED</td><td>6.4952 (11), 6.4512 (10), 6.3926 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.2628 (11), 5.1835 (10), 5.0902 (9)</td><td>14</td><td>ED</td><td>6.5952 (11), 6.5290 (10), 6.4445 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.4462 (11), 5.3831 (10), 5.3140 (9)</td><td>15</td><td>ED</td><td>6.6323 (11), 6.5844 (10), 6.5385 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.5997 (11), 5.5653 (10), 5.5233 (9)</td><td>16</td><td>ED</td><td>6.7117 (11), 6.6471 (10), 6.5685 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.7517 (11), 5.6916 (10), 5.6219 (9)</td><td>17</td><td>ED</td><td>6.8309 (11), 6.7654 (10), 6.6925 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.8299 (11), 5.8112 (10), 5.7940 (9)</td><td>18</td><td>ED</td><td>6.8487 (11), 6.7993 (10), 6.7316 (9)</td></tr><tr><td>8</td><td>ED</td><td>6.1144 (11), 6.0665 (10), 6.0151 (9)</td><td>19</td><td>ED</td><td>6.8640 (11), 6.8071 (10), 6.7399 (9)</td></tr><tr><td>9</td><td>ED</td><td>6.2333 (11), 6.1669 (10), 6.0837 (9)</td><td>20</td><td>ED</td><td>6.9338 (11), 6.8663 (10), 6.7949 (9)</td></tr><tr><td>10</td><td>ED</td><td>6.2663 (11), 6.2031 (10), 6.1291 (9)</td><td>21</td><td>ED</td><td>6.9636 (11), 6.8885 (10), 6.8056 (9)</td></tr><tr><td>11</td><td>ED</td><td>6.3345 (11), 6.2712 (10), 6.2002 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LIII: Scaling dimension for the sector $S = 3^+$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.1174 (11), 5.0098 (10), 4.8893 (9)</td><td>12</td><td>ED</td><td>6.7103 (11), 6.6133 (10), 6.4993 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.4233 (11), 5.3038 (10), 5.1670 (9)</td><td>13</td><td>ED</td><td>6.7602 (11), 6.6386 (10), 6.5192 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.7542 (11), 5.6214 (10), 5.4693 (9)</td><td>14</td><td>ED</td><td>6.8205 (11), 6.7278 (10), 6.6150 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.8986 (11), 5.8242 (10), 5.7364 (9)</td><td>15</td><td>ED</td><td>6.8243 (11), 6.7501 (10), 6.6273 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.0168 (11), 5.9267 (10), 5.8263 (9)</td><td>16</td><td>ED</td><td>6.8878 (11), 6.8026 (10), 6.7174 (9)</td></tr><tr><td>6</td><td>ED</td><td>6.2272 (11), 6.1312 (10), 6.0075 (9)</td><td>17</td><td>ED</td><td>7.0123 (11), 6.8877 (10), 6.7470 (9)</td></tr><tr><td>7</td><td>ED</td><td>6.2796 (11), 6.1931 (10), 6.0988 (9)</td><td>18</td><td>ED</td><td>7.0150 (11), 6.9196 (10), 6.8263 (9)</td></tr><tr><td>8</td><td>ED</td><td>6.3884 (11), 6.3128 (10), 6.2228 (9)</td><td>19</td><td>ED</td><td>7.0829 (11), 6.9801 (10), 6.8580 (9)</td></tr><tr><td>9</td><td>ED</td><td>6.4439 (11), 6.3782 (10), 6.3037 (9)</td><td>20</td><td>ED</td><td>7.1378 (11), 7.0365 (10), 6.9127 (9)</td></tr><tr><td>10</td><td>ED</td><td>6.5593 (11), 6.4875 (10), 6.4028 (9)</td><td>21</td><td>ED</td><td>7.1558 (11), 7.0568 (10), 6.9419 (9)</td></tr><tr><td>11</td><td>ED</td><td>6.6803 (11), 6.5876 (10), 6.4647 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LIV: Scaling dimension for the sector $S = 3^+$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.4337 (11), 5.2879 (10), 5.1254 (9)</td><td>6</td><td>ED</td><td>6.4582 (11), 6.3422 (10), 6.2031 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.7405 (11), 5.5757 (10), 5.3907 (9)</td><td>7</td><td>ED</td><td>6.6106 (11), 6.4504 (10), 6.2682 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.0486 (11), 5.8683 (10), 5.6670 (9)</td><td>8</td><td>ED</td><td>6.7391 (11), 6.6344 (10), 6.4909 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.2320 (11), 6.1197 (10), 5.9886 (9)</td><td>9</td><td>ED</td><td>6.8074 (11), 6.6858 (10), 6.5395 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.3823 (11), 6.2392 (10), 6.0776 (9)</td><td>10</td><td>ED</td><td>6.8519 (11), 6.7076 (10), 6.5608 (9)</td></tr></tbody></table>
<table><caption>TABLE LV: Scaling dimension for the sector $S = 3^+$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.5940 (11), 5.4146 (10), 5.2192 (9)</td><td>6</td><td>ED</td><td>6.5566 (10), 6.3501 (9), 6.3585 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.7069 (10), 5.4875 (9), 5.2429 (8)</td><td>7</td><td>ED</td><td>6.6423 (10), 6.4303 (9), 6.3700 (8)</td></tr><tr><td>3</td><td>ED</td><td>5.9901 (10), 5.7552 (9), 5.9160 (8)</td><td>8</td><td>ED</td><td>6.8330 (10), 6.6173 (9), 6.5561 (8)</td></tr><tr><td>4</td><td>ED</td><td>6.3153 (10), 6.1283 (9), 6.0142 (8)</td><td>9</td><td>ED</td><td>6.8604 (10), 6.6542 (9), 6.6044 (8)</td></tr><tr><td>5</td><td>ED</td><td>6.4143 (10), 6.2245 (9), 6.1975 (8)</td><td>10</td><td>ED</td><td>6.9074 (10), 6.7227 (9), 6.6847 (8)</td></tr></tbody></table>

<table><caption>TABLE LVI: Scaling dimension for the sector $S = 3^+$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.4563 (10), 5.2451 (9), 5.0191 (8)</td><td>6</td><td>ED</td><td>6.6431 (10), 6.5757 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.7579 (10), 5.5176 (9)</td><td>7</td><td>ED</td><td>6.7253 (10), 6.6438 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.0420 (10), 6.1802 (9)</td><td>8</td><td>ED</td><td>6.8440 (10), 6.7890 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.3849 (10), 6.2685 (9)</td><td>9</td><td>ED</td><td>6.9469 (10), 6.8824 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.4865 (10), 6.4767 (9)</td><td>10</td><td>ED</td><td>6.9666 (10), 6.9509 (9)</td></tr></tbody></table>

<table><caption>TABLE LVII: Scaling dimension for the sector $S = 3^-$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (σ<sub>(3)</sub>)</td><td>ED</td><td>2.1731 (12), 2.1759 (11), 2.1791 (10)</td><td>4</td><td>ED</td><td>5.3274 (11), 5.2847 (10), 5.2401 (9)</td></tr><tr><td>1 (σ<sub>(3)</sub>)</td><td>DMRG</td><td>2.1249 (26), 2.1341 (24), 2.1362 (22)</td><td>5</td><td>ED</td><td>6.1466 (11), 6.1310 (10), 6.1156 (9)</td></tr><tr><td>1 (σ<sub>(3)</sub>)</td><td>ε</td><td>3.0000 (0), 2.0455 (1), 2.0652 (2), 2.0034 (3), 2.1447 (4), 1.7171 (5), 3.1800 (6)</td><td>6</td><td>ED</td><td>6.6312 (11), 6.5719 (10), 6.4973 (9)</td></tr><tr><td>1 (σ<sub>(3)</sub>)</td><td>1/N</td><td>1.5000 (0), 2.4457 (1), 2.3355 (2)</td><td>7</td><td>ED</td><td>6.6535 (11), 6.6543 (10), 6.6468 (9)</td></tr><tr><td>2 (∂∂σ<sup>(0)</sup><sub>(3)</sub>)</td><td>ED</td><td>4.4096 (12), 4.4265 (11), 4.4452 (10)</td><td>8</td><td>ED</td><td>6.7481 (11), 6.7580 (10), 6.7673 (9)</td></tr><tr><td>2 (∂∂σ<sup>(0)</sup><sub>(3)</sub>)</td><td>ε</td><td>5.0000 (0), 5.1364 (1)</td><td>9</td><td>ED</td><td>6.7747 (11), 6.7956 (10), 6.8445 (9)</td></tr><tr><td>2 (∂∂σ<sup>(0)</sup><sub>(3)</sub>)</td><td>1/N</td><td>-1.0000 (0), 3.5000 (1), 5.1662 (2)</td><td>10</td><td>ED</td><td>6.7842 (11), 7.0157 (10), 6.9621 (9)</td></tr><tr><td>3</td><td>ED</td><td>4.8890 (11), 4.8834 (10), 4.8803 (9)</td><td>11</td><td>ED</td><td>6.7984 (11), 7.1141 (10), 7.0896 (9)</td></tr></tbody></table>
<table><caption>TABLE LVIII: Scaling dimension for the sector $S = 3^-$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂σ<sub>(3)</sub>)</td><td>ED</td><td>3.2219 (12), 3.2238 (11), 3.2252 (10)</td><td>8</td><td>ED</td><td>6.1798 (11), 6.1232 (10), 6.0602 (9)</td></tr><tr><td>1 (∂σ<sub>(3)</sub>)</td><td>DMRG</td><td>3.1851 (24), 3.1871 (20), 3.1959 (18)</td><td>9</td><td>ED</td><td>6.3537 (11), 6.3369 (10), 6.3207 (9)</td></tr><tr><td>2 (∂∂∂σ<sub>(3)</sub><sup>(1)</sup>)</td><td>ED</td><td>5.0620 (11), 5.0350 (10), 5.0045 (9)</td><td>10</td><td>ED</td><td>6.4716 (11), 6.4475 (10), 6.4231 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.3659 (11), 5.4007 (10), 5.4352 (9)</td><td>11</td><td>ED</td><td>6.5481 (11), 6.5360 (10), 6.4591 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.5018 (11), 5.5083 (10), 5.5179 (9)</td><td>12</td><td>ED</td><td>6.6170 (11), 6.5832 (10), 6.5596 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.8074 (11), 5.7346 (10), 5.6588 (9)</td><td>13</td><td>ED</td><td>6.6347 (10), 6.6540 (9), 6.6845 (8)</td></tr><tr><td>6</td><td>ED</td><td>5.8847 (11), 5.8764 (10), 5.8696 (9)</td><td>14</td><td>ED</td><td>6.8152 (10), 6.8073 (9), 6.7606 (8)</td></tr><tr><td>7</td><td>ED</td><td>6.0274 (11), 5.9980 (10), 5.9636 (9)</td><td>15</td><td>ED</td><td>6.8582 (10), 6.8156 (9), 6.8208 (8)</td></tr></tbody></table>

<table><caption>TABLE LIX: Scaling dimension for the sector $S = 3^-$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (σ<sub>(3)μν</sub>)</td><td>ED</td><td>4.0222 (12), 4.0265 (11), 4.0307 (10)</td><td>11</td><td>ED</td><td>6.2076 (11), 6.1702 (10), 6.1207 (9)</td></tr><tr><td>1 (σ<sub>(3)μν</sub>)</td><td>DMRG</td><td>3.9825 (24), 3.9962 (18), 4.0021 (16)</td><td>12</td><td>ED</td><td>6.2924 (11), 6.2540 (10), 6.2116 (9)</td></tr><tr><td>2 (∂∂∂σ<sub>(3)</sub><sup>(2)</sup>)</td><td>ED</td><td>4.2159 (12), 4.2037 (11), 4.1875 (10)</td><td>13</td><td>ED</td><td>6.3360 (11), 6.3056 (10), 6.2576 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.0898 (11), 5.0840 (10), 5.0705 (9)</td><td>14</td><td>ED</td><td>6.4285 (11), 6.3657 (10), 6.2679 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.2459 (11), 5.2120 (10), 5.1775 (9)</td><td>15</td><td>ED</td><td>6.4458 (11), 6.3886 (10), 6.3304 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.4107 (11), 5.3625 (10), 5.3108 (9)</td><td>16</td><td>ED</td><td>6.5049 (11), 6.4713 (10), 6.4640 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.6220 (11), 5.5595 (10), 5.4922 (9)</td><td>17</td><td>ED</td><td>6.7178 (11), 6.6532 (10), 6.5568 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.8806 (11), 5.8305 (10), 5.7654 (9)</td><td>18</td><td>ED</td><td>6.7292 (11), 6.6875 (10), 6.6323 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.8867 (11), 5.8764 (10), 5.8118 (9)</td><td>19</td><td>ED</td><td>6.7475 (11), 6.7107 (10), 6.6620 (9)</td></tr><tr><td>9</td><td>ED</td><td>6.0328 (11), 5.9675 (10), 5.9590 (9)</td><td>20</td><td>ED</td><td>6.7571 (11), 6.7223 (10), 6.6763 (9)</td></tr><tr><td>10</td><td>ED</td><td>6.1030 (11), 6.0776 (10), 6.0445 (9)</td><td>21</td><td>ED</td><td>6.7488 (10), 6.6905 (9), 6.6407 (8)</td></tr></tbody></table>
<table><caption>TABLE LX: Scaling dimension for the sector $S = 3^-$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂∂∂σ<sup>(3)</sup><sub>(3)</sub>)</td><td>ED</td><td>4.5599 (11), 4.5501 (10), 4.5382 (9)</td><td>12</td><td>ED</td><td>6.5107 (11), 6.4147 (10), 6.2996 (9)</td></tr><tr><td>2</td><td>ED</td><td>4.9559 (11), 4.9120 (10), 4.8461 (9)</td><td>13</td><td>ED</td><td>6.5330 (11), 6.4469 (10), 6.3312 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.0220 (11), 4.9792 (10), 4.9345 (9)</td><td>14</td><td>ED</td><td>6.5431 (11), 6.4739 (10), 6.4025 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.5929 (11), 5.5560 (10), 5.4907 (9)</td><td>15</td><td>ED</td><td>6.5958 (11), 6.5119 (10), 6.4241 (9)</td></tr><tr><td>5</td><td>ED</td><td>5.6589 (11), 5.5946 (10), 5.5351 (9)</td><td>16</td><td>ED</td><td>6.6754 (11), 6.6072 (10), 6.5233 (9)</td></tr><tr><td>6</td><td>ED</td><td>5.7942 (11), 5.7277 (10), 5.6574 (9)</td><td>17</td><td>ED</td><td>6.7496 (11), 6.6743 (10), 6.5826 (9)</td></tr><tr><td>7</td><td>ED</td><td>5.8916 (11), 5.8228 (10), 5.7383 (9)</td><td>18</td><td>ED</td><td>6.7689 (11), 6.7090 (10), 6.6511 (9)</td></tr><tr><td>8</td><td>ED</td><td>5.9851 (11), 5.8996 (10), 5.8110 (9)</td><td>19</td><td>ED</td><td>6.8423 (11), 6.7816 (10), 6.6691 (9)</td></tr><tr><td>9</td><td>ED</td><td>6.0099 (11), 5.9377 (10), 5.8500 (9)</td><td>20</td><td>ED</td><td>6.8844 (11), 6.8009 (10), 6.7027 (9)</td></tr><tr><td>10</td><td>ED</td><td>6.1373 (11), 6.0644 (10), 5.9816 (9)</td><td>21</td><td>ED</td><td>6.9056 (11), 6.8494 (10), 6.7389 (9)</td></tr><tr><td>11</td><td>ED</td><td>6.3247 (11), 6.2378 (10), 6.1401 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXI: Scaling dimension for the sector $S = 3^-$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.3703 (11), 5.3143 (10), 5.2431 (9)</td><td>12</td><td>ED</td><td>6.5148 (11), 6.3878 (10), 6.2426 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.5568 (11), 5.4399 (10), 5.3025 (9)</td><td>13</td><td>ED</td><td>6.5819 (11), 6.4778 (10), 6.3608 (9)</td></tr><tr><td>3</td><td>ED</td><td>5.6178 (11), 5.5270 (10), 5.4168 (9)</td><td>14</td><td>ED</td><td>6.6539 (11), 6.5374 (10), 6.3966 (9)</td></tr><tr><td>4</td><td>ED</td><td>5.9889 (11), 5.9427 (10), 5.8665 (9)</td><td>15</td><td>ED</td><td>6.7070 (11), 6.6641 (10), 6.5420 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.1320 (11), 6.0265 (10), 5.9120 (9)</td><td>16</td><td>ED</td><td>6.8221 (11), 6.6951 (10), 6.6033 (9)</td></tr><tr><td>6</td><td>ED</td><td>6.1874 (11), 6.0790 (10), 5.9576 (9)</td><td>17</td><td>ED</td><td>6.8533 (11), 6.7530 (10), 6.6233 (9)</td></tr><tr><td>7</td><td>ED</td><td>6.2494 (11), 6.1562 (10), 6.0347 (9)</td><td>18</td><td>ED</td><td>6.9284 (11), 6.7891 (10), 6.6432 (9)</td></tr><tr><td>8</td><td>ED</td><td>6.2755 (11), 6.1892 (10), 6.1020 (9)</td><td>19</td><td>ED</td><td>6.9423 (11), 6.8125 (10), 6.6634 (9)</td></tr><tr><td>9</td><td>ED</td><td>6.3623 (11), 6.2464 (10), 6.1169 (9)</td><td>20</td><td>ED</td><td>6.9705 (11), 6.8466 (10), 6.7305 (9)</td></tr><tr><td>10</td><td>ED</td><td>6.4381 (11), 6.3255 (10), 6.1938 (9)</td><td>21</td><td>ED</td><td>7.0090 (11), 6.9183 (10), 6.7867 (9)</td></tr><tr><td>11</td><td>ED</td><td>6.4759 (11), 6.3605 (10), 6.2291 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXII: Scaling dimension for the sector $S = 3^-$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.8511 (11), 5.7018 (10), 5.5238 (9)</td><td>6</td><td>ED</td><td>6.5455 (11), 6.3992 (10), 6.2203 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.8751 (11), 5.7321 (10), 5.5742 (9)</td><td>7</td><td>ED</td><td>6.5771 (11), 6.4323 (10), 6.2353 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.0151 (11), 5.8631 (10), 5.6879 (9)</td><td>8</td><td>ED</td><td>6.6122 (11), 6.4619 (10), 6.3257 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.4079 (11), 6.3092 (10), 6.1607 (9)</td><td>9</td><td>ED</td><td>6.6932 (11), 6.5552 (10), 6.3918 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.5004 (11), 6.3421 (10), 6.1813 (9)</td><td>10</td><td>ED</td><td>6.7594 (11), 6.6399 (10), 6.4528 (9)</td></tr></tbody></table>
<table><caption>TABLE LXIII: Scaling dimension for the sector $S = 3^-$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.9807 (11), 5.7998 (10), 5.6077 (9)</td><td>6</td><td>ED</td><td>6.5423 (10), 6.3281 (9), 6.2829 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.8998 (10), 5.6874 (9), 5.5467 (8)</td><td>7</td><td>ED</td><td>6.5962 (10), 6.4015 (9), 6.3273 (8)</td></tr><tr><td>3</td><td>ED</td><td>6.0552 (10), 5.8206 (9), 5.9873 (8)</td><td>8</td><td>ED</td><td>6.7411 (10), 6.5270 (9), 6.3703 (8)</td></tr><tr><td>4</td><td>ED</td><td>6.4552 (10), 6.2301 (9), 6.0985 (8)</td><td>9</td><td>ED</td><td>6.7614 (10), 6.5448 (9), 6.4707 (8)</td></tr><tr><td>5</td><td>ED</td><td>6.5259 (10), 6.2857 (9), 6.1790 (8)</td><td>10</td><td>ED</td><td>6.7832 (10), 6.5679 (9), 6.5165 (8)</td></tr></tbody></table>

<table><caption>TABLE LXIV: Scaling dimension for the sector $S = 3^-$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.8279 (10), 5.6285 (9), 5.4157 (8)</td><td>6</td><td>ED</td><td>6.5838 (10), 6.5677 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.9809 (10), 5.8179 (9)</td><td>7</td><td>ED</td><td>6.6831 (10), 6.6311 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.1176 (10), 6.2433 (9)</td><td>8</td><td>ED</td><td>6.8103 (10), 6.6773 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.4800 (10), 6.3567 (9)</td><td>9</td><td>ED</td><td>6.8270 (10), 6.7781 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.5622 (10), 6.4502 (9)</td><td>10</td><td>ED</td><td>6.8830 (10), 6.8075 (9)</td></tr></tbody></table>
E. $S = 4$:

<figure><img src="image_15.png" /><figcaption>FIG. 9: Spectrum in the $S = 4^+$ sector.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>FIG. 10: Spectrum in the $S = 4^-$ sector.</figcaption></figure>

<table><caption>TABLE LXV: Scaling dimension for the sector $S = 4^+$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (t<sub>(4)</sub>)</td><td>ED</td><td>3.2686 (11), 3.2765 (10), 3.2870 (9)</td><td>2</td><td>1/N</td><td>2.0000 (0), 4.0000 (1), 6.8820 (2)</td></tr><tr><td>1 (t<sub>(4)</sub>)</td><td>DMRG</td><td>3.1646 (26), 3.1819 (24), 3.1877 (22)</td><td>3</td><td>ED</td><td>6.2537 (11), 6.2485 (10), 6.2446 (9)</td></tr><tr><td>1 (t<sub>(4)</sub>)</td><td>ε</td><td>4.0000 (0), 3.0909 (1), 2.9113 (2), 3.1912 (3), 2.4223 (4), 4.9354 (5), -4.5299 (6)</td><td>4</td><td>ED</td><td>6.5854 (11), 6.5528 (10), 6.5202 (9)</td></tr><tr><td>1 (t<sub>(4)</sub>)</td><td>1/N</td><td>2.0000 (0), 3.8013 (1), 3.4470 (2)</td><td>5</td><td>ED</td><td>7.4387 (11), 7.3971 (10), 7.2985 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.5250 (11), 5.5430 (10), 5.5628 (9)</td><td>6</td><td>ED</td><td>7.4892 (11), 7.4265 (10), 7.4110 (9)</td></tr><tr><td>2</td><td>ε</td><td>6.0000 (0), 6.7273 (1)</td><td>7</td><td>ED</td><td>7.6229 (11), 7.6788 (10), 7.6864 (9)</td></tr></tbody></table>
<table><caption>TABLE LXVI: Scaling dimension for the sector $S = 4^+$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (∂t<sup>(1)</sup><sub>(4)</sub>)</td><td>ED</td><td>4.3166 (11), 4.3186 (10), 4.3203 (9)</td><td>9</td><td>ED</td><td>7.6307 (11), 7.6176 (10), 7.5986 (9)</td></tr><tr><td>1 (∂t<sup>(1)</sup><sub>(4)</sub>)</td><td>DMRG</td><td>4.2450 (24)</td><td>10</td><td>ED</td><td>7.7246 (11), 7.7313 (10), 7.6872 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.3498 (11), 6.3360 (10), 6.3154 (9)</td><td>11</td><td>ED</td><td>7.7980 (11), 7.7599 (10), 7.7145 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.5207 (11), 6.5247 (10), 6.5235 (9)</td><td>12</td><td>ED</td><td>7.8571 (11), 7.8218 (10), 7.8300 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.6342 (11), 6.6646 (10), 6.7074 (9)</td><td>13</td><td>ED</td><td>7.8875 (10), 7.8640 (9), 7.8158 (8)</td></tr><tr><td>5</td><td>ED</td><td>6.9957 (11), 6.9358 (10), 6.8758 (9)</td><td>14</td><td>ED</td><td>8.0409 (10), 7.9119 (9), 7.8810 (8)</td></tr><tr><td>6</td><td>ED</td><td>7.2065 (11), 7.1858 (10), 7.1614 (9)</td><td>15</td><td>ED</td><td>8.1002 (10), 8.0414 (9), 7.9184 (8)</td></tr><tr><td>7</td><td>ED</td><td>7.3241 (11), 7.2925 (10), 7.2519 (9)</td><td>16</td><td>ED</td><td>8.1563 (10), 8.0632 (9), 8.0176 (8)</td></tr><tr><td>8</td><td>ED</td><td>7.4493 (11), 7.3941 (10), 7.3307 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXVII: Scaling dimension for the sector $S = 4^+$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1 (t<sub>(4)</sub><sub>μν</sub>)</td><td>ED</td><td>5.1776 (11), 5.1871 (10), 5.1831 (9)</td><td>11</td><td>ED</td><td>7.2551 (11), 7.2564 (10), 7.1746 (9)</td></tr><tr><td>1 (t<sub>(4)</sub><sub>μν</sub>)</td><td>DMRG</td><td>5.0691 (24)</td><td>12</td><td>ED</td><td>7.3767 (11), 7.3167 (10), 7.2845 (9)</td></tr><tr><td>2 (∂∂t<sup>(2)</sup><sub>(4)</sub>)</td><td>ED</td><td>5.2537 (11), 5.2320 (10), 5.2181 (9)</td><td>13</td><td>ED</td><td>7.3845 (11), 7.3431 (10), 7.3539 (9)</td></tr><tr><td>2 (∂∂t<sup>(2)</sup><sub>(4)</sub>)</td><td>DMRG</td><td>5.2959 (24)</td><td>14</td><td>ED</td><td>7.4704 (11), 7.4298 (10), 7.3618 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.2174 (11), 6.2153 (10), 6.2081 (9)</td><td>15</td><td>ED</td><td>7.5709 (11), 7.4725 (10), 7.3831 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.5038 (11), 6.4741 (10), 6.4422 (9)</td><td>16</td><td>ED</td><td>7.5910 (11), 7.5410 (10), 7.4878 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.6984 (11), 6.6531 (10), 6.6041 (9)</td><td>17</td><td>ED</td><td>7.6292 (11), 7.5904 (10), 7.5009 (9)</td></tr><tr><td>6</td><td>ED</td><td>6.8450 (11), 6.7917 (10), 6.7344 (9)</td><td>18</td><td>ED</td><td>7.7063 (11), 7.6481 (10), 7.5871 (9)</td></tr><tr><td>7</td><td>ED</td><td>7.0731 (11), 7.0651 (10), 7.0082 (9)</td><td>19</td><td>ED</td><td>7.7237 (11), 7.6682 (10), 7.6558 (9)</td></tr><tr><td>8</td><td>ED</td><td>7.1663 (11), 7.1188 (10), 7.0564 (9)</td><td>20</td><td>ED</td><td>7.8706 (10), 7.8014 (9), 7.6906 (8)</td></tr><tr><td>9</td><td>ED</td><td>7.1878 (11), 7.1563 (10), 7.1366 (9)</td><td>21</td><td>ED</td><td>7.9077 (10), 7.8539 (9), 7.7356 (8)</td></tr><tr><td>10</td><td>ED</td><td>7.2216 (11), 7.1869 (10), 7.1643 (9)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE LXVIII: Scaling dimension for the sector $S = 4^+$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.7669 (11), 5.7649 (10), 5.7604 (9)</td><td>12</td><td>ED</td><td>7.5330 (11), 7.4454 (10), 7.3415 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.9959 (11), 5.9239 (10), 5.8356 (9)</td><td>13</td><td>ED</td><td>7.5645 (11), 7.5549 (10), 7.5082 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.1118 (11), 6.0886 (10), 6.0567 (9)</td><td>14</td><td>ED</td><td>7.7013 (11), 7.6274 (10), 7.5539 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.7711 (11), 6.7368 (10), 6.6864 (9)</td><td>15</td><td>ED</td><td>7.7738 (11), 7.6786 (10), 7.5796 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.8796 (11), 6.8197 (10), 6.7527 (9)</td><td>16</td><td>ED</td><td>7.7827 (11), 7.6960 (10), 7.6015 (9)</td></tr><tr><td>6</td><td>ED</td><td>6.9910 (11), 6.9239 (10), 6.8422 (9)</td><td>17</td><td>ED</td><td>7.8138 (11), 7.7269 (10), 7.6316 (9)</td></tr><tr><td>7</td><td>ED</td><td>7.0566 (11), 6.9981 (10), 6.9291 (9)</td><td>18</td><td>ED</td><td>7.8432 (11), 7.7539 (10), 7.6447 (9)</td></tr><tr><td>8</td><td>ED</td><td>7.1942 (11), 7.1185 (10), 7.0423 (9)</td><td>19</td><td>ED</td><td>7.9109 (11), 7.7897 (10), 7.6635 (9)</td></tr><tr><td>9</td><td>ED</td><td>7.2429 (11), 7.1722 (10), 7.0890 (9)</td><td>20</td><td>ED</td><td>7.8583 (10), 7.7738 (9), 7.6541 (8)</td></tr><tr><td>10</td><td>ED</td><td>7.3925 (11), 7.3086 (10), 7.2133 (9)</td><td>21</td><td>ED</td><td>7.8911 (10), 7.8164 (9), 7.6791 (8)</td></tr><tr><td>11</td><td>ED</td><td>7.4751 (11), 7.3776 (10), 7.2621 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXIX: Scaling dimension for the sector $S = 4^+$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.1455 (11), 6.1236 (10), 6.0964 (9)</td><td>12</td><td>ED</td><td>7.4718 (11), 7.3564 (10), 7.2226 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.5020 (11), 6.3847 (10), 6.2387 (9)</td><td>13</td><td>ED</td><td>7.4910 (11), 7.3878 (10), 7.2891 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.5729 (11), 6.5069 (10), 6.4308 (9)</td><td>14</td><td>ED</td><td>7.6218 (11), 7.5094 (10), 7.3833 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.7208 (11), 6.6298 (10), 6.5199 (9)</td><td>15</td><td>ED</td><td>7.6653 (11), 7.5519 (10), 7.4191 (9)</td></tr><tr><td>5</td><td>ED</td><td>7.1030 (11), 7.0502 (10), 6.9548 (9)</td><td>16</td><td>ED</td><td>7.7562 (11), 7.6308 (10), 7.4852 (9)</td></tr><tr><td>6</td><td>ED</td><td>7.1387 (11), 7.0771 (10), 6.9892 (9)</td><td>17</td><td>ED</td><td>7.7971 (11), 7.6842 (10), 7.5540 (9)</td></tr><tr><td>7</td><td>ED</td><td>7.1725 (11), 7.1040 (10), 7.0517 (9)</td><td>18</td><td>ED</td><td>7.8525 (11), 7.7260 (10), 7.5748 (9)</td></tr><tr><td>8</td><td>ED</td><td>7.3275 (11), 7.2303 (10), 7.0994 (9)</td><td>19</td><td>ED</td><td>7.8650 (11), 7.7401 (10), 7.5890 (9)</td></tr><tr><td>9</td><td>ED</td><td>7.3641 (11), 7.2503 (10), 7.1244 (9)</td><td>20</td><td>ED</td><td>7.9207 (11), 7.7983 (10), 7.6427 (9)</td></tr><tr><td>10</td><td>ED</td><td>7.3750 (11), 7.2790 (10), 7.1766 (9)</td><td>21</td><td>ED</td><td>7.9390 (11), 7.8501 (10), 7.6942 (9)</td></tr><tr><td>11</td><td>ED</td><td>7.4393 (11), 7.3326 (10), 7.2136 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXX: Scaling dimension for the sector $S = 4^+$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.8107 (11), 6.6733 (10), 6.4845 (9)</td><td>6</td><td>ED</td><td>7.5489 (11), 7.4164 (10), 7.2480 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.8417 (11), 6.7240 (10), 6.6164 (9)</td><td>7</td><td>ED</td><td>7.5823 (11), 7.4624 (10), 7.2724 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.9982 (11), 6.8620 (10), 6.7016 (9)</td><td>8</td><td>ED</td><td>7.6307 (11), 7.4726 (10), 7.3094 (9)</td></tr><tr><td>4</td><td>ED</td><td>7.0935 (11), 6.9460 (10), 6.7770 (9)</td><td>9</td><td>ED</td><td>7.6654 (11), 7.4915 (10), 7.3317 (9)</td></tr><tr><td>5</td><td>ED</td><td>7.5039 (11), 7.3690 (10), 7.2101 (9)</td><td>10</td><td>ED</td><td>7.6972 (11), 7.5907 (10), 7.4372 (9)</td></tr></tbody></table>
<table><caption>TABLE LXXI: Scaling dimension for the sector $S = 4^+$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>7.0335 (11), 6.8505 (10), 6.7309 (9)</td><td>6</td><td>ED</td><td>7.5652 (10), 7.4226 (9), 7.3472 (8)</td></tr><tr><td>2</td><td>ED</td><td>6.9214 (10), 6.8290 (9), 6.6469 (8)</td><td>7</td><td>ED</td><td>7.6158 (10), 7.4771 (9), 7.3830 (8)</td></tr><tr><td>3</td><td>ED</td><td>7.0307 (10), 6.9333 (9), 7.0778 (8)</td><td>8</td><td>ED</td><td>7.6258 (10), 7.5712 (9), 7.4206 (8)</td></tr><tr><td>4</td><td>ED</td><td>7.1666 (10), 7.3156 (9), 7.2052 (8)</td><td>9</td><td>ED</td><td>7.6498 (10), 7.5892 (9), 7.4528 (8)</td></tr><tr><td>5</td><td>ED</td><td>7.5251 (10), 7.3695 (9), 7.2778 (8)</td><td>10</td><td>ED</td><td>7.7569 (10), 7.6097 (9), 7.5140 (8)</td></tr></tbody></table>

<table><caption>TABLE LXXII: Scaling dimension for the sector $S = 4^+$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.9373 (10), 6.7469 (9), 6.5453 (8)</td><td>6</td><td>ED</td><td>7.6760 (10), 7.6397 (9)</td></tr><tr><td>2</td><td>ED</td><td>7.1036 (10), 6.9156 (9)</td><td>7</td><td>ED</td><td>7.7736 (10), 7.6995 (9)</td></tr><tr><td>3</td><td>ED</td><td>7.2453 (10), 7.3257 (9)</td><td>8</td><td>ED</td><td>7.8656 (10), 7.7313 (9)</td></tr><tr><td>4</td><td>ED</td><td>7.5594 (10), 7.4585 (9)</td><td>9</td><td>ED</td><td>7.8780 (10), 7.7761 (9)</td></tr><tr><td>5</td><td>ED</td><td>7.6459 (10), 7.5583 (9)</td><td>10</td><td>ED</td><td>7.9368 (10), 7.8175 (9)</td></tr></tbody></table>

<table><caption>TABLE LXXIII: Scaling dimension for the sector $S = 4^-$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.0076 (11), 6.0079 (10), 6.0103 (9)</td><td>4</td><td>ED</td><td>7.8189 (11), 7.8145 (10), 7.8153 (9)</td></tr><tr><td>2</td><td>ED</td><td>7.4723 (11), 7.4649 (10), 7.4600 (9)</td><td>5</td><td>ED</td><td>7.9333 (11), 7.9604 (10), 7.9509 (9)</td></tr><tr><td>3</td><td>ED</td><td>7.5500 (11), 7.5561 (10), 7.5613 (9)</td><td>6</td><td>ED</td><td>7.9347 (11), 8.0147 (10), 7.9704 (9)</td></tr></tbody></table>
<table><caption>TABLE LXXIV: Scaling dimension for the sector $S = 4^-$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>4.8514 (11), 4.8410 (10), 4.8308 (9)</td><td>8</td><td>ED</td><td>7.3742 (11), 7.3586 (10), 7.3409 (9)</td></tr><tr><td>1</td><td>DMRG</td><td>4.8718 (26), 4.8785 (24), 4.8763 (22)</td><td>9</td><td>ED</td><td>7.5629 (11), 7.5404 (10), 7.5240 (9)</td></tr><tr><td>1</td><td>ε</td><td>6.0000 (0), 4.8636 (1)</td><td>10</td><td>ED</td><td>7.7388 (10), 7.7076 (9), 7.6619 (8)</td></tr><tr><td>1</td><td>1/N</td><td>-4.0000 (0), 3.5000 (1), 5.5264 (2)</td><td>11</td><td>ED</td><td>7.8064 (10), 7.8043 (9), 7.7811 (8)</td></tr><tr><td>2</td><td>ED</td><td>5.8805 (11), 5.8734 (10), 5.8649 (9)</td><td>12</td><td>ED</td><td>7.8746 (10), 7.8525 (9), 7.7965 (8)</td></tr><tr><td>3</td><td>ED</td><td>6.4004 (11), 6.3895 (10), 6.3806 (9)</td><td>13</td><td>ED</td><td>7.9030 (10), 7.8661 (9), 7.8428 (8)</td></tr><tr><td>4</td><td>ED</td><td>6.7058 (11), 6.7062 (10), 6.7135 (9)</td><td>14</td><td>ED</td><td>7.9228 (10), 7.9090 (9), 7.8981 (8)</td></tr><tr><td>5</td><td>ED</td><td>6.9276 (11), 6.9113 (10), 6.8891 (9)</td><td>15</td><td>ED</td><td>8.0128 (10), 8.0014 (9), 7.9826 (8)</td></tr><tr><td>6</td><td>ED</td><td>6.9609 (11), 6.9437 (10), 6.9208 (9)</td><td>16</td><td>ED</td><td>8.1029 (10), 8.1042 (9), 8.0770 (8)</td></tr><tr><td>7</td><td>ED</td><td>7.0434 (11), 7.0418 (10), 7.0470 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXXV: Scaling dimension for the sector $S = 4^-$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.2972 (11), 5.2569 (10), 5.2130 (9)</td><td>9</td><td>ED</td><td>7.1688 (11), 7.1515 (10), 7.1306 (9)</td></tr><tr><td>2</td><td>ED</td><td>5.7320 (11), 5.6955 (10), 5.6522 (9)</td><td>10</td><td>ED</td><td>7.2809 (11), 7.2470 (10), 7.2168 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.2039 (11), 6.1638 (10), 6.1204 (9)</td><td>11</td><td>ED</td><td>7.4904 (11), 7.4703 (10), 7.4535 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.6972 (11), 6.6809 (10), 6.6360 (9)</td><td>12</td><td>ED</td><td>7.5538 (11), 7.5192 (10), 7.4766 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.7286 (11), 6.6989 (10), 6.6677 (9)</td><td>13</td><td>ED</td><td>7.6579 (11), 7.6285 (10), 7.5575 (9)</td></tr><tr><td>6</td><td>ED</td><td>6.7842 (11), 6.7581 (10), 6.7528 (9)</td><td>14</td><td>ED</td><td>7.6829 (11), 7.6345 (10), 7.6002 (9)</td></tr><tr><td>7</td><td>ED</td><td>7.1262 (11), 7.0812 (10), 7.0259 (9)</td><td>15</td><td>ED</td><td>7.7319 (11), 7.6856 (10), 7.6202 (9)</td></tr><tr><td>8</td><td>ED</td><td>7.1531 (11), 7.1179 (10), 7.0859 (9)</td><td>16</td><td>ED</td><td>7.7459 (11), 7.6967 (10), 7.6502 (9)</td></tr></tbody></table>
<table><caption>TABLE LXXVI: Scaling dimension for the sector $S = 4^-$, $L = 3$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.6999 (11), 5.6284 (10), 5.5498 (9)</td><td>11</td><td>ED</td><td>7.4271 (11), 7.3479 (10), 7.2788 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.0400 (11), 5.9643 (10), 5.8768 (9)</td><td>12</td><td>ED</td><td>7.4431 (11), 7.3673 (10), 7.2832 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.4239 (11), 6.3387 (10), 6.2381 (9)</td><td>13</td><td>ED</td><td>7.5556 (11), 7.5223 (10), 7.4861 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.5666 (11), 6.5010 (10), 6.4284 (9)</td><td>14</td><td>ED</td><td>7.5975 (11), 7.5578 (10), 7.5027 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.7642 (11), 6.7378 (10), 6.7002 (9)</td><td>15</td><td>ED</td><td>7.6759 (11), 7.6201 (10), 7.5246 (9)</td></tr><tr><td>6</td><td>ED</td><td>6.8875 (11), 6.8130 (10), 6.7318 (9)</td><td>16</td><td>ED</td><td>7.7392 (11), 7.6533 (10), 7.5821 (9)</td></tr><tr><td>7</td><td>ED</td><td>7.0939 (11), 7.0349 (10), 6.9715 (9)</td><td>17</td><td>ED</td><td>7.7693 (11), 7.7167 (10), 7.6496 (9)</td></tr><tr><td>8</td><td>ED</td><td>7.1570 (11), 7.1346 (10), 7.1085 (9)</td><td>18</td><td>ED</td><td>7.8065 (11), 7.7502 (10), 7.6946 (9)</td></tr><tr><td>9</td><td>ED</td><td>7.3274 (11), 7.2873 (10), 7.1827 (9)</td><td>19</td><td>ED</td><td>7.8798 (11), 7.8042 (10), 7.7274 (9)</td></tr><tr><td>10</td><td>ED</td><td>7.3731 (11), 7.3044 (10), 7.2597 (9)</td><td>20</td><td>ED</td><td>7.9634 (11), 7.9151 (10), 7.8567 (9)</td></tr></tbody></table>

<table><caption>TABLE LXXVII: Scaling dimension for the sector $S = 4^-$, $L = 4$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.0281 (11), 5.9231 (10), 5.8069 (9)</td><td>12</td><td>ED</td><td>7.5677 (11), 7.5056 (10), 7.4530 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.2705 (11), 6.1598 (10), 6.0345 (9)</td><td>13</td><td>ED</td><td>7.6587 (11), 7.5870 (10), 7.4903 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.5736 (11), 6.4468 (10), 6.3006 (9)</td><td>14</td><td>ED</td><td>7.6920 (11), 7.6181 (10), 7.5265 (9)</td></tr><tr><td>4</td><td>ED</td><td>6.8771 (11), 6.7551 (10), 6.5923 (9)</td><td>15</td><td>ED</td><td>7.7662 (11), 7.6823 (10), 7.5523 (9)</td></tr><tr><td>5</td><td>ED</td><td>6.8974 (11), 6.7853 (10), 6.6828 (9)</td><td>16</td><td>ED</td><td>7.8107 (11), 7.6869 (10), 7.5896 (9)</td></tr><tr><td>6</td><td>ED</td><td>7.0286 (11), 6.9598 (10), 6.8737 (9)</td><td>17</td><td>ED</td><td>7.8594 (11), 7.7445 (10), 7.6218 (9)</td></tr><tr><td>7</td><td>ED</td><td>7.1372 (11), 7.0328 (10), 6.9171 (9)</td><td>18</td><td>ED</td><td>7.8873 (11), 7.7879 (10), 7.6448 (9)</td></tr><tr><td>8</td><td>ED</td><td>7.3652 (11), 7.2496 (10), 7.1010 (9)</td><td>19</td><td>ED</td><td>7.9059 (11), 7.8033 (10), 7.7043 (9)</td></tr><tr><td>9</td><td>ED</td><td>7.3912 (11), 7.3204 (10), 7.2417 (9)</td><td>20</td><td>ED</td><td>7.9714 (11), 7.8714 (10), 7.7583 (9)</td></tr><tr><td>10</td><td>ED</td><td>7.4762 (11), 7.4242 (10), 7.3475 (9)</td><td>21</td><td>ED</td><td>7.9777 (11), 7.8975 (10), 7.7938 (9)</td></tr><tr><td>11</td><td>ED</td><td>7.5281 (11), 7.4555 (10), 7.3709 (9)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXXVIII: Scaling dimension for the sector $S = 4^-$, $L = 5$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.3735 (11), 6.2345 (10), 6.0811 (9)</td><td>6</td><td>ED</td><td>7.3381 (11), 7.2133 (10), 7.0745 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.6294 (11), 6.4726 (10), 6.2981 (9)</td><td>7</td><td>ED</td><td>7.4875 (11), 7.3327 (10), 7.1532 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.9082 (11), 6.7320 (10), 6.5329 (9)</td><td>8</td><td>ED</td><td>7.5802 (11), 7.4550 (10), 7.3082 (9)</td></tr><tr><td>4</td><td>ED</td><td>7.1846 (11), 6.9934 (10), 6.7806 (9)</td><td>9</td><td>ED</td><td>7.7009 (11), 7.5259 (10), 7.3240 (9)</td></tr><tr><td>5</td><td>ED</td><td>7.2465 (11), 7.1236 (10), 6.9794 (9)</td><td>10</td><td>ED</td><td>7.7216 (11), 7.6008 (10), 7.4621 (9)</td></tr></tbody></table>
<table><caption>TABLE LXXIX: Scaling dimension for the sector $S = 4^-$, $L = 6$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.5574 (11), 6.3855 (10), 6.1992 (9)</td><td>6</td><td>ED</td><td>7.3648 (10), 7.2843 (9), 7.2883 (8)</td></tr><tr><td>2</td><td>ED</td><td>6.6307 (10), 6.4229 (9), 6.1977 (8)</td><td>7</td><td>ED</td><td>7.4979 (10), 7.3482 (9), 7.3823 (8)</td></tr><tr><td>3</td><td>ED</td><td>6.8791 (10), 6.6384 (9), 6.8981 (8)</td><td>8</td><td>ED</td><td>7.5466 (10), 7.5323 (9), 7.5212 (8)</td></tr><tr><td>4</td><td>ED</td><td>7.1167 (10), 7.1054 (9), 6.9728 (8)</td><td>9</td><td>ED</td><td>7.6777 (10), 7.6202 (9), 7.6584 (8)</td></tr><tr><td>5</td><td>ED</td><td>7.2924 (10), 7.1799 (9), 7.1265 (8)</td><td>10</td><td>ED</td><td>7.7445 (10), 7.7142 (9), 7.7305 (8)</td></tr></tbody></table>

<table><caption>TABLE LXXX: Scaling dimension for the sector $S = 4^-$, $L = 7$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.4406 (10), 6.2376 (9), 6.0194 (8)</td><td>6</td><td>ED</td><td>7.5890 (10), 7.5950 (9)</td></tr><tr><td>2</td><td>ED</td><td>6.7022 (10), 6.4784 (9)</td><td>7</td><td>ED</td><td>7.6695 (10), 7.6082 (9)</td></tr><tr><td>3</td><td>ED</td><td>6.9379 (10), 7.1637 (9)</td><td>8</td><td>ED</td><td>7.8547 (10), 7.7571 (9)</td></tr><tr><td>4</td><td>ED</td><td>7.3659 (10), 7.2488 (9)</td><td>9</td><td>ED</td><td>7.8643 (10), 7.9659 (9)</td></tr><tr><td>5</td><td>ED</td><td>7.4594 (10), 7.4375 (9)</td><td>10</td><td>ED</td><td>7.9983 (10), 7.9805 (9)</td></tr></tbody></table>
F. $S = 5$:

<figure><img src="image_17.png" /><figcaption>FIG. 11: Spectrum in the $S = 5^+$ sector.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>FIG. 12: Spectrum in the $S = 5^-$ sector.</figcaption></figure>

<table><caption>TABLE LXXXI: Scaling dimension for the sector $S = 5^+$, $L = 0$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>5.9237 (11), 7.3398 (10), 7.3421 (9)</td><td>4</td><td>DMRG</td><td>8.0792 (14)</td></tr><tr><td>1</td><td>DMRG</td><td>6.1069 (14)</td><td>5</td><td>ED</td><td>9.2480 (10), 9.2552 (9), 9.3241 (8)</td></tr><tr><td>2</td><td>ED</td><td>7.3378 (11), 8.7746 (10), 8.7514 (9)</td><td>5</td><td>DMRG</td><td>8.2728 (14)</td></tr><tr><td>2</td><td>DMRG</td><td>7.5954 (14)</td><td>6</td><td>ED</td><td>9.3486 (10), 9.3418 (9), 9.7205 (8)</td></tr><tr><td>3</td><td>ED</td><td>8.8908 (10), 8.9026 (9), 8.9226 (8)</td><td>7</td><td>ED</td><td>9.5466 (10), 9.7291 (9), 9.7724 (8)</td></tr><tr><td>3</td><td>DMRG</td><td>7.9110 (14)</td><td>8</td><td>ED</td><td>9.6012 (10), 9.9380 (9), 9.8007 (8)</td></tr><tr><td>4</td><td>ED</td><td>9.2275 (10), 9.1784 (9), 9.2815 (8)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE LXXXII: Scaling dimension for the sector $S = 5^+$, $L = 1$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.1937 (11), 6.1913 (10), 6.1904 (9)</td><td>5</td><td>ED</td><td>8.2119 (10), 8.1811 (9), 8.1463 (8)</td></tr><tr><td>1</td><td>DMRG</td><td>6.1688 (20), 6.1821 (16), 6.4375 (14)</td><td>5</td><td>DMRG</td><td>8.3480 (14)</td></tr><tr><td>2</td><td>ED</td><td>6.9310 (11), 7.2178 (10), 7.2059 (9)</td><td>6</td><td>ED</td><td>8.2293 (10), 8.1963 (9), 8.2061 (8)</td></tr><tr><td>2</td><td>DMRG</td><td>7.1534 (14)</td><td>7</td><td>ED</td><td>8.4446 (10), 8.4647 (9), 8.4886 (8)</td></tr><tr><td>3</td><td>ED</td><td>7.2259 (11), 7.7167 (10), 7.7063 (9)</td><td>8</td><td>ED</td><td>8.6483 (10), 8.6170 (9), 8.5695 (8)</td></tr><tr><td>3</td><td>DMRG</td><td>7.4804 (14)</td><td>9</td><td>ED</td><td>8.9240 (10), 8.9225 (9), 8.9025 (8)</td></tr><tr><td>4</td><td>ED</td><td>7.7276 (11), 8.1270 (10), 8.1455 (9)</td><td>10</td><td>ED</td><td>9.0317 (10), 8.9947 (9), 8.9691 (8)</td></tr><tr><td>4</td><td>DMRG</td><td>8.0254 (14)</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE LXXXIII: Scaling dimension for the sector $S = 5^+$, $L = 2$. For the ED and DMRG data, the numbers in bracket is the corresponding system size. For the $\epsilon$-expansion and large-$N$ expansion, the numbers in the bracket indicate the order of the expansion.</caption><thead><tr><th>Index</th><th>Method</th><th>Δ</th><th>Index</th><th>Method</th><th>Δ</th></tr></thead><tbody><tr><td>1</td><td>ED</td><td>6.5813 (11), 6.5484 (10), 6.5132 (9)</td><td>7</td><td>ED</td><td>8.3534 (10), 8.2808 (9), 8.1868 (8)</td></tr><tr><td>1</td><td>DMRG</td><td>6.9040 (14)</td><td>8</td><td>ED</td><td>8.4480 (10), 8.4194 (9), 8.3458 (8)</td></tr><tr><td>2</td><td>ED</td><td>7.0568 (11), 7.0209 (10), 6.9773 (9)</td><td>9</td><td>ED</td><td>8.4812 (10), 8.4441 (9), 8.4414 (8)</td></tr><tr><td>2</td><td>DMRG</td><td>7.3740 (14)</td><td>10</td><td>ED</td><td>8.6334 (10), 8.6384 (9), 8.6450 (8)</td></tr><tr><td>3</td><td>ED</td><td>7.5006 (11), 7.4613 (10), 7.4170 (9)</td><td>11</td><td>ED</td><td>8.8395 (10), 8.7516 (9), 8.7082 (8)</td></tr><tr><td>3</td><td>DMRG</td><td>7.8609 (14)</td><td>12</td><td>ED</td><td>8.8671 (10), 8.7763 (9), 8.8505 (8)</td></tr><tr><td>4</td><td>ED</td><td>7.7617 (11), 7.9706 (10), 7.9098 (9)</td><td>13</td><td>ED</td><td>8.8772 (10), 8.8134 (9), 8.9500 (8)</td></tr><tr><td>4</td><td>DMRG</td><td>8.0762 (14)</td><td>14</td><td>ED</td><td>8.9008 (10), 8.8684 (9), 8.9981 (8)</td></tr><tr><td>5</td><td>ED</td><td>8.0152 (10), 7.9690 (9), 7.9207 (8)</td><td>15</td><td>ED</td><td>9.0068 (10), 8.9809 (9), 9.0657 (8)</td></tr><tr><td>5</td><td>DMRG</td><td>8.0783 (14)</td><td>16</td><td>ED</td><td>9.0669 (10), 9.0466 (9), 9.1808 (8)</td></tr><tr><td>6</td><td>ED</td><td>8.1529 (10), 8.1704 (9), 8.1831 (8)</td><td></td><td></td><td></td></tr></tbody></table>