should be marginally irrelevant for the conjecture to hold). Furthermore, $\epsilon = \epsilon_*$ marks the boundary of the region of analyticity of the long-range fixed point, and for $\epsilon > \epsilon_*$ the long-range fixed point with real couplings seizes to exist. It would be extremely interesting to provide rigorous evidence for these phenomena.

### 8.1.6 Extension to non-integer $N$? To non-integer $d$?

In quantum field theory, one often likes to continue the number of fields from a positive integer, as it nominally should be, to an arbitrary real or even complex value. For lattice models, such continuation often have geometric meaning, as for the $O(N)$ and Potts models when it can be interpreted respectively in terms of loops and Fortuin-Kastelein clusters. Symmetry meaning of such continuations in terms of Deligne categories was recently discussed in [99]. We wish to discuss how such a continuation can be rigorously performed for the model studied here. First one has to factor out explicitly the products of $\Omega$-tensors out of the couplings, i.e. write

$$H_{2k}(\mathbf{A}, \mathbf{x}) = \Omega_{a_1 a_2} \Omega_{a_3 a_4} \cdots \Omega_{a_{2k-1} a_{2k}} \tilde{H}_{2k}(\tilde{\mathbf{A}}, \mathbf{x}) \pm \text{permutations},$$

where the kernels $\tilde{H}_{2k}$ are “$\Omega$-free”, i.e. no longer depend on the $a$ indices, the sequence $\tilde{\mathbf{A}}$ containing only $\mu$ indices. The RG equations can be rewritten in terms of such $\Omega$-free kernels, with contractions of $\Omega$-tensors giving rise in each order to some factors depending polynomially on $N$. One should then study such $\Omega$-free fixed point equations. It is tempting to conjecture that one can prove fixed point existence for any $N$ and its analytic dependence on $N$.<fnref n="47" />

More speculatively, one could also try to perform analytic continuation in the space dimension $d$. One would have to use rotation invariance to come up with a parametrization of the kernel in terms of scalar functions depending on distances between points, times polynomials in point differences. Expressing the RG equations in terms of scalar functions only, dimension $d$ becomes just a parameter which can in principle be continued to non-integer values. Controlling this continuation will likely require major changes in our arguments (much more so than the continuation in $N$), since we relied on the existence of the physical position space carrying a positive integration measure in several crucial points. But the stakes are high: if one could prove non-perturbative analyticity in $d$, it would be the first rigorous result of this kind in almost 50 years since the space of 3.99 dimensions was ushered in by Wilson and Fisher [4].

<footnote>the SR (short-range) fixed point. However it's still not conserved by continuity, so its divergence represents extra states not present in SR. (3) At the coalescence of LR and SR fixed points there must be a marginal operator, on general grounds and because logarithmic corrections are seen in Monte Carlo simulations. The SR fixed point by itself does not have a marginal operator; it can however be constructed as $\psi\zeta$. More arguments are given in [58, 98] where this picture was proposed and thoroughly tested for consistency.</footnote>

<footnote n="47">⁴⁷If this is achieved, the coefficient $N-8$ of the one-loop beta function becomes a new small parameter for $N$ near 8. One could then work for $\epsilon = 0$ where the quartic is marginal, and construct a fixed point with $\lambda = O(N-8)$ balancing the one-loop term against the two-loop term which has an $O(1)$ coefficient. We are grateful to Dario Benedetti for this comment. This could then be done even in $d = 4$, for a theory of local symplectic fermions (8.1). This would be a rigorous version of the Banks-Zaks fixed points in 4d gauge theories [100–102].</footnote>
### 8.1.7 Connections to Functional Renormalization Group

FRG represents the most systematic attempt to implement Wilsonian RG in absence of small parameter; see references in the introduction. We are not aware of any FRG studies of specifically symplectic fermions (local or long-range), although other fermionic models (such as Gross-Neveu, Thirring, or Nambu-Jona-Lasinio), or mixed fermion-boson models with Yukawa interactions have been studied via FRG-like techniques; see e.g. [103–106].

Let us compare our results to FRG calculations. In our theorems, all irrelevant couplings were included, and an RG fixed point was rigorously located in a Banach space of interactions, with a provably convergent way to approach it (for a wide range of cutoff procedures). Any FRG calculation always truncates the space of interactions, so that only a subset of irrelevant couplings is included (even if an infinite one). To our knowledge, there are no rigorous results about the best way to exhaust the space of interactions. What is done instead often looks like a matter of prejudice or of convenience. E.g., for bosonic models with the field $\varphi$ one typically allows an arbitrary potential $V(\varphi)$ but only a handful of derivatives of $\varphi$, because the former is believed (although unproven) to be more important, but also because an arbitrary potential is easy (the so called local potential approximation), while derivatives are hard. This state of affairs is both an invitation to mathematical physicists to weigh in and provide rigorous criteria, and to FRG practitioners to explore different exhaustion schemes.

FRG experts may find instructive that our construction used nonlocal interactions terms parametrized by kernels having fast decay at infinity.<fnref n="48" /> In principle, our interactions could be expanded in the basis of local monomials with fields carrying an arbitrarily high number of derivatives (the expansion coefficients would be all finite because of the stretched exponential decay of the kernels). However, we have not found such an expansion necessary. It is an open question if rigorous RG analysis can be carried out with interactions expanded in local monomials, and what would be the appropriate Banach space.

Another difference between our result and FRG is that we work with the full Wilsonian effective action, while most FRG calculations are nowadays performed in terms of the one-particle irreducible (1PI) effective action, which flows according to the Wetterich equation [112], as opposed to Polchinski’s equation.<fnref n="49" /> Empirically, this seems to give better results. The 1PI effective action may be expected to be a somewhat more local object than the Wilsonian effective action, but it too cannot be fully local. We are not aware of any rigorous fixed point results in terms of the 1PI effective action.

### 8.1.8 Bosonic fixed points

While this paper deals with fermionic fixed point, most fixed points of interest to physics do contain bosonic fields. A few available rigorous bosonic fixed points are listed in appendix K. Notably, they include the bosonic analogues of the models that we studied here, i.e. long-range bosonic $O(N)$ field theories with weakly relevant quartic bare interactions. Unfortunately, these rigorous constructions remain rather daunting, in spite of serious ped-

<footnote n="48">⁴⁸See also [107–111] for FRG setups allowing nonlocal momentum dependence in the vertices.</footnote>

<footnote n="49">⁴⁹See also a non-partisan review in [113], chapter 5.</footnote>
agogical work which went into trying to simplify them (e.g. [77]). Further simplification is desirable, however unlikely. A very accessible review can be found in [114].

A major complication in the bosonic case, compared to the fermionic one, is that, in defining the RG map $H \mapsto H'$, the terms involving fluctuation fields $\varphi$ that are (on a local scale) large compared with their standard deviation must be treated in a distinguished way: rather than dealing with them via resummations of perturbation theory, they are controlled via apriori bounds on the probability of such “large fields” configurations in combination with a “polymer expansion”.<fnref n="50" /> These additional small/large fields decomposition and polymer expansion add a whole new combinatorial level to the construction, which inevitably leads to several technical complications. To date, essentially all the rigorous works on the construction of bosonic fixed points use a parameterization of the full probability distribution of the form

$$e^H + \mathcal{P}, \qquad (8.2)$$

rather than of the more standard Gibbs form $e^H$: in (8.2), $H$ includes the relevant and marginal parts of the interaction, which are exponentiated, while $\mathcal{P}$ includes the “non-perturbative” large field contributions, which are kept non-exponentiated; this mixed form turns out to be optimal for proving that the RG map preserves the space of interactions. Assuming (8.2), RG fixed point equation becomes $H' = H, \mathcal{P} = \mathcal{P}'$, whose form is quite different from (and quite more involved than) the standard “Exact RG equations”, such as Polchinski’s or Wetterich’s, which typically neglect the contribution from the polymer expansion of the large fields contributions (the $\mathcal{P}$-term).

## Acknowledgments

We warmly thank Giuseppe Benfatto for his generous contributions to this work in its preliminary phases. We thank David Brydges and Manfred Salmhofer for communications about the rigorous status of Polchinski’s equation, and Marco Serone for discussions about renormalons. We thank Malek Abdesselam, Dario Benedetti, David Brydges, Krzysztof Gawedzki, Antti Kupiainen, Pronob Mitter, Gordon Slade, Kay Wiese for comments on the draft.

SR is grateful to Accademia Nazionale dei Lincei for organizing the conference “Advances in Mathematics and Mathematical Physics” (Rome, September 2017) where the first seeds of this project were planted, and to the organizers of the program “Renormalisation in quantum field theory” (Isaac Newton Institute, Cambridge, Fall 2018) where part of the work was done. AG is grateful to IHES for hospitality in January-February 2020, where part of this work was completed.

<footnote n="50">⁵⁰ Loosely speaking, one proceeds as follows: the volume is paved into boxes of typical length comparable with the correlation length of the fluctuation field; each box is called “good” or “bad” depending on whether the typical size of $\varphi$ in the box is smaller or larger than a large multiple of the standard deviation, respectively; the probability of a bad box is bounded apriori and proved to be very small in the perturbative parameter: therefore, bad boxes are typically far apart from each other; in other words, they form a rarefied gas, whose partition function can be computed via an analogue of the low fugacity expansion for the pressure of lattice gases (this is the polymer expansion which we referred to in the main text).</footnote>
The work of AG and VM has been supported by: the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC CoG UniCoSM, grant agreement n.724939), and by MIUR, PRIN 2017 project MaQuMA cod. 2017ASFLJR, which are gratefully acknowledged. SR is supported by the Simons Foundation grant 488655 (Simons Collaboration on the Nonperturbative Boot-strap), and by Mitsubishi Heavy Industries as an ENS-MHI Chair holder.

# A Gevrey classes and fluctuation propagator bounds

In this appendix we give an explicit example of a compactly supported cutoff function satisfying the Gevrey condition, and prove the stretched exponential bound (4.15) for the fluctuation propagator.

## A.1 Cutoff function of Gevrey class

Here we will explain that the set of cutoff functions satisfying conditions (2.2) and (2.3) is not empty. Bump $C^\infty$ functions being standard, we will explain how to satisfy in addition to (2.2) the condition (2.3) $\Leftrightarrow$ (4.14), which we copy here:

$$ \sup_{k \in \mathbb{R}^d} |\partial^\alpha \chi(k)| \le C^n n^{ns}, \quad n = |\alpha| = 0, 1, 2 \dots \quad (A.1) $$

We will not assume any knowledge about Gevrey classes; see e.g. [115, 116] and [37], appendix C.

Recall the following classic result for analytic function. Let $F(k)$ be a function which allows an analytic continuation from real $k \in \mathbb{R}^d$ to a polydisk $D_R$, i.e. the region of complex $z \in \mathbb{C}^d$ such that $|z_i - (k_0)_i| \le R$ ($i = 1, \dots, d$). Then, by the Cauchy integral representation, the derivatives of $F(k)$ at the midpoint of the polydisk are bounded by ($n = |\alpha|$)

$$ |\partial^\alpha F(k_0)| \le n! R^{-n} A, \quad A = \max_{D_R} |F(z)|. \quad (A.2) $$

By (A.2), an analytic $\chi(k)$ would satisfy (A.1) with $s = 1$. However, by (2.2) our $\chi(k)$ is compactly supported, hence cannot be analytic. The best we can hope for is (A.1) with $s > 1$.

Let us first construct a $d = 1$ example of a compactly supported Gevrey-class function. Fix $r > 0$ and consider a $C^\infty$ function (see figure 3)

$$ X_0(t) = \begin{cases} 0 & t \le 0 \\ e^{-1/t}, & t > 0. \end{cases} \quad (A.3) $$

This function is not compactly supported, but this will be corrected below. For now let us check that it is Gevrey class, namely that it satisfies the $d = 1$ analogue of (A.1):

$$ |\partial^n X_0(t)| \le C^n n^{ns}, \quad n = 0, 1, 2, \dots \quad (A.4) $$

with $s = 1 + 1/r$. ($C$ stands for various positive constants which can change from one line to the next.) Indeed, consider the function

$$ X_0(t+z) = e^{-1/(t+z)^r}. \quad (A.5) $$
<figure><img src="image_4.png" /><figcaption>Figure 3. (a) Function $X_0(t)$ for $r = 1$; (b,c) The corresponding functions $X_1(t)$ and $X(t)$.</figcaption></figure>

In the disk of complex $|z| < \kappa t$, where $\kappa > 0$ is sufficiently small, this function is analytic and bounded in absolute value by $e^{-C/t^r}$.<fnref n="51" /> By the $d = 1$ case of the Cauchy estimate (A.2), we have:

$$|\partial^n X_0(t)| \le n!(\kappa t)^{-n}e^{-C/t^r}, \qquad (A.6)$$

from where (A.4) follows via elementary maximization of the r.h.s. over $t$.

From $X_0(t)$ which vanishes at $t \le 0$, we pass to a function of compact support $[1/2, 1]$:

$$X_1(t) = X_0(t - 1/2)X_0(1 - t). \qquad (A.7)$$

By the Leibniz rule, it's easy to verify that $X_1(t)$ also satisfies (A.4). Finally, we put

$$X(t) = \int_{|t|}^{\infty} X_1(t')dt' \qquad (A.8)$$

which is constant for $|t| \le 1/2$, vanishes for $|t| \ge 1$, and still satisfies (A.4). We rescale it so that $X(0) = 1$. See figure 3.

The function $X(t)$ is an explicit example of a cutoff function satisfying conditions (2.2) and (2.3) in $d = 1$. Infinitely many examples of this sort can be given multiplying $X_0(t)$ in (A.3) by an analytic function and repeating the construction.

The function $\chi(k)$ in $d$ dimensions will be given in terms of the 1d function by

$$\chi(k) = X(|k|). \qquad (A.9)$$

While eq. (A.1) can be verified using the chain rule, we will instead give a more robust argument via analytic continuation and the Cauchy estimate. The function $f(k) = |k|$ is real analytic for $|k| \in [1/2, 1]$ where derivatives of $X(t)$ are nonzero. Generally, a composition $X(f(k))$ of a Gevrey class function $X(t)$ and a real analytic function $f(k)$ remains in the Gevrey class. For the proof, let $f(z)$ be analytic continuation into a polydisk $|z_i - k_i| \le R$ (we can choose $R = 1/4$ for $f(k) = |k|$, $1/2 \le |k| \le 1$). Writing $X(f(z)) = X(f(k) + [f(z) - f(k)])$ and Taylor-expanding we have

$$X(f(z)) = \sum_{i=0}^{\infty} \frac{1}{i!} \partial^i X(f(k))[f(z) - f(k)]^i. \qquad (A.10)$$

<footnote n="51">⁵¹ $\kappa$ here depends on $r$. We can choose it so that $\text{Re}[(1 + \zeta)^{-r}] > 1/2$ for $|\zeta| < \kappa$.</footnote>
Suppose we want to compute $\partial^\alpha[X(f(k))], |\alpha| = n$. We can compute this derivative by differentiating the Taylor series (A.10) truncated to $i \le n$, since all terms with $i > n$ are anyway higher order:

$$ \partial_k^\alpha [X(f(k))] = \partial_z^\alpha \Phi(z)|_{z=k}, \quad \Phi(z) = \sum_{i=0}^{n} \frac{1}{i!} \partial^i X(f(k)) [f(z) - f(k)]^i. \quad (A.11) $$

The function $\Phi(z)$ is analytic. It can be bounded in the polydisk by

$$ |\Phi(z)| \le C^n n^{n(s-1)}, \quad (A.12) $$

using (A.4) for $X(t)$, and that $f(z)$ is bounded in the polydisk. From here using (A.2) we get (A.1).

## A.2 Fluctuation propagator bounds

### A.2.1 k-space

Recall that the Fourier transform of $g(x)$ is given by eq. (2.10) which we copy here:

$$ \hat{g}(k) = [\chi(k) - \chi(\gamma k)]/|k|^{d/2+\epsilon}. \quad (A.13) $$

In this subsection we will show, using (A.1), that, for any $k \in \mathbb{R}^d$ and any $n = |\alpha| \ge 0$,

$$ |\partial^\alpha \hat{g}(k)| \le C \frac{(C\gamma)^n n^{ns}}{|k|^{d/2+\epsilon}}. \quad (A.14) $$

(C will denote $\gamma$- and $n$-independent constants which may change from one equation to the next.)

We first estimate the derivatives of $1/|k|^{d/2+\epsilon}$. Consider the analytic continuation of $F(k) = 1/|k|^{d/2+\epsilon}$ into the polydisk centered at $k \ne 0$ of radius $R = \frac{1}{2} \max_i |k_i|$. The maximum of $|F(z)|$ in this polydisk is bounded by $CF(k)$. We conclude by (A.2) that$^{52}$

$$ \left| \partial^{\alpha} \frac{1}{|k|^{d/2+\epsilon}} \right| \le n! R^{-n} C F(k) \le \frac{C^n n!}{|k|^{d/2+\epsilon+n}}. \quad (A.15) $$

Finally since $|k| \ge 1/(2\gamma)$ on supp$[\chi(k) - \chi(\gamma k)]$ we conclude

$$ \left| \partial^{\alpha} \frac{1}{|k|^{d/2+\epsilon}} \right| \le \frac{(C\gamma)^n n!}{|k|^{d/2+\epsilon}} \quad \text{on supp}[\chi(k) - \chi(\gamma k)]. \quad (A.16) $$

Now let us prove (A.14). By the Leibniz rule we have

$$ \partial^\alpha \hat{g}(k) = \sum_{\beta \le \alpha} Q_\beta \partial^\beta \frac{1}{|k|^{d/2+\epsilon}} \times \partial^{\alpha-\beta} [\chi(k) - \chi(\gamma k)], \quad (A.17) $$

<footnote>$^{52}$For $d=1$ bound (A.16) would be easy to get by repeated differentiation deriving an explicit formula for the l.h.s. The analytic continuation argument is more robust to show that the same estimate is true for any $d$.</footnote>
<figure><img src="image_5.png" /><figcaption>**Figure 4.** The Fourier transform of the function $X(t)$ from figure 3(c), plotted in log scale against $|x|^{1/s}$ where $s = 1 + 1/r = 2$. The expected $\exp(-C|x|^{1/s})$ decay is visible.</figcaption></figure>

where $Q_β = \prod_{i=1}^d (\alpha_i)$. We estimate the $\partial^β$ derivative on the support of $\chi(k) - \chi(\gamma k)$ via (A.16), while the $\partial^{\alpha-\beta}$ factor by (A.1) with $C \to C\gamma$. Combining these two estimates via $(n = |\alpha|)$

$$ (C\gamma)^{|\beta|} (C\gamma)^{|\alpha-\beta|} = (C\gamma)^n, \\ |\beta|^{|\beta|} |\alpha - \beta|^{s|\alpha-\beta|} \le n^{s|\beta|} n^{s|\alpha-\beta|} = n^{sn}, \qquad (A.18) $$

and using that $\sum_β Q_β = 2^n$, we get (A.14).

### A.2.2 x-space

Finally we show (4.15). Consider first the bound for $g(x)$. We use the standard trick that the Fourier transform of $(-ix)^\alpha g(x)$ is $\partial^\alpha \hat{g}(k)$, hence

$$ \sup_x |x^\alpha g(x)| \le (2\pi)^{-d} \|\partial^\alpha \hat{g}\|_{L^1}. \qquad (A.19) $$

Bound (A.14) then implies (note that $\hat{g}(k)$ has compact support and that $\int_{|k| \le 1} \frac{d^d k}{|k|^{d/2+\epsilon}} < \infty$)

$$ \sup_x |x^\alpha g(x)| \le C(C\gamma)^n n^{ns}, \qquad (A.20) $$

which can be rewritten as

$$ |g(x)| \le C u^{-n} n^{ns} \equiv C e^{ns \log \frac{n}{u^{1/s}}}, \quad u = C|x|/\gamma. \qquad (A.21) $$

From here (4.15) for $g(x)$ follows by choosing $n$ optimally as $n = \lfloor u^{1/s}/e \rfloor$.

The Fourier transforms of the first and second derivatives of $g(x)$ are $g_1(k) = k_\mu \hat{g}(k)$ and $g_2(k) = k_\mu k_\nu \hat{g}(k)$. Using (A.14) for $\hat{g}(k)$, it's easy to see that $g_1(k), g_2(k)$ satisfy the same type of bounds. Thus the bounds for the first and second derivatives of $g(x)$ follow by the same argument. In fact derivative of any order will have the same kind of decay, only the constants will degrade.

As an illustration, we plot in figure 4 the numerically computed Fourier transform of the function $X(t)$ from figure 3(c). The plot shows the expected $\exp(-C|x|^{1/2})$ decay.
<figure><img src="image_6.png" /><figcaption>**Figure 5.** This figure represents a term in $H_{\text{eff}}$ corresponding to $n = 1$ in (B.1): just one vertex with $|A| = 6$ (gray oval). Empty circles are the external fields and filled circles are the internal ones. The generated $H_{\text{eff}}$ term has $l = |B| = 2$. The 4 internal fields are contracted (just one possible contraction is shown).</figcaption></figure>

## B Details about $H_{\text{eff}}$

In this appendix we give more details about the derivation of eq. (5.3). Plugging (5.2) into (5.1), we represent $H_{\text{eff}}(\psi)$ as

$$ H_{\text{eff}}(\psi) = \sum_{n=1}^{\infty} \frac{1}{n!} \sum_{\mathbf{A}_1, \dots, \mathbf{A}_n} \sum_{\mathbf{B}_1, \dots, \mathbf{B}_n} (-)^{\#} \qquad (B.1) \\ \times \int d^d\mathbf{x} \Psi(\mathbf{B}_1, \mathbf{x}_{\mathbf{B}_1}) \dots \Psi(\mathbf{B}_n, \mathbf{x}_{\mathbf{B}_n}) \prod_{i=1}^{n} H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) \langle \Phi(\bar{\mathbf{B}}_1, \mathbf{x}_{\bar{\mathbf{B}}_1}); \dots; \Phi(\bar{\mathbf{B}}_n, \mathbf{x}_{\bar{\mathbf{B}}_n}) \rangle_c $$

Let us explain this in words. We sum over even-length sequences $\mathbf{A}_i$ indexing terms in $H$. We introduce a coordinate sequence $\mathbf{x}$ of length $|\mathbf{A}_1| + \dots + |\mathbf{A}_n|$ to be integrated over. We further sum over subsequences $\mathbf{B}_i \subset \mathbf{A}_i$ selecting which fields inside the $H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})$ term are external. The fields from the complements $\bar{\mathbf{B}}_i = \mathbf{A}_i \setminus \mathbf{B}_i$ are internal, to be contracted in the connected expectation. We use $\mathbf{x}_{\mathbf{A}_i}, \mathbf{x}_{\mathbf{B}_i}, \mathbf{x}_{\bar{\mathbf{B}}_i}$ to denote the part of the vector $\mathbf{x}$ for the corresponding subsequence. The $(-)^{\#}$ is the sign, which we don't need to track, of the permutation reordering sequence $\mathbf{A}_1 + \dots + \mathbf{A}_n$ to $\mathbf{B}_1 + \dots + \mathbf{B}_n + \bar{\mathbf{B}}_1 + \dots + \bar{\mathbf{B}}_n$.

There are several distinguished groups of terms in (B.1):

* Terms with $n = 1$ and $\mathbf{B}_1 = \mathbf{A}_1$. They involve no contractions and their sum gives back $H$.

* Terms with $n = 1$ and $\mathbf{B}_1 \neq \mathbf{A}_1$. These involve a single $H(\mathbf{A}, \mathbf{x})$ vertex with several fields identified as internal and contracted among themselves, while the rest remaining external (see figure 5).

* Terms with $n \ge 2$, which therefore correspond to contractions of several vertices. Since (B.1) involves connected expectation, we have to sum over contractions such that the graph becomes connected when every interaction vertex is shrunk to a point (see figure 6).

* Terms for which all $\mathbf{B}_k$ are empty, meaning that all fields are contracted. These sum up to a $\psi$-independent constant (infinite when working in infinite volume as we are). As mentioned in section 2.1, footnote 12, this constant will be dropped.

Finally, eq. (5.3) follows by rewriting (B.1) in the form (4.2).
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Left: a contraction of three vertices, $n = 3$ in (B.1). The generated $H_{\text{eff}}$ term has $l = 4$ (the total number of empty circles). Right: the graph of contractions obtained when every gray oval is shrunk to a point (denoted by a fat dot). This graph is connected, as it should be because we are considering connected expectations.</figcaption></figure>

### C Trimming details

This appendix deals with the trimming map introduced in section 5.2.2, and with how it behaves with respect to the weighted norms. This map takes the interaction $H_{\text{eff}}$ in a general representation and returns an equivalent trimmed representation. Consider the parts of $\mathcal{H} = H_{\text{eff}}$ which need to be set to zero: $\mathcal{H}_{4,0}$, $\mathcal{H}_{2,0}$, $\mathcal{H}_{2,1}$. Recall that $\mathcal{H}_{l,p}$ corresponds to $l$-leg interactions with $p$ derivatives:

$$ \mathcal{H}_{l,p} \leftrightarrow \sum_{|\mathbf{A}|=l, d(\mathbf{A})=p} \int d^d\mathbf{x} \mathcal{H}(\mathbf{A}, \mathbf{x}) \Psi(\mathbf{A}, \mathbf{x}). \qquad (C.1) $$

The dependence on $Sp(N)$ indices should be given by possible invariant tensors:

$$ \mathcal{H}_{4,0} \leftrightarrow \mathbf{A} = (a, b, c, e), \quad \mathcal{H}(\mathbf{A}, \mathbf{x}) = \Omega_{ab}\Omega_{ce}F_1(\mathbf{x}) - \Omega_{ac}\Omega_{be}F_2(\mathbf{x}) + \Omega_{ae}\Omega_{bc}F_3(\mathbf{x}) \qquad (C.2) $$

$$ \mathcal{H}_{2,0} \leftrightarrow \mathbf{A} = (a, b), \quad \mathcal{H}(\mathbf{A}, \mathbf{x}) = \Omega_{ab}G(\mathbf{x}) \qquad (C.3) $$

$$ \mathcal{H}_{2,1} \leftrightarrow \mathbf{A} = ((a, \mu), b), \quad \mathcal{H}(\mathbf{A}, \mathbf{x}) = \Omega_{ab}K_1^\mu(\mathbf{x}), \qquad (C.4) $$

$$ \mathbf{A} = (a, (b, \mu)), \quad \mathcal{H}(\mathbf{A}, \mathbf{x}) = \Omega_{ab}K_2^\mu(\mathbf{x}). \qquad (C.5) $$

Recall that kernels $\mathcal{H}(\mathbf{A}, \mathbf{x})$ are antisymmetric, $\mathcal{H}(\pi\mathbf{A}, \pi\mathbf{x}) = (-)^{\pi}\mathcal{H}(\mathbf{A}, \mathbf{x})$. This implies various symmetry relations for the functions $F, G, K$. E.g. $G$ is symmetric, $F_1, F_2, F_3$ are all related by permutations of their arguments, and finally $K_1^\mu(x_1, x_2) = K_2^\mu(x_2, x_1)$.

By spatial parity (see footnote 15) we have $K_1^\mu(-x_1, -x_2) = -K_1^\mu(x_1, x_2)$. Combined with translational and rotational invariance this implies that $K_1^\mu(x_1, x_2) = (x_1 - x_2)^\mu \bar{K}(|x_1 - x_2|)$.

The map $T_{2R}^{2,1}$ takes $\mathcal{H}_{2,1}$ and returns an equivalent interaction of $H_{2,2}$ type. Consider the part of $\mathcal{H}_{2,1}$ with $\mathbf{A} = ((a, \mu), b)$. Using the interpolation identity

$$ \psi_b(x_2) = \psi_b(x_1) + \int_0^1 dt \partial_t [\psi_b(x_1 + t(x_2 - x_1))], \qquad (C.6) $$

this term is mapped onto the sum of two terms. The first one is

$$ \Omega_{ab} \int d^d x_1 \left[ \int d^d x_2 K_1^\mu(x_1, x_2) \right] (\partial_\mu \psi_a \psi_b)(x_1), \qquad (C.7) $$
which vanishes because as mentioned above $K_1^\mu(x_1, x_2)$ is odd in $x_1 - x_2$. So no local terms are generated in the case at hand. The second term is

$$ \Omega_{ab} \int d^d x_1 d^d x_2 K_1^\mu(x_1, x_2) \partial_\mu \psi_a(x_1)(x_2 - x_1)_\nu \int_0^1 dt \partial_\nu \psi_b(x_1 + t(x_2 - x_1)). \quad (C.8) $$

Changing integration variables from $d^d x_2$ to $d^d y$ with $y = x_1 + t(x_2 - x_1)$ and doing the integral over $t$ we have an identity

$$ \int d^d x_2 K_1^\mu(x_1, x_2)(x_2 - x_1)_\nu \int_0^1 dt \partial_\nu \psi_b(x_1 + t(x_2 - x_1)) = \int d^d y L^{\mu\nu}(x_1, y) \partial_\nu \psi_b(y), \quad (C.9) $$

with the help of which we rewrite (C.8) as

$$ \Omega_{ab} \int d^d x_1 d^d y L^{\mu\nu}(x_1, y) \partial_\mu \psi_a(x_1) \partial_\nu \psi_b(y). \quad (C.10) $$

which is an interaction of type $H_{2,2}$ as promised, and we associate it with $T_{2R}^{2,1}(\mathcal{H}_{2,1})$. The action on the part of $\mathcal{H}_{2,1}$ with $\mathbf{A} = (a, (\mu, b))$ is analogous and we have to add it to the previous result.

The maps $T_{2L}^{2,0}$ and $T_{2R}^{2,0}$ take $\mathcal{H}_{2,0}$ and return an equivalent interaction which is a sum of $H_{2L}$ and $H_{2,2}$ type interactions. Using the interpolation identity (C.6), an $\mathcal{H}_{2,0}$ interaction is mapped to a sum of two terms. The first term is the local quadratic interaction and we associate it with $T_{2L}^{2,0}$:

$$ T_{2L}^{2,0}(\mathcal{H}_{2,0}) = \nu \Omega_{ab} \int d^d x_1 (\psi_a \psi_b)(x_1), \quad \nu = \int d^d x_2 G(x_1, x_2). \quad (C.11) $$

(By translational invariance the integral $\int d^d x_2 G(x_1, x_2)$ is $x_1$-independent). The second term is

$$ \Omega_{ab} \int d^d \mathbf{x} G(x_1, x_2) \psi_a(x_1)(x_2 - x_1)_\nu \int_0^1 dt \partial_\nu \psi_b(x_1 + t(x_2 - x_1)), \quad (C.12) $$

and similarly to (C.10) we can write it after a change of variable and t-integration as

$$ \Omega_{ab} \int d^d x_1 d^d y G^\nu(x_1, y) \psi_a(x_1) \partial_\nu \psi_b(y). \quad (C.13) $$

This is of type $H_{2,1}$ which we already considered. Acting on it with the map $T_{2R}^{2,1}$ we will get an equivalent interaction of type $H_{2,2}$. This final result is $T_{2R}^{2,0}(\mathcal{H}_{2,0})$.

The maps $T_{4L}^{4,0}$ and $T_{4R}^{4,0}$ are constructed with the help of the interpolation identity (5.17), which maps $\mathcal{H}_{4,0}$ to an equivalent sum of two interactions, the first of which defines $T_{4L}^{4,0}$ as it is a local quartic interaction with the coupling

$$ \lambda = \int_{x_1=0} d^d \mathbf{x} [F_1(\mathbf{x}) + F_2(\mathbf{x}) + F_3(\mathbf{x})] = 3 \int_{x_1=0} d^d \mathbf{x} F_1(\mathbf{x}), \quad (C.14) $$

while the second term is an interaction of $H_{4,1}$ type which is associated with $T_{4R}^{4,0}$.

We now consider **weighted norm estimates** for the introduced maps. Since $w \ge 1$, the localization maps have simply norm one (factor 3 in (C.14) cancels with $1/3$ in (4.18)):

$$ \|T_{2L}^{2,0}(\mathcal{H}_{2,0})\|_w \le \|\mathcal{H}_{2,0}\|_w, \quad \|T_{4L}^{4,0}(\mathcal{H}_{4,0})\|_w \le \|\mathcal{H}_{4,0}\|_w. \quad (C.15) $$
On the other hand, due to the factors like $(x_2 - x_1)$ in (C.8), the interpolation maps will not preserve the norm $\| \cdot \|_w$. Let us aim instead for an inequality of the type $\|T(\mathcal{H})\|_{w'} \le \text{Const.} \|\mathcal{H}\|_{w''}$ where $w'$ is a slightly weaker weight than $w''$ (i.e. growing slower than $w''$ at infinity). Eventually we will choose $w' = w(\cdot/\gamma)$, $w'' = w$.

For $T_{2R}^{2,1}$, we need to estimate the $w'$-norm of $L^{\mu\nu}$ in (C.10) in terms of the $w''$-norm of $K_1^\mu$. The relation between $L^{\mu\nu}$ and $K_1^\mu$ is encoded by the identity (C.9) which by translational invariance and renaming $\partial_\nu \psi^b$ by $f$ can be written equivalently as

$$ \int d^dy L^{\mu\nu}(0, y)f(y) = \int d^dx K_1^\mu(0, x)x^\nu \int_0^1 dt f(tx), \qquad (C.16) $$

where $f(y)$ is an arbitrary function. The actual expression for $L^{\mu\nu}$ in terms of $K_1^\mu$ can be written by e.g. choosing $f(y) = \delta(y - y_0)$ but we don't need it. We write the norm of $L^{\mu\nu}$ as

$$ \begin{aligned} \|L^{\mu\nu}\|_{w'} &= \int d^dy |L^{\mu\nu}(0, y)| w'(0, y) = \int d^dy L^{\mu\nu}(0, y) \Sigma(y) w'(0, y) \\ &= \int d^dx K_1^\mu(0, x) x^\nu \int_0^1 dt [\Sigma(y) w'(0, y)]_{y=tx} \\ &\le \int d^dx |K_1^\mu(0, x)| |x| w'(0, x), \qquad (C.17) \\ &\le C_1 \|K_1^\mu\|_{w''}, \end{aligned} $$

where in the first line we defined $\Sigma(y) = \text{sign}L^{\mu\nu}(0, y)$, in the second line we used identity (C.16) with $f(y) = \Sigma(y)w'(0, y)$, in the third line we used that $|\Sigma| = 1$ and assumed that the weight $w'$ is monotonically increasing. Finally, in the last line we assumed the inequality:

$$ |x|w'(0, x) \le C_1 w''(0, x) \quad (x \in \mathbb{R}^d). \qquad (C.18) $$

Multiplying the bound (C.17) by 2 to account for the contribution of $K_2^\mu$, we conclude

$$ \|T_{2R}^{2,1}(\mathcal{H}_{2,1})\|_{w'} \le 2C_1 \|\mathcal{H}_{2,1}\|_{w''} \qquad (C.19) $$

For $T_{2R}^{2,0}$, very similar considerations will apply. Recall that we have to apply the interpolation identity twice, and each time we will pay a factor of $|x|$ in the weight function. So we get a bound

$$ \|T_{2R}^{2,0}(\mathcal{H}_{2,0})\|_{w'} \le C_2 \|\mathcal{H}_{2,0}\|_{w''} \qquad (C.20) $$

under the condition

$$ |x|^2 w'(0, x) \le C_2 w''(0, x) \quad (x \in \mathbb{R}^d). \qquad (C.21) $$

For $T_{4R}^{4,0}$, the interpolation identity (5.17) will give rise to an extra factor $|x_2| + |x_3| + |x_4|$ in the weight function. We will therefore obtain:

$$ \|T_{4R}^{4,0}(\mathcal{H}_{4,0})\|_{w'} \le C_3 \|\mathcal{H}_{4,0}\|_{w''}, \qquad (C.22) $$

assuming

$$ (|x_2| + |x_3| + |x_4|) w'(0, x_2, x_3, x_4) \le C_3 w''(0, x_2, x_3, x_4) \quad (x_2, x_3, x_4 \in \mathbb{R}^d). \qquad (C.23) $$
Finally, we specialize to the case of interest for us: $w'' = w$, $w' = w(\cdot/\gamma)$ where $w$ is our weight (4.16). We leave it as an elementary exercise to show that, for $\gamma \ge 2$, inequali- ties (C.18), (C.21), (C.23) hold with $C_1, C_3 = O(\gamma)$, $C_2 = O(\gamma^2)$ and the constants in the $O$ bounds depend only on $C_w$ and $\sigma$ in (4.16). [Here $\gamma \ge 2$ is useful as $C_i$ would blow up in the limit $\gamma \to 1$, $w' \to w''$.] Bounds (5.43) follow.

## D Determinant bounds for fermionic expectations

This appendix discusses the determinant bounds (Gram-Hadamard and Gawedzki-Kupia- inen-Lesniewski) for the simple and connected<sup>53</sup> fermionic expectations. These are standard in mathematical physics, but will be unfamiliar to most theoretical physicists, usually concerned with computing, not bounding. These bounds are closely related to the Pauli principle and, at a formal level, to the fermionic expectation being a determinant (hence the name). We will also review the Brydges-Battle-Federbush (BBF) formula, a clever integral representation for connected expectations, useful to derive bounds (and perhaps for other things).

Classic sources (citing previous literature) are [117] for the bounds and [118] for the BBF formula. Other presentations are in [119–121].

### D.1 Simple expectations

We are interested in the expectations of the fluctuation field $\phi_a(x)$ which is a gaussian Grassmann field with the propagator $\langle\phi_a(x)\phi_b(y)\rangle = \Omega_{ab}g(x-y)$. We use the notation $\Phi(\mathbf{A}, \mathbf{x})$ for field products as in (4.1). The simple fermionic expectations are given by, see (2.5),

$$ \langle \Phi(\mathbf{A}, \mathbf{x}) \rangle \equiv \langle \Phi_{A_1}(x_1) \dots \Phi_{A_{2s}}(x_r) \rangle = \sum (-)^p \times \text{Wick contractions}, \quad (D.1) $$

where a Wick contraction is a product of $s$ propagators like

$$ \langle \Phi_{A_1}(x_1) \Phi_{A_2}(x_2) \rangle \dots \langle \Phi_{A_{2s-1}}(x_{r-1}) \Phi_{A_{2s}}(x_r) \rangle, \quad (D.2) $$

or any other pairing where fields are ordered as $p(1) \dots p(r)$ and $(-)^p = \pm 1$ in (D.1) is the sign of the corresponding permutation $p$.

Eq. (D.1) contains factorially many terms, but there are cancellations because of the signs. To see this, we rewrite (D.1) as a determinant [122]. As a model, take gaussian Grassmann fields $\xi$ and $\bar{\xi}$ with propagator

$$ \langle \xi(x) \bar{\xi}(y) \rangle = g(x-y). \quad (D.3) $$

Then,

$$ \langle \xi(x_1) \dots \xi(x_s) \bar{\xi}(y_1) \dots \bar{\xi}(y_s) \rangle = \pm \det M, \quad M_{ij} = g(x_i - y_j). \quad (D.4) $$

In the general case (D.1), fields $\phi_a$ carry indices $a = 1 \dots N$, and propagator is $\Omega_{ab}g(x-y)$. Renaming odd-$a$ fields as $\xi_\alpha$, even-$a$ as $\bar{\xi}_\alpha$, the $\xi_\alpha$-$\bar{\xi}_\alpha$ pairs $(\alpha = 1 \dots N/2)$ are decoupled,

<footnote n="53"><sup>53</sup>Connected expectations are referred to as ‘truncated’ in mathematical physics.</footnote>
with propagator $\propto \delta_{\alpha\alpha'}$. The number of $\xi$ and $\bar{\xi}$ fields in the non-vanishing expectation must be the same, let $x_i$ and $y_j$ be their coordinates. Then the expectation (D.1) is, up to a sign, the determinant of the $s \times s$ matrix:

$$ \det \mathcal{M}, \quad \mathcal{M}_{ij} = \delta_{\alpha_i \alpha_j} \Gamma_{ij}(x_i - y_j). \qquad (D.5) $$

Here $\Gamma_{ij}$ is either $g(x-y)$ or its derivative if some fields carry derivatives. We will estimate it with the help of

**Lemma D.1 (Gram-Hadamard inequality)** For $(f_i)$, $(h_i)$ ($i=1...s$) two lists of vectors in a Hilbert space, let $M_{ij}$ be the $s \times s$ matrix of their inner products: $M_{ij} = (f_i, h_j)$. Then $D(f, h) = \det M$ satisfies an upper bound:

$$ |D(f, h)| \le \prod_{i=1}^{s} \|f_i\| \|h_i\|. \qquad (D.6) $$

*Proof.* For $h_i$ orthonormal, this holds interpreting the determinant as the volume of parallelepiped formed by $f_i$ (this case is known as Hadamard's inequality). By rescaling, the inequality remains true for $h_i$ orthogonal of arbitrary length. We will next reduce the general case to this special case.

Out of general $h_i$, we build $\tilde{h}_i$ by Gram-Schmidt: $\tilde{h}_1 = h_1$, $\tilde{h}_2 = h_2 - \alpha h_1 \perp \tilde{h}_1$ (projection of $h_2$ on the subspace orthogonal to $\tilde{h}_1$), $\tilde{h}_3$ projection of $h_3$ on the subspace orthogonal to $\tilde{h}_1, \tilde{h}_2$ etc. By properties of determinants:

$$ D(f, h) = D(f, \tilde{h}). \qquad (D.7) $$

From the special case, the r.h.s. is bounded by $\prod \|f_i\| \|\tilde{h}_i\|$, and $\|\tilde{h}_i\| \le \|h_i\|$ since it's a projection. Q.E.D.

To apply this result, we have to write the matrix elements (D.5) as products of vectors in a Hilbert space. Without indices and derivatives we have

$$ g(x_i - y_j) = (f_i, h_j), \qquad (D.8) $$

introducing two families of $L^2$ functions in momentum space (a trick due to [123]):$^{54}$

$$ f_i(k) = e^{-ikx_i} |\hat{g}(k)|^{1/2} \frac{\hat{g}(k)}{|\hat{g}(k)|}, \quad h_j(k) = e^{-iky_j} |\hat{g}(k)|^{1/2}, \qquad (D.9) $$

When some fields carry derivatives, we just include a factor $ik_\mu$ into the corresponding function. Finally with indices, we view $f_i$ and $h_j$ as vector functions, multiplying (D.9) by the unit vectors in the directions $\alpha_i, \alpha_j$, whose inner product reproduces the Kronecker $\delta_{\alpha_i \alpha_j}$. We just proved

**Lemma D.2 (Gram-Hadamard bound)** We have the bound

$$ |\langle \Phi_{A_1}(x_1) \dots \Phi_{A_{2s}}(x_{2s}) \rangle| \le (C_{GH})^s \quad \text{with} \\ C_{GH} = \max \left( \int \frac{d^d k}{(2\pi)^d} |\hat{g}(k)|, \int \frac{d^d k}{(2\pi)^d} (k_1)^2 |\hat{g}(k)| \right). \qquad (D.10) $$

<footnote>$^{54}$ $\hat{g}(k)$ will be non-negative if $\chi(k)$ is non-negative and monotonic, but we state this part of the argument for a general complex $\hat{g}(k)$.</footnote>
This bound is related to the Pauli principle, as can be seen from the following alternative proof. We can represent $\Phi_A(x)$ as operators acting on a Hilbert space (fermionic Fock space). Fermionic occupation numbers being either zero or one, operators $\Phi_A(x)$ turn out to have a finite norm $\|\Phi_A\|$, and expectation then grow at most as a power $\|\Phi_A\|^{2s}$. This should be contrasted with the bosonic case, when the operator norm would have been infinite (even for a simple harmonic oscillator).

**Lemma D.3** Let $\hat{g}(k)$ be as in (2.10), with $\chi(k)$ satisfying (2.2). Then $C_{GH}$ is uniformly bounded over $\gamma \ge 2$ and $\varepsilon \in T$ where $T \subset \mathbb{C}$ is any compact subset of the complex half-plane (7.1).

*Proof.* We have $C_{GH} \le \int_{|k|\le 1} d^d k (2\pi|k|)^{-(d/2+\text{Re}\,\varepsilon)}$, uniformly bounded since $\max_T \text{Re}\,\varepsilon < d/6$ by (7.1).

## D.2 Connected expectations

Dividing the points $\mathbf{x}$ into $n$ groups $\mathbf{x}_1, \dots, \mathbf{x}_n$, connected (also called ‘truncated’) expectations are given by

$$ \langle \Phi(\mathbf{A}_1, \mathbf{x}_1); \dots; \Phi(\mathbf{A}_n, \mathbf{x}_n) \rangle_c = \sum (-)^p \times \text{connected Wick contractions.} \qquad (\text{D.11}) $$

Connected Wick contractions form a subset of terms from (D.1), those for which the graph of propagators becomes connected when each group of points $\mathbf{x}_i$ is collapsed into one point. The signs $(-)^p$ are the same as in (D.1).

Because $g(x)$ decays at infinity, connected expectations are small when any two groups $\mathbf{x}_i$ and $\mathbf{x}_j$ get far apart. We need a bound incorporating both this fact and the cancellations due to signs. This will be done via a clever generalization of the determinant representation to connected expectations.

To begin with, via (D.1) and (D.11), the simple and connected expectations satisfy two relations $(\Phi_i \equiv \Phi(\mathbf{A}_i, \mathbf{x}_i))$. First, they coincide for a single group of points:

$$ \langle \Phi_i \rangle_c = \langle \Phi_i \rangle. \qquad (\text{D.12}) $$

Second, simple expectation can be computed by partitioning $n$ group of fields into subsets, taking products of connected expectations within each subset, and summing over all ways of partitioning:<fnref n="55" />

$$ \langle \prod_{i=1}^{n} \Phi_k \rangle = \sum_{\Pi \in \text{partitions of } \{1 \dots n\}} (-1)^{\pi} \prod_{Y \in \Pi} \langle \Phi_{Y_1} ; \Phi_{Y_2} ; \dots \rangle_c. \qquad (\text{D.13}) $$

The $(-)^\pi$ is the parity of the permutation bringing fields in the r.h.s. into the original order in the l.h.s. (it's not the same permutation as in (D.11).

<footnote n="55">⁵⁵This formula is more rapid that (D.11) to compute connected expectations: one recursively expresses them via the usual expectations, which in turn are evaluated by the determinant formula (D.5) [124]. This observation speeds up the Diagrammatic Monte Carlo algorithm from footnote 24 [124] achieving polynomial complexity [125]. One wonders if the BBF formula (D.47) below could give an alternative practical way to evaluate connected expectations.</footnote>
Reading (D.13) from right to left, one can recursively computes connected expectations from simple ones. E.g. for $n = 2, 3$ we have

$$ \langle \Phi_1 \Phi_2 \rangle = \langle \Phi_1; \Phi_2 \rangle_c + \langle \Phi_1 \rangle \langle \Phi_2 \rangle, \qquad (D.14) $$

$$ \begin{aligned} \langle \Phi_1 \Phi_2 \Phi_3 \rangle = & \langle \Phi_1; \Phi_2; \Phi_3 \rangle_c + \langle \Phi_1; \Phi_2 \rangle_c \langle \Phi_3 \rangle + \langle \Phi_1 \rangle \langle \Phi_2; \Phi_3 \rangle_c \\ & + (-)^{N_2 N_3} \langle \Phi_1; \Phi_3 \rangle_c \langle \Phi_2 \rangle + \langle \Phi_1 \rangle \langle \Phi_2 \rangle \langle \Phi_3 \rangle. \end{aligned} \qquad (D.15) $$

In the r.h.s. we replaced $\langle \Phi_k \rangle_c = \langle \Phi_k \rangle$ by (D.12). $N_k$ is the number of fields in $\Phi_k$. From (D.14) we find $\langle \Phi_1; \Phi_2 \rangle_c = \langle \Phi_1 \Phi_2 \rangle - \langle \Phi_1 \rangle \langle \Phi_2 \rangle$; substituting this into (D.15) we find $\langle \Phi_1; \Phi_2; \Phi_3 \rangle_c$; etc. So (D.13) provides an alternative definition of connected expectations, a useful starting point for what follows.

Let $m = \sum_{k=1}^n |\mathbf{x}_k|$ be the total number of points. In section D.1 we wrote the simple expectation as a determinant of the matrix $M$ defined in (D.5). Introducing auxiliary Grassmann variables $\eta_i$ and $\bar{\eta}_j$ ($m/2$ of each type), we write it then as a Grassmann integral

$$ \langle \prod_{k=1}^{n} \Phi(\mathbf{A}_k, \mathbf{x}_k) \rangle = \int \prod d\eta_i d\bar{\eta}_j e^V, \qquad (D.16) $$

where we defined the potential function

$$ V = \sum_{i,j} \mathcal{M}_{ij} \eta_i \bar{\eta}_j. \qquad (D.17) $$

Here $i, j$ index the fields classified in section D.1 as $\xi_{\alpha_i}(x_i)$ and $\bar{\xi}_{\alpha_j}(y_j)$. Depending in which group $\mathbf{x}_k$ their positions $x_i$ and $y_j$ belong, we subdivide $V$ as

$$ V = \frac{1}{2} \sum_{k,l=1}^{n} V_{kl}, \qquad (D.18) $$

$$ V_{kl} = \sum_{i,j: x_i \in \mathbf{x}_k, y_j \in \mathbf{x}_l} \mathcal{M}_{ij} \eta_i \bar{\eta}_j + (k \leftrightarrow l). \qquad (D.19) $$

Define $V(X)$ and $\psi(X)$ on any finite subset $X \subset \{1 \dots n\}$ by

$$ V(X) = \frac{1}{2} \sum_{k,l \in X} V_{kl}, \quad \psi(X) = e^{V(X)}. \qquad (D.20) $$

We can think of $V(X)$ as the total potential energy for a group of points with pairwise interactions. Define connected part $\psi_c(X)$ recursively by the following equations:

$$ \psi_c(X) = \psi(X) \quad \text{if } |X| = 1, \qquad (D.21) $$

$$ \psi(X) = \sum_{\Pi \in \text{partitions of } X} \prod_{Y \in \Pi} \psi_c(Y). \qquad (D.22) $$

Crucially, the form these equations is such that integrating them in $\eta_i, \bar{\eta}_j$, we land precisely on (D.12), (D.13) (including the $(-)^\pi$ sign), provided we identify

$$ \langle \Phi_{Y_1}; \Phi_{Y_2}; \dots \rangle_c = \int \prod d\eta_i d\bar{\eta}_j \psi_c(Y), \qquad (D.23) $$
where the integral is over the subset of Grassmann variables belonging to Y (which means $x_i \in \mathbf{x}_k, y_j \in \mathbf{x}_l$, where $k, l \in Y$). Computing connected expectations is thus reduced to finding $\psi_c(Y)$ in terms of $V$. We will consider this problem in general, for an arbitrary symmetric $V_{kl}$. That our $V_{kl}$ is given by eq. (D.19) will become important again only in section D.4.

There is a standard formula for $\psi_c$:

$$ \psi_c(X) = \sum_{G \in \text{connected graphs on } X, |G| = |X|} \prod_{kl \in G} (e^{V_{kl}} - 1) \prod_{k \in X} e^{\frac{1}{2} V_{kk}}. \quad (D.24) $$

But this is not very useful for our purposes: plugging it into (D.23) just gives back eq. (D.11) (perhaps not surprisingly as both (D.11) and (D.24) involve connected graphs; we leave the proof as an exercise.) The number of terms in (D.24) is asymptotically $2^{\binom{n}{2}}$, since for large $n$ almost all graphs on $n$ points are connected (e.g. [126], example II.15).

## D.3 Brydges-Battle-Federbush (BBF) formula

We will now derive a remarkable formula for $\psi_c(X)$ with much fewer terms. First two simplifying remarks: 1) The diagonal interactions $V_{kk}$ enter into $\psi_c(X)$ as a trivial common factor $\exp(\frac{1}{2} \sum_{k \in X} V_{kk})$. So we will set them to zero and reinstate in the final result. We consider a complete graph on $n$ points with pairs $kl$ as the graph edges $e$, and we write

$$ V(X) = \sum_{e} V(e). \quad (D.25) $$

2) It is enough to aim for the equation

$$ e^{V(X)} = \sum_{Y \ni 1} e^{V(X \setminus Y)} \psi_c(Y), \quad (D.26) $$

from which (D.22) follows by iterating. Every term in the r.h.s. has $Y$ and $X \setminus Y$ “decoupled” in the sense that no edges linking them are involved. Let us describe a general decoupling mechanism.

Let $F(X)$ be any sum of pairwise interactions $F(e)$, and $Z \subset X$ a subset we wish to decouple. We say that an edge $e$ “exits $Z$” (written $e \vdash Z$) if it is of the form $e = kl$ where $k \in Z, l \in X \setminus Z$. In other words, an edge exits $Z$ if it connects $Z$ to $X \setminus Z$. Introduce a variable $s \in [0, 1]$ and a new pairwise interaction with exiting edges rescaled by $s$, others left intact:

$$ F\left[\begin{array}{c} s \\ Z \end{array}\right](e) = \begin{cases} sF(e), & \text{if } e \vdash Z, \\ F(e) & \text{otherwise.} \end{cases} \quad (D.27) $$

We also define $F\left[\begin{array}{c} s \\ Z \end{array}\right](X)$ summing this new interaction:

$$ F\left[\begin{array}{c} s \\ Z \end{array}\right](X) = \sum_{e \in X} F\left[\begin{array}{c} s \\ Z \end{array}\right](e), \quad (D.28) $$
Then $F\begin{bmatrix} 1 \\ Z \end{bmatrix}(X) = F(X)$ and $F\begin{bmatrix} 0 \\ Z \end{bmatrix}(X) = F(X \setminus Z) + F(Z)$ is decoupled. Therefore we have

$$e^{F(X)} = e^{F\begin{bmatrix} 0 \\ Z \end{bmatrix}(X)} + \int_{0}^{1} ds \partial_{s} \exp F\begin{bmatrix} s \\ Z \end{bmatrix}(X) = e^{F(X \setminus Z)} e^{F(Z)} + \sum_{e \vdash Z} \int_{0}^{1} ds F(e) \exp F\begin{bmatrix} s \\ Z \end{bmatrix}(X). \quad (D.29)$$

The first term is decoupled. In the second one we have a sum over exiting edges. Below, each of these summands will be decoupled further with respect to the sets $Z \sqcup e$ obtained by joining to $Z$ the outside vertex of the edge $e$. ($\sqcup$ denotes “vertex union”).

Think of $\begin{bmatrix} s \\ Z \end{bmatrix}$ as an operation, which can be applied repeatedly. E.g. $F\begin{bmatrix} s_1 \\ Z_1 \end{bmatrix}\begin{bmatrix} s_2 \\ Z_2 \end{bmatrix}$ means first rescale by $s_1$ all edges $e_1 \vdash Z_1$, then by $s_2$ all edges $e_2 \vdash Z_2$ by $s_2$. In general these operations don't commute.

To derive (D.26), we first apply this general mechanism with $Z = \{1\}$ consisting of one point, and $F(X) = V(X)$. We obtain

$$e^{V(X)} = e^{V(X \setminus \{1\})} + \sum_{e_1 \vdash \{1\}} \int_{0}^{1} ds_1 V(e_1) \exp V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}(X). \quad (D.30)$$

The first term gives the first term in the r.h.s. of (D.26), with $Y = \{1\}$ and $\psi_c(Y) = 1$. For each term in the sum over $e_1$, we define the set $Z_{e_1} = \{1\} \sqcup e_1$ and decouple with respect to $Z_{e_1}$, i.e. apply (D.29) for $F = V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}$ and the rescaled interaction $F\begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix} = V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}\begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix}$:

$$e^{V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}(X)} = e^{V(X \setminus Z_{e_1})} e^{V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}(Z_{e_1})} + \sum_{e_2 \vdash Z_{e_1}} \int_{0}^{1} ds_2 V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}(e_2) \exp V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}\begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix}(X). \quad (D.31)$$

In the first term we used $F(X \setminus Z_{e_1}) = V(X \setminus Z_{e_1})$, since $F = V$ outside $Z_{e_1}$. Plugging (D.31) into (D.30) we get:

$$e^{V(X)} = e^{V(X \setminus \{1\})} + \sum_{e_1 \vdash \{1\}} e^{V(X \setminus Z_{e_1})} \int_{0}^{1} ds_1 V(e_1) \exp V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}(Z_{e_1}) + \sum_{e_1 \vdash \{1\}} \sum_{e_2 \vdash Z_{e_1}} \int_{0}^{1} ds_1 ds_2 V(e_1) V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}(e_2) \exp V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}\begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix}(X). \quad (D.32)$$

To compare with (D.26), we rewrite the second line as

$$\sum_{e_1 \vdash \{1\}} e^{V(X \setminus Z_{e_1})} \psi_c(Z_{e_1}), \quad \psi_c(Z_{e_1}) = \int_{0}^{1} ds_1 V(e_1) \exp V\begin{bmatrix} s_1 \\ \{1\} \end{bmatrix}(Z_{e_1}). \quad (D.33)$$

This equation defines $\psi_c$ for sets of two points.
We then continue iterating. For each term in the third line of (D.32), we define $Z_{e_1e_2} = Z_{e_1} \cup e_2$ and decouple with respect to this set. This allows us to rewrite the third line as:

$$ \sum_{e_1 \notin \{1\}} \sum_{e_2 \notin Z_{e_1}} e^{V(X \setminus Z_{e_1 e_2})} \int_0^1 ds_1 ds_2 V(e_1) V \begin{bmatrix} s_1 \\ \{1\} \end{bmatrix} (e_2) \exp V \begin{bmatrix} s_1 \\ \{1\} \end{bmatrix} \begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix} (Z_{e_1 e_2}), \quad (D.34) $$

plus a triple integral term which we don't write. To compare this with (D.26), we write it as

$$ \sum_{Z \ni \{1\}, |Z|=3} e^{V(X \setminus Z)} \psi_c(Z). \quad (D.35) $$

We should define $\psi_c(Z)$ summing over all possible orders of adding edges so that the final set $Z_{e_1e_2} = Z$, which means the two added edges should be picked from $Z$. We thus have

$$ \psi_c(Z) = \sum_{e_i \subset Z, e_1 \notin \{1\}, e_2 \notin Z_{e_1}} \int_0^1 ds_1 ds_2 V(e_1) V \begin{bmatrix} s_1 \\ \{1\} \end{bmatrix} (e_2) \exp V \begin{bmatrix} s_1 \\ \{1\} \end{bmatrix} \begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix} (Z). \quad (D.36) $$

Continuing to iterate, we obtain the following general formula for $|Z| = n$, $Z \ni \{1\}$, which involves $n-1$ added edges and integrations:

$$ \psi_c(Z) = \sum_{e_i \subset Z, e_1 \notin \{1\}, e_2 \notin Z_{e_1}, e_3 \notin Z_{e_1 e_2}, \dots} \quad (D.37) $$

$$ \times \int_0^1 \prod_{k=1}^{n-1} ds_k \left\{ V(e_1) V \begin{bmatrix} s_1 \\ \{1\} \end{bmatrix} (e_2) V \begin{bmatrix} s_1 \\ \{1\} \end{bmatrix} \begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix} (e_3) \cdots \right\} \exp V \begin{bmatrix} s_1 \\ \{1\} \end{bmatrix} \begin{bmatrix} s_2 \\ Z_{e_1} \end{bmatrix} \cdots \begin{bmatrix} s_{n-1} \\ Z_{e_1 e_2 \cdots e_{n-2}} \end{bmatrix} (Z). $$

This is valid under the simplifying assumption $V_{kk} = 0$ made above, while for general $V$ we must multiply by $\exp(\frac{1}{2} \sum_{k \in Z} V_{kk})$. Another remark: definition (D.21), (D.22) shows that $\psi_c(Z)$, like $\psi(Z)$, must be symmetric with respect to permutations. Eq. (D.37) is not manifestly symmetric as it selects $\{1\} \in Z$, although of course it produces symmetric results, see an example below. This lack of manifest symmetry will not be a problem in our applications.

**Example D.1** It's instructive to check this formula for three points: $Z = \{1, 2, 3\}$. We thus have three edges $e = 12, 13, 23$. The rescaled interactions are $(s_1 = s, s_2 = t)$:

$$ V \begin{bmatrix} s \\ \{1\} \end{bmatrix} : \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.5] \node (v1) at (0,1) {$sV_{13}$}; \node (v2) at (0,0) {$sV_{12}$}; \draw[->] (v1) -- (v2); \node[above] at (v1) {$3$}; \node[below] at (v2) {$2$}; \end{tikzpicture} \quad V \begin{bmatrix} s \\ \{1\} \end{bmatrix} \begin{bmatrix} t \\ \{12\} \end{bmatrix} : \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.5] \node (v1) at (0,1) {$tsV_{13}$}; \node (v2) at (0,0) {$tsV_{12}$}; \draw[->] (v1) -- (v2); \node[above] at (v1) {$3$}; \node[below] at (v2) {$2$}; \end{tikzpicture} \quad (D.38) $$

Thus the term in (D.36) corresponding to $e_1 = 12$ reads:

$$ \int_0^1 ds dt V_{12}(sV_{13} + V_{23}) e^{sV_{12} + tsV_{13} + tV_{23}}. \quad (D.39) $$

For $e_1 = 13$ we get the same integral with $2 \leftrightarrow 3$. Doing the two integrals and summing, we obtain the same expression as by expanding out (D.24):

$$ \psi_c(123) = 2 - e^{V_{12}} - e^{V_{13}} - e^{V_{23}} + e^{V_{12}+V_{13}+V_{23}}. \quad (D.40) $$
Continuing with the general case, it is not hard to write out all parts of (D.37) explicitly. Define a function $g: \{1, \dots, n\} \to \{1, \dots, n\}$ which tells us the order of adding vertices when adding edges $e_l$: $g(1) = 1$ and $g(l)$ is the endpoint of the edge $e_{l-1}$. So $e_{l-1} = g(k)g(l)$ with some $k < l$. In the exponential of (D.37) the contribution $V_e$ of this edge appears rescaled by

$$r_e = s_k s_{k+1} \cdots s_{l-1}, \qquad (D.41)$$

while in the prefactor $\{\dots\} V_{e_{l-1}}$ appears rescaled by

$$s_k s_{k+1} \cdots s_{l-2} \qquad (D.42)$$

(or 1 if $k = l - 1$). This expression is consistent with applying $\partial_{s_{l-1}}$ to (D.41), as it must be by (D.29).

It follows from this discussion that every summand in (D.37) can be written as

$$\prod_{k=1}^{n-1} V_{e_k} \times \int_0^1 \prod_{k=1}^{n-1} ds_i f(\mathbf{s}) \exp \sum_e r_e V_e, \qquad (D.43)$$

where $r_e$ are given by (D.41) and $f(\mathbf{s})$ is some product of $s$'s. Its dependence on the choice of added edges can be made completely explicit, but for us it will suffice to know that $f \ge 0$. Furthermore, we would like to view the integral in (D.43) as performed over $r_e$'s, not over $s$'s, writing it as

$$\prod_{k=1}^{n-1} V_{e_k} \times \int d\mu(\mathbf{r}) \exp \sum_e r_e V_e, \qquad (D.44)$$

where $d\mu(\mathbf{r})$ is some nonnegative measure, push-forward of the measure $\prod_{k=1}^{n-1} ds_k f(\mathbf{s})$ to $r_e$'s. This singular, delta-function-like, measure is concentrated on $r_e$'s which can be represented as (D.41).

For the final repackaging, consider the graph which is the union of all added edges:

$$T = e_1 \cup e_2 \cup \dots \cup e_{n-1}. \qquad (D.45)$$

By construction, this is a tree with $n$ vertices $\{1 \dots n\}$. Note that the same tree $T$ may appear from different sequences of edges, all satisfying the constraints $e_1 \vdash \{1\}, e_2 \vdash Z_{e_1}$, etc in (D.37). E.g. the following tree may arise from $\{e_1, e_2, e_3\} = \{13, 12, 24\}$ or $\{12, 13, 14\}$ or $\{12, 24, 13\}$:

$$\begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75, every node/.style={scale=0.75}] 
\node (1) at (0,0) {$1$};
\node (2) at (1,0) {$2$};
\node (3) at (2,0) {$3$};
\node (4) at (3,0) {$4$};
\draw[->] (1) -- (2);
\draw[->] (2) -- (3);
\draw[->] (3) -- (4);
\node at (1.2,0) {$V_{e_1}$};
\node at (2.2,0) {$V_{e_2}$};
\node at (3.2,0) {$V_{e_3}$};
\node at (4.2,0) {$V_{e_4}$};
\end{tikzpicture} \qquad (D.46)$$

Terms corresponding to the same tree $T$ have the same prefactor $\prod_{k=1}^{n-1} V_{e_k}$ in (D.44), although the measures $d\mu(\mathbf{r})$ will be different. Let us group all such terms into one term per tree $T$, summing their measures into some total measure. Reintroducing as well the trivial $\exp(\frac{1}{2} \sum V_{kk})$ factor, we obtain the remarkable BBF formula [127, 128]; our presentation followed [118].
**Lemma D.4 (BBF formula)** Let $T$ run through all trees with $n$ vertices $\{1...n\}$. There exist non-negative measures $d\mu_T(\mathbf{r})$ so that, for any pairwise interaction $V$:

$$ \psi_c(1...n) = \sum_T \prod_{e \in T} V_e \int d\mu_T(\mathbf{r}) \exp\left(\sum_e r_e V_e + \frac{1}{2} \sum_{k=1}^n V_{kk}\right). \quad (D.47) $$

In addition, these measures have the following two properties:

* For each $\mathbf{r}$ in the support of $d\mu_T(\mathbf{r})$, there exists a bijection $g: \{1,...,n\} \to \{1,...,n\}$ and a set of $n-1$ numbers $s_k \in [0,1]$ such that for all $k < l$

$$ r_{g(k)g(l)} = s_k s_{k+1} \cdots s_{l-1}. \quad (D.48) $$

* The measures $d\mu_T(\mathbf{r})$ are probability measures, i.e. they have total weight 1: $\int d\mu_T(\mathbf{r}) = 1$.

The first property is clear, since $d\mu_T(\mathbf{r})$ were obtained as sums of such measures. The second property can be checked by a trick: apply the general formula to a particular conveniently chosen potential $V$. Pick a tree $T$ and consider $V$ such that $V_e = \epsilon$ for $e \in T$ and $V = 0$ otherwise. That $\int d\mu_T(\mathbf{r}) = 1$ follows by comparing, for small $\epsilon$, the equations

$$ \psi_c(1...n) = \begin{cases} \epsilon^n \int d\mu_T(\mathbf{r}) + \text{higher order from (D.47)}, & \epsilon^n + \text{higher order from (D.24)}. \end{cases} \quad (D.49) $$

**Remark D.1** The measure $d\mu_T$ may be thought as having several components, corresponding to different bijections $g$, whose number equals the number of ways to grow the tree $T$ by adding edges. Each component can be pulled back to an integral over $s_k$'s, and the total weight of all components is 1. This can also be verified by an explicit computation (Lemma A.4 in [119] or Lemma 2.3 in [121]).

The number of terms in the BBF formula is much smaller than in the standard formula (D.24): it grows as the number of trees on $n$ points, which is $n^{n-2}$ (Cayley).

We will use the BBF formula to prove bounds on the connected fermionic expectations. One wonders if this formula can also be useful to evaluate connected expectations, e.g. performing the integral numerically, rather than just prove bounds. We are not aware of such applications (see also footnote 55).

## D.4 Gawedzki-Kupiainen-Lesniewski (GKL) bound

We will now present a bound on fermionic connected expectations due to Gawedzki and Kupiainen [129]. Its physical origin, like for the Gram-Hadamard bound (D.10), is the Pauli principle. Our exposition follows Lesniewski [117], who gave an elegant proof based on the BBF formula.<fnref n="56" />

<footnote n="56">⁵⁶Original proofs [123, 129] were based on an improved Gram-Hadamard inequality for the simple expectations, transferred to connected expectations via cluster expansion techniques. See also [75] for an alternative approach.</footnote>
We start with eq. (D.23) copied here for $Y = \{1, \dots, n\}$:

$$ \langle \Phi(\mathbf{A}_1, \mathbf{x}_1); \dots; \Phi(\mathbf{A}_n, \mathbf{x}_n) \rangle_c = \int \prod_{i \in T} d\eta_i d\bar{\eta}_j \psi_c(1 \dots n). \quad (D.50) $$

Recall that we introduced the potential function (D.17) (see (D.5) for matrix $M$) subdivided in symmetric pairwise interactions $V_{kl}$, $1 \le k, l \le n$, see (D.18), (D.19). The BBF formula (D.47) gives a general expression for $\psi_c(1 \dots n)$ in terms of $V_{kl}$.

What happens with every term in the BBF formula when we plug in expressions (D.19) for $V_{kl}$ and do the Grassmann integral? First of all let us look at the prefactor $\prod_{e \in T} V_e$. By (D.19), every $V_e = V_{kl}$ is a sum of terms $M_{ij}\eta_i\bar{\eta}_j$ where $x_i \in \mathbf{x}_k$, $y_j \in \mathbf{x}_l$ or vice versa and $M_{ij}$ is a propagator, see (D.5). We choose in each $V_e$, $e \in T$, one of such possible propagator terms. The graph with all chosen propagators as edges is called an “anchored tree $\mathcal{T}$ on the $n$ groups of points $\mathbf{x}_i$”. When each group of points $\mathbf{x}_i$ is contracted to one points, the anchored tree $\mathcal{T}$ becomes a tree (the tree $T$ in the case at hand). We will say that “$\mathcal{T}$ comes from $T$”.

Let us look at a term corresponding to a fixed anchored tree $\mathcal{T}$ coming from $T$. The variables $\eta$ and $\bar{\eta}$ of the vertices along the anchored tree are saturated by the prefactor. Doing the Grassmann integral over these “tree-saturated” variables gives a product of $n-1$ propagators along $\mathcal{T}$:

$$ \prod_{\substack{\mathcal{T} \\ \text{along } \mathcal{T}}} \Gamma_{ij}(x_i - y_j). \quad (D.51) $$

We are left with the integral of the exponent over the remaining $m-2(n-1)$ variables.<fnref n="57" /> (By the rules of Grassmann integration, the tree-saturated variables can now be set to zero in the exponent.) The potential function in the exponent of (D.47) will have the form

$$ \sum_{ij} \mathcal{N}_{ij} \eta_i \bar{\eta}_j, \quad \mathcal{N}_{ij} = r_{k(i)k(j)} \mathcal{M}_{ij}, \quad (D.52) $$

summing over the remaining variables, so $\mathcal{N} = \mathcal{N}(\mathbf{r})$ is an $s \times s$ matrix with $s = \frac{1}{2}(m-2(n-1))$. If variables $i, j$ belong to two different groups of points $\mathbf{x}_{k(i)}, \mathbf{x}_{k(j)}$, then $r_{k(i)k(j)} = r_e$ in (D.52), where $e = k(i)k(j)$ is the edge of the tree $T$, progenitor of the anchored tree, and $r_e$ is the rescaling factor in the BBF formula. Terms from the same group, coming from $\frac{1}{2} \sum V_{kk}$ in (D.47), should not be rescaled: so we set $r_{kk} = 1$. The Grassmann integral over the remaining variables is then the determinant of the so defined matrix $\mathcal{N}$. To summarize, the connected expectation (D.50) can be represented as

$$ \sum_{\mathcal{T}} \prod_{\substack{\mathcal{T} \\ \text{along } \mathcal{T}}} \Gamma_{ij}(x_i - y_j) \int d\mu_T(\mathbf{r}) \det \mathcal{N}. \quad (D.53) $$

Aiming to bound $\det \mathcal{N}$ by the Gram-Hadamard inequality (D.6), we wish to represent $\mathcal{N}_{ij}$ as a product of vectors in a Hilbert space. For $M_{ij}$ such a representation was given in section D.1. To deal with the extra factor $r_{k(i)k(j)}$ we will use

<footnote n="57">⁵⁷Recall that $m = \sum_{k=1}^n |\mathbf{x}_k|$ is the total number of points, and we have one Grassmann variable per point.</footnote>
**Lemma D.5** Let $s_k \in [0, 1], k = 1...n-1$. There exist n unit vectors $u_k = u_k(s) \in \mathbb{R}^n$ such that $(u_k, u_l) = s_k...s_{l-1}$ for all $1 \le k < l \le n$.

*Proof.* Let $v_k$ be the standard orthonormal basis in $\mathbb{R}^n$. We put $u_1 = v_1$. Take $u_2$ the unit-length linear combination of $u_1$ and $v_2$ which has projection $s_1 u_1$ on $V_1 = \text{span}(v_1)$, explicitly $u_2 = s_1 u_1 + (1 - s_1^2)^{1/2} v_2$. Take $u_3$ the unit-length linear combination of $u_2$ and $v_3$ which has projection $s_2 u_2$ on $V_2 = \text{span}(v_1, v_2)$, explicitly $u_3 = s_2 u_2 + (1 - s_2^2)^{1/2} v_3$. Continuing in this fashion, we end up with a sequence of unit vectors $u_k \in V_k$ whose orthogonal projections on the previous $V_{k-1}$ are

$$P_{V_{k-1}}(u_k) = s_{k-1}u_{k-1}. \qquad (D.54)$$

Computing $(u_k, u_l)$, $k < l$, via orthogonal projections $u_l \to V_{l-1} \to V_{l-2}...$ gives precisely $s_k...s_{l-1}$. Q.E.D.

Now for any component of $d\mu_T$ measure, i.e. one particular bijection $g$ in the BBF formula, we satisfy (D.48) via

$$r_{kl} = (u_{g^{-1}(k)}, u_{g^{-1}(l)}), \qquad (D.55)$$

which is also symmetric in $k, l$ and $r_{kk} = 1$. Finally the rescaling factor in the $\mathcal{N}$ matrix:

$$r_{k(i)k(j)} = (u_{g^{-1}(k(i))}, u_{g^{-1}(k(j))}). \qquad (D.56)$$

In section D.1 we showed that $M_{ij} = (f_i, h_j)$ where $f, h$ are vectors in a Hilbert space. Considering the tensor product of those vectors with $u_{g^{-1}(k(i))} \in \mathbb{R}^n$, we obtain an inner product representation for $N_{ij}$ elements. Since $u$'s have unit length, by the same argument which led to (D.10) we obtain a bound with the same constant $C_{GH}$:

$$|\det \mathcal{N}| \le (C_{GH})^s, \quad s = \frac{1}{2}(m - 2(n - 1)). \qquad (D.57)$$

This is true for any $r_e$ lying in the support of the measure $d\mu_T$. We can also integrate this bound since $d\mu_T$ has weight 1. We conclude that the connected expectation is bounded by

$$(C_{GH})^s \sum_T \sum_{\mathcal{T} \text{ comes from } T} |\text{eq. (D.51)}| = (C_{GH})^s \sum_{\mathcal{T}} |\text{eq. (D.51)}|. \qquad (D.58)$$

where we used that every anchored tree comes from one and only one tree. We finally get:

**Lemma D.6 (GKL bound)** Fermionic connected expectations are bounded by

$$|\langle \Phi(\mathbf{A}_1, \mathbf{x}_1); \dots; \Phi(\mathbf{A}_n, \mathbf{x}_n) \rangle_c| \le (C_{GH})^s \sum_{\mathcal{T}} \prod_{\text{along } \mathcal{T}} |\Gamma_{ij}(x_i - y_j)|, \qquad (D.59)$$

where $s = \frac{1}{2} \sum_{i=1}^n |\mathbf{x}_i| - (n-1)$, $C_{GH}$ is from (D.10), the sum is over all anchored trees $\mathcal{T}$ on n groups of points $\mathbf{x}_k$, and the product of propagators is along $\mathcal{T}$.

The $\Gamma_{ij}$ here are either propagators, or their first derivatives with respect to $x_i$ and/or $y_j$. Since we are assuming (4.15), we can replace $|\Gamma_{ij}(x_i - y_j)|$ by $M(x_i - y_j)$ in the r.h.s. of the bound. See also figure 7 for an illustration. For $n = 1$ the GKL bound reduces to the Gram-Hadamard bound (D.10).
<figure><img src="image_8.png" /><figcaption>**Figure 7.** This illustrates the $n = 3$ case of the connected expectation. (a) Three groups of points. (b) A particular connected Wick contraction. (c) Red: an anchored tree consisting of $n-1$ propagators. Blue: remaining $s$ propagators.</figcaption></figure>

### D.5 Bound on the number of anchored trees

We will prove that the number of anchored trees on $n$ groups of points $\mathbf{x}_i$ is bounded by

$$N_T \le n! 4 \sum_{i=1}^{n} |\mathbf{x}_i|. \qquad (D.60)$$

Recall that an anchored tree is a graph which becomes a tree when each group $\mathbf{x}_i$ is collapsed to a point. The graphs are labeled (i.e. the vertices are distinguishable).

By Cayley's formula, the number of labeled trees $T$ on $n$ points is $n^{n-2}$. Its proof via Prüfer sequences [130] simultaneously gives a finer result: the number of labeled trees specifying degrees $d_k$ of each vertex $k$ is

$$\frac{(n-1)!}{(d_1-1)! \cdots (d_n-1)!} \qquad (D.61)$$

To get an anchored tree $\mathcal{T}$ out of $T$, we choose a propagator for each edge $e = kl$. There are at most $m_k m_l$ choices per edge, with $m_k = |\mathbf{x}_k|$ the number of points in the group $k$, so at most $\prod_{k=1}^n (m_k)^{d_k}$ choices in total. Multiplying by (D.61) and summing over all possible degrees, gives an upper bound on the number of anchored trees:

$$N_T \le (n-1)! \sum_{(d_k)_k^n} \prod_{k=1}^n \frac{(m_k)^{d_k}}{(d_k-1)!} \qquad (D.62)$$

Summing over each $d_k$ independently from 1 to $\infty$ (which is an overestimate since e.g. degrees are limited from above by $n-1$), results in a further bound

$$N_T \le (n-1)! \prod_{k=1}^{n} F(m_k), \qquad F(x) = \sum_{p=1}^{\infty} \frac{x^p}{(p-1)!} = xe^x. \qquad (D.63)$$

We have $F(x) \le B^x$ with $B = e^{1+1/e} \approx 3.93$, and so (D.60) follows. For a proof not relying on (D.61) see [121], Lemma 2.4.

### E Proof of $S_l^{\ell_1, \dots, \ell_n}$ norm bound

Our goal here is to prove the bound (5.38). Let $h_i \in H_{\ell_i}$ and denote $\tilde{H}_l = S_l^{\ell_1, \dots, \ell_n}$ ($h_1, \dots, h_n \in B_l$). Unpacking the definition in section 5.1, the kernels of $\tilde{H}_l$ are

$$\tilde{H}_l(\mathbf{B}, \mathbf{x}_B) = A \frac{1}{n!} \sum_{\substack{\mathbf{B}_1, \dots, \mathbf{B}_n \\ \sum \mathbf{B}_i = \mathbf{B}}} \sum_{\substack{\mathbf{A}_1, \dots, \mathbf{A}_n \\ \mathbf{A}_i \supset \mathbf{B}_i}} (-)^{\# K_{(\mathbf{B}_i, \mathbf{A}_i)_1^n}} (\mathbf{x}_B), \qquad (E.1)$$

$$K_{(\mathbf{B}_i, \mathbf{A}_i)_1^n} (\mathbf{x}_B) = \int d^d \mathbf{x}_{\overline{\mathbf{B}}} C (\mathbf{x}_{\overline{\mathbf{B}}}) \prod_{i=1}^{n} h_i (\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}).$$
Let us count the terms in the sum (E.1) corresponding to a fixed **B** and fixed lengths $|\mathbf{A}_i| = l_i$. It is easy to see that these terms are in one-to-one correspondence with sequences **R** of total length $l_1 + l_2 + \cdots + l_n$ extending the sequence **B**. [Every such sequence **R** can be cut into sequences $\mathbf{A}_1, \dots, \mathbf{A}_n$, uniquely since the lengths $|\mathbf{A}_i|$ are kept fixed, and then $\mathbf{B}_i$ can be extracted, uniquely, as the part of **B** falling into $\mathbf{A}_i$.] It follows that the number of terms in (E.1) is bounded by:

$$ \text{Number of terms in (E.1)} = \left(\sum l_i\right) (Nd+N) \sum l_i - l \le 2 \sum l_i \times (Nd+N) \sum l_i, \quad (\text{E.2}) $$

where in the first equality $Nd+N$ is the maximal number of choices for every element of the sequence $\mathbf{R} \setminus \mathbf{B}$, see (4.1), assuming that they are assigned independently. [This counting does not take into account that some of these terms would vanish by constraints imposed by the rotation and $\text{Sp}(N)$ invariances.]

We next pick some $\mathbf{B}_i$, $\mathbf{A}_i$ and consider the corresponding term $K(\mathbf{x}_{\mathbf{B}}) = K_{(\mathbf{B}_i, \mathbf{A}_i)_1^n}(\mathbf{x}_{\mathbf{B}})$ in (E.1). Its integration kernel $C(\mathbf{x}_{\bar{\mathbf{B}}})$, eq. (5.4), is bounded by the GKL bound (D.59) with $\mathbf{x}_i = \mathbf{x}_{\bar{\mathbf{B}}_i}$:

$$ |C(\mathbf{x}_{\bar{\mathbf{B}}})| = \left|\left\langle \Phi(\bar{\mathbf{B}}_1, \mathbf{x}_{\bar{\mathbf{B}}_1}); \dots; \Phi(\bar{\mathbf{B}}_n, \mathbf{x}_{\bar{\mathbf{B}}_n}) \right\rangle_c\right| \le (C_{\text{GH}})^s \sum_{\mathcal{T}} \prod_{(xx') \in \mathcal{T}} M(x-x'), \quad (\text{E.3}) $$

where $s = \frac{1}{2} \sum |\bar{\mathbf{B}}_i| - (n-1) \le \frac{1}{2} \sum l_i$. We wish to bound the norm of $K(\mathbf{x}_{\mathbf{B}})$:

$$ \|K\|_w = \int_{x_1=0} d^d\mathbf{x}_{\mathbf{B}} |K(\mathbf{x}_{\mathbf{B}})| w(\mathbf{x}_{\mathbf{B}}) \quad (\text{E.4}) $$

with the weight function (4.16). Let $\mathcal{T}$ be any anchored tree in (E.3). If $\tau_i$ are trees connecting points of $\mathbf{x}_{\mathbf{A}_i}$, then $\mathcal{T} \cup \tau_1 \cup \dots \cup \tau_n$ connects points in $\mathbf{x}_{\mathbf{B}}$. Therefore, we have a bound:

$$ \text{St}(\mathbf{x}_{\mathbf{B}}) \le \sum_{i=1}^{n} \text{St}(\mathbf{x}_{\mathbf{A}_i}) + \prod_{(xx') \in \mathcal{T}} |x - x'|. \quad (\text{E.5}) $$

Raising this to the power $\sigma$ and using the elementary inequality

$$ (\sum p_i)^\sigma \le \sum p_i^\sigma \quad (p_i \ge 0, 0 < \sigma \le 1), \quad (\text{E.6}) $$

we conclude

$$ w(\mathbf{x}_{\mathbf{B}}) \le \prod_{i=1}^{n} w(\mathbf{x}_{\mathbf{A}_i}) \prod_{(xx') \in \mathcal{T}} w(\{x, x'\}). \quad (\text{E.7}) $$

Using in (E.4) this bound, the definition of $K(\mathbf{x}_{\mathbf{B}})$, and the bound (E.3), we get

$$ \begin{aligned} \|K\|_w &\le (C_{\text{GH}})^s \sum_{\mathcal{T}} \int_{x_1=0} d^d\mathbf{x}_{\mathbf{B}\cup\bar{\mathbf{B}}} \prod_{(xx') \in \mathcal{T}} M(x-x')w(\{x, x'\}) \prod_{i=1}^n h_i(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})w(\mathbf{x}_{\mathbf{A}_i}) \\ &= (C_{\text{GH}})^s N_{\mathcal{T}} \|M\|_w^{n-1} \prod_{i=1}^n \|h_i\|_w. \end{aligned} \quad (\text{E.8}) $$
The latter equality is shown via an “amputating tree leaves” argument. Given an anchored tree $\mathcal{T}$, we can find a leaf: a group of points $x_{\overline{B}_k} \subset x_{A_k}$ connected to the rest by just one edge of $\mathcal{T}$, call it $(zz')$ where $z \in x_{A_k}$. Amputating the leaf consists of two steps. First, keep $z$ fixed and integrate over all the other leaf points, which gives a factor $\|h_k\|_w$. Second, integrate over $z$ (keeping $z'$ fixed), which gives a factor $\|M\|_w$. Then find the next leaf and continue the amputation. We get the same result for each $\mathcal{T}$. By appendix D.5, the number of anchored trees

$$N_{\mathcal{T}} \le n! 4^{\sum |x_{\overline{B}_i}|} = n! 4 \sum l_i - l \le n! 4 \sum l_i.$$

To summarize, the number of terms in (E.1) is bounded by (E.2), the norm of each individual term by (E.8), $s$ by $\frac{1}{2} \sum l_i$ and $N_{\mathcal{T}}$ by $n! 4 \sum l_i$. The latter $n!$ cancels with $\frac{1}{n!}$ in (E.1). A further useful fact is that the antisymmetrization operator $\mathcal{A}$ (footnote 15) does not increase the norm (it averages over all permutations with signs, and the norm is defined as the maximum over all permutations). Taking all of these into account, we get a bound

$$\| \tilde{H}_l \|_w \le C_{\gamma}^{n-1} \prod_{i=1}^{n} C_0^{l_i} \| h_i \|_w \quad (E.9)$$

with $C_{\gamma} = \|M\|_w$, $C_0 = 8(Nd + N)\sqrt{C_{GH}}$. This is the bound (5.38) in the case $\sum l_i \ge l + 2(n-1)$. In the opposite case $S_l^{\ell_1, \dots, \ell_n}$ vanishes (section 5.1), so there is nothing to prove.

**Remark E.1** We can see how some of the above steps justify choices made in the main text. The Steiner diameter is tailor-made for (E.5). Stretched exponential weight is handy because of (E.6). The $L_1$ norm (as opposed to any other $L_p$) works great when recursively amputating tree leaves.

## F Estimates of $\Delta_k^{(1)}, \Delta_k^{(2)}$

In this appendix we prove Lemma 7.2. To estimate $\Delta_k^{(1)}$, we use $b_k \le A\delta^k$ for $k \ge 1$, sum the geometric progression, and use $C\delta \le 1/2$, which gives what we need:

$$\Delta_k^{(1)} \le C^{k+2} A \delta^{k+1} \frac{1}{1 - C\delta} \le 2C^{k+2} A \delta^{k+1} \quad (k \ge 0). \quad (F.1)$$

The estimate of $\Delta_k^{(2)}$ is more subtle and will require several steps. First we introduce the tool of **extending by zeros**. For any sequence $\varkappa = (k_i)_1^n$, we say that a sequence $\varkappa'$ “extends $\varkappa$ by $m$ zeros” if it is obtained from $\varkappa$ inserting $m$ zeros in arbitrary places. This definition is also valid if $\varkappa$ itself already contains some zeros (this will be useful). For $m = 0$ we have $\varkappa' = \varkappa$. E.g. (0,2,0) extends (2) by 2 zeros, and (0,1,0,1) extends (0, 1, 1) by 1 zero.

We define $F_{\text{ext}}[\varkappa]$ as the sum of $F[\varkappa']$ over all $\varkappa'$ extending $\varkappa$ by an arbitrary number of zeros $m \ge 0$:

$$F_{\text{ext}}[\varkappa] = \sum_{m=0}^{\infty} \sum_{\varkappa' : \text{extends } \varkappa \text{ by } m \text{ zeros}} F[\varkappa'] \quad (F.2)$$
For a fixed $m$, the number of sequences $\varkappa'$ is $\le (m+n/m)$ where $n$ is the length of $\varkappa$. $[=(m+n/m)$ if the original sequence $\varkappa$ does not have zeros.] Since they all have $F[\varkappa'] = F[\varkappa](C_\gamma C b_0)^m$, we obtain

$$F_{\text{ext}[\varkappa]} \le F[\varkappa] \sum_{m=0}^{\infty} \binom{m+n}{m} (C_\gamma C b_0)^m = \frac{1}{(1 - C_\gamma C b_0)^{n+1}} F[\varkappa].$$

Recall $b_0 \le A\delta$. Thus using $C_\gamma C A\delta \le 1/2$ we have

$$F_{\text{ext}[\varkappa]} \le 2^{n(\varkappa)+1} F[\varkappa]. \qquad (F.3)$$

Next we will sum over sequences with a fixed $\sum k_i$. Namely we define

$$\Phi_k = \sum_{n=2}^{\infty} \sum_{(k_i)_1^n, \sum k_i=k} F[(k_i)_1^n] \quad (k \ge 0) \qquad (F.4)$$

We first estimate $\Phi_k$'s and then convert into the estimate for $\Delta_k^{(2)} = \sum_{k'=k}^{\infty} \Phi_{k'}$.

Consider first $k \ge 2$ (see below for the simpler $k = 0, 1$). We can obtain all sequences in $\Phi_k$ extending by zeros sequences which satisfy $\sum k_i = k$, $k_i \ge 1$ (and whose length is therefore at most $k$). Thus

$$\Phi_k \le \sum_{n=1}^{k} \sum_{(k_i)_1^n, \sum k_i=k} F_{\text{ext}}[(k_i)_1^n] \le \sum_{n=1}^{k} 2^{n+1} \sum_{(k_i)_1^n, \sum k_i=k} F[(k_i)_1^n], \qquad (F.5)$$

where we used (F.3). Note that although $n \ge 2$ in (F.4), we need $n \ge 1$ in (F.5) to include the one-term sequence ($k$) whose two-term extensions ($k, 0$) and ($0, k$) appear in (F.4).$^{58}$

Using $b_{k_i} \le A\delta^{k_i}$ and $\sum k_i = k$ we have, for sequences with all $k_i \ge 1$,

$$F[(k_i)_1^n] \le (C_\gamma)^{n-1} \prod_{i=1}^{n} C^{k_i+1} A \delta^{k_i} = (C_\gamma)^{n-1} C^{k+n} A^n \delta^k = (C_\gamma C A)^{n-1} C^{k+1} A \delta^k. \qquad (F.6)$$

Finally, by elementary combinatorics the number of $n$-term sequences $(k_i)_1^n$ satisfying $\sum k_i = k, k_i \ge 1$ is $\binom{k-1}{n-1}$. Plugging all this information into (F.5), we have

$$\Phi_k \le 4C^{k+1} A \delta^k \sum_{n=1}^{k} \binom{k-1}{n-1} (2C_\gamma C A)^{n-1} = 4C^{k+1} A \delta^k (1 + 2C_\gamma C A)^{k-1}, \qquad (F.7)$$

which implies

$$\Phi_k \le 2C(2C\delta)^k A \quad (k \ge 2). \qquad (F.8)$$

once we use

$$C_\gamma C A \le 1/2. \qquad (F.9)$$

For future use, let us derive a bound on $F[(k_i)]$, $F_{\text{ext}}[(k_i)]$ under the condition (F.9), this time allowing for $k_i \ge 0$. Using $b_k \le A\delta^{\max(k,1)}$ we then have the same bound (F.6) but

<footnote>$^{58}$Their contribution is suppressed by $b_0$ compared to ($k$), but taking this suppression into account does not lead to a better estimate because of other terms present in (F.5).</footnote>
with an extra factor $\delta^m$ where $m$ is the number of zeros in the sequence. Using (F.9), (F.3) we have

$$F[(k_i)_{i=1}^n] \le 2^{1-n} C^{k+1} A \delta^{k+m} \quad (\sum k_i = k, k_i \ge 0, m = \#\{k_i = 0\}), \quad (F.10)$$

$$F_{\text{ext}}[(k_i)_1^n] \le 4C^{k+1}A\delta^{k+m}. \quad (F.11)$$

Finally let us bound $\Phi_k$ for $k = 0, 1$. For $k = 0$, eq. (F.4) involves the sequence $(0, 0)$ and its extensions by zeros. We then have by (F.11):

$$\Phi_0 = F_{\text{ext}}[(0,0)] \le 4CA\delta^2. \quad (F.12)$$

For $k = 1$, eq. (F.4) involves the sequences $(1,0)$, $(0,1)$ and their extensions by zeros. By (F.11):

$$\Phi_1 = 2F_{\text{ext}}[(1,0)] \le 8C^2 A \delta^2. \quad (F.13)$$

Finally we estimate $\Delta_k^{(2)} = \sum_{k'=k}^{\infty} \Phi_{k'}$ summing (F.8) in geometric progression, which is possible since $2C\delta \le 1/2$, and adding (F.12), (F.13) when needed. We thus obtain:

$$\Delta_k^{(2)} \le C(2C\delta)^k A \quad (k \ge 2), \quad (F.14)$$

$$\Delta_1^{(2)} = \Phi_1 + \Delta_2^{(2)} \le (8C^2 + 16C^3)A\delta^2,$$

$$\Delta_0^{(2)} = \Phi_0 + \Phi_1 + \Delta_2^{(2)} \le (4C + 8C^2 + 16C^3)A\delta^2.$$

## G One-loop coefficients $I_1$ and $I_2$

Here we evaluate the coefficients $I_1$, $I_2$ in the fixed-point equations from section 5.5, and prove their needed properties. Thinking in momentum space, computing the contributions to the effective $\nu$ and $\lambda$ shown in (5.31) involves setting to zero the external momenta in the corresponding Feynman diagrams (see footnote 26). To compute $I_1$ we consider the one-loop diagram $\mathcal{O}$, where the vertex is the local quartic coupling and the field propagating in the loop is the fluctuating component $\varphi$. This gives

$$I_1 = 2(N-2) \int \frac{d^d k}{(2\pi)^d} \frac{\rho(k)}{|k|^{d/2+\epsilon}}, \quad (G.1)$$

where we denoted $\rho(k) = \chi(k) - \chi(\gamma k)$. To work out the combinatorial prefactor $2(N-2)$, rewrite the local quartic as $Q(\psi) = \frac{1}{3}q_{abcd}\psi_a\psi_b\psi_c\psi_d$ (integration over $x$ understood) with $q_{abcd} = \Omega_{ab}\Omega_{cd} - \Omega_{ac}\Omega_{bd} + \Omega_{ad}\Omega_{bc}$ totally antisymmetric. Then $Q(\psi + \phi)$ contains the quadratic in $\phi$ term $2q_{abcd}\psi_a\psi_b\phi_c\phi_d$, from where we get the term $2q_{abcd}\Omega_{cd}\psi_a\psi_b = 2(N-2)\Omega_{ab}\psi_a\psi_b$ in the effective action (times the $k$-integral). Identities $\Omega_{ac}\Omega_{bc} = \delta_{ab}$, $\Omega_{ab}\Omega_{ab} = N$ are helpful.

Similarly, to compute $I_2$ we consider the diagrams $\mathcal{O} \sim \mathcal{O}(k^2)$ (with two local quartic vertices and two $\phi$-propagators) and $\mathcal{O} \sim \mathcal{O}(k^2)$ (where we start with the diagram $\mathcal{O} \sim \mathcal{O}(k^2)$ describing the $H_{6SL}$ interaction, with the wavy line denoting $\hat{x}_*(k)$, and contract two out of six external vertices
with the $\varphi$-propagator). Their contributions to $I_2$ are given by

$$ 
\begin{aligned}
I_2^{\varphi} &= -4(N-8) \int \frac{d^d k}{(2\pi)^d} \frac{\rho(k)^2}{|k|^{d+2\varepsilon}}, \\
I_2^{\mathcal{X}} &= -8(N-8) \int \frac{d^d k}{(2\pi)^d} \frac{\rho(k) R(k)}{|k|^{d+2\varepsilon}},
\end{aligned} 
\qquad (G.2) 
$$

where we wrote the Fourier transform of $\mathcal{X}_*(x)$ as

$$ \hat{\mathcal{X}}_*(k) = -8\lambda^2 \frac{R(k)}{|k|^{\frac{d}{2}+\varepsilon}}, \qquad R(k) = \sum_{n=1}^{\infty} \gamma^{4\varepsilon n} \rho(k/\gamma^n). \qquad (G.3) $$

To compute the prefactor in $I_2^{\varphi}$, we use $Q(\psi + \phi) = 2q_{abcd}\psi_a\psi_b\phi_c\phi_d$ and the identity

$$ q_{abcd}q_{a'b'c'd'}\Omega_{cc'}\Omega_{dd'} = (N-4)\Omega_{ab}\Omega_{a'b'} + 2\Omega_{aa'}\Omega_{bb'} + 2\Omega_{ab'}\Omega_{a'b}. \qquad (G.4) $$

The prefactor in $I_2^{\mathcal{X}}$ can be computed similarly.$^{59}$ Thus

$$ I_2 = I_2^{\varphi} + I_2^{\mathcal{X}} = -4(N-8) \int \frac{d^d k \rho(k)^2 + 2\rho(k)R(k)}{(2\pi)^d |k|^{d+2\varepsilon}}. \qquad (G.5) $$

Now that we computed $I_1$ and $I_2$, let us discuss their properties. $I_1 \propto N - 2$ is explained by the fact that for $N = 2$ the quartic interaction vanishes (see the Introduction). That $I_2 \neq 0$ for $N = 2$ is not a contradiction, since for a vanishing quartic interaction the change in $\lambda$ is anyway unphysical.

Note that perturbative beta-functions of symplectic fermion models can be obtained from the beta-functions of bosonic $O(N_b)$ models by setting formally $N_b = -N$. This is valid for local models [44, 48] and extends to nonlocal (long-range) models considered here. The $(N-8)$ factor in $I_2$ is thus related to the well-known $(N_b + 8)$ factor in the bosonic $O(N_b)$ model one-loop beta-function [131]. The long-range bosonic $O(N_b)$ beta function is known at three loops [78].$^{60}$ The two-loop term in [78], (3.16) is proportional to $5N_b + 22$ and does not vanish for $N_b \to -8$.

Therefore, vanishing of $I_2$ for $N = 8$ is an accident unrelated to any symmetry, which does not repeat in higher orders. With vanishing $\lambda^2$ and nonzero $\lambda^3$ term in $\lambda'$, there will be a perturbative fixed point with $\lambda = O(\sqrt{\varepsilon})$ at $N = 8$, and it should be possible to justify its existence non-perturbatively (see section 8.1.1). In this paper we stick to the generic case $N \neq 8$.

Because the integrals in (G.1) and (G.5) are cutoff at both UV and IR momenta, $I_1$ and $I_2$ depend analytically on $\varepsilon$. At small $\varepsilon$ they behave as follows:

<footnote>$^{59}$The following argument explains why it is the double of $I_2^{\varphi}$. Let $Q_a = (\delta/\delta\psi_a)Q$, $Q_{ab} = (\delta^2/\delta\psi_a\delta\psi_b)Q$ be the first and second functional derivatives of $Q(\psi)$ at $\psi = 0$. The $\varphi$ diagram is $-\frac{1}{4}Q_{ab}Q_{cd}\Omega_{ac}\Omega_{bd}$ times the loop integral. The $\mathcal{X}$-term can be written as $-\frac{1}{2} \int d^d x d^d y Q_a(x)Q_c(y)\Omega_{ac}\mathcal{X}(x-y)$. It is then clear that the $I_2^{\mathcal{X}}$ diagram gives $-\frac{1}{2}Q_{ab}Q_{cd}\Omega_{ac}\Omega_{bd}$ times the corresponding loop integral.</footnote>

<footnote>$^{60}$From the two-loop level it differs from the local bosonic $O(N_b)$ beta function (e.g. [132], (11.98)). Even the dependence on $N_b$ is different, due to the absence of wavefunction renormalization diagrams in the nonlocal model.</footnote>
**Lemma G.1** We have, for $|\epsilon| \le 1/\log \gamma$,

$$I_1 = 2(N-2) \left[ (1 - \gamma^{-d/2}) \int \frac{d^d k}{(2\pi)^d} \frac{\chi(k)}{|k|^{d/2}} + O(\epsilon \log \gamma) \right], \quad (G.6)$$

$$I_2 = -4(N-8) \left[ \frac{S_d}{(2\pi)^d} \log \gamma + O(\epsilon(\log \gamma)^2) \right], \quad (G.7)$$

where $S_d$ is the area of the unit sphere in $\mathbb{R}^d$ and the constants in Oare $\gamma$- and $\epsilon$-independent.

Note that $I_2$ is $\chi$-independent as $\epsilon \to 0$, even though $I_2^{\infty}$ and $I_2^x$ separately depend on $\chi$. This is not accidental, see remark 6.5.

*Proof.* The $|k|^{-d/2-\epsilon}$ factor in the integrand in (G.1) can be estimated as follows:

$$\frac{1}{|k|^{d/2+\epsilon}} = \frac{1}{|k|^{d/2}} + \frac{|k|^{-\epsilon}-1}{|k|^{d/2}} = \frac{1}{|k|^{d/2}} + \frac{O(\epsilon \log \gamma)}{|k|^{d/2}} \quad (1/(2\gamma) \le |k| \le 1). \quad (G.8)$$

The integral in (G.1) is thus given, modulo $O(\epsilon \log \gamma)$ error, by

$$\int \frac{d^d k}{(2\pi)^d} \frac{\rho(k)}{|k|^{d/2}} = (1 - \gamma^{-d/2}) \int \frac{d^d k}{(2\pi)^d} \frac{\chi(k)}{|k|^{d/2}}, \quad (G.9)$$

thus proving (G.6). Let us next prove (G.7). Recall that $\gamma \ge 2$. We have

$$\begin{gathered} \supp \rho \subset \{1/(2\gamma) \le |k| \le 1\}, \\ \rho \equiv 1 \quad \text{on} \quad \{1/\gamma \le |k| \le 1/2\}. \end{gathered} \quad (G.10)$$

From here it follows that $R(k) = \gamma^{4\epsilon}\rho(k/\gamma) = \gamma^{4\epsilon}(1 - \chi(k))$ on the support of $\rho$. We then rewrite the numerator of the integrand in (G.5) as:

$$\begin{aligned} \rho(k)^2 + 2\rho(k)R(k) &= \rho(k)^2 + 2\rho(k)(1 - \chi(k))[1 + (\gamma^{4\epsilon} - 1)] \\ &= \rho(k)^2 + 2\rho(k)(1 - \chi(k)) + O(\epsilon \log \gamma), \\ &= F(k) - F(\gamma k) + O(\epsilon \log \gamma), \quad F(k) = 2\chi(k) - \chi(k)^2, \end{aligned} \quad (G.11)$$

and the $|k|^{-d-2\epsilon}$ factor similarly to (G.8). Collecting the error terms, the integral in (G.5) is

$$\int \frac{d^d k}{(2\pi)^d} \frac{F(k) - F(\gamma k)}{|k|^d} + O(\epsilon(\log \gamma)^2). \quad (G.12)$$

Like $\chi(k)$, the function $F(k) \equiv 1$ for $|k| \le 1/2$ and vanishes for $|k| \ge 1$. The integral in (G.12) can now be computed:

$$\int \frac{d^d k}{(2\pi)^d} \frac{F(k) - F(\gamma k)}{|k|^d} = \frac{S_d}{(2\pi)^d} \log \gamma, \quad (G.13)$$

e.g. by separating the integration region into the region close to the origin plus the rest, and using the properties of $F(k)$ given above. In particular, the answer is $\chi$-independent. Q.E.D.
## H Finite volume and non-perturbative validity of $H_{eff}$

In this appendix we will provide details mentioned in Remarks 2.1 and 5.1, regarding a definition of our model in finite volume, and a rigorous derivation of eq. (5.3). Our plan is as follows. First we consider general aspects of gaussian and interacting Grassmann fields in finite volume and with a UV cutoff, and explain that because the effective number of Grassmann variables is finite, all path integrals are manifestly well defined. Then we apply this to the effective action in finite volume, and show that perturbation theory, if convergent, gives the correct answer. Finally we pass to the infinite volume limit and show that it agrees with eq. (5.3).

**General aspects.** Working in the finite volume $V = [-V/2, V/2]^d$ with periodic boundary conditions, we Fourier-expand the fields as

$$ \psi_a(x) = \frac{1}{V^d} \sum_{k \in K_V} \psi_{a,k} e^{ikx}, \qquad (H.1) $$

where $K_V = (2\pi/V)\mathbb{Z}^d \cap \text{supp } \chi$ is the finite set of Fourier momenta which belong to supp $\chi$. We truncate away all other momenta because they have zero propagator. The finite volume gaussian measure $d\mu_{P,V}$ is a finite-dimensional measure over Grassmann variables $\psi_{a,k}$:

$$ d\mu_{P,V}(\psi) = \text{Pf}^{-1} \prod_{\psi_{a,k}, k \in K_V} d\psi_{a,k} e^{S_{2,V}(\psi)}, \qquad (H.2) $$

$$ S_{2,V} = \frac{1}{2V^d} \sum_{k \in K_V} \hat{P}(k)^{-1} \Omega_{ab} \psi_{a,k} \psi_{b,-k}, $$

where the normalization factor $\text{Pf} > 0$ is the Pfaffian of $S_{2,V}$ which is an antisymmetric quadratic form in $\psi_{a,k}$'s. This is a meaningful finite volume version of the formal eq. (2.4) in infinite volume. The propagator $\langle \psi_a(x) \psi_b(y) \rangle = \Omega_{ab} P_V(x-y)$ where $P_V$ is the periodic version of (2.1):

$$ P_V(x) = \frac{1}{V^d} \sum_{k \in (2\pi/V)\mathbb{Z}^d} \hat{P}(k) e^{ikx}, \quad \hat{P}(k) \text{ as in (2.1)}. \qquad (H.3) $$

For fixed $x$ and $V \to \infty$, we have $P_V(x) \to P(x)$, just as for the higher-order expectations. In this sense we can say that $d\mu_{P,V} \to d\mu_P$, the infinite volume measure defined in section 2. The interacting Grassmann measure is then defined as

$$ Z_V^{-1} d\mu_{P,V}(\psi) e^{sH_V(\psi)}, \qquad (H.4) $$

where $H_V$ is given by the finite volume analogue of (4.2):

$$ H_V(\psi) = \sum_{\mathbf{A}} \int_{\mathcal{V}^l} d^d\mathbf{x} H_V(\mathbf{A}, \mathbf{x}) \Psi(\mathbf{A}, \mathbf{x}), \qquad (H.5) $$

and $Z_V = \int d\mu_{P,V}(\psi) e^{sH_V(\psi)}$ is the partition function. Factor $s$ multiplying $H_V(\psi)$ in (H.4) is for further convenience, eventually we will set $s = 1$. Using Fourier expansion (H.1),
we express $H_V(\psi)$ as a series in $\psi_{a,k}$'s with finite coefficients (assuming that the kernels $H_V(\mathbf{A}, \mathbf{x})$ are in $L_1$). By eq. (H.2) and the usual rules of Grassmann integration, $Z_V$ and $d\mu_{P,V}(\psi)e^{sH_V(\psi)}$ are well defined and are polynomials in $s$, because there are only finitely many $\psi_{a,k}$'s and only finitely many terms from the Taylor expansion of $e^{sH_V(\psi)}$ will contribute. In particular $Z_V$ is finite. The measure (H.4) will therefore be well defined as long as $Z_V \neq 0$.

**Effective action in finite volume.** Consider next eq. (2.13) in finite volume. Defining $d\mu_{g,V}(\phi)$ as $d\mu_{P,V}(\psi)$, we consider

$$I(s, \psi) = \int d\mu_{g,V}(\phi)e^{sH_V(\psi+\phi)}. \qquad (H.6)$$

By the arguments as above, we have $I(s, \psi) = e^{sH_V(\psi)}p(s, \psi)$ where $p(s, \psi)$ is a polynomial in $s$.

We would like to find $H_{\text{eff}}^V(s, \psi)$ so that

$$e^{H_{\text{eff}}^V(s, \psi)} = I(s, \psi). \qquad (H.7)$$

Let us define $H_{\text{eff}}^V(s, \psi)$ by the perturbative expansion (see eqs. (5.1), (5.3)):

$$H_{\text{eff}}^{V}(s, \psi) = \sum_{\mathbf{B}} \int_{V|\mathbf{B}|} d^{d}\mathbf{x} H_{\text{eff}}^{V}(s, \mathbf{B}, \mathbf{x}_{\mathbf{B}}) \Psi(\mathbf{B}, \mathbf{x}_{\mathbf{B}}), \qquad (H.8)$$

where $H_{\text{eff}}^V(s, \mathbf{B}, \mathbf{x_B})$ are given by the finite volume analogue of eq. (5.3) replacing $H \to sH_V$:

$$H_{\text{eff}}^{V}(s, \mathbf{B}, \mathbf{x_B}) = \mathcal{A} \sum_{n=1}^{\infty} \frac{s^n}{n!} \sum_{\sum \mathbf{B}_i = \mathbf{B}, \mathbf{A}_i \supset \mathbf{B}_i} (-)^{\#} \int_{V|\mathbf{B}|} d^d \mathbf{x}_{\mathbf{B}} C_V(\mathbf{x}_{\mathbf{B}}) \prod_{i=1}^{n} H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}), \qquad (H.9)$$

where $C_V$ is as in (5.4) only with finite-volume propagators. From the arguments like in Key lemma, we will be able to show that this series converges and defines $H_{\text{eff}}^V(s, \mathbf{B}, \mathbf{x_B})$ as analytic $L_1$-valued functions in the disk $|s| < 2$ (Lemma H.1 below, Part (b)). Since, by perturbation theory, $e^{H_{\text{eff}}^V(s, \psi)}$ and $I(s, \psi)$ have the same Taylor series in $s$, we conclude eq. (H.7) is satisfied in the disk $|s| < 2$ where they are both analytic, in particular at $s = 1$. This proves that (H.9) gives the correct effective action in finite volume.

**Effective action in infinite volume.** For an infinite volume interaction $H(\psi)$ given by (4.2), we consider the corresponding finite-volume interaction (H.5) with kernels given by periodization (we are assuming translational invariance):

$$H_V(\mathbf{A}, (0, x_2, \dots, x_l)) = \sum_{r_i \in \mathbb{Z}^d, i=2\dots l} H(\mathbf{A}, (0, x_2 + r_2 V, \dots, x_l + r_l V)). \qquad (H.10)$$

To prove that (5.3) is the correct effective action in infinite volume, we will show that it can be obtained as a $V \to \infty$ limit of the kernels of $H_{\text{eff}}^V$, in the precise sense of Part (c) of the following lemma. (Part (b) was used above to justify the effective action in finite volume.)
**Lemma H.1** There exists $A > 0$ and $\delta > 0$ such that, for any infinite volume interaction satisfying

$$ \|H_l\|_w \le A\delta^{\min(1,l/2-1)} \quad (l \ge 2), \qquad (H.11) $$

and defining the finite volume interactions by (H.10) for any $V \ge 1$, we have

(a) the kernels of $H_{\text{eff}}$ and of $H_{\text{eff}}^V$ given by eqs. (5.3) and by (H.9) with $s = 1$ are well defined (the series is convergent in $L_1$);

(b) the kernels of $H_{\text{eff}}^V(s)$ defined by (H.9) are well defined and analytic $L_1$-valued functions in the disk $|s| < 2$;

(c) for any $\mathbf{B}$ we have $H_{\text{eff}}^V(\mathbf{B}, \mathbf{x}) \to H_{\text{eff}}(\mathbf{B}, \mathbf{x})$ as $V \to \infty$ in the sense of $L_1$ norm on any fixed bounded subset of $(\mathbb{R}^d)^l$.

*Proof.* Claim (a) for $H_{\text{eff}}^V$ is a consequence of (b), which we prove as follows: consider the $L_1$ norm (with, as usual, one of the points fixed to the origin) of the $n$-th term of the series for $H_{\text{eff}}^V$,

$$ \sum_{\sum \mathbf{B}_i = \mathbf{B}, \mathbf{A}_i \supset \mathbf{B}_i} \frac{|s|^n}{n!} \int_{V^{|\mathbf{A}|}, x_1=0} d^d\mathbf{x}_A |C_V(\mathbf{x}_{\overline{\mathbf{B}}})| \prod_{i=1}^n |H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})| \qquad (H.12) $$

where $\mathbf{A} = \mathbf{A}_1 + \cdots + \mathbf{A}_n$. Recall that $C_V(\mathbf{x}_{\overline{\mathbf{B}}})$ is as in (5.4) with finite volume propagator $g_V$ replacing $g$. Here:

$$ g_V(x) = \frac{1}{V^d} \sum_{k \in (2\pi/V)\mathbb{Z}^d} \hat{g}(k) e^{ikx} = \sum_{r \in \mathbb{Z}^d} g(x + rV). \qquad (H.13) $$

From the Fourier representation of $g_V$ (first equality in (H.13)), we see that $g_V$ can be written in Gram form, as in (D.8), with $f_i$ and $h_i$ as in (D.9), with the only difference that the finite volume scalar product between $f_i$ and $h_j$ should be interpreted as $(f_i, h_j) = \frac{1}{V^d} \sum_{k \in (2\pi/V)\mathbb{Z}^d} \hat{f}_i(k) \hat{h}_j(k)$. Therefore, the Gram-Hadamard bound (D.10) holds, with $C_{\text{GH}}$ replaced by $C_{\text{GH},V}$, which is defined by the same expression as $C_{\text{GH}}$, modulo the replacement of $\int \frac{d^d k}{(2\pi)^d}$ by the corresponding Riemann sum. Moreover, from the real space representation of $g_V$ (second equality in (H.13)) and (4.15), we see that $g_V$ satisfies a bound analogous to (4.15) itself, with $|x/\gamma|$ replaced by $\|x\|/\gamma$ and $\|x\| = \min_{r \in \mathbb{Z}^d} |x + rV|$ the norm on the torus, and with the constant $C_{\chi 1}$ replaced by a larger one, but still independent of $V$ and $\gamma$. We denote by $M_V(x)$ the analogue of the right side of (4.15) with these two replacements. From these considerations, we see that $C_V(\mathbf{x}_{\overline{\mathbf{B}}})$ is bounded as in (E.3),

$$ |C_V(\mathbf{x}_{\overline{\mathbf{B}}})| \le (C_{\text{GH},V})^{\frac{1}{2} \sum_i l_i} \sum_{\mathcal{T}} \prod_{(xx') \in \mathcal{T}} M_V(x - x'). \qquad (H.14) $$

Thanks to these considerations, proceeding as in appendix E, we get the analogue of (E.8), namely

$$ \int_{V^{|\mathbf{A}|}, x_1=0} d^d \mathbf{x}_A |C_V(\mathbf{x}_{\overline{\mathbf{B}}})| \prod_{i=1}^n |H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})| \qquad (H.15) $$

$$ \le (C_{\text{GH},V})^{\frac{1}{2} \sum_i l_i} \sum_{\mathcal{T}} \int_{V^{|\mathbf{A}|}, x_1=0} d^d \mathbf{x}_A \prod_{(xx') \in \mathcal{T}} M_V(x - x') \prod_{i=1}^n |H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})|, $$
which, by computing the integrals, can be further bounded as

$$ (H.15) \le (C_{\text{GH},V})^{\frac{1}{2}} \sum_{i} l_i N_{\mathcal{T}} \|M_V\|_1^{n-1} \prod_{i=1}^{n} \|H_{V,l_i}\|_1. \qquad (H.16) $$

Now, by using (H.10), we see that $\|H_{V,l_i}\|_1 \le \|H_{l_i}\|_w$, which is bounded by $A\delta^{\min(1,l/2-1)}$, thanks to (H.11). Recalling also (E.2) and the fact that $\mathcal{N}_{\mathcal{T}} \le n!4\sum l_i$, we find

$$ (H.12) \le |s|^n A^n \|M_V\|_1^{n-1} \sum_{(l_i)_1^n} \prod_{i=1}^{n} (C_V) \sum_{l_i} \delta^{\min(1, l_i/2-1)} \qquad (H.17) $$

where $C_V = 16N(d+1)C_{\text{GH},V}^{1/2}$. Note that both $C_V$ and $\|M_V\|_1$ are uniformly bounded in $V$. Positive even integers $l_i$ satisfy $\sum_i l_i \ge l+2(n-1)$, but here it will be enough to extend the sum to arbitrary $l_i \ge 2$. Therefore, with a suitable $V$-independent constant $C$, we get a bound

$$ (H.17) \le \left( \frac{C|s|\delta}{1-C\delta} \right)^n, \qquad (H.18) $$

from which summability in $n$ follows, for all $|s| < 2$, if $\delta$ is sufficiently small. Of course, item (a) for $H_{\text{eff}}$ in infinite volume follows from the same argument.<fnref n="61" />

**Remark H.1** According to the discussion after eq. (H.5), we also need to make sure that the finite volume partition function is nonzero. The constant, $\psi$-independent term in the effective action can be estimated by the same argument as above, and it is given by the torus volume times a convergent series, in particular it is finite. Hence the partition function, which is its exponential, is nonzero. The effective action is thus well defined. Once we know that the $\psi$-independent term is finite, we may drop it as we did throughout.

Let us now prove (c) (cf. [65], appendix D). We fix a bounded subset of $(\mathbb{R}^d)^l$ that, without loss of generality, we assume to be centered in the origin, and we call it $V_0$. We want to prove that the sum over $n$ of

$$ \sum_{\mathbf{A}_i \supset \mathbf{B}_i} \frac{1}{n!} \int_{V_0^{\mathbf{B}|\mathbf{|}}} d^d \mathbf{x}_{\mathbf{B}} \left| \int_{V_{\mathbf{|B|}}} d^d \mathbf{x}_{\overline{\mathbf{B}}} C_V(\mathbf{x}_{\overline{\mathbf{B}}}) \prod_{i=1}^{n} H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) - \int_{\mathbb{R}^{\mathbf{|B|}d}} d^d \mathbf{x}_{\overline{\mathbf{B}}} C(\mathbf{x}_{\overline{\mathbf{B}}}) \prod_{i=1}^{n} H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) \right| \qquad (H.19) $$

goes to zero as $V \to \infty$. We will in fact prove that the sum of (H.19) over $n$ goes to zero exponentially fast in $V$ as $V \to \infty$. We rewrite the integral over $V^{\mathbf{|B|}}$ by multiplying the integrand by $1=(\text{St}_V(\mathbf{x}_{\mathbf{A}}) \le V/4)+1(\text{St}_V(\mathbf{x}_{\mathbf{A}}) > V/4)$ [here, if $\mathbf{x} = (x_1, \dots, x_l)$, the finite volume Steiner diameter $\text{St}_V(\mathbf{x})$ is the length of the shortest tree on the torus (possibly with extra vertices) which connects all the points in $\mathbf{x}$. Note that $\text{St}_V(\mathbf{x}) \le \min_{r \in \mathbb{Z}^d} \text{St}(\mathbf{x}+\mathbf{r}V)$],

<footnote n="61">⁶¹This case is also a consequence of Key Lemma but we preferred to give an independent argument to demonstrate how much simpler it is to show the convergence than, as in Key Lemma, to get an optimal bound on the sum.</footnote>
and similarly for the integral over $\mathbb{R}^{\overline{|\mathbf{B}|^d}}$, with the finite volume Steiner diameter replaced by the standard, infinite volume, one. In view of this manipulation, we bound (H.19) from above by $|\mathcal{V}_0|(R_{1,n}+R_{2,n}+R_{3,n})$, where, letting $x_1$ being the first coordinate in the list $\mathbf{x}_B$:

$$R_{1,n} = \sum_{\sum_{\mathbf{A}_i \supset \mathbf{B}_i} \mathbf{B}_i = \mathbf{B}} \frac{1}{n!} \int_{\mathcal{V}^{|\mathbf{A}|}, x_1=0} d^d\mathbf{x}_A |C_V(\mathbf{x}_{\overline{\mathbf{B}}})| \prod_{i=1}^n |H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})| (\mathrm{St}_V(\mathbf{x}_A) > V/4),$$

$$R_{2,n} = \sum_{\sum_{\mathbf{A}_i \supset \mathbf{B}_i} \mathbf{B}_i = \mathbf{B}} \frac{1}{n!} \int_{\mathbb{R}^{|\mathbf{A}|}, x_1=0} d^d\mathbf{x}_A |C_V(\mathbf{x}_{\overline{\mathbf{B}}})| \prod_{i=1}^n |H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})| 1(\mathrm{St}(\mathbf{x}_A) > V/4), \quad (\mathrm{H.20})$$

$$R_{3,n} = \sum_{\sum_{\mathbf{A}_i \supset \mathbf{B}_i} \mathbf{B}_i = \mathbf{B}} \frac{1}{n!} \int_{\mathbb{R}^{|\mathbf{A}|}, x_1=0} d^d\mathbf{x}_A \left| C_V(\mathbf{x}_{\overline{\mathbf{B}}}) \prod_{i=1}^n H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) - C(\mathbf{x}_{\overline{\mathbf{B}}}) \prod_{i=1}^n H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) \right| 1(\mathrm{St}(\mathbf{x}_A) \le V/4),$$

where, in the definition of $R_{3,n}$, we used the fact that $1(\mathrm{St}_V(\mathbf{x}_A) \le V/4)$ is the same as $1(\mathrm{St}(\mathbf{x}_A) \le V/4)$, provided we identify the points of the torus $\mathcal{V}$ closer than $V/4$ to the origin with the corresponding points in $\mathbb{R}^d$.

In order to bound $R_{1,n}$ we proceed as we did above for (H.12), with only a few differences: consider the analogue of (H.15) that, compared with that equation, has the additional constraint $1(\mathrm{St}_V(\mathbf{x}_A) > V/4)$ under the integral sign. In the second line, we multiply and divide each factor $M_V(x-x')$ by $w_V(x, x')$ and each factor $H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})$ by $w_V(\mathbf{x}_{\mathbf{A}_i})$, where $w_V(\mathbf{x})$ is the finite volume analogue of $w(\mathbf{x})$, namely $w_V(\mathbf{x}) = w(\mathbf{x}) = e^{C_w(\mathrm{St}_V(\mathbf{x})/\gamma)^\sigma}$. We collect together all the factors $1/w_V(x, x')$ and $1/w_V(\mathbf{x}_{\mathbf{A}_i})$ and note that, on the support of $1(\mathrm{St}_V(\mathbf{x}_A) > V/4)$,

$$\left( \prod_{(xx') \in \mathcal{T}} \frac{1}{w_V(x, x')} \right) \left( \prod_{i=1}^{n} \frac{1}{w_V(\mathbf{x}_{\mathbf{A}_i})} \right) \le e^{-C_w(\mathrm{St}_V(\mathbf{x}_A)/\gamma)^\sigma} \le e^{-C_w(V/(4\gamma))^\sigma}. \quad (\mathrm{H.21})$$

Therefore, we can bound the analogue of (H.15) by the analogue of the right side of (H.16), that is

$$(C_{GH,V})^{\frac{1}{2} \sum_i l_i N_T} e^{-C_w(V/(4\gamma))^\sigma} \|M_V\|_{w_V}^{n-1} \prod_{i=1}^{n} \|H_{V,l_i}\|_{w_V}. \quad (\mathrm{H.22})$$

Note also that, thanks to (H.10) and (H.11), $\|H_{V,l_i}\|_{w_V} \le \|H_{l_i}\|_w \le A\delta^{\min(1,l_1/2-1)}$. Putting things together, we get the analogue of (H.18):

$$R_{1,n} \le e^{-C_w(V/(4\gamma))^\sigma} \left( \frac{C\delta}{1-C\delta} \right)^n, \quad (\mathrm{H.23})$$

for a suitable $V$-independent constant $C$. Clearly, for $\delta$ small enough, the sum over $n$ of $R_{1,n}$ converges and goes to zero exponentially as $V \to \infty$. Analogous discussion and bounds are valid for $R_{2,n}$.<fnref n="62" />

<footnote n="62">⁶²This discussion also makes clear that finite-volume convergence statements in Parts (a) and (b) can be easily generalized to weighted $L_1$ norm with weight $w_V$.</footnote>
Let us now consider $R_{3,n}$. We rewrite the difference

$$ C_V(\mathbf{x}_{\overline{B}}) \prod_{i=1}^{n} H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) - C(\mathbf{x}_{\overline{B}}) \prod_{i=1}^{n} H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}). \qquad (\text{H.24}) $$

in telescopic form as the sum of $n+1$ terms, in each of which either a difference $C_V(\mathbf{x}_{\overline{B}}) - C(\mathbf{x}_{\overline{B}})$ or $H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) - H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})$ appears. The terms with $H_V - H$ can be bounded via an analogue of (H.16), with the important difference that one of the factors $\|H_{V,l_i}\|_1$ is replaced by (denoting $\mathbf{x}_{\mathbf{A}_i} = (x_1, \dots, x_{l_i})$ and $\mathbf{r} = (r_1, \dots, r_{l_i})$)

$$ \int_{\mathbb{R}^{|\mathbf{A}_i|, x_1=0}} d^d\mathbf{x}_{\mathbf{A}_i} |H_V(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) - H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})| 1(\mathrm{St}(\mathbf{x}_{\mathbf{A}_i}) \le V/4) \qquad (\text{H.25}) \\ \le \sum_{\substack{\mathbf{r} \in \mathbb{Z}^{d l_i} \\ r_1 = 0, \mathbf{r} \ne \mathbf{0}}} \int_{\mathbb{R}^{d l_i, x_1=0}} d^d\mathbf{x}_{\mathbf{A}_i} |H(\mathbf{A}, \mathbf{x}_{\mathbf{A}_i} + \mathbf{r}V)| \frac{w(\mathbf{x}_{\mathbf{A}_i} + \mathbf{r}V)}{w(\mathbf{x}_{\mathbf{A}_i} + \mathbf{r}V)} 1(\mathrm{St}(\mathbf{x}_{\mathbf{A}_i}) \le V/4), $$

where in passing from the first to the second line we used the definition (H.10) and we multiplied and divided by $w(\mathbf{x}_{\mathbf{A}_i} + \mathbf{r}V)$. Now, note that, on the support of $1(\mathrm{St}(\mathbf{x}_{\mathbf{A}_i}) \le V/4)$, $|\mathbf{x}_{\mathbf{A}_i} + \mathbf{r}V| > V/2$ for any $\mathbf{r} \ne \mathbf{0}$. Therefore, the second line of (H.25) can be bounded from above by $\|H_{l_i}\|_w/w(V/2)$, where $1/w(V/2) = e^{-C_w(V/(2\gamma))^{\sigma}}$ represents the desired exponentially small gain as $V \to \infty$.

Consider now the contribution to $R_{3,n}$ associated with the difference $C_V(\mathbf{x}_{\overline{B}}) - C(\mathbf{x}_{\overline{B}})$. Recall that both $C(\mathbf{x}_{\overline{B}})$ and $C_V(\mathbf{x}_{\overline{B}})$ can be written in terms of the BBF formula that, see (D.53), can be written as

$$ C(\mathbf{x}_{\overline{B}}) = \sum_{\mathcal{T}} \prod_{(xx') \in \mathcal{T}} g(x - x') \int d\mu_T(\mathbf{r}) \det \mathcal{N}, \qquad (\text{H.26}) $$

where $\mathcal{N} = \mathcal{N}(\mathbf{r})$ is a Gram matrix, i.e., with elements represented as a suitable scalar product. Of course, $C_V(\mathbf{x}_{\overline{B}})$ admits a representation analogous to (H.26), with $g$ replaced by $g_V$ and $\mathcal{N}$ replaced by $\mathcal{N}_V$. Using (H.26) and the analogous representation for $C_V(\mathbf{x}_{\overline{B}})$, we write the difference $C_V(\mathbf{x}_{\overline{B}}) - C(\mathbf{x}_{\overline{B}})$ in telescopic form, as the sum of terms in each of which either a difference $g_V(x - x') - g(x - x')$ or $\det \mathcal{N}_V - \det \mathcal{N}$ appears. In the former terms, recalling (H.13), we write $g_V(x) - g(x) = - \sum_{r \ne 0} g(x + rV)$ and, proceeding as we did for the bound of the terms with $H_V - H$, we see that they are exponentially small in $V$, and their sum over $n$ too.

We are left with the term involving the difference $\det \mathcal{N}_V - \det \mathcal{N}$, which we rewrite once again in telescopic form as

$$ \det \mathcal{N}_V - \det \mathcal{N} = \sum_{i,j=1}^{s} (\det \mathcal{N}_V^{(i,j)} - \det \mathcal{N}_V^{(i,j)'}) \qquad (\text{H.27}) $$

where $s$ is the linear size of the matrices, $\mathcal{N}_V^{(i,j)}$ is the matrix whose elements with label smaller or equal to (resp. larger than) $(i, j)$ in the lexicographic order are equal to the elements of $\mathcal{N}_V$ (resp. $\mathcal{N}$), and $(i, j)'$ is the label immediately preceding $(i, j)$ in the
lexicographic order (if $(i, j) = (1, 1)$, we interpret $\mathcal{N}_V^{(1,1)'} \equiv \mathcal{N}$). Since $\mathcal{N}_V^{(i,j)}$ and $\mathcal{N}_V^{(i,j)'}$ differ in only one element, expanding in minors along row $i$ we have

$$ \det \mathcal{N}_V^{(i,j)} - \det \mathcal{N}_V^{(i,j)'} = (-1)^{i+j} ((\mathcal{N}_V)_{i,j} - \mathcal{N}_{i,j}) \det \tilde{\mathcal{N}}_V^{(i,j)}, \quad (H.28) $$

where $\tilde{\mathcal{N}}_V^{(i,j)}$ denotes the matrix $\mathcal{N}_V^{(i,j)}$ with both the $i$-th row and the $j$-th column removed. Recall that both $\mathcal{N}_V$ and $\mathcal{N}$ are Gram matrices; in particular, they can be written as $(\mathcal{N}_V)_{k,l} = (f_{V,k}, h_{V,l})$ and $\mathcal{N}_{k,l} = (f_k, h_l)$ for appropriate vectors $f_V, h_V, f, h$ in two apriori different Hilbert spaces $\mathcal{H}_V$ and $\mathcal{H}$. Remarkably, also $\tilde{\mathcal{N}}_V^{(i,j)}$ is in Gram form, that is, for any $k \in \{1, \dots, s\} \setminus \{i\}$ and any $l \in \{1, \dots, s\} \setminus \{j\}$, we can write $(\tilde{\mathcal{N}}_V^{(i,j)})_{k,l} = (F_k, H_l)$, where $(\cdot, \cdot)$ denotes the scalar product in $\mathcal{H}_V \oplus \mathcal{H}$, and $F_k, H_l$ are the following vectors in $\mathcal{H}_V \oplus \mathcal{H}$:

$$ F_k = \begin{cases} (f_{V,k}, 0) & \text{if } k < i \\ (0, f_k) & \text{if } k > i \end{cases} \quad \text{and} \quad H_l = (h_{V,l}, h_l). \quad (H.29) $$

Therefore, $\det \tilde{\mathcal{N}}_V^{(i,j)}$ can be bounded qualitatively in the same way as $\det \mathcal{N}_V$ or $\det \mathcal{N}$, so that, using (H.28) into (H.27), and recalling that $(\mathcal{N}_V)_{i,j} - \mathcal{N}_{i,j}$ is proportional to $g_V - g$, we find that the term in $R_{3,n}$ involving the difference $\det \mathcal{N}_V - \det \mathcal{N}$ is bounded qualitatively as all the other terms, that is, they are exponentially small in $V$, and their sum over $n$ too. This concludes the proof of Lemma H.1. $\square$

Consider e.g. $H(\psi)$ corresponding to the fixed point whose existence we proved. By corollary 6.1, this interaction satisfies bounds (6.43) which for sufficiently small $\epsilon$ are stronger than (H.11). Therefore, the effective action is indeed given by (5.3) as we assumed all along.

# I Fixed point in a formal power series expansion

In this appendix we will show that eq. (5.32) $f(y) = 0$ can be solved in a formal power series expansion in $\epsilon$. This is rather easy, compared to the proof of the existence of an actual solution given in section 6. We introduce a positive grading function on the couplings $y_i \in \{\nu, \lambda, u_{2R}, u_{4R}, u_{6R}, (u_\ell)_{\ell \ge 8}\}$:

$$ \mathrm{gr}(\nu) = \mathrm{gr}(\lambda) = 1, \quad \mathrm{gr}(u_{2R}) = \mathrm{gr}(u_{4R}) = 2, \quad \mathrm{gr}(u_{6R}) = 3, \quad \mathrm{gr}(u_\ell) = k(l) = \frac{l}{2} - 1(l \ge 8). \quad (I.1) $$

We also define grading of a product as a sum of gradings. It is then easy to check that each function $e_{y_i}$ is a sum of terms whose grading is $\ge \mathrm{gr}(y_i)$, with strict inequality for $\nu, \lambda$ (table 1). This motivates the following

**Theorem I.1** Equation $f(y) = 0$ has a unique solution where couplings are formal power series in $\epsilon$ starting from:

$$ \nu = \frac{a}{b}\epsilon + O(\epsilon^2), \quad \lambda = -\frac{1}{b}\epsilon + O(\epsilon^2), \quad y_i = O(\epsilon^{\mathrm{gr}(y_i)}) \quad (y_i \in \{u_{2R}, u_{4R}, u_{6R}, (u_\ell)_{\ell \ge 8}\}). \quad (I.2) $$
<table><caption>**Table 1.** Grading of terms in functions $e_{y_i}$. We only show the variables on which the terms depend. E.g. $u_{2R}$ and $\lambda^2$ in $e_{2R}$ stand for $R_{2R}^{2R}(u_{2R}) = Du_{2R}$ and $R_{2R}^{4L,4L}(\lambda, \lambda)$, respectively.</caption><thead><tr><th>Function</th><th>Grading</th><th>Notable present terms</th><th>Notable absent terms</th></tr></thead><tbody><tr><td>e<sub>&nu;</sub></td><td>&ge; 2</td><td>&lambda;<sup>2</sup>, &nu;&lambda;, u<sub>4R</sub></td><td>&nu;, u<sub>2R</sub>, &lambda;</td></tr><tr><td>e<sub>&lambda;</sub></td><td>&ge; 3</td><td>&nu;&lambda;<sup>2</sup>, &lambda;<sup>3</sup>, &nu;<b>X</b><sub>&lambda;</sub>, &lambda;<b>X</b><sub>&lambda;</sub>, &lambda;u<sub>4R</sub>, u<sub>6R</sub></td><td>&lambda;, u<sub>4R</sub>, &lambda;<sup>2</sup>, <b>X</b><sub>&lambda;</sub>, &nu;&lambda;, &nu;u<sub>4R</sub></td></tr><tr><td>e<sub>2R</sub></td><td>&ge; 2</td><td>u<sub>2R</sub>, u<sub>4R</sub>, &lambda;<sup>2</sup></td><td>&nu;, &lambda;</td></tr><tr><td>e<sub>4R</sub></td><td>&ge; 2</td><td>u<sub>4R</sub>, &lambda;<sup>2</sup></td><td>&lambda;</td></tr><tr><td>e<sub>6R</sub></td><td>&ge; 3</td><td>u<sub>6R</sub>, &lambda;<sup>3</sup>, &nu;&lambda;<sup>2</sup>, &nu;<b>X</b><sub>&lambda;</sub>, u<sub>4R</sub>&lambda;, u<sub>8</sub></td><td><b>X</b><sub>&lambda;</sub>, &lambda;<sup>2</sup></td></tr><tr><td>e<sub>&ell;</sub>, &ell; &ge; 8</td><td>&ge; k(l)</td><td>u<sub>&ell;</sub>, &lambda;<sup>k(l)</sup>, &sum;<sub>l<sub>1</sub>+...+l<sub>n</sub> &ge; l+2(n&minus;1)</sub> H<sub>&ell;<sub>1</sub></sub> &middot;&middot;&middot; H<sub>&ell;<sub>n</sub></sub></td><td></td></tr></tbody></table>

*Proof.* Parameter $\epsilon$ enters (5.32) through the explicit term $\epsilon\lambda$. In addition, all the other coefficients such as $a, b$ and the multilinear kernels from the r.h.s. of (5.29) also depend on $\epsilon$. This dependence originates from the fluctuation propagator $g(x)$ defined in (2.10), and it is nonsingular as $\epsilon \to 0$ in our setup involving the UV and IR cutoffs. Below we will keep track only of the explicit dependence on $\epsilon$ from the $\epsilon\lambda$ term, which we denote $\epsilon$. All other coefficients will be treated as constants. We will give an algorithm to expand the solution as a formal power series in $\epsilon$. To produce a power series in $\epsilon$, one would have to set $\epsilon \to \epsilon$ and additionally expand all coefficients in $\epsilon$.

We start by rescaling the couplings $y_i \to \epsilon^{\text{gr}(y_i)}y_i$. We will abuse notation denoting the rescaled couplings by the same letters. We have to show that the rescaled couplings have unique power series expansions starting at $O(1)$. The equations $f(y) = 0$ in terms of the rescaled couplings can be written as (the explanations and the definition of $e_{y_i,k}$ are given after (I.6))

$$ \text{couplings of grading 1:} \quad \left\{ \begin{aligned} -\nu - a\lambda &= \sum_{k \ge 1} \epsilon^k e_{\nu,k} \\ -\lambda - b\lambda^2 &= \sum_{k \ge 1} \epsilon^k e_{\lambda,k+1} \end{aligned} \right. \qquad (I.3) $$

$$ \text{couplings of grading 2:} \quad \left\{ \begin{aligned} (1-D)u_{2R} - R_{2R}^{4R}u_{4R} &= \sum_{k \ge 0} \epsilon^k e_{2R,k} \\ (1-D)u_{4R} &= \sum_{k \ge 0} \epsilon^k e_{4R,k} \end{aligned} \right. \qquad (I.4) $$

$$ \text{couplings of grading 3:} \quad \left\{ \begin{aligned} (1-D)u_{6R} - R_{6R}^{8}u_{8} &= \sum_{k \ge 0} \epsilon^k e_{6R,k} \\ (1-D)u_{8} &= \sum_{k \ge 0} \epsilon^k e_{8,k} \end{aligned} \right. \qquad (I.5) $$

$$ \text{couplings of grading } \ge 4: (1-D)u_{\ell} = \sum_{k \ge 0} \epsilon^k e_{\ell,k} \quad (\ell \ge 10), \qquad (I.6) $$

For each coupling, $e_{y_i,k}$ denotes the part of $e_{y_i}$ which contains the terms of grading exactly $\text{gr}(y_i) + k$. In addition we separated the linear terms $Du_i$ as well as $R_{2R}^{4R}u_{4R}$ and $R_{6R}^{8}u_{8}$ from $e_{u_{i,0}}$ in (I.4)-(I.6). With this definition the remaining $e_{u_{i,0}}$ are at least quadratic in its arguments. Note that the r.h.s. of (I.3) are $O(\epsilon)$, while the other equations have r.h.s. $O(1)$. Note also the shift $k \to k+1$ in $e_{\lambda,k+1}$ in (I.3).

**Step 1.** The $O(1)$ parts of $\nu$ and $\lambda$ known, $\nu = \frac{a}{b} + O(\epsilon)$, $\lambda = -\frac{1}{b} + O(\epsilon)$, let us solve for the $O(1)$ parts of the other couplings. Firstly, note that the $O(1)$ parts of the r.h.s. of (I.4)-(I.6), $e_{u_{i,0}}$, having grading exactly $\text{gr}(u_i)$ and being at least quadratic, are computable in
terms of the $O(1)$ parts of the couplings with smaller grading. Secondly, the linear operators in the l.h.s. of (1.4)-(1.6) are invertible. Indeed, the operator $(1-D)$ is invertible on each irrelevant coupling subspace as is clear from definition (5.20).$^{63}$ eqs. (1.4) and (1.5) involve a matrix-triangular operator with $(1-D)$ on the diagonal, hence also invertible. By these two observations, $O(1)$ parts of all couplings are uniquely determined starting from $\nu$ and $\lambda$ and going recursively up in grading.

**Step 2a.** Now suppose we computed expansions of all couplings up to and including $O(\epsilon^N)$ (call it “inductive hypothesis 1”), and we want to solve for the $\epsilon^{N+1}$ terms. For any quantity $\alpha = \sum \epsilon^n \alpha_n$ we denote by $[\alpha]_n = \alpha_n$ the $\epsilon^n$ coefficient. We start with (1.3) and take its $\epsilon^{N+1}$ part:

$$ -[\nu]_{N+1} - a[\lambda]_{N+1} = \sum_{k=1}^{N+1} [e_{\nu,k}]_{N+1-k}, \qquad (I.7) $$

$$ (-1 - 2b[\lambda]_0)[\lambda]_{N+1} = b \sum_{k=1}^{N} [\lambda]_k[\lambda]_{N+1-k} + \sum_{k=1}^{N+1} [e_{\lambda,k+1}]_{N+1-k}. \qquad (I.8) $$

All the terms in the r.h.s. are computable by inductive hypothesis 1. Since $-1 - 2b[\lambda]_0 = 1$ we can compute first $[\lambda]_{N+1}$ and then $[\nu]_{N+1}$.

**Step 2b.** The remaining couplings are treated recursively going up in grading as before. Suppose all couplings of grading lower than $u_i$ are already known up to and including $O(\epsilon^{N+1})$ (call it “inductive hypothesis 2”). Consider the equation for $u_i$ (if there are two couplings having the same grading we should study their equations together as in Step 1) and take its $\epsilon^{N+1}$ part. In the l.h.s. we have an invertible linear operator, same as in Step 1, acting on $[u_i]_{N+1}$, while in the r.h.s. we have

$$ \sum_{k=0}^{N+1} [e_{u_i,k}]_{N+1-k}. \qquad (I.9) $$

For $k \ge 1$ this is computable by inductive hypothesis 1, and for $k = 0$ by inductive hypothesis 2, since $e_{u_i,0}$ has grading $\text{gr}(u_i)$ and is at least quadratic, which means it involves only lower-grading couplings. Therefore, we can compute $[u_i]_{N+1}$ and continue the induction.$^{64}$ This finishes the proof of the theorem.

**Remark 1.1** At the level of formal series expansions bosonic and fermionic fixed point are quite analogous. Consider e.g. the bosonic model (K.1). In perturbation theory, we could parametrize its fixed point by an interaction written in terms of kernels, like in (4.2). We could derive a perturbative renormalization map acting on the sequence of kernels

<footnote>$^{63}$We have $(1-D)^{-1} = 1 + D + D^2 + \dots$ and the series converges in $L_1$ if $\gamma^{-D_1-p} < 1$ which is the condition for irrelevance.</footnote>

<footnote>$^{64}$Steps 2a, 2b can be unified, at the price of rendering the argument less explicit, by moving the nonlinear functions $e_{u_i,0}$ to the l.h.s. and noting that the Jacobian of the resulting nonlinear infinite matrix function of $y$ in the l.h.s. is invertible at the point $(\nu, \lambda, u_i) = \left(\frac{a}{b}, -\frac{1}{b}, u_i^{(0)}\right)$ where $u_i^{(0)}$ are the $O(1)$ values of $u_i$ computed in Step 1.</footnote>
in the trimmed representation, similarly to section 5. We could then find, exactly as in theorem I.1, a fixed point in a formal power series in $\epsilon$.

This analogy breaks down beyond perturbation theory. As stated in remark 5.1 and proved in appendix H, perturbative expansion captures the full fermionic effective action at small coupling. This does not hold for bosons due to large field effects (“instantons”) at arbitrarily small couplings. Furthermore, rigorous non-perturbative studies of bosonic models parametrize irrelevant interactions not by kernels as in (4.2), but by a more complicated “polymer expansion” (see [114] and section 8.1.8).

Another difference is that for bosons, the formal power series solution in $\epsilon$ is expected to be only asymptotic, like the $\epsilon$-expansion series for the Wilson-Fisher fixed point in $d = 4 - \epsilon$ dimensions [133], necessitating Borel resummations for the critical exponents [5]. The same considerations should apply to the long-range bosonic model (K.1). On the other hand, for fermions we have established in section 6.5 that the fixed point depends analytically on $\epsilon$. This implies that the formal power series solution will be convergent for small $\epsilon$. For a direct proof of convergence of the fermionic power series expansion via tree expansion, see the next appendix.

## J Fixed point via the tree expansion

In the main sections of this paper we provided an explicit rigorous construction of a non-trivial RG fixed point, by finding the appropriate Banach space, which the RG map acts on, and by proving its contractivity in an appropriate neighborhood of this space. In the case of fermionic theories, as in the case at hand, the fixed point can also be found by a different strategy, which bypasses the construction of the Banach space and the contractivity argument, and is based on an expansion in tree diagrams, sometimes referred to as “Gallavotti-Nicolò” trees [134], see also [51, 71] and [119] for a review in the context of interacting fermionic theories. While the two constructions build on the same general foundations from sections 1–5 (trimmed representation of the interaction, weighted norm $\| \cdot \|_w$ for measuring the size of couplings, and the norm bounds on $D$ and $R_\ell^{\ell_1, \dots, \ell_n}$), the tree expansion is closer in spirit to a direct combinatorial proof convergence of the formal $\epsilon$-expansion of appendix I. Let us briefly describe here the construction of the fixed point via trees.

The starting point is the fixed point equation for the irrelevant couplings $u_\ell$, with $\ell = 2R, 4R, 6R, 8, 10, \dots$, which we rewrite, extracting the term $(n; (\ell_1, \dots, \ell_n)) = (1; \ell)$ explicitly:

$$ (1 - D)u_{\ell} = \sum_{n \ge 1} \sum_{(\ell_i)_1^n}^{*} R_{\ell}^{\ell_1, \dots, \ell_n} (H_{\ell_1}, \dots, H_{\ell_n}), \qquad (J.1) $$

where the $*$ on the sum indicates the constraint that, if $n = 1$, then $\ell_1 \neq \ell$. We look for a solution in power series in $\nu$ and $\lambda$, $u_\ell = \sum_{k_1, k_2 \ge 0} u_\ell^{(k_1, k_2)} \nu^{k_1} \lambda^{k_2}$, with $u_\ell^{(k_1, k_2)} = 0$ unless $k_1 + k_2 \ge |\ell|/2 - 1$ and, moreover, $k_1 + k_2 \ge 2$ for $\ell = 2R, 4R$ and $k_1 + k_2 \ge 3$ for $\ell = 6R$.
By plugging this ansatz in (J.1), we get$^{65}$

$$ u_{\ell}^{(k_1, k_2)} = \sum_{n \ge 1} \sum_{(\ell_i)_1^n} \sum_{\{k_{1,i}, k_{2,i}\}_{i=1,2}^{(k_1, k_2)}} (1-D)^{-1} R_{\ell}^{\ell_1, \dots, \ell_n} (H_{\ell_1}^{(k_{1,1}, k_{2,1})}, \dots, H_{\ell_n}^{(k_{1,n}, k_{2,n})}), \quad (J.2) $$

where the third sum runs over integers $k_{1,i}, k_{2,i} \ge 0$, with $i = 1, \dots, n$, such that $k_{1,1} + \dots + k_{1,n} = k_1$ and $k_{2,1} + \dots + k_{2,n} = k_2$. Moreover, the arguments $H_{\ell_i}^{(k_{1,i}, k_{2,i})}$ should be interpreted as being equal to $u_{\ell_i}^{(k_{1,i}, k_{2,i})}$ if $\ell_i \ne 2L, 4L, 6SL$, to $\nu$ if $\ell_i = 2L$ (in which case we set $(k_{1,i}, k_{2,i}) = (1,0)$), to $\lambda$ if $\ell_i = 4L$ (in which case we set $(k_{1,i}, k_{2,i}) = (0,1)$), and to $\mathcal{X}_*$ if $\ell_i = 6SL$ (in which case we set $(k_{1,i}, k_{2,i}) = (0,2)$). Eq. (J.2) can be graphically represented as follows:

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

where the vertex labelled $v_0$ in the right side represents the action of $(1-D)^{-1} R_{\ell}^{\ell_1, \dots, \ell_n}$; we shall say that the $n$ lines (or “branches”) labelled $\ell_1, \dots, \ell_n$ “enter the vertex $v_0$”; similarly, we’ll say that the branch to the left of $v_0$ “exits” from $v_0$: it carries the label $\ell$ and its left endpoint is called *root*. In the special cases $(\ell_i, (k_{1,i}, k_{2,i})) = (2L, (1,0)), (4L, (0,1)), (6SL, (0,2))$, the big dots in the right side will be reinterpreted as small dots with labels $\nu, \lambda, \mathcal{X}_*$, respectively:

$$ \bullet 2L, (1,0) \equiv \bullet^{\nu}, \quad \bullet 4L, (0,1) \equiv \bullet^{\lambda}, \quad \bullet 6SL, (0,2) \equiv \bullet^{\mathcal{X}} $$

Iterating the graphical equation above until the endpoints are all small dots with labels $\nu, \lambda$, or $\mathcal{X}_*$, we obtain an expansion in tree diagrams of the following form:

<figure><img src="image_10.png" /></figure>

<footnote>$^{65}$Note that since $u_{\ell}$ is irrelevant, $(1-D)$ is an invertible operator, see footnote 63.</footnote>
In this example, the branch labels are left implicit; each vertex $v_r$, with $r = 1, \dots, 5$, is associated with the action of an operator $(1 - D)^{-1} R_{\ell}^{\ell_1, \dots, \ell_n}$, with $\ell$ the label of the line exiting from $v_r$, $n$ the number of lines entering it, and $\ell_1, \dots, \ell_n$ their labels. If $m_\nu$, $m_\lambda$ and $m_\chi$ are the numbers of endpoints of type $\nu, \lambda$ and $\chi_*$, respectively (in the example, $m_\nu = 2$, $m_\lambda = 6$ and $m_\chi = 2$), and $\ell_0$ is the label of the line exiting from $v_0$, then the tree contributes to $u_{\ell_0}^{(m_\nu, m_\lambda + 2m_\chi)}$.

Let us now use the norm bounds on $D$ and $R_{\ell}^{\ell_1, \dots, \ell_n}$, see sections 5.3 and 5.6, to bound the value of any such tree in the $\| \cdot \|_w$ norm, assuming $|\nu|, |\lambda| \le \delta$. The norm bounds on $D$ imply that the norm of $(1 - D)^{-1}$ is bounded by some $d_\gamma > 1$ (uniformly in $\ell$). Using this and eq. (5.35), the norm of the value of a tree is bounded by the product of the norms of its endpoints, which is bounded by

$$ \delta^{m_\nu+m_\lambda} (B_\gamma \delta^2)^{m_\chi} \le (B_\gamma \delta)^k, \quad k = m_\nu + m_\lambda + 2m_\chi, \qquad (J.3) $$

times the product over vertices $v_r$ (i.e. all vertices $v$ which are not endpoints):

$$ \prod_{v \text{ not e.p.}} d_\gamma C_\gamma^{n_v-1} \gamma^{-D_{l_v}} C_0^{\sum_{i=1}^{n_v} |\ell_i(v)|}, \qquad (J.4) $$

where $l_v = |\ell_v|$, with $\ell_v$ the label of the line exiting from $v$, and $\ell_i(v)$ the label of the $i$-th line entering $v$. Note that $\ell_i(v) = \ell_{v'}$ for some $v'$, which may be an endpoint. Therefore, (J.4) can be equivalently rewritten as

$$ C_0^{-|\ell_{v_0}|} \times \prod_{v \text{ not e.p.}} d_\gamma C_\gamma^{n_v-1} \gamma^{-D_{l_v}} C_0^{l_v} \times \prod_{v \text{ e.p.}} C_0^{l_v}. \qquad (J.5) $$

Replacing the first factor by 1, and including the factor (J.3) associated with the endpoints, the value of a single tree is bounded by

$$ (B_\gamma C_0^4 \delta)^k \prod_{v \text{ not e.p.}} d_\gamma C_\gamma^{n_v-1} \gamma^{-D_{l_v}} C_0^{l_v}, \qquad (J.6) $$

where we also used that $\sum_{v \text{ e.p.}} l_v = 2m_\nu + 4m_\lambda + 6m_\chi \le 4k$.

As the next step we have to sum over all possible values of $\ell_i$'s which label tree branches. We take $\gamma$ sufficiently large so that $\gamma^{-(d/4-\epsilon/2)} C_0 < 1$, implying (recall $D_l = l(d/4-\epsilon/2)-d$)

$$ \sum_{l=2}^{\infty} \gamma^{-D_l} (C_0)^l < \infty. \qquad (J.7) $$

Then the sum $\sum_{\ell_v}$ of (J.6) is bounded by

$$ (B_\gamma C_0^4 \delta)^k \prod_{v \text{ not e.p.}} \tilde{d}_\gamma C_\gamma^{n_v-1}. \qquad (J.8) $$

Furthermore, it is easy to see that

$$ \sum_{v \text{ not e.p.}} (n_v - 1) = m - 1, \qquad (J.9) $$
where $m_\chi = m_\nu + m_\lambda + m_\chi \le k$ is the total number of the endpoints, and

$$ \sum_{v \text{ not e.p.}} 1 \le \frac{1}{2} \sum_{v \text{ e.p.}} l_v, \qquad (J.10) $$

as implied by $|\ell| \le \sum_i |\ell_i| - 2$ in each vertex $v$ (note the constraint (5.36) for $n \ge 2$, and that for $n = 1$ the sum in (J.1) starts from $l_1 = l+2$). Using the last two relations, we can estimate (J.8) by

$$ (B_\gamma C_0^4 C_\gamma \tilde{d}_\gamma^2 \delta)^k. \qquad (J.11) $$

Finally we have to sum this expression over all trees subject to the constraint on the endpoints $m_\nu + m_\lambda + 2m_\chi = k$. The number of such trees is smaller than $4^k$ (see, e.g., Lemma A.1 of [119]). We conclude that the contribution to $u_\ell$ of order $k$ is bounded in the $\| \cdot \|_w$ norm by $(A_\gamma \delta)^k$, for some $\gamma$-dependent constant $A_\gamma$. This implies that the tree expansion for $u_\ell$ is convergent for $\delta \le \delta_0(\gamma)$ small enough. This concludes the construction of the irrelevant couplings $u_\ell$ in terms of the fixed point relevant couplings $\nu$ and $\lambda$, which implies automatically that $u_\ell$ are analytic in $\nu$ and $\lambda$ in the neighborhood $|\nu|, |\lambda| \le \delta_0(\gamma)$ of the origin in $\mathbb{C}^2$. The same argument also establishes analyticity of $u_\ell$ in $\varepsilon$ as long as it belongs to the complex half-plane $\text{Re} \varepsilon < d/6$ where couplings $u_\ell$ are all irrelevant.

We are left with the beta function equations for the relevant couplings. Via the same strategy, we find that they are given by the first two equations of (5.30), with $e_\nu^{(0)}$ and $e_\lambda^{(0)}$ expressed in terms of two tree expansions, convergent for $\delta$ sufficiently small. In conclusion, at the fixed point, $\nu$ and $\lambda$ satisfy

$$ \nu = \gamma^{\frac{d}{2}+\varepsilon}(\nu + I_1\lambda) + e_{\nu}^{(0)}(\nu, \lambda, \varepsilon), \qquad \lambda = \gamma^{2\varepsilon}(\lambda + I_2\lambda^2) + e_{\lambda}^{(0)}(\nu, \lambda, \varepsilon), \qquad (J.12) $$

with $e_\nu^{(0)}$ and $e_\lambda^{(0)}$ two analytic functions of $\nu, \lambda, \varepsilon$ for $|\nu|, |\lambda| \le \delta_0(\gamma)$, $\text{Re} \varepsilon < d/6$. Moreover, we have $e_\nu^{(0)}$ and $e_\lambda^{(0)}$ of order $\delta^2$ and $\delta^3$, respectively where $\delta = \max(|\nu|, |\lambda|)$ and for $\varepsilon$ in any compact subset of $\text{Re} \varepsilon < d/6$. $I_1$ and $I_2$ also depend analytically on $\varepsilon$. By the analytic implicit function theorem, these equations have a unique solution $\nu_*(\varepsilon)$, $\lambda_*(\varepsilon)$ which is defined in the disk $|\varepsilon| \le \varepsilon_0(\gamma)$ and is analytically close to the lowest order approximated solution $\lambda_0 = (1 - \gamma^{2\varepsilon})/I_2$, $\nu_0 = I_1\lambda_0/(1 - \gamma^{\frac{d}{2}+\varepsilon})$.

**Remark J.1** The condition of $\gamma$ large is not truly required for the tree expansion to converge; in fact it converges for any $\gamma > 1$ (for $\lambda, \nu$ sufficiently small). In the proof above, large $\gamma$ was needed due to the pessimistic way in which we bounded some combinatorial factors in the previous sections. Note that the origin of the factor $(C_0)^l$ in (J.7) has to be traced back to the factors $C_0^{|\ell_i|}$ in (5.36). A critical rereading of the proof leading to those factors shows that the factor $C_0^{\sum_{i=1}^{n_\nu} |\ell_i(v)|}$ in (J.4) can be replaced by $C_R(\sum_{i=1}^{n_\nu} |\ell_i(v)|)C_0^{\sum_{i=1}^{n_\nu} |\ell_i(v)| - l_v}$. The product of the factors $C_0^{\sum_{i=1}^{n_\nu} |\ell_i(v)| - l_v}$ over the vertices $v$ that are not endpoints gives

<footnote>$^{66}$See in particular the first equality in (E.2) and the second one in (E). The factor $C_R > 1$ is the constant in (5.43); it appears only if $l_v = 2, 4$. In the main text it was absorbed into $C_0$ (footnote 31), but now we keep it explicit, since it blows up as $\gamma \to 1$, due to the blowup of constants $C_{1,2,3}$ from the end of appendix C.</footnote>
$C_0 \sum_{v \text{ e.p.}} (l_v - l_{v_0})$, with $l_0$ the vertex attached to the root; this does not need any condition on $\gamma$. The product of $C_R$'s is bounded by $C_R^{2k}$ because of (J.10). Finally, the sum over the branch labels, given the type of endpoints, reduces to $\sum_{\{\ell_v\}} \prod_{v \text{ not e.p.}} \gamma^{-D_{l_v}} (\sum_{i=1}^{n_v} |\ell_i(v)|)$, which can be bounded as explained in appendix A.6.1 of [119], leading to a factor smaller than $(\frac{1}{1-\gamma^{-\alpha}})^{4k}$, for some $\alpha > 0$ and any $\gamma > 1$.

**Remark J.2** In Wilsonian RG, the single RG step contains all the information about the fixed point, so that we should feel free to forget about what happened in the RG past. The fact that the tree expansion represents the fixed point kernels by tree diagrams with several levels may superficially seem to go against this standard idea. This is not the case: the tree expansion as presented in this appendix is just a way to solve the fixed point equation for a single RG step.<fnref n="67" /> In general, trees with several levels are exponentially suppressed, as compared to “short” ones: this is the so called short-memory property (see e.g. the remark after (7.26) in [28]).

The tree expansion can be easily adapted to the construction of the full Wilsonian RG flow of the effective couplings from the ultraviolet to the infrared, see section J.1 below for a brief discussion of this fact, and to the computation of correlation functions and critical exponents. It has been used to construct them in several 1D fermionic theories [28–31] and 2D statistical mechanics models at criticality [32–39], whose nontrivial fixed points are all in the Luttinger liquid universality class. In these cases, the construction of the fixed point requires a proof that the beta function for the quartic coupling is asymptotically vanishing in the infrared limit; historically, the proof of vanishing beta functions in these models has first been proved via a comparison with the Luttinger model exact solution [28], and later via a combined use of Ward Identities and Schwinger-Dyson equations [30]. We are not aware of a construction of nontrivial fixed points in the Luttinger liquid universality class via methods different from the tree expansion; it would be an interesting exercise to reproduce their construction via a contraction argument in a suitable Banach space of interactions, as done in this paper for long-range symplectic fermions with quartic interaction.

The fermionic nature of models such as the one studied in this paper makes the approach based on the tree expansion an extremely efficient tool for constructing the RG fixed point, arguably simpler than the one based on the contraction argument in a Banach space. However, we do not expect that the tree expansion is as a general scheme as the other, which is, to date, the only available technique for constructing nontrivial bosonic fixed points, see appendix K, and it looks the most promising (at least conceptually) for approaching the non-perturbative problem of constructing very non-Gaussian fixed points in the vicinity of approximate fixed points (possibly computed via the truncation of some other alternative scheme, such as numerical FRG). This explains the reason why in the main sections of this paper we decided to follow the scheme based on the contraction argument in a Banach space: it provides a benchmark for other approaches, like the Functional

<footnote n="67">⁶⁷To make an analogy with something already seen, consider the construction of the fixed point sextic semilocal term in section 3: $\mathcal{X}_*$ is the solution to the single step equation (3.11), and its explicit expression in terms of $\lambda$ involves a sum over many “levels”, i.e., the integers $n$ in (3.12).</footnote>
Renormalization Group (whose conceptual similarity allows for a direct comparison of results), and it displays general features which do not depend on the specific, fermionic, nature of the problem.

## J.1 On the flow of the effective couplings

A tree expansion analogous to the one described above for the construction of the fixed point can be used to compute the whole sequence of effective potentials associated with the Wilsonian RG flow from the ultraviolet to the infrared scales. Such a generalized tree expansion was described in several previous reviews on the subject, see in particular [119]. Suppose that we interested in constructing the model formally defined by the interacting Grassmann measure $Z^{-1}d\mu_P(\psi)e^{H^{(0)}(\psi)}$ at all distances (rather than being interested “just” in the construction of its infrared fixed point, as done in this paper), where $d\mu_P(\psi)$ is the Grassmann Gaussian integration with the propagator $P(x)$ in (2.1) (with fixed ultraviolet cutoff but without any infrared one) and $H^{(0)}$ is a local interaction like the one in (2.7), with bare couplings $\nu_0, \lambda_0$. The partition function $\int d\mu_P(\psi)e^{H^{(0)}(\psi)}$ and the closely related generating function of correlations can be computed iteratively, by first integrating out momenta in the annulus of radii $\gamma^{-1}$ and 1, then in the one of radii $\gamma^{-2}$ and $\gamma^{-1}$, and so on. In formulae, this means rewriting the propagator $P(x)$ as $\sum_{h \le 0} g^{(h)}(x)$, with $g^{(h)}(x) = \gamma^{h(d/2-\epsilon)}g(\gamma^h x)$ and $g(x)$ the same as in (2.10); correspondingly, the fluctuation field $\psi$ is decomposed as $\psi = \sum_{h \le 0} \psi^{(h)}$ and the Grassmann Gaussian integration $d\mu_P(\psi)$ as $\prod_{h \le 0} d\mu_{g^{(h)}}(\psi^{(h)})$ (cf. with (2.11)), in terms of which we define the sequence (cf. with (2.13)):

$$e^{H_{\text{eff}}^{(h-1)}(\psi)} = \int d\mu_{g^{(h)}}(\psi^{(h)})e^{H_{\text{eff}}^{(h)}(\psi+\psi^{(h)})}, \quad h \le 0,$$

with $H_{\text{eff}}^{(0)} \equiv H^{(0)}$. After appropriate rescaling, we obtain the effective potentials (cf. with (2.16))

$$H^{(h)}(\psi) = H_{\text{eff}}^{(h)}(\gamma^{h[\psi]}\psi(\cdot, \gamma^h)),$$

which satisfy the RG equation

$$R[H^{(h)}] = H^{(h-1)},$$

with $R = R(\epsilon, \gamma)$ the same renormalization map introduced after (2.16). A mild generalization of the construction of this paper allows us to prove that, for $\lambda_0$ positive and sufficiently small (we are taking here $\epsilon$ positive and small, as well), there exists a choice of $\nu_0$ such that the whole sequence of effective potentials $\{H^{(h)}\}_{h \le 0}$ is well defined, they all belong to the same Banach space (the same we used to construct the fixed point) and $\lim_{h \to -\infty} H^{(h)} = H_*$, where $H_*$ is the fixed point constructed above. Correspondingly, the local part of $H^{(h)}$ is parametrized by two running coupling constants $\nu_h, \lambda_h$, which interpolate between the bare values $\nu_0, \lambda_0$ and the fixed point values $\nu = \lim_{h \to -\infty} \nu_h, \lambda = \lim_{h \to -\infty} \lambda_h$. In particular, the sequence $\{\nu_h, \lambda_h\}_{h \le 0}$ is small, uniformly in the scale index $h$.

Remarkably, the effective potentials $H^{(h)}$ can be expressed in terms of a convergent tree expansion, analogous to the one described above, with the important difference that now the endpoints carry a scale label and are, therefore, associated with couplings $\nu_k$ or
$\lambda_k$, computed at scales $h < k \le 0$. From the knowledge of the effective potentials, one can reconstruct all the observables one is interested in, including the correlation functions computed at arbitrary finite distance (before any scaling limit is taken): these will be expressed as convergent expansions in the whole sequence of running couplings {$\nu_h, \lambda_h$}$_{h \le 0}$. See, e.g., chapters 12, 13, 14 in [119]. Note that the existence of such a convergent expansion does not imply convergence of the naive perturbation theory in the bare couplings: schematically, the relation between $\lambda_h$ and $\lambda_0$ has the same features as the one between $\lambda(t)$ and $\lambda_0$ in (1.6); in particular, $\lambda_h$ is analytic in $\lambda_0$ non-uniformly in $h$, while it is Borel summable in $\lambda_0$ uniformly in $h$. Therefore, pre-scaling-limit observables are expected to be, at best, Borel summable in $\lambda_0$. On the contrary, observables at the fixed point, such as critical exponents, are expressed as convergent expansions in the fixed point couplings $\nu_*, \lambda_*$ only, and, therefore, recalling that $\nu_*$ and $\lambda_*$ are analytic in $\varepsilon$, they can be proved to be analytic functions of $\varepsilon$, as well.

## K Rigorously constructed bosonic fixed points

In this appendix we will mention some existing rigorous constructions of non-gaussian bosonic fixed points. Earlier works not directly focusing on such fixed points, but instrumental for acquiring rigorous RG control in bosonic theories, include [135–139].

In 1998, Brydges, Dimock and Hurd [22] gave the first construction of a fixed point in a bosonic scalar field theory with a long-range interactions. In analogy to (1.1), their bare action can be written schematically as

$$MFT(\varphi) + \nu \int d^d x \varphi^2 + \lambda \int d^d x \varphi^4, \qquad (K.1)$$

i.e. a gaussian scale-invariant Mean Field Theory of a bosonic field $\varphi$ in $\mathbb{R}^d$ of dimension $[\varphi] = d/4 - \varepsilon/2$ with a quadratic and quartic interactions. They considered the model in $d=4$, which necessitated adding to (K.1) one more relevant local interaction $\int d^d x (\partial \varphi)^2$. Contrary to what the title of [22] may suggest, it does not provide a rigorous definition of the Wilson-Fisher fixed point in $d = 4 - \varepsilon$. The two models differ already in their perturbative critical exponents. E.g. the scaling dimension of $\varphi$ gets corrections at $O(\varepsilon^2)$ in the Wilson-Fisher model, while such corrections are absent in the model of [22] at any order in $\varepsilon$.

In 2000, Mitter and Scoppola [140] studied a different model perturbing MFT by a $\delta$-function interaction:

$$MFT(\varphi) + g \int dx \delta^{(N)}(\varphi(x)), \qquad (K.2)$$

where $\varphi$ is an $N$-component field in $\mathbb{R}^{d=1}$. The $\delta$-function penalizes configurations when $\varphi(x)$ passes through zero, which physically describes repulsion of a polymer from an impurity. The scaling dimension of this interaction is $-N[\varphi]$. They constructed a fixed point of this model in the case when $[\varphi]$ is negative and small while $N$ is large so that $-N[\varphi] = 1 - \varepsilon$ is close to marginality.
In 2003, Brydges, Mitter, and Scoppola [23] constructed a fixed point of exactly the model (K.1). Following [140], they used fluctuation covariance of finite support in $x$-space, simplifying the proof compared to the construction in [22].<fnref n="68" /> Although nominally $d = 3$ in [23], the proof should apply also for $d = 1, 2$ [25, 141]. See also the nice review in [114]. Further work in this direction was done by Abdesselam [24] who constructed a full renormalization group trajectory from MFT at short distances to the fixed point of [23] at long distances. More recently, Slade [26] considered an analogous fixed point for an $n$-component field $\varphi$. He also considered the case $n = 0$, corresponding to the self-avoiding random walk. This formal limit is analyzed rigorously by considering a theory of two scalar bosons and two scalar fermions whose global symmetry is $OSp(2|2)$. Unlike in our model, there is no quartic interactions for fermions in [26] because there are only two of them. Such a “supersymmetric” model was also studied earlier by Mitter and Scoppola [142].

Physically, model (K.1) should describe the critical point of the long-range Ising model [93] (see also [143]), and much is known or conjectured about it. The critical point is expected to have conformal invariance [82]. At $\varepsilon = \varepsilon_c(d)$ the critical point should cross over to the local Wilson-Fisher fixed point plus a decoupled Gaussian sector [58]. See also [78] for higher-loop perturbative computations of critical exponents.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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