# Hilbert space fragmentation imposed real spectrum of non-Hermitian systems

Somsubhra Ghosh¹, K. Sengupta¹, and Indranil Paul²

¹School of Physical Sciences, Indian Association for the Cultivation of Science, Kolkata 700032, India.

²Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, 75205 Paris, France.

(Dated: July 12, 2024)

We show that constraints imposed by strong Hilbert space fragmentation (HSF) along with the presence of certain global symmetries can ensure the reality of eigenspectrum of non-Hermitian quantum systems; such a reality cannot be guaranteed by global symmetries alone. We demonstrate this insight for two interacting finite chains, namely the fermionic Nelson-Hatano and the Su-Schrieffer-Heeger models, none of which has a *PT* symmetry. We show analytically that strong HSF and real spectrum are both consequences of the same dynamical constraints in the limit of large interaction, provided the systems have sufficient global symmetries. We also show that a local equal-time correlation function can detect the many-body exceptional point at a finite critical interaction strength above which the eigenspectrum is real.

## I. INTRODUCTION

Non-Hermitian many-body Hamiltonians are of great current interest for their relevance to open quantum systems, and also for their novel properties without Hermitian analog [1–20]. These features include novel topological properties including the presence of so called exceptional points. At these points, certain energy eigenvalues are degenerate and the corresponding eigenfunctions coalesce. Across such points, these eigenvalues can transform from being real to complex [21–27]. Moreover, the non-equilibrium dynamics of such systems are of interest since they posses qualitatively different characteristics compared to that of their closed Hermitian counterparts [15–20]. Some such characteristics include the presence of athermal steady states and entanglement transitions following a quench or in the presence of a periodic drive.

The purpose of the current work is to investigate an important related feature of these systems. Namely, why the spectra of certain non-Hermitian Hamiltonians are entirely real in some parameter regimes. Note, this question cannot be addressed completely by invoking global symmetries. For example, pseudo-Hermiticity only guarantees that complex eigenvalues, if they appear, come in complex conjugate pairs [28, 29]. Likewise, a so-called *PT*-symmetric system, where *P* and *T* refer to parity and time reversal operators, respectively, has completely real eigenvalues only in the regime where all the energy eigenfunctions are also simultaneously eigenfunctions of the *PT* operator [30–32], and the question remains as to what guarantees the latter. In fact, as discussed later, the models that we study in this work are not *PT*-symmetric and, as such, *PT* symmetry plays no role.

In this work we show that the combination of dynamical constraints imposed in the limit of large interaction along with global symmetries together can protect a “phase” where the spectrum is entirely real. We illustrate our idea with finite chains of the fermionic Hatano-Nelson (HN) and the non-Hermitian Su-Schrieffer-Heeger (SSH) models with nearest neighbor interaction.

An important ingredient in what follows is strong Hilbert space fragmentation in the limit of infinitely large interaction, where the Fock space breaks up into dynamically disjoint pieces whose number scales exponentially with the system size [33, 34]. This phenomena is the focus of intense research at present in Hermitian systems, since it leads to non-ergodicity and the ability to generate exotic non-equilibrium states [35–44].

<figure><img src="image_1.png" /><figcaption>FIG. 1. (Color Online) (a) Schematic of Hatano-Nelson model with non-reciprocal hopping, and nearest neighbor interaction $V_1$. Filled sites are marked red. (b), (c) show the maximum imaginary and the real parts of the spectrum, respectively, as a function of $V_1$, for system size $L = 10$ and $\gamma = 0.2J$. At $V_1^c$ the system encounters a many-body exceptional point. For $V_1 < V_1^c$ complex conjugate pairs of eigenvalues first appear. In (c) the two eigenvalues which coalesce at the exceptional point are delineated in blue. (d) $V_1^c$ diverges as $\gamma \to J$, and also with $L$.</figcaption></figure>

Our main results are the following. (1) We show analytically that in the fragmented limit there are many-body similarity transformations that map the above non-Hermitian systems to Hermitian ones. Thus, in this limit, the spectra are real. In fact, fragmentation and real spectrum are both shown to be consequences of the same dynamical constraints which emerge in the limit of infinitely large interaction. (2) We show that the spectra persist to be real for arbitrarily large but finite interaction, provided the systems have sufficient global symmetry protection. These symmetries impose a hidden Hermiticity in those subspaces where the reality of the spectrum is not guaranteed by the fragmentation limit. Results (1) and (2) together imply that the spectra are real for interactions above finite critical values where the systems encounter many-body
exceptional points. (3) We compute a local equal-time density-density fermionic correlation function and show that it can be used to detect the exceptional point location. Overall, our work provides the first analysis of the role of dynamical constraints in determining the spectrum of a non-Hermitian system.

The plan of the rest of the paper is as follows. In Sec. II, we discuss the role of HSF on reality of the eigenspectra of the Hatano-Nelson model and demonstrate the presence of a "phase" with entirely real eigenvalues. This is followed by Sec. III where analogous phenomena is discussed for the SSH model. Next, in Sec. IV, we discuss the behavior of the equal-time correlation function of the Hatano-Nelson model. Finally, we conclude in Sec. V. Some details of the calculations are elaborated in the appendices.

## II. HATANO-NELSON MODEL

### A. Model

For pedagogical reason we demonstrate our principle in detail first for the HN model, and argue later that the same holds for the SSH model. In terms of fermionic creation and annihilation operators ($c_i^\dagger, c_i$) at site $i$ the HN Hamiltonian is

$$ \mathcal{H}_{HN} = \sum_{i=1}^{L} \left[ (J - \gamma)c_i^\dagger c_{i+1} + (J + \gamma)c_{i+1}^\dagger c_i + V_1 \hat{n}_i \hat{n}_{i+1} \right], \quad (1) $$

where $\hat{n}_i \equiv c_i^\dagger c_i$ is the number operator at site $i$, $L$ is the system size, and $\gamma > 0$ is the non-reciprocal hopping parameter, see Fig. 1(a). We study the system at half-filling with $\sum_i n_i = L/2$, and we impose (anti) periodic boundary condition for total particle number (even) odd. This choice ensures that the system is translationally invariant, as discussed below and in Appendix D. For open boundary condition and $\gamma < J$ the problem is trivial because there is a one-body similarity transformation which makes $\mathcal{H}_{HN}$ Hermitian [24]. As discussed in detail in the next section, $\mathcal{H}_{HN}$ is pseudo-Hermitian, while its global symmetries are $G = (PC, R)$ with $[\mathcal{H}_{HN}, G] = 0$, where $(PC, R)$ are parity, charge conjugation and translation by one site, respectively. Furthermore, since integrability plays no role, our results are valid even in the presence of next-nearest neighbor interaction.

The spectral properties of $\mathcal{H}_{HN}$ are summarized in Fig. 1, (b) - (d). Panels (b) and (c) show that, for $\gamma < J$, the spectrum is real for $V_1 > V_1^c$, a critical value. As shown in (c), at $V_1^c$ one pair (or two pairs) of eigenvalues and eigenvectors coalesce at a many-body exceptional point, and they become complex conjugate pairs for $V_1 < V_1^c$, as dictated by the pseudo-Hermiticity of $\mathcal{H}_{HN}$. Panel (d) shows that $V_1^c$ diverges as $\gamma \to J$, and also with system size $L$. Note, while these features were reported recently [45], the link between Hilbert space fragmentation with the reality of the spectrum, which is the focus of this work, has not been explored earlier.

### B. Global Symmetries

We mention two transformations which establish that the above Hamiltonian is pseudo-Hermitian. First, parity transformation $P$, such that the site number $i \to (L - i + 1)$. Since, this maps right hops to left hops and vice versa, we get $\mathcal{P}\mathcal{H}_{HN}\mathcal{P} = \mathcal{H}_{HN}^\dagger$. Second, particle-hole transformation $C$, such that $C c_i C = (-1)^i c_i^\dagger$. Under this transformation, $c_i^\dagger c_{i+1} \to -c_i c_{i+1}^\dagger = c_{i+1}^\dagger c_i$ and $c_{i+1}^\dagger c_i \to c_i^\dagger c_{i+1}$. Thus, the strengths of the right and the left hops are interchanged. The density-density interaction $n_i n_{i+1} \to (1 - n_i)(1 - n_{i+1})$. However, due to half-filling, this term remains invariant when summed over all sites. Thus, at half-filling we get $\mathcal{C}\mathcal{H}_{HN}\mathcal{C} = \mathcal{H}_{HN}^\dagger$. However, by themselves these relations do not guarantee that at large interaction $V_1$, the energy eigenvalues are real. It is well-known that pseudo-Hermiticity simply guarantees that complex eigenvalues, if they appear, come in complex conjugate pairs.

Combining these two relations one gets a symmetry of the Hamiltonian $\mathcal{H}_{HN}$, namely $\mathcal{P}\mathcal{C}$, since $(\mathcal{C}\mathcal{P})\mathcal{H}_{HN}(\mathcal{P}\mathcal{C}) = \mathcal{H}_{HN}$. It is due to this symmetry that some energy eigenvalues are doubly degenerate (the non-degenerate ones being eigenstates of $\mathcal{P}\mathcal{C}$) leading to situations when two pairs of eigenvalues coalesce at the exceptional point. In addition to this, $\mathcal{H}_{HN}$ respects the translational symmetry $\mathcal{R}$ ($i \to (i+1)$ modulo $L$), which plays an important role in preserving the reality of the eigenspectrum for finite $V_1$, as explained in a later section.

Note, in the absence of spin degrees of freedom, the Hamiltonian of Eq. (1) is trivially time reversal symmetric, with $\mathcal{T}\mathcal{H}_{HN}\mathcal{T} = \mathcal{H}_{HN}$. This implies that $(\mathcal{P}\mathcal{T})\mathcal{H}_{HN}(\mathcal{P}\mathcal{T}) = \mathcal{H}_{HN}^\dagger$. In other words, the model is not $\mathcal{P}\mathcal{T}$-symmetric.

### C. Limit of Fragmentation

For large interaction, we keep terms to linear order in $(J, \gamma)$ and ignore those of order $(J, \gamma)^2/V_1$ and smaller. This gives [46],

$$ \mathcal{H}_{HN} \approx \mathcal{H}_{HN,f} = \sum_{i=1}^{L} \left[ \hat{P}_i \left( (J - \gamma)c_i^\dagger c_{i+1} + (J + \gamma)c_{i+1}^\dagger c_i \right) \hat{P}_i + V_1 \hat{n}_i \hat{n}_{i+1} \right], \quad (2) $$

where the projector $\hat{P}_i \equiv 1 - (\hat{n}_{i-1} - \hat{n}_{i+2})^2$ ensures that the hopping is constrained, and is allowed only if the process does not change the total number of nearest neighbor occupations $\hat{N}_d \equiv \sum_i \hat{n}_i \hat{n}_{i+1}$.

The Hermitian version of $\mathcal{H}_{HN,f}$ has been shown to display strong Hilbert space fragmentation [37, 38]. Since fragmentation is independent of whether the hopping mediated connectivity between the many-body Fock states is reciprocal or not, the non-Hermitian $\mathcal{H}_{HN,f}$ shows fragmentation as well. Below we prove that the dynamical constraints that give rise to fragmentation ensures the existence of a many-body similarity transformation that maps $\mathcal{H}_{HN,f}$ into a Hermitian form for $\gamma < J$.
D. Many-body Similarity Transformation

The first step of the proof is to label the many-body states. Traditionally, this is done using “spins and movers” [37, 38, 46]. Here we take a different strategy, and we label them by “defects”. A “particle-defect” and a “hole-defect” are two occupied or two unoccupied nearest-neighbor sites, respectively. Due to half-filling, particle- and hole-defects appear in pairs, and their numbers are conserved, since $[\mathcal{H}_{HN,f}, \hat{N}_d] = 0$. Thus, the Hilbert space factorizes into sectors with eigenvalue $N_d = 0, 1, \dots, L/2 - 1$.

All dynamically frozen (i.e., zero connectivity) states, which includes the $N_d = 0$ sector, have real energies. For $N_d \neq 0$ we label a defect position by the location of the leftmost of the two nearest-neighbor sites. Thus, any state with $N_d = 1$ has label $|(i)(j)\rangle$, where i and j are locations of the particle- and hole-defect, respectively. Likewise, a state with $N_d = 2$ is labeled by $|(i_1, i_2)(j_1, j_2)\rangle$, and $N_d = n$ by $|(i_1, i_2, \dots, i_n)(j_1, j_2, \dots, j_n)\rangle$. Since the fermions are indistinguishable, permutations of i and of j imply the same state. Thus, the state $|(5)(7)\rangle$ shown in Fig. 2(a), belongs to $N_d = 1$, and has a particle- and a hole-defect at sites i = 5, 7, respectively.

Due to half-filling the defect locations are not arbitrary, but follow certain rules. (a) If two particle-defects at $i_1$ and $i_2$ are “adjacent”, then $(i_1, i_2)$ can only be (odd, even) or (even, odd). The same applies for two adjacent hole-defects. Here, “adjacent” does not imply defects located right next to one another. Two defects are adjacent if there is no third defect in between the two while traversing in one of the two directions. (b) If a particle-defect at $i_1$ is adjacent to a hole-defect at $j_1$, then $(i_1, j_1)$ can only be (even, even) or (odd, odd). One can verify that the wavefunctions in Fig. 2(a) satisfy these rules.

The second step is to determine the defect dynamics which, due to the constrained hopping, obey the following rules. (i) An allowed fermion hop changes i or j by $\pm 2$ modulo L. (ii) Since second nearest neighbor hopping is absent, two defects cannot cross each other. It is shown in Appendix A that due to rules (i) and (ii) each sector of $N_d$ breaks into exponentially large number of disjoint subsectors, i.e. fragments, that scale as $e^{N_d}$ [34].

The third step is to establish the constrained hopping induced connectivity between the many-body wavefunctions within each non-trivial subsector. There is no general pattern for these connectivities, and they need to be worked out case by case, even though the proof below holds for all the connectivities. To show few examples, Fig. 2(b) is the connectivity for $L = 10$, $N_d = 2$ with $(i_1, i_2)(j_1, j_2) = (\text{odd}, \text{odd})(\text{odd}, \text{odd})$, while Fig. 2(c) is for $L = 10$, $N_d = 1$ with $(i)(j) = (\text{odd})(\text{odd})$. The dashed and solid arrows denote fermions hopping to the left and right (amplitudes $J_{1,2} \equiv J \mp \gamma$), respectively. Reversing an arrow implies $J_1 \leftrightarrow J_2$. A fermion hopping left can result either a particle-defect to move left, i.e., $i \rightarrow (i-2) \mod L$, or a hole-defect to move right, i.e., $j \rightarrow (j+2) \mod L$. Thus, each connectivity diagram can be viewed as a single “particle” hopping in the abstract space of many-body wavefunctions in a non-reciprocal manner.

The fourth and final step of the proof is to establish the ex-

<figure><img src="image_2.png" /><figcaption>FIG. 2. (Color Online) (a) Examples of labeling many-body wavefunctions by the location of “defects”. Two nearest neighbor sites form a particle- or a hole-defect when they are both occupied or both unoccupied, respectively. The defect position is the location of the leftmost of the two sites. (b) and (c) are examples of connectivities for $L = 10$, $N_d = 2, 1$, respectively, see text. Solid (green) and dashed (brown) arrows denote fermions hopping to the right and left, respectively. Reversing an arrow direction implies exchanging solid $\leftrightarrow$ dashed. (d) Three examples of non-reciprocal hopping over four sites. (i) has open boundary, while (ii) and (iii) have periodic boundary conditions. $1, r, r^2, \text{et cetera}$ (in blue) in (b)-(d) are the scaling factors $\lambda$, such as in Eq. 3, of the wavefunctions next to them, which define the similarity transformation, wherever possible. In (d, ii) one of the sites, indicated by a question mark, cannot be scaled consistently. For a closed loop a similarity transformation is only possible if there are equal number of solid and dashed arrows while traversing the loop in a given direction, as in (d, iii).</figcaption></figure>

existence of the similarity transformation in each sub-sector. For pedagogical reason we first consider few examples of non-reciprocal hopping of a single particle in a four-site system. Fig. 2(d, i) is a linear chain with open boundary condition. This can be mapped to a Hermitian form for $\gamma < J$ by the scaling

$$|i\rangle \rightarrow \lambda_i|i\rangle, \quad \langle i| \rightarrow (1/\lambda_i)\langle i|, \qquad (3)$$

where $\lambda_i = 1, r, r^2, r^3$, for $i = 1, \dots, 4$, respectively, and $r \equiv \sqrt{J_2/J_1}$ [24]. However, for periodic boundary condition, as in Fig. 2(d, ii), the transformation will not work since the new link 4-1 of the closed loop cannot be made Hermitian. This exemplifies that finding similarity transformations is non-trivial where the connections form closed loops, which is precisely our case as seen in, e.g., Fig. 2(b, c). Now consider Fig. 2(d, iii) which is also a closed loop, but where the hops are $J_2, J_2, J_1, J_1$ moving clockwise. In this case, once again, a similarity mapping exist, with $\lambda_i = 1, r, r^2, r$, respectively. This example illustrates the crucial property that a closed loop with equal number of $J_1$ and $J_2$ hops, while traversing along a direction, can be mapped to a Hermitian form. This is because a $J_2$ link requires an additional scaling of $r$ for the second site compared to the first, which is compensated by a $J_1$ link which requires a $1/r$ scaling. This is exactly the case for the connectivities of Fig. 2 (b, c), where the scalings associated with
the wavefunctions are marked in blue. Additional examples of such scalings are discussed in the Appendix B. We prove below that all the connections of $\mathcal{H}_{HN,f}$ are such that each and every possible loop has this property.

Starting from any $|(i_1, i_2, \dots, i_n)(j_1, j_2, \dots, j_n)\rangle$, a closed loop is obtained in three basic ways.

(1) If one or more of the site indices change as, say, $i \to i' \to i''$ and so on, and then reverse the path to go back to $i$, while obeying the rules (i) and (ii). Since the reverse of a $J_1$ hop is a $J_2$ hop, and vice versa, traversing the loop along one direction will necessarily have equal number of $J_1$ and $J_2$ hops. The loop $(1)(7) \to (1)(5) \to (3)(5) \to (3)(7) \to (1)(7)$ in Fig. 2(c) is an example.

(2) If a defect does not retrace its path, but moves across the chain, traversing $L$ sites, and gets back to its original position using the periodic boundary condition. However, according to rule (ii) this can happen only if all other defects perform the same circular motion in the same direction and regain their original positions, each having traversed $L$ sites. Since a particle-defect moving to the right is a $J_2$ hop, while a hole-defect moving to the right is a $J_1$ hop, and since there are equal number of particle- and hole-defects, this loop, too, will have equal number of $J_1$ and $J_2$ hops. Starting from state (1)(3) on the left side of Fig. 2(c) and ending again at (1)(3) on the right side of the figure is an example of such a loop.

(3) In some cases, such as in Fig. 2(b), it is possible for the defects to exchange positions such that $i_1 \to i_2 \to i_3 \dots \to i_n \to i_1$, and $j_1 \to j_2 \to j_3 \dots \to j_n \to j_1$. In this case a loop is completed by permuting the indices, while the defects neither retrace their paths nor complete the circle. Here the sum of the sites traversed by all the particle-defects is $L$ and the same is true for all the hole-defects, and they are along the same direction. Thus, again here the loop has equal number of $J_1$ and $J_2$ hops.

This completes the proof that $\mathcal{H}_{HN,f}$ can be mapped into a Hermitian form for $\gamma < J$; this feature guarantees reality of eigenspectrum of $\mathcal{H}_{HN,f}$ in this limit.

### E. Finite $V_1^c$ and Symmetry Protection

The above conclusion is not sufficient for our purpose once $V_1$ is large but finite. To understand why, consider two eigenstates of $\mathcal{H}_{HN,f}$ from the same sub-sector of $\hat{N}_d$. Measuring energies from the average eigenvalue, the sub-system has the structure

$$ \mathcal{H}_{eff} = \begin{pmatrix} l & m_1 + m_2 \\ m_1 - m_2 & -l \end{pmatrix}, $$

with eigenvalues $\pm\sqrt{l^2 + m_1^2 - m_2^2}$. Since $m_{1,2} \sim \mathcal{O}((J, \gamma)^2/V_1)$ or smaller, for finite $l$ the reality of the eigenvalues is guaranteed for $V_1$ sufficiently large. But, this argument fails in case of degeneracy when $l = 0$. Nevertheless, the reality of the spectrum can still be protected if the two degenerate states are connected by a symmetry $G$ of the full Hamiltonian $\mathcal{H}_{HN}$, which squares to identity in the degenerate subspace, as we show below.

Let $|\psi\rangle$ and $|\phi\rangle$ represent two degenerate right eigenstates of $\mathcal{H}_{HN,f}$ connected by a symmetry $G$ of both $\mathcal{H}_{HN,f}$ and $\mathcal{H}_{HN}$, i.e. $|\phi\rangle = G|\psi\rangle$. The corresponding right eigenbras are $\langle\psi|$ and $\langle\phi| = \langle\psi|G^\dagger$. Further, let us assume that $G^2|\psi\rangle = |\psi\rangle$ and $G^2|\phi\rangle = |\phi\rangle$. The off-diagonal matrix elements are seen to be equal as follows.

$$ \begin{aligned} \langle\psi|\mathcal{H}_{HN}|\phi\rangle &= \langle\phi|\mathcal{G}\mathcal{H}_{HN}\mathcal{G}|\psi\rangle = \langle\phi|\mathcal{H}_{HN}\mathcal{G}^2|\psi\rangle \\ &= \langle\phi|\mathcal{H}_{HN}|\psi\rangle \end{aligned} \quad (4) $$

Since the eigenvectors of $\mathcal{H}_{HN,f}$ can be chosen to be real when its eigenvalues are real, these matrix elements are also real. This implies that the full Hamiltonian $\mathcal{H}_{HN}$ effectively has a Hermitian structure within this degenerate subspace. The degeneracy of the two states $|\psi\rangle$ and $|\phi\rangle$ at the level of $\mathcal{H}_{HN,f}$ is lifted by $\delta\mathcal{H}_{HN} = \mathcal{H}_{HN} - \mathcal{H}_{HN,f}$ in a hermitian manner at most. This leads to level repulsion which is opposite to what is required for two eigenstates to coalesce. This also implies that eigenstates, or their linear combinations, that are related by symmetry of the type of $G$ cannot coalesce. Thus, there will be a finite range of $V_1$ for which the spectrum of $\mathcal{H}_{HN}$ would be real, until one (or two) pair(s) of eigenstates, not related by any symmetry (of the type of $G$), coalesce at an exceptional point. We illustrate this using one concrete example in Appendix C.

As discussed in Appendix D, the above symmetry protection can be destroyed by a choice of boundary condition that breaks translation symmetry. In this case one has complex eigenvalues for any finite value of $V_1$, even though the spectrum is real in the fragmented limit.

We estimate $l \sim \sqrt{J^2 - \gamma^2}/e^{cL}$ by the average level spacing of a sub-sector, where the constant $c > 0$ depends on the sub-sector size. Empirically, we find that, for the pair that coalesce, $m_1$ is at least one order of magnitude smaller than $m_2$, while $m_2 \propto V_1^{-\alpha}(J^2 - \gamma^2)^{-\beta/2}$ where the exponents $(\alpha, \beta)$ are $L$-dependent. This implies that $V_1^c \sim J e^{cL/\alpha}/(1 - (\gamma/J)^2)^{(\beta+1)/(2\alpha)}$. Thus, as shown in Fig. 1(d), $V_1^c$ diverges exponentially with $L$, and as a power-law with a $L$-dependent exponent for $\gamma \to J$.

Note, in passing, that for certain values of $L$ the two coalescing levels at $V_1^c$ are each doubly degenerate, so that below $V_1^c$ there are two pairs of complex conjugate eigenvalues. This degeneracy is related to $\mathcal{P}C$ invariance.

### III. GENERALIZATION TO NON-HERMITIAN SU-SCHRIEFFER-HEEGER MODEL

#### A. Model

Next we discuss the non-Hermitian interacting SSH model as one possible generalization of the concept presented in the previous section. We consider a fermionic SSH chain with left (right) intracellular hopping $J_1 = J - \gamma_1 (J_2 = J + \gamma_1)$, left (right) intercellular hopping $K_1 = K - \gamma_2 (K_2 = K + \gamma_2)$ and a nearest neighbor interaction $V_1$, as shown in Fig. 3(a). The
Hamiltonian of the system reads

$$ \mathcal{H}_{SSH} = \sum_{i=1}^{L/2} \left[ J_1 c_{2i-1}^{\dagger} c_{2i} + J_2 c_{2i}^{\dagger} c_{2i-1} + K_1 c_{2i}^{\dagger} c_{2i+1} + K_2 c_{2i+1}^{\dagger} c_{2i} \right] + \sum_{i=1}^{L} V_1 \hat{n}_i \hat{n}_{i+1}. \quad (5) $$

Each unit cell is composed of two non-equivalent sites $2i-1$ and $2i$ so that there are $L$ sites in total. The nearest neighbor interaction $V_1$ does not distinguish between the two non-equivalent sites. The non-Hermiticity of $\mathcal{H}_{SSH}$ arises due to the non-reciprocal intracellular and intercellular hoppings.

As we show here, there exists a one-body similarity transformation by virtue of which for any parameter set $(J, K, \gamma_1, \gamma_2)$, $\gamma_1$ can be made equal to $\gamma_2$, while keeping different values of $J$ and $K$. To do so, we scale the particle creation and annihilation operator on every odd site by

$$ c_{2j-1}^{\dagger} \rightarrow \lambda c_{2j-1}^{\dagger}, \quad c_{2j-1} \rightarrow \frac{1}{\lambda} c_{2j-1} $$

respectively. In terms of this transformed operators, the Hamiltonian reads

$$ \mathcal{H}_{SSH} = \sum_{i=1}^{L/2} \left[ \tilde{J}_1 c_{2i-1}^{\dagger} c_{2i} + \tilde{J}_2 c_{2i}^{\dagger} c_{2i-1} + \tilde{K}_1 c_{2i}^{\dagger} c_{2i+1} + \tilde{K}_2 c_{2i+1}^{\dagger} c_{2i} \right] + \sum_{i=1}^{L} V_1 \hat{n}_i \hat{n}_{i+1}. \quad (6) $$

where $\tilde{J}_1 = \tilde{J} + \tilde{\gamma}_1 = \lambda J_1$, $\tilde{J}_2 = \tilde{J} - \tilde{\gamma}_1 = J_2/\lambda$, $\tilde{K}_1 = \tilde{K} + \tilde{\gamma}_2 = K_1/\lambda$ and $\tilde{K}_2 = \tilde{K} - \tilde{\gamma}_2 = \lambda K_2$. Choosing

$$ \lambda = \sqrt{\frac{J + K + \gamma_2 - \gamma_1}{J + K - \gamma_2 + \gamma_1}}, $$

one gets $\tilde{\gamma}_1 = \tilde{\gamma}_2$ in this transformed frame, while $\tilde{J} \neq \tilde{K}$. For our numerical results, we choose our parameter set to obey this and vary $\delta = K - J$, without any loss of generality.

We consider the chain at half-filling, $\sum_i n_i = L/2$ with (anti-)periodic boundary condition for (even) odd $L/2$. Fig. 3(c) shows that in this case too, there is a strength of the nearest neighbor interaction, $V_1^c$ above which, the spectrum is completely real within numerical precision. For $V_1 < V_1^c$, islands appear in the spectrum where the imaginary part is orders of magnitudes smaller than the real part of the eigenvalues. The islands progressively shorten in height before hitting zero within numerical precision at $V_1^c$. The inset of Fig. 3(c) shows such islands. The value of this interaction strength is seen to increase with system size as well as with $\delta$.

<figure><img src="image_3.png" /><figcaption>FIG. 3. (a) Schematic of the interacting SSH model with non-reciprocal intra- and inter-cellular hops and nearest neighbor interaction $V_1$. The blue boxes indicate one unit cell of the SSH chain and the filled sites are marked in red. (b) Defects at odd site can only move by an intercellular hop (blue dashed and red dotted in (i) and (ii)), while defects at even site can move by an intracellular hop (green solid in (iii)). (c) Variation of the maximum imaginary part of the eigenvalues with the interaction strength $V_1$ for three system sizes $L = 8, 10, 12$ corresponding to $\gamma_1 = \gamma_2 = 0.1, J = 0.9$ and $K = 1.0$. For all the system sizes, beyond a certain interaction strength, the imaginary parts of the eigenvalues become zero within numerical precision. The inset shows islands at $V_1 < V_1^c$ (for instance, $V_1^c = 23.8$ for $L = 10$) for $L = 10$ (red) and $L = 12$ (black), where the imaginary part is finite, but orders of magnitudes smaller than the real part. (d) Variation of the maximum imaginary part of the eigenvalues with $\delta = K - J$ for three system sizes $L = 8, 10, 12$ with $\gamma_1 = \gamma_2 = 0.1, K = 1.0$ and $V_1 = 1000$. The presence of a critical value of $\delta$ ($\delta_c = 0.156$ for $L = 10$, for example), below which the spectrum is completely real is seen. Also, the value of $\delta_c$ at a given $V_1$ reduces with increase in system size. See text for details.</figcaption></figure>

## B. Fragmentation and Similarity Transformation

We consider the large $V_1$ limit, $V_1 \gg J, K, \gamma_1, \gamma_2$ and focus on the first-order effective Hamiltonian

$$ \mathcal{H}_{SSH,f} = \sum_{i=1}^{L/2} \left[ \hat{P}_{2i} (J_1 c_{2i-1}^{\dagger} c_{2i} + J_2 c_{2i}^{\dagger} c_{2i-1}) \hat{P}_{2i} + \hat{P}_{2i+1} (K_1 c_{2i}^{\dagger} c_{2i+1} + K_2 c_{2i+1}^{\dagger} c_{2i}) \hat{P}_{2i+1} + V_1 \sum_{i=1}^{L} \hat{n}_i \hat{n}_{i+1} \right] \quad (7) $$

where $\hat{P}_i = 1 - (\hat{n}_{i-2} - \hat{n}_{i+1})^2$ is a projector that only allows those hops which preserve the number of nearest neighbor pairs (defects), $N_d = \sum_i n_i n_{i+1}$. It is useful to remember here that a particle-defect refers to two particles sitting on neighboring sites and a hole-defect refers to two neighboring vacant sites. In addition, since a particle hop changes the position of a defect by $\pm 2$ modulo $L$ sites, therefore the number of defects at odd sites ($N_{d,odd} = \sum'_i n_i n_{i+1}$) and even sites ($N_{d,even} = \sum''_i n_i n_{i+1}$) are individually conserved by $\mathcal{H}_{SSH,f}$. Here, single (double) prime implies sum over odd
(even) sites only. It is easy to see now that these constraints are similar to those which appear in the Hatano-Nelson model in the large $V_1$ limit. Since the connectivity of states in the Hilbert space does not depend on the hopping strengths (as long as they are much smaller than $V_1$), therefore $H_{SSH,f}$ also exhibits strong Hilbert space fragmentation in spite of having different intercellular and intracellular hoppings.

In addition, it is important to note the following. (i) At half-filling, the number of particle defects at the odd sites is equal to the number of hole defects at odd sites and the same holds true for defects at even sites. (ii) A defect at an odd site can only move by a intercellular hop and a defect at an even site can only move by a intracellular hop (Fig. 3(b)). (iii) Two defects cannot cross each other.

It remains to understand whether the similarity transformation goes through in the fragmented limit. For this purpose, we need to label the states first. The labelling scheme is similar to that used in the case of Hatano-Nelson model. Thus, in the $N_d = 1$ sector, $|(i)(j)⟩$ denotes a state having a particle-defect at site $i$ and a hole-defect at site $j$. Similarly, in the $N_d = 2$ sector, a state is labelled as $|(i_1, i_2)(j_1, j_2)⟩$ and a state in the $N_d = n$ sector is $|(i_1, i_2, ..., i_n)(j_1, j_2, ..., j_n)⟩$.

Next, it is again useful to remember here that for the similarity transformation to be consistent along a closed loop (in Fock space), the number of right non-reciprocal hops should be equal to the number of left non-reciprocal hops along the loop. Since in this particular case, the intercellular and intracellular hops are individually non-reciprocal, this criterion translates to having equal number of right and left intercellular hops and equal number of right and left intracellular hops separately along any closed loop. To check whether this is the case, we again consider the three ways by which a state $|(i_1, i_2, ..., i_n)(j_1, j_2, ..., j_n)⟩$ can close on to itself in Fock space.

(1) **The defects retrace their path:** In this case, one or more defects hop following $i \to i' \to i'' \to ...$ and then retrace their path to return to the original configuration and close the loop. Since a right intercellular (intracellular) hop will necessarily require a left intercellular (intracellular) hop for its reversal, therefore in this case, it trivially holds that a closed loop will have equal number of right and left intercellular (intracellular) hops.

(2) **The defect moves across the chain making L hops and returns to its original position using the periodic boundary condition:** Due to rule (iii) above, this implies that all other defects would also have to return to their initial positions after going across the chain in the same direction. Each such defect will have to make L hops across the chain. From rule (i) and (ii), this implies that there will be equal number of particle-hole-defects undergoing intercellular (intracellular) hops in the same direction. Since a particle-defect hopping to the right is equivalent to a particle hopping to the right and a hole-defect hopping to the right is equivalent to a particle hopping to the left, this suggests that in this case too, there will be equal number of right and left intercellular hops and equal number of right and left intracellular hops separately along a closed loop. This is shown in Fig. 4.

(3) **The defects permute their positions to reach the ini-

<figure><img src="image_4.png" /><figcaption>FIG. 4. Connectivity diagram of $H_{SSH,f}$ in Fock space showing type (2) connectivity in which the defects traverse through the entire chain to return to their original configuration. Solid green and yellow arrows refer to right and left intracellular hops respectively while red dotted and blue dashed arrows refer to right and left intercellular hops respectively. See text for details.</figcaption></figure>

**tial configuration:** In this case, a sequence of defects exchange their positions as $i_1 \to i_2 \to i_3 \to \dots \to i_n \to i_1$ and $j_1 \to j_2 \to j_3 \to \dots \to j_n \to j_1$ to reach the initial state and complete the loop. Thus, the sum of the sites traversed by all the particle-defects is $L$, and the same is true for all the hole-defects. However this type of connectivity is only possible if all the defects are present either on the odd sites or on the even sites. If the defects are on odd (even) sites and the motion is along the anticlockwise direction, then all the hops of the particle-defects will be of the type $K_2$ ($J_2$), while those of the hole-defects will be of the type $K_1$ ($J_1$). Thus, here too there will be an equal number of $K_{1,2}$ hops and no $J_{1,2}$ hop, or vice versa.

Thus, a consistent similarity transformation can always be carried out along any closed loop with each right and left intercellular (intercellular) hop being scaled by a factor $r(r')$ and $1/r(1/r')$ respectively, where $r = \sqrt{J_2/J_1}$ and $r' = \sqrt{K_2/K_1}$.

### C. Symmetry Protection

In this section, we discuss the protection of the real eigenvalues once we move out of the limit of fragmentation $H_{SSH,f}$ by considering large, but finite, $V_1$. As discussed for the HN model, the nondegenerate eigenvalues are guaranteed to stay real once $V_1$ is finite, while the degenerate ones are not. In the HN model we found that the degenerate subspaces are symmetry protected which enforces Hermiticity in these subspaces. The case of the SSH model is more complex because, due to period doubling with $J \neq K$, translations by odd number of sites are no longer symmetry operations. Thus, we find certain pairs of degenerate eigenvalues in the fragmented limit which are not symmetry connected. These pairs are symmetry connected only if $J = K$ when the full translation symmetry of the chain is restored, i.e., in the limit of the HN model. As expected, we find that in such projected non-symmetry connected two-dimensional subspaces the effective Hamiltonian is non-Hermitian with the structure

$$H_{eff} = \begin{pmatrix} l & m_1 + m_2 \\ m_1 - m_2 & -l \end{pmatrix},$$
<figure><img src="image_5.png" /><figcaption>FIG. 5. (Color Online) (a)-(c) Time evolution of the correlation function $\chi(t)$ for $V_1 = 3J$, 4.285J, and 6J, respectively, for $L = 14$ and $\gamma = 0.2J$. In (b) the system is very close to the exceptional point at $V_1^c \approx 4.2863$. (d) Variation of the relaxation timescale $\tau$ with $V_1$ showing a one-sided divergence as $V_1 \to V_1^c$ from below.</figcaption></figure>

with $m_{1,2} \neq 0$. However, if we define $\delta \equiv (K - J)$ as the deviation from the HN limit, we are guaranteed that the non-Hermitian component $m_2 \to 0$ as $\delta \to 0$. Thus, for $V_1$ sufficiently large (larger than $V_1^c$ of the HN model) there is a finite critical value $\delta_c$ below which $m_1 > m_2$ for all such non-symmetry connected degenerate pairs of eigenvalues, which guarantees that the eigenvalues stay real even if the subspace is non-Hermitian. This is shown in Fig. 3(d) for $L = 8, 10, 12$. The value of $\delta_c$ is seen to decrease with increase in system size for a given value of $V_1$. Beyond $\delta_c$, island-like features again emerge suggesting that the behavior of $m_2/m_1$ is non-monotonic as a function of $\delta$.

#### IV. CORRELATION FUNCTION AND DETECTION OF EXCEPTIONAL POINT

In this section, we study the time evolution of a correlation function which, in principle, can be measured to identify the location of a many-body exceptional point. We choose the Hatano Nelson model for this purpose. One such example is $\chi(t) = \langle\psi(t)|\hat{N}_d|\psi(t)\rangle/L$ starting from a random Fock state $|\psi(0)\rangle = \sum_m c_m|\phi_m\rangle$, expanded in the basis of the right eigenvectors $|\phi_m\rangle$ of $\mathcal{H}_{HN}$. The time-evolved wavefunction, suitably normalized to account for the non-Hermiticity of $\mathcal{H}_{HN}$, is

$$|\psi(t)\rangle = \frac{e^{-i\mathcal{H}_{HN}t/\hbar}|\psi(0)\rangle}{||e^{-i\mathcal{H}_{HN}t/\hbar}|\psi(0)\rangle||} = \frac{\sum_m c_m(t)|\phi_m\rangle}{\sqrt{\sum_{m,n} c_m^*(t)c_n(t)\langle\phi_m|\phi_n\rangle}},$$

where $c_m(t) = c_m e^{-i\epsilon_m t/\hbar}$.

Fig. 5, panels (a)-(c) show the time evolution of $\chi(t)$ for $V_1$ less than, nearly equal to, and greater than $V_1^c$, respectively, for the HN model. For $V_1 < V_1^c$ the time evolution is dominated by the eigenvalue with the largest imaginary component. Consequently, after a timescale $\tau \sim 1/\text{Max}[\text{Im }\epsilon]$, the correlation function attains a steady state value $\chi(t) \gg \tau) \sim 1/L \langle\phi_m^*|\hat{N}_d|\phi_m^*\rangle$, where $|\phi_m^*\rangle$ is the eigenvector with the largest Im $\epsilon$. This implies that $\tau$ diverges as $V_1 \to V_1^c$ from below, as seen in Fig. 5(d). For $V_1 \ge V_1^c$ all the eigenvalues are real and the system quickly attains a diagonal ensemble, and $\chi(t)$ fluctuates about an average value $\chi(t) \gg \tau) \sim 1/L \sum_m |c_m|^2 \langle\phi_m|\hat{N}_d|\phi_m\rangle$ [47, 48], implying that the peak of $\tau(V_1)$ in Fig. 5(d) is one-sided. This peak can be used to detect the exceptional point.

#### V. DISCUSSION

In this work, we have studied the Hatano-Nelson and the Schrieffer-Heeger models with non-reciprocal hops and nearest neighbor interactions. We have shown that these models provide two examples of a class of non-Hermitian systems which have a “phase” with entirely real spectrum for sufficiently large interactions.

We note that none of these models have PT symmetry. Thus the generic expectation in the existing literature for the existence of real eigenvalues in presence of such symmetry is not applicable to the present models. Our work explicitly points out that the presence of global symmetries can not, by itself, explain the reality of eigenspectrum of the model at large interaction strength.

In contrast, as we show in this work, such a phase with entirely real eigenspectrum is a consequence of two ingredients. First, the dynamical constraints in the infinitely large interaction limit which also fragments the Hilbert space of the models. This Hilbert space fragmentation leads to restriction on the matrix elements of the Hamiltonian between Fock states and, as we show, allows for an existence of a many-body similarity transformation which allows us to prove the reality of the eigenspectrum in this limit. Second, moving away from the fragmented limit, we show that the presence of global symmetries allows one to retain the reality of eigenspectrum up to a critical interaction strength. The effective Hamiltonian between two states related by such symmetries (and which are therefore degenerate in the limit of large $V$) can be shown to be Hermitian for finite $V \le V_c$; this allows for persistence of real eigenvalues till a critical interaction strength is reached. Interestingly, this mechanism seems to be effective even when such symmetries are weakly broken as seen from the analysis of the SSH model. Thus our work points out the role of HSF for the presence of real eigenspectrum; to the best of our knowledge, such application of HSF in the context of non-Hermitian system has not been pointed out earlier.

In addition we also provide a method of detection of the first exceptional point in these systems. Our analysis indicates that time taken by an equal-time correlator of such a system to reach its steady state value following a quench diverges at the first exceptional point. This observation, which only relies on the presence of an infinitesimal complex component of a pair of eigenvalues at this point, provides a link between presence of exceptional points and the behavior of experimentally measurable equal-time correlation functions in such non-Hermitian systems.
through pair diffusion. Since P pairs can be put in $2N_d$ intermediate positions in between the defects, and since translation of a given configuration by two sites also generates new configurations (note, defects can move only by two sites), the size of the largest fragment is

$$N_f = \frac{L}{2} \left( 2N_d + P - 1 \right) = \frac{L}{2} \left( \frac{3L/4 - 2}{L/4 - 1} \right).$$

On the other hand, the total size of the symmetry sector with $N_d$ defects is [38]

$$N_s = 2 \binom{L/2}{L/4} \binom{L/2 - 1}{L/4}.$$

In the limit of large chain length $L$, this implies that the ratio $\lambda \equiv N_f/N_s \approx (0.8)^L$, which vanishes exponentially with the system size. In other words, even the largest fragment samples only a vanishing portion of the total possible states. This establishes strong Hilbert space fragmentation.

## Appendix B: More on Connectivity Diagrams and Similarity Transformation

In this section, we provide an illustration of the similarity transformation in a simple case and explicit examples of the three possible types of closed loops as mentioned in the main text.

To begin with, we consider a chain of length $L = 6$ having 3 fermions with PBC. We focus on the $N_d = 1$ sector and since the Hamiltonian $H_{HN,f}$ changes the positions of the hole and the particle defects by $\pm 2$ (mod $L$) only, we consider the (odd)(odd) and the (even)(even) sectors separately. The basis states for the (odd)(odd) sector are enumerated as $\{|(1)(3)\rangle, |(1)(5)\rangle, |(3)(5)\rangle, |(3)(1)\rangle, |(5)(1)\rangle, |(5)(3)\rangle\}$. It's worthwhile to note that the labelling scheme for the states followed here are the same as outlined in the main text. In this basis, the Hamiltonian $H_{HN,f}^{\text{odd}}$ reads as

$$H_{HN,f}^{\text{odd}} = \begin{pmatrix} V_1 & J_2 & 0 & 0 & 0 & J_2 \\ J_1 & V_1 & J_1 & 0 & 0 & 0 \\ 0 & J_2 & V_1 & J_2 & 0 & 0 \\ 0 & 0 & J_1 & V_1 & J_1 & 0 \\ 0 & 0 & 0 & J_2 & V_1 & J_2 \\ J_1 & 0 & 0 & 0 & J_1 & V_1 \end{pmatrix} \quad (B1)$$

where $J_1 \equiv J - \gamma$ and $J_2 \equiv J + \gamma$. The connectivity diagram corresponding to $H_{HN,f}^{\text{odd}}$ is shown in Fig. 7(a). This is, in fact, an example of the second type of connectivity mentioned in the previous section wherein a particle defect moves across the entire chain to return to its original position. This structure is reminiscent of a single-particle hopping problem on a one-dimensional ring in Fock space but with different hopping strengths on alternate links. To map this to our familiar hop-

ping problem, we define a similarity transformation

$$S = \begin{pmatrix} r & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & r & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & r & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} \quad (B2)$$

with $r = \sqrt{J_2/J_1}$. The scalings are also shown in Fig. 7(a) in blue. It's straightforward to see that under this transformation, $H_{HN,f}^{\text{odd}}$ assumes a hermitian structure given by

$$ \begin{aligned} \tilde{H}_{HN,f}^{\text{odd}} &= S^{-1} H_{HN,f}^{\text{odd}} S \\ &= \begin{pmatrix} V_1 & \alpha & 0 & 0 & 0 & \alpha \\ \alpha & V_1 & \alpha & 0 & 0 & 0 \\ 0 & \alpha & V_1 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & V_1 & \alpha & 0 \\ 0 & 0 & 0 & \alpha & V_1 & \alpha \\ \alpha & 0 & 0 & 0 & \alpha & V_1 \end{pmatrix} \end{aligned} \quad (B3) $$

where $\alpha = \sqrt{J^2 - \gamma^2}$.

The corresponding connectivity structure in Fock space is shown in Fig. 7(b). This, of course, admits of a real spectrum and the eigenvalues can be readily written down as $\mathcal{E}_k = V_1 + 2\sqrt{J^2 - \gamma^2} \cos(k)$. Here the fictitious lattice spacing is taken to be unity and $k = -\pi + m\pi/3$, with $m = 0, 1, \dots, 5$. The eigenvectors $|k\rangle$ of the original problem can be recovered using the similarity transformation $S$.

It's useful to note here that the (even)(even) sector is related to the (odd)(odd) sector through a translation by one lattice spacing $\mathcal{R}$. This is evident on comparing the connection diagrams in Fig. 7(a) and (c). Hence $H_{HN,f}^{\text{even}}$ has exactly the same structure and spectrum as that of $H_{HN,f}^{\text{odd}}$. Equivalently, $H_{HN,f}^{N_d=1} = H_{HN,f}^{\text{odd}} \oplus H_{HN,f}^{\text{even}}$ is invariant under $\mathcal{R}$. This property would be significant in the next section.

Before closing this section, we chalk out examples of the other types of connectivities discussed in the main text. Fig. 7(d) illustrates the situation where a particle defect completes a loop by retracing back its path. This gives an example of the first type of connectivity discussed in the main text. Fig. 8 shows a case where multiple defects exchange their positions in order to complete the loop. This is an example of the third type of connectivity. It's worth noting that in all these three cases, there are an equal number of right hops and left hops, which makes the similarity transformation possible. As the system size increases, more complicated connection diagrams emerge. However, any closed loop in these connection diagrams will fall in one of these three classes or will be some combination of these three and hence will admit of this general feature.

## Appendix C: Symmetry Protection and Hidden Hermiticity

We use this section to illustrate with a concrete example, the symmetry protection of the real eigenvalues once we move out
<figure><img src="image_6.png" /><figcaption>FIG. 7. (a) Connectivity diagram corresponding to $\mathcal{H}_{HN,f}^{\text{odd}}$ showing both real space hoppings and the Fock space structure. Solid (green) and dashed (brown) arrows represent right hops with strength $J_2$ and left hops with strength $J_1$ respectively. The numbers in blue show the scaling factors of the states of the states, required for the similarity transformation $S$. This gives an example of the second type of connectivity. (b) Connection diagram in Fock space of $\tilde{\mathcal{H}}_f^{\text{odd}}$ after the similarity transformation is carried out. Both the right and left hops, indicated by double-headed blue arrows, have strengths $\sqrt{J_2^2 - \gamma^2}$ and hence the structure is hermitian. (c) Same as (a) but now for $\mathcal{H}_{HN,f}^{\text{even}}$ in the (even)(even) sector. It can be seen that (c) is related to (a) by $R$. After carrying out the similarity transformation $S$, one gets a hermitian structure like that of (b) for the (even)(even) sector. (d) A closed loop from the $N_d = 1$ sector of the half-filled $L = 10$ chain in which the defects retrace back their paths to return to their initial position. This is an example of the first type of connectivity. (a)-(c) correspond to $N_d = 1$ sector of a half-filled $L = 6$ chain.</figcaption></figure>

of the limit of fragmentation of $\mathcal{H}_{HN,f}$ (i.e., infinite $V_1$), and consider large but finite $V_1$ effects.

We again consider the $N_d = 1$ sector of the half-filled chain of length $L = 6$, whose spectrum was discussed in the previous section. Corresponding to the eigenvalue $\mathcal{E}_{\pi/3}$, there would be 4 degenerate states of $\mathcal{H}_{HN,f}$, namely $|\psi_{1,2}\rangle_o$ and $|\psi_{1,2}\rangle_e$.

$$ \begin{aligned} |\psi_1\rangle_o &= \frac{1}{\sqrt{3}}(rC|(1)(3)\rangle - C|(1)(5)\rangle - r|(3)(5)\rangle - C|(3)(1)\rangle \\ &\quad + rC|(5)(1)\rangle + |(5)(3)\rangle) \\ |\psi_2\rangle_o &= \frac{1}{\sqrt{3}}S(r|(1)(3)\rangle + |(1)(5)\rangle - |(3)(1)\rangle - r|(5)(1)\rangle) \end{aligned} $$

$$ \begin{aligned} |\psi_1\rangle_e &= \frac{1}{\sqrt{3}}(rC|(2)(4)\rangle - C|(2)(6)\rangle - r|(4)(6)\rangle - C|(4)(2)\rangle \\ &\quad + rC|(6)(2)\rangle + |(6)(4)\rangle) \\ |\psi_2\rangle_e &= \frac{1}{\sqrt{3}}S(r|(2)(4)\rangle + |(2)(6)\rangle - |(4)(2)\rangle - r|(6)(2)\rangle) \end{aligned} $$

where $r = (J_2/J_1)^{1/2}$, $C = \cos \frac{\pi}{3}$ and $S = \sin \frac{\pi}{3}$.

The first two states are from the (odd)(odd) sector and lie in the subspace spanned by $|k = \pm \frac{\pi}{3}\rangle$, which are related by $\mathcal{P}C$ symmetry. This can be seen easily considering that under $\mathcal{P}C$, $\{|(1)(3)\rangle \leftrightarrow |(3)(5)\rangle, |(3)(1)\rangle \leftrightarrow |(5)(3)\rangle, |(1)(5)\rangle \leftrightarrow |(1)(5)\rangle, |(5)(1)\rangle \leftrightarrow |(5)(1)\rangle\}$. The same can be said about

the last two states which lie in the (even)(even) sector. States from the (odd)(odd) sector and (even)(even) sector are related by $R$, viz $|\psi_{1(2)}\rangle_e = R|\psi_{1(2)}\rangle_o$. It is useful to note here that in addition $(\mathcal{P}C)^2 = I$ and $R^2|\psi_{1(2)}\rangle_{o,e} = |\psi_{1(2)}\rangle_{o,e}$.

We consider the matrix representation of the second order correction, $\mathcal{H}_{HN}^{(2)}$ in this degenerate subspace. Enumerating the basis as $\{|\psi_1\rangle_o, |\psi_2\rangle_o, |\psi_1\rangle_e, |\psi_2\rangle_e\}$, $\mathcal{H}_{HN}^{(2)}$ reads

$$ \mathcal{H}_{HN}^{(2)} = \begin{pmatrix} \mu & 0 & v & \lambda \\ 0 & \mu & -\lambda & v \\ v & -\lambda & \mu & 0 \\ \lambda & v & 0 & \mu \end{pmatrix} \quad (C1) $$

where $\mu = -\frac{J_1 J_2}{V_1}(1 + r^2)$, $v = -\frac{r}{4V_1}(J_1^2 + J_2^2)$ and $\lambda = \frac{\sqrt{3}}{4V_1}r(J_1^2 + J_2^2)$. This is Hermitian and admits of real spectrum. This hermiticity is expected to hold order-by-order in perturbation since Eq. 4 in the min text is true for exact $\mathcal{H}_{HN}$. Thus, the reality of the spectrum is guaranteed within this subspace.

There remains a second possibility in which these degeneracies might be accidental, i.e. not related by any symmetry, at some point in parameter space. However, we would argue here that such accidental degeneracies are not possible in generic integrable or non-integrable systems. In case of generic finite-sized non-integrable systems, one generally expects level repulsion leading to finite gaps in the spectrum, which protects the reality of the eigenspectrum at finite $V_1$. These gaps de-
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 8. (a) A closed loop from the $N_d = 3$ sector of the half-filled $L = 14$ chain where the defects permute their positions to reach the initial configuration. This is an example of the third kind of connectivity as specified in the main text. Solid (green) and dashed (brown) arrows represent right hops with strength $J_2$ and left hops with strength $J_1$ respectively. The numbers in blue show the scaling factors of the states of the states, required for the similarity transformation. (b) The same connectivity diagram in the Fock space with the states labelled according to the labelling scheme used in the main text.</figcaption></figure>

crease with increasing size as pointed out in the main text; consequently, we expect $V_c$ to diverge in the thermodynamic limit. And in case of an integrable model, a generic crossing between two eigenstates involves eigenstates from different conserved charge sectors. The full Hamiltonian, being integrable, cannot connect between states in different sectors at any order in $1/V_1$. Thus, there is no matrix element between them once $V_1$ is lowered. For our case, we have not observed any occurrence of accidental degeneracy.

## Appendix D: Boundary Condition, Symmetry Breaking and Complex Eigenvalues

So far, we have considered PBC for odd filling and anti-PBC for even filling. In this section, we show that choosing the opposite set of boundary conditions breaks the translational invariance of the full Hamiltonian $\mathcal{H}_{HN}$ and results in complex eigenvalues for any finite value of $V_1$. This happens even though $\mathcal{H}_{HN,f}$ has real eigenvalues just as in the case of the standard boundary condition. This is an example where there is no sufficient symmetry protection in the degenerate subspace of $\mathcal{H}_{HN,f}$, and therefore $V_c$ is trivially infinite, because any finite $V_1$ will lead to complex eigenvalues.

For illustration purposes, we consider a half-filled chain

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 9. (a) Fock space connectivity diagram corresponding to the full Hamiltonian $\mathcal{H}_{HN}$ for $L = 4$ with periodic boundary condition. Letters in blue above the arrows show the hopping strength for that particular hop. (b) The fate of the connectivity diagram on application of the one-site lattice translation operator $R$. Comparison of the two figures clearly show that $\mathcal{H}_{HN}$ with this boundary condition is not invariant under $R$. See text for details.</figcaption></figure>
of length $L = 4$ with PBC. Choosing the basis as $\{|0\rangle, |\bar{0}\rangle, |(1)(3)\rangle, |(2)(4)\rangle, |(3)(1)\rangle, |(4)(2)\rangle\}$, the Hamiltonian $\mathcal{H}_{HN}$ can be written as

$$ \mathcal{H}_{HN} = \begin{pmatrix} 0 & 0 & -J_1 & J_2 & J_1 & J_2 \\ 0 & 0 & J_2 & J_1 & -J_2 & J_1 \\ -J_2 & J_1 & V_1 & 0 & 0 & 0 \\ J_1 & J_2 & 0 & V_1 & 0 & 0 \\ J_2 & -J_1 & 0 & 0 & V_1 & 0 \\ J_1 & J_2 & 0 & 0 & 0 & V_1 \end{pmatrix} \quad (\text{D1}) $$

where $|0\rangle \equiv |\circ \circ \circ \circ\rangle$ and $|\bar{0}\rangle \equiv |\circ \circ \circ \circ\rangle$. The corresponding connection diagram is shown in Fig. 9(a). It can be readily verified from the matrix structure as well as from the connection diagram that $\mathcal{H}$ is not invariant under $\mathcal{R}$ which maps $|0\rangle \leftrightarrow |\bar{0}\rangle$ and $|(1)(3)\rangle \rightarrow |(2)(4)\rangle \rightarrow |(3)(1)\rangle \rightarrow |(4)(2)\rangle \rightarrow |(1)(3)\rangle$.

However, $\mathcal{H}_{HN,f}$ in this basis reads

$$ \mathcal{H}_{HN,f} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & V_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & V_1 & 0 & 0 \\ 0 & 0 & 0 & 0 & V_1 & 0 \\ 0 & 0 & 0 & 0 & 0 & V_1 \end{pmatrix} \quad (\text{D2}) $$

This is trivially Hermitian in this case because all the basis states are annihilated by the constrained hopping terms in $\mathcal{H}_{HN,f}$. Such states are termed as frozen states in the field of fragmentation physics. The energy eigenstates can be chosen to be the basis states and they are doubly degenerate in the $N_d = 0$ sector and exhibit 4-fold degeneracy in the $N_d = 1$ sector.

We now consider the representation of the second-order cor-

rection to $\mathcal{H}_{HN}$ in both these degenerate sub-spaces.

$$ \mathcal{H}_{HN}^{(2)} = \begin{pmatrix} 4\mu & -2\delta & 0 & 0 & 0 & 0 \\ 2\delta & 4\mu & 0 & 0 & 0 & 0 \\ 0 & 0 & -2\mu & -\delta & 2\mu & -\delta \\ 0 & 0 & \delta & -2\mu & -\delta & -2\mu \\ 0 & 0 & 2\mu & \delta & -2\mu & \delta \\ 0 & 0 & \delta & -2\mu & -\delta & -2\mu \end{pmatrix} \quad (\text{D3}) $$

where $\mu = -\frac{J_1 J_2}{V_1}$ and $\delta = \frac{1}{V_1}(J_2^2 - J_1^2)$. This is non-Hermitian, implying that for any finite value of $V_1$, the eigen-spectrum will be complex. Explicitly, the eigenvalues of $\mathcal{H}_{HN,f} + \mathcal{H}_{HN}^{(2)}$ are

$$ \epsilon = \left( V_1, V_1, -\frac{4}{V_1}(J \pm i\gamma)^2, V_1 + \frac{4}{V_1}(J \pm i\gamma)^2 \right) \quad (\text{D4}) $$

This makes it impossible to have a real eigenspectrum for any finite $V_1$; this fact has been numerically found in Ref. [45].

It is important to note here that $\mathcal{H}_{HN,f}$ yields a real spectrum irrespective of the chosen boundary condition. In the case shown, it is trivially real; in other cases, where the connection diagrams are more complicated, a similarity transformation like the above will smoothly go through giving rise to a real spectrum. The hidden Hermiticity argument becomes invalid because Eq. 4 of the main text fails, owing to the lack of translational symmetry $\mathcal{R}$ of the full Hamiltonian $\mathcal{H}_{HN}$. This results in the higher-order corrections to be manifestly non-Hermitian in the degenerate sub-spaces, thereby immediately destroying the reality of the spectrum for any finite $V_1$.

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