SISSA 02/2024/FISI

$\mathcal{N}=2$ SYK models with dynamical
bosons and fermions

Francesco Benini1,2,3, Tomás Reis1,2, Saman Soltani1,2, Ziruo Zhang1,2,4

1 SISSA, Via Bonomea 265, 34136 Trieste, Italy
2 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
3 ICTP, Strada Costiera 11, 34151 Trieste, Italy
4 Kavli Institute for Theoretical Sciences (KITS), University of the Chinese
Academy of Sciences, Beijing 100190, China

Abstract

We study a class of SYK models with $\mathcal{N}=2$ supersymmetry, described
by $N$ fermions in chiral Fermi multiplets, as well as $\alpha N$ first-order
bosons in chiral multiplets. The interactions are characterised by two
integers $(p,q)$. We focus on the large $N$ and low energy limit of these
models. Despite the presence of dynamical bosons, we find conformal behaviour
akin to the standard SYK model. We use $\mathcal{I}$-extremization of a Witten
index to study the supersymmetric solutions. In particular, we find an exact
expression for the entropy, which matches the numerical solutions to the
Schwinger–Dyson equations. We further solve the model both in the large $p$
and large $p,q$ limits. Numerically, we verify our analytical results and
obtain estimates for the Schwarzian coupling in the near zero-temperature
limit. We also study the low-lying spectrum of operators to determine the
parameter ranges where the Schwarzian mode dominates the IR dynamics. Lastly,
we study out-of-time-ordered correlators to show that the model is maximally
chaotic.

###### Contents

  1. 1 Introduction and summary
  2. 2 $\boldsymbol{\mathcal{N}=2}$ SYK models with dynamical bosons
  3. 3 Solution in the annealed approximation
    1. 3.1 Equations of motion at large $N$
    2. 3.2 Conformal solutions
    3. 3.3 Superconformal solutions
    4. 3.4 Existence of superconformal solutions at fixed $p$ and $q$
    5. 3.5 Solutions at large $p$ and fixed $q$
    6. 3.6 Solutions at large $p$ and $q$
    7. 3.7 Luttinger–Ward relation
    8. 3.8 Non-conformal solutions
    9. 3.9 Witten index and $\mathcal{I}$-extremization
  4. 4 Numerical results
    1. 4.1 Summary of the numerical method
    2. 4.2 Conformal behaviour for $(p,q)=(1,2)$
    3. 4.3 Charge and entropy for $(p,q)=(1,2)$
    4. 4.4 Results for other values of $(p,q)$
  5. 5 Spectrum of low-lying operators
    1. 5.1 Expanding the action
    2. 5.2 Computing the spectrum
  6. 6 Chaos exponents
    1. 6.1 Integral equations for component 4-point functions
    2. 6.2 OTOCs and chaos exponents
    3. 6.3 The retarded kernel
    4. 6.4 Exponents from the fermionic and bosonic retarded kernels
  7. 7 Discussion
  8. A Superspace conventions
    1. A.1 Super-reparametrizations
    2. A.2 Superconformal algebra
  9. B Large $\boldsymbol{N}$ action and EOMs in components
  10. C IR conformal ansatz and UV limit
    1. C.1 Reality of Euclidean $2$-point functions
    2. C.2 Conformal ansatz with chemical potential
    3. C.3 UV limits
  11. D Luttinger–Ward identities
  12. E Kernel components and action on eigenfunctions
    1. E.1 Euclidean kernel components
    2. E.2 Retarded kernel components
  13. F Ladder diagrams for 4-point functions

##  1 Introduction and summary

The quantum-mechanical models introduced by Sachdev, Ye, and Kitaev (SYK) [1,
2] have the properties of being solvable at strong coupling in the large $N$
limit, exhibiting maximally chaotic behaviour, and developing an emergent
conformal symmetry in the infrared (IR) [1, 3, 4, 2, 5, 6, 7]. Their IR
physics is dominated by a pseudo Goldstone boson, the Schwarzian mode, which
is dual to the two-dimensional JT gravity in AdS2. Given the universal
appearance of near-AdS2 throats in the near-horizon region of near-extremal
black holes, the SYK model poses itself as a model of quantum black-hole
physics: it captures the dynamics of a universal “breathing” mode that lives
in the near-horizon region [8, 9]. Many variations on the SYK model have been
studied, including [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28] which are close to the subject of this paper.

An interesting question is whether one can microscopically derive a certain
type of SYK-like models directly from gravity or string theory [10]. A step in
that direction was made in [29] in the context of holography and black holes
in higher-dimensional AdS space. The quantum mechanical model found in [29]
presents many peculiar features, e.g., it has $\mathcal{N}=2$ supersymmetry,
it describes both dynamical fermions and bosons belonging to many species with
unequal abundance in a large $N$ limit, it enjoys multiple Abelian symmetries.

With that motivation in mind, in this paper we study a class of SYK-like
models with $\mathcal{N}=2$ supersymmetry, made of $N$ regular fermions in
chiral Fermi multiplets, as well as $\alpha N$ bosons in chiral multiplets
with one-derivative kinetic terms (a.k.a. first-order bosons), in the large
$N$ limit. The parameter $\alpha$ controls the relative abundance. The
interactions are described by “superpotential” $J$-terms (in the language of
[30]), are controlled by two positive integers $(p,q)$ (where $p$ is odd), and
may include — depending on $p,q$ — a scalar potential and Yukawa interactions.
As in SYK, the interactions are all-to-all and given by random variables that
represent quenched disorder.

Our model has some similarities to the $\mathcal{N}=2$ two-fermion model of
[26]. Indeed our model has both a $U(1)_{R}$ R-symmetry and a $U(1)_{F}$
flavor symmetry, and we study the model with chemical potentials turned on,
finding similar physical properties. We are particularly interested in the
role played by dynamical bosons, which could lead to a variety of phenomena
such as spin-glass phases or condensation, see e.g. [31, 32, 33, 34, 35, 28].
In our model, instead, bosons turn out to be tamed and lead to conformal
solutions — both with and without supersymmetry — similarly to the standard
SYK model. This implies that our $(p,q)$ models retain the main features
expected from near-BPS black hole horizons, yet accommodating the presence of
dynamical bosons, and make us confident that also the more convoluted, but
also more realistic, quantum mechanical models of [29] might exhibit the same
properties.

Let us summarize our results. In Section 2 we present our model. In Section 3
we solve the model in the annealed approximation and at leading order in
$1/N$, in terms of Schwinger–Dyson equations. In particular we verify that the
system admits both supersymmetric and non-supersymmetric conformal solutions
in the IR. For certain values of the chemical potentials, we also find some
non-conformal solutions in which the correlators behave as those of gapped
free fields. Supersymmetric solutions at zero temperature can also be studied
through a symmetry-refined Witten index. From it, employing
$\mathcal{I}$-extremization [36, 37] we extract the zero-temperature entropy
and the R-symmetry charge of the vacuum. Both are compatible with our
numerical computations and with the Luttinger–Ward relation [38]. The entropy
computation gives support to the claim that the model is conformal in the IR,
rather than in a spin-glass phase. We analyse the system at large $p$, as well
as at large $p$ and $q$. In these limits we are able to reach an intermediate
low-energy regime which is conformal, but not the very low-energy behaviour,
which is also conformal but with different spectrum. The two agree only in the
supersymmetric case.

In Section 4 we perform various numerical analyses of our model, both for
supersymmetric and non-supersymmetric values of the chemical potentials, for
different values of $(p,q)$. In particular we solve the Schwinger–Dyson
equations, confirming the IR conformal behaviour found analytically. We test
the Luttinger–Ward relations and the computation of the zero-temperature
entropy. Going to non-zero temperature allows us to extract the Schwarzian
coupling.

In Section 5 we study the low-lying spectrum of operator dimensions. Besides
observing the presence of the Schwarzian mode, the two currents, and their
superpartners, we determine in which window of parameters we expect the
Schwarzian mode to dominate the IR dynamics, and identify windows in which
other modes are expected to dominate. In Section 6 we extract the Lyapunov
exponents from the out-of-time-ordered (OTOC) 4-point functions, finding that
the model is maximally chaotic (it saturates the bound of [39]).

We conclude in Section 7 by presenting a few puzzling results we hope to
resolve in future work. We provide many appendices with the technical details
of computations.

##  2 $\boldsymbol{\mathcal{N}=2}$ SYK models with dynamical bosons

We are interested in a quantum mechanical model with $\mathcal{N}=2$
supersymmetry (one complex supercharge) described by $N$ Fermi multiplets
$\mathcal{Y}_{a}$ ($a=1,\dots,N$) and $\alpha N$ chiral multiplets $\Phi_{b}$
($b=1,\dots,\alpha N$), where $\alpha$ is a positive real constant that we
call the abundance parameter.111At finite $N$ the parameter $\alpha$ is
quantized such that $\alpha N\in\mathbb{Z}$, but at large $N$, $\alpha$ is
essentially continuous. The field content of a Fermi multiplet $\mathcal{Y}$
is a dynamical fermion $\eta$ and an auxiliary scalar $f$, while that of a
chiral multiplet $\Phi$ is, in our model, a dynamical boson $\phi$ (with one-
derivative kinetic term) and an auxiliary fermion $\psi$. In the Lagrangian
formulation and in Lorentzian signature, the model is described by the action:

$\displaystyle S_{\text{L}}$
$\displaystyle=\int\\!dt\,\biggl{[}i\eta_{a}^{\dagger}\bigl{(}\partial_{t}+i\mu_{\eta}\bigr{)}\eta_{a}+f_{a}^{\dagger}f_{a}+i\phi_{b}^{\dagger}\bigl{(}\partial_{t}+i\mu_{\phi}\bigr{)}\phi_{b}+\psi_{b}^{\dagger}\psi_{b}$
(2.1) $\displaystyle\qquad\quad-J_{a_{1}\ldots a_{p}b_{1}\ldots
b_{q}}\Bigl{(}p\,f_{a_{1}}\eta_{a_{2}}\cdots\eta_{a_{p}}\phi_{b_{1}}\cdots\phi_{b_{q}}+q\,\eta_{a_{1}}\cdots\eta_{a_{p}}\psi_{b_{1}}\phi_{b_{2}}\cdots\phi_{b_{q}}\Bigr{)}+\text{h.c.}\biggr{]}.$

We use the convention that repeated indices are summed over. These so-called
J-term interactions in the second line are inspired from the ones obtained in
[29], corresponding here to the $p=1$, $q=2$ case. The couplings
$J_{a_{1}\ldots a_{p}b_{1}\ldots b_{q}}$ are complex Gaussian random variables
with zero mean and variance

$\bigl{\langle}J^{\phantom{*}}_{a_{1}\ldots a_{p}b_{1}\ldots
b_{q}}J^{*}_{c_{1}\ldots c_{p}d_{1}\ldots
d_{q}}\bigr{\rangle}=\frac{J}{p\,q\,N^{p+q-1}}\,\delta_{a_{1}\ldots
a_{p}}^{[c_{1}\ldots c_{p}]}\,\delta_{b_{1}\ldots b_{q}}^{(d_{1}\ldots
d_{q})}\;.$ (2.2)

Here $J$ has dimension of mass, and is kept fixed in the large $N$ limit so
that the averaged partition function has uniform scaling. The couplings are
antisymmetric in the first $p$ indices and symmetric in the next $q$ indices.
Note that $p$ is odd, while $q$ is any positive integer. In (2.1) we have also
included the chemical potentials $\mu_{\phi}$, $\mu_{\eta}$. Under charge
conjugation

$\eta_{a}\mapsto\eta_{a}^{\dagger}\;,\quad f_{a}\mapsto
f_{a}^{\dagger}\;,\quad\phi_{b}\mapsto\phi_{b}^{\dagger}\;,\quad\psi_{b}\mapsto\psi_{b}^{\dagger}\;,\quad\mu_{\eta}\mapsto-\mu_{\eta}\;,\quad\mu_{\phi}\mapsto-\mu_{\phi}\;,$
(2.3)

the Fermi multiplet kinetic terms are invariant, while the chiral multiplet
kinetic terms are odd and thus they break charge conjugation symmetry.

Integrating out the auxiliary fields, the action becomes

$\displaystyle S_{\text{L}}$
$\displaystyle=\int\\!dt\,\biggl{[}i\eta_{a}^{\dagger}\bigl{(}\partial_{t}+i\mu_{\eta}\bigr{)}\eta_{a}+i\phi_{b}^{\dagger}\bigl{(}\partial_{t}+i\mu_{\phi}\bigr{)}\phi_{b}$
(2.4) $\displaystyle\qquad\quad-\Bigl{\lvert}p\,J_{a_{1}\ldots
a_{p}b_{1}\ldots
b_{q}}\eta_{a_{2}}\cdots\eta_{a_{p}}\phi_{b_{1}}\cdots\phi_{b_{q}}\Bigr{\rvert}^{2}-\Bigl{\lvert}q\,J_{a_{1}\ldots
a_{p}b_{1}\ldots
b_{q}}\eta_{a_{1}}\cdots\eta_{a_{p}}\phi_{b_{2}}\cdots\phi_{b_{q}}\Bigr{\rvert}^{2}\biggr{]}\,.$

In the Hamiltonian formulation, the bosonic conjugate momentum to $\phi$ is
$\Pi_{\phi}=i\phi^{\dagger}$ and therefore the (anti)commutation relations are
$\\{\eta_{a},\eta^{\dagger}_{c}\\}=\delta_{ac}$ and
$[\phi_{b},\phi^{\dagger}_{d}]=\delta_{bd}$ (all other ones vanishing). The
Hamiltonian is $\widetilde{H}=H+\mu_{\eta}Q_{\eta}+\mu_{\phi}Q_{\phi}$, where

$H=\bigl{\lvert}p\,J_{a_{1}\ldots a_{p}b_{1}\ldots
b_{q}}\eta_{a_{2}}\cdots\eta_{a_{p}}\phi_{b_{1}}\cdots\phi_{b_{q}}\bigr{\rvert}^{2}+\bigl{\lvert}q\,J_{a_{1}\ldots
a_{p}b_{1}\ldots
b_{q}}\eta_{a_{1}}\cdots\eta_{a_{p}}\phi_{b_{2}}\cdots\phi_{b_{q}}\bigr{\rvert}^{2}$
(2.5)

is the interacting Hamiltonian with chemical potentials turned off, while
$Q_{\eta}=\frac{1}{2}[\eta_{a}^{\dagger},\eta_{a}]$ and
$Q_{\phi}=\frac{1}{2}\\{\phi_{b}^{\dagger},\phi_{b}\\}$ are the charge
operators. The ordering ambiguities in $H$ are fixed by supersymmetry by
requiring that $H=\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}$ in terms of the
supercharge

$\mathcal{Q}\,\sim\,J_{a_{1}\ldots a_{p}b_{1}\ldots
b_{q}}\eta_{a_{1}}\cdots\eta_{a_{p}}\phi_{b_{1}}\cdots\phi_{b_{q}}\;.$ (2.6)

Each bosonic degree of freedom $\phi$ describes a particle in magnetic field
(see, e.g., [40]). Upon canonical quantization, $Q_{\phi}\geq\frac{1}{2}$ and
the bosonic Hilbert space is a Fock space with lowest weight state $|0\rangle$
defined by $\phi|0\rangle=0$ and generated by the creation operator
$\phi^{\dagger}$. For $p=1$ the Hamiltonian $H$ includes a positive-definite
scalar potential for $\phi$, and $\widetilde{H}$ is bounded from below for any
given value of $\mu_{\phi}$ and finite $N$. However $J_{a_{1}\ldots
a_{p}b_{1}\ldots b_{q}}$ goes to zero at large $N$, and thus one should take
$\mu_{\phi}>0$ in order to avoid instabilities (as we will see, the limit
$\mu_{\phi}\to 0$ is stable after taking the large $N$ limit). For $p>1$, $H$
has flat directions and thus one should restrict to $\mu_{\phi}>0$ even at
finite $N$.

To go to Euclidean signature, we set $\tau=it$ and define
$iS_{\text{L}}=-S_{\text{E}}$ (we also use a bar in place of ${\dagger}$).
Then the Euclidean Lagrangian in components reads:

$\displaystyle\mathcal{L}_{\text{E}}$
$\displaystyle=\overline{\eta}_{a}\bigl{(}\partial_{\tau}+\mu_{\eta}\bigr{)}\eta_{a}-\overline{f}_{a}f_{a}+\overline{\phi}_{b}\bigl{(}\partial_{\tau}+\mu_{\phi}\bigr{)}\phi_{b}-\overline{\psi}_{b}\psi_{b}$
(2.7) $\displaystyle\quad+J_{a_{1}\ldots a_{p}b_{1}\ldots
b_{q}}\Bigl{(}p\,f_{a_{1}}\eta_{a_{2}}\cdots\eta_{a_{p}}\phi_{b_{1}}\cdots\phi_{b_{q}}+q\,\eta_{a_{1}}\cdots\eta_{a_{p}}\psi_{b_{1}}\phi_{b_{2}}\cdots\phi_{b_{q}}\Bigr{)}+\text{h.c.}\;.$

It will be convenient to work in superspace with coordinates
$(\tau,\theta,\overline{\theta})$ and use the superfields

$\Phi=\phi+\theta\psi+\frac{1}{2}\theta\overline{\theta}\,(\partial_{\tau}+\mu_{\phi})\phi\;,\qquad\qquad\mathcal{Y}=\eta-\theta
f+\frac{1}{2}\theta\overline{\theta}\,(\partial_{\tau}+\mu_{\eta})\eta\;.$
(2.8)

Our conventions for supersymmetry and superspace are in Appendix A, while we
refer to [29, 41] for more detailed presentations. Using superspace integrals,
the Euclidean action is

$S_{\text{E}}=-\int\\!d^{3}T\,\bigl{(}\overline{\mathcal{Y}}_{a}\mathcal{Y}_{a}+\overline{\Phi}_{b}\Phi_{b}\bigr{)}+J_{a_{1}\ldots
a_{p}b_{1}\ldots
b_{q}}\int\\!d^{2}T\;\mathcal{Y}_{a_{1}}\cdots\mathcal{Y}_{a_{p}}\Phi_{b_{1}}\cdots\Phi_{b_{q}}+\text{h.c.}\;.$
(2.9)

In the following we will mostly work in Euclidean signature and drop the
subscript E.

$\begin{array}[]{c | c c | c c}\hline\cr\hline\cr&Q_{\eta}&Q_{\phi}&Q_{F}&R\\\ \hline\cr\eta&1&0&q&R[\eta]\\\ f&1-p&-q&q&R[\eta]-1\\\\[1.99997pt] \phi&0&1&-p&R[\phi]\\\ \psi&-p&1-q&-p&R[\phi]-1\\\ \hline\cr\hline\cr\end{array}$ Table 1: Charge assignments under the continuous global symmetries $\mathrm{U}(1)^{2}$, using two alternative parametrizations. The R-charge assignments satisfy $p\,R[\eta]+q\,R[\phi]=1$.

The group of continuous global symmetries is $\mathrm{U}(1)^{2}$, which we can
parametrize by the two charges $Q_{\eta}$, $Q_{\phi}$ with charge assignments
as in Table 1. An alternative parametrization is
$\mathrm{U}(1)_{F}\times\mathrm{U}(1)_{R}$ with charges
$Q_{F}=q\,Q_{\eta}-p\,Q_{\phi}$ and $R=R[\eta]\,Q_{\eta}+R[\phi]\,Q_{\phi}$
that satisfy $p\,R[\eta]+q\,R[\phi]=1$. Here $Q_{F}$ is a flavour symmetry
that commutes with supersymmetry, whilst $R$ is a generic R-symmetry. For
future reference, $Q_{\eta}$ and $Q_{\phi}$ can be written in terms of $Q_{F}$
and $R$ as

$Q_{\eta}=p\,R+R[\phi]\,Q_{F}\;,\qquad\qquad Q_{\phi}=q\,R-R[\eta]\,Q_{F}\;.$
(2.10)

When $d\equiv\gcd(p,q)$ is bigger than 1, there is also a discrete flavour
symmetry $\mathbb{Z}_{d}$. Indeed consider the discrete symmetry $g$ that acts
as

$g:\qquad\mathcal{Y}_{a}\;\mapsto\;e^{\frac{2\pi
i}{p}}\;\mathcal{Y}_{a}\;,\qquad\quad\Phi_{b}\;\mapsto\;\Phi_{b}\;.$ (2.11)

One can check222Indeed let $\ell=\bigl{(}q/d\bigr{)}{}^{-1}_{\text{mod }p/d}$,
then a rotation $\mathrm{U}(1)_{F}$ by an angle $\alpha=2\pi\ell/p$ acts as
$g^{d}$. that $g^{d}\in\mathrm{U}(1)_{F}$. Besides, one can write the fermion
parity operator as

$(-1)^{F}=g^{\frac{p-1}{2}}e^{\pi iR_{0}}\;,$ (2.12)

where $R_{0}$ is a reference R-symmetry with assignments
$R_{0}[\eta]=\frac{1}{p}$, $R_{0}[\phi]=0$. When $d=1$ this is $e^{\pi iR}$
for some R-symmetry $R$ that is a linear combination of $R_{0}$ and $Q_{F}$,
but otherwise it is not.

The relations between chemical potentials, according to Table 1, are

$\mu_{F}=R[\phi]\,\mu_{\eta}-R[\eta]\,\mu_{\phi}\;,\qquad\qquad\mu_{R}=p\,\mu_{\eta}+q\,\mu_{\phi}\;.$
(2.13)

In particular, whenever we want to impose $\mu_{R}=0$ because of
supersymmetry, we also fix $\mu_{\eta}=-q\mu_{\phi}/p$ and
$\mu_{F}=-\mu_{\phi}/p$, and therefore we should restrict to $\mu_{\eta}<0$
and $\mu_{F}<0$.

##  3 Solution in the annealed approximation

Having established the model, we want to explore its large $N$ dynamics. When
averaging over the couplings, particularly when focusing on correlators and
4-point functions, it is in principle preferable to first calculate the
observables in one instance of the model, and then average. This is usually
called quenched disorder, which is analytically very hard to control. The
approach we follow here is that of annealed disorder, in which we calculate
the averaged action first and then derive the observables from it. For
quantities such as correlators and the entropy, those two approaches can lead
to different results. In the standard SYK model, many observables are found to
be “self-averaging” which means that the differences between distinct
averaging schemes are subleading in $1/N$. We expect a similar behaviour in
our model. A crucial counterexample would be if the entropy calculation were
dominated by a replica-breaking solution, which is not captured by the
annealed approximation. The fact that our model contains fermions might be
crucial to rule this out, as argued in [33]. In the rest of the paper we will
use annealed disorder, assuming that it approximates the quenched system to
leading order in $1/N$.

###  3.1 Equations of motion at large $\boldsymbol{N}$

We follow the standard steps performed in [11, 26] to derive the equations of
motion at large $N$. For simplicity, we start with no chemical potentials
turned on, and perform our manipulations in superspace. The corresponding
expressions in components, and including non-vanishing chemical potentials,
can be found in Appendix B.

After averaging over $J_{a_{1}\ldots a_{p}b_{1}\ldots b_{q}}$, the partition
function is

$\displaystyle\langle
Z\rangle=\int\\!\mathcal{D}\mathcal{Y}_{a}\,\mathcal{D}\Phi_{b}\,\exp\biggl{[}\,\int\\!d^{3}T\,\Bigl{(}\overline{\mathcal{Y}}_{a}\mathcal{Y}_{a}+\overline{\Phi}_{b}\Phi_{b}\Bigr{)}+\frac{J}{p\,q\,N^{p+q-1}}\int\\!d^{2}T_{2}\,d^{2}\overline{T}_{1}\times{}$
(3.1) $\displaystyle\hskip
50.00008pt{}\times\overline{\mathcal{Y}}_{a_{1}}(T_{1})\,\mathcal{Y}_{a_{1}}(T_{2})\,\cdots\,\overline{\mathcal{Y}}_{a_{p}}(T_{1})\,\mathcal{Y}_{a_{p}}(T_{2})\,\overline{\Phi}_{b_{1}}(T_{1})\,\Phi_{b_{1}}(T_{2})\,\cdots\,\overline{\Phi}_{b_{q}}(T_{1})\,\Phi_{b_{q}}(T_{2})\biggr{]}\;.$

We define the bilocal fields

$\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})\,\equiv\,\frac{1}{N}\sum\nolimits_{a}\overline{\mathcal{Y}}_{a}(T_{1})\,\mathcal{Y}_{a}(T_{2})\;,\qquad\quad\mathcal{G}_{\Phi}(T_{1},T_{2})\,\equiv\,\frac{1}{N}\sum\nolimits_{b}\overline{\Phi}_{b}(T_{1})\,\Phi_{b}(T_{2})\;,$
(3.2)

which are chiral in $T_{2}$ and anti-chiral in $T_{1}$. They encode the two-
point functions of the model. We introduce them — together with the bilocal
Lagrange multipliers $\Sigma_{\mathcal{Y}}$ and $\Sigma_{\Phi}$ — into the
action by inserting the following identity in the path integral:

$1=\\!\int\\!\mathcal{D}\mathcal{G}_{\mathcal{Y}}\mathcal{D}\Sigma_{\mathcal{Y}}\mathcal{D}\mathcal{G}_{\Phi}\mathcal{D}\Sigma_{\Phi}\exp\biggl{\\{}{-}N\\!\int\\!d^{2}T_{2}\,d^{2}\overline{T}_{1}\;\Sigma_{\mathcal{Y}}(T_{1},T_{2})\biggl{[}\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})-\frac{1}{N}\overline{\mathcal{Y}}_{a}(T_{1})\mathcal{Y}_{a}(T_{2})\biggr{]}\\\
-N\\!\int\\!d^{2}T_{2}\,d^{2}\overline{T}_{1}\;\Sigma_{\Phi}(T_{1},T_{2})\biggl{[}\mathcal{G}_{\Phi}(T_{1},T_{2})-\frac{1}{N}\overline{\Phi}_{b}(T_{1})\Phi_{b}(T_{2})\biggr{]}\biggr{\\}}\,.$
(3.3)

At this stage, $\mathcal{G}_{\mathcal{Y},\Phi}(T_{1},T_{2})$ and
$\Sigma_{\mathcal{Y},\Phi}(T_{1},T_{2})$ are generic superfields which are
chiral in $T_{2}$ and anti-chiral in $T_{1}$. Each of them is specified by
four independent functions of $\tau_{1,2}$ according to the expansions in
(B.1). The components of $\Sigma_{\mathcal{Y},\Phi}$ encode the self-energies
of the fields in the model. The kinetic term of $\mathcal{Y}_{a}$ can be
rewritten in a bilocal way using the identity

$\int\\!d^{3}T\;\overline{\mathcal{Y}}_{a}\mathcal{Y}_{a}=\int\\!d^{2}T_{2}\,d^{2}\overline{T}_{1}\;\overline{\mathcal{Y}}_{a}(T_{1})\,D_{1}\overline{D}_{2}\delta^{3}(T_{1}-T_{2})\,\mathcal{Y}_{a}(T_{2})\;,$
(3.4)

and similarly for $\Phi_{b}$. Here
$\delta^{3}(T_{1}-T_{2})\equiv(\overline{\theta}_{1}-\overline{\theta}_{2})(\theta_{1}-\theta_{2})\delta(\tau_{1}-\tau_{2})$
is the superspace delta function, while $D_{1}$ and $\overline{D}_{2}$ are
(anti-)chiral superspace derivatives with respect to $T_{1}$ and $T_{2}$,
respectively (see Appendix A). For concreteness, we have

$\displaystyle D_{1}\overline{D}_{2}\delta^{3}(T_{1}-T_{2})$
$\displaystyle=\delta(\tau_{1}-\tau_{2})-\frac{\theta_{1}\overline{\theta}_{1}+\theta_{2}\overline{\theta}_{2}-2\theta_{2}\overline{\theta}_{1}}{2}\partial_{\tau_{1}}\delta(\tau_{1}-\tau_{2})-\frac{\theta_{1}\theta_{2}\overline{\theta}_{1}\overline{\theta}_{2}}{4}\partial_{\tau_{1}}^{2}\delta(\tau_{1}-\tau_{2})$
$\displaystyle=\delta\Bigl{(}\tau_{1}-\tau_{2}-\tfrac{1}{2}\,\theta_{1}\overline{\theta}_{1}-\tfrac{1}{2}\,\theta_{2}\overline{\theta}_{2}+\theta_{2}\overline{\theta}_{1}\Bigr{)}\;.$
(3.5)

The fields $\mathcal{Y}_{a}$ and $\Phi_{b}$ only appear quadratically in the
action and can be integrated out, producing Berezinian (a.k.a.
superdeterminant) factors in the path integral. We are left with an expression
for the annealed partition function in terms of the bilocal fields:

$\displaystyle\langle Z\rangle$
$\displaystyle=\int\\!\mathcal{D}\mathcal{G}_{\mathcal{Y}}\,\mathcal{D}\Sigma_{\mathcal{Y}}\,\mathcal{D}\mathcal{G}_{\Phi}\,\mathcal{D}\Sigma_{\Phi}\,\exp\biggl{\\{}-N\biggl{[}-\log\operatorname{Ber}\Bigl{(}D_{1}\overline{D}_{2}\delta^{3}(T_{1}-T_{2})+\Sigma_{\mathcal{Y}}(T_{1},T_{2})\Bigr{)}$
$\displaystyle\quad+\alpha\log\operatorname{Ber}\Bigl{(}D_{1}\overline{D}_{2}\delta^{3}(T_{1}-T_{2})+\Sigma_{\Phi}(T_{1},T_{2})\Bigr{)}+\\!\int\\!d^{2}T_{2}\,d^{2}\overline{T}_{1}\,\biggl{(}\Sigma_{\mathcal{Y}}(T_{1},T_{2})\,\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})$
$\displaystyle\quad+\Sigma_{\Phi}(T_{1},T_{2})\,\mathcal{G}_{\Phi}(T_{1},T_{2})-\frac{J}{pq}\,\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})^{p}\,\mathcal{G}_{\Phi}(T_{1},T_{2})^{q}\biggr{)}\biggr{]}\biggr{\\}}\;.$
(3.6)

Here Ber is the Berezinian (or superdeterminant), whose definition is given in
(A.13).

Let us write down the equations of motion for the bilocal fields in terms of
superfields. Extremizing (3.6) with respect to
$\mathcal{G}_{\mathcal{Y},\Phi}$ leads to the algebraic equations:

$\Sigma_{\mathcal{Y}}(T_{1},T_{2})=\frac{J}{q}\,\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})^{p-1}\,\mathcal{G}_{\Phi}(T_{1},T_{2})^{q}\,,\;\;\Sigma_{\Phi}(T_{1},T_{2})=\frac{J}{p}\,\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})^{p}\,\mathcal{G}_{\Phi}(T_{1},T_{2})^{q-1}\,.$
(3.7)

Extremizing with respect to $\Sigma_{\mathcal{Y},\Phi}$ leads to the integro-
differential equations:

$\displaystyle
D_{2}\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})+\int\\!d^{2}T_{3}\;\Sigma_{\mathcal{Y}}(T_{2},T_{3})\,\mathcal{G}_{\mathcal{Y}}(T_{1},T_{3})$
$\displaystyle=\overline{\delta}^{2}(T_{1},T_{2})\;,$ (3.8) $\displaystyle
D_{2}\mathcal{G}_{\Phi}(T_{1},T_{2})+\int\\!d^{2}T_{3}\;\Sigma_{\Phi}(T_{2},T_{3})\,\mathcal{G}_{\Phi}(T_{1},T_{3})$
$\displaystyle=-\alpha\,\overline{\delta}^{2}(T_{1},T_{2})\;.$

If the derivative terms $D_{2}\mathcal{G}_{\mathcal{Y},\Phi}$ are dropped, one
can show that (3.7)–(3.8) are invariant under super-reparametrizations
$\text{Diff}^{+}(S^{1|2})$ defined in (A.11), where the superfields transform
as

$\displaystyle\mathcal{G}_{\mathcal{Y},\Phi}(T_{1},T_{2})$
$\displaystyle\mapsto\;\bigl{(}D\theta^{\prime}_{1}\bigr{)}{}^{2\Delta_{\eta,\phi}}\;\bigl{(}-\overline{D}\,\overline{\theta}{}_{2}^{\prime}\bigr{)}{}^{2\Delta_{\eta,\phi}}\;\mathcal{G}_{\mathcal{Y},\Phi}(T^{\prime}_{1},T^{\prime}_{2})$
(3.9) $\displaystyle\Sigma_{\mathcal{Y},\Phi}(T_{1},T_{2})$
$\displaystyle\mapsto\;\bigl{(}D\theta^{\prime}_{1}\bigr{)}{}^{1-2\Delta_{\eta,\phi}}\bigl{(}-\overline{D}\,\overline{\theta}{}_{2}^{\prime}\bigr{)}{}^{1-2\Delta_{\eta,\phi}}\,\Sigma_{\mathcal{Y},\Phi}(T^{\prime}_{1},T^{\prime}_{2})$

with $p\,\Delta_{\eta}+q\,\Delta_{\phi}=\frac{1}{2}$. The chiral measure and
the (anti)chiral delta functions transform as in (A.12) and (A.15). This shows
that $\eta$ and $\phi$ are chiral primary operators under
$\mathrm{SU}(1,1|1)\subset\text{Diff}^{+}(S^{1|2})$. While (3.8) is anti-
chiral in $T_{1,2}$, there is an equivalent way to write the equations using
chiral superfields:

$\displaystyle-\overline{D}_{1}\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})+\int\\!d^{2}\overline{T}_{3}\;\Sigma_{\mathcal{Y}}(T_{3},T_{1})\,\mathcal{G}_{\mathcal{Y}}(T_{3},T_{2})$
$\displaystyle=\delta^{2}(T_{1},T_{2})\;,$ (3.10)
$\displaystyle-\overline{D}_{1}\mathcal{G}_{\Phi}(T_{1},T_{2})+\int\\!d^{2}\overline{T}_{3}\;\Sigma_{\Phi}(T_{3},T_{1})\,\mathcal{G}_{\Phi}(T_{3},T_{2})$
$\displaystyle=-\alpha\,\delta^{2}(T_{1},T_{2})\;.$

The equations (3.8) and (3.10) are derived by extremizing Ber with respect to
$\Sigma_{\mathcal{Y},\Phi}$ in (3.6). Indeed, treating it formally like any
other operator determinant, one gets:

$\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})=\bigl{(}D\overline{D}\delta^{3}+\Sigma_{\mathcal{Y}}\bigr{)}^{-1}(T_{1},T_{2})\,,\quad\mathcal{G}_{\Phi}(T_{1},T_{2})=-\alpha\,\bigl{(}D\overline{D}\delta^{3}+\Sigma_{\Phi}\bigr{)}^{-1}(T_{1},T_{2})\,.$
(3.11)

By convoluting with either the chiral or anti-chiral coordinate of
$\bigl{(}D\overline{D}\delta^{3}+\Sigma_{\mathcal{Y},\Phi}\bigr{)}$ and given
the property (A.7), one obtains (3.8) or (3.10).

###  3.2 Conformal solutions

We now assume that, at low energies, the kinetic terms are negligible and thus
that the integro-differential equations (3.8) can be approximated by dropping
the first term on the left-hand-side. Besides, assuming that the fermion
number symmetry remains unbroken, we search for solutions with vanishing mixed
fermionic components in $\mathcal{G}$ and $\Sigma$, i.e., we impose

$G_{\bar{\eta}f}=G_{\bar{f}\eta}=G_{\bar{\phi}\psi}=G_{\bar{\psi}\phi}=\Sigma_{\bar{\eta}f}=\Sigma_{\bar{f}\eta}=\Sigma_{\bar{\phi}\psi}=\Sigma_{\bar{\psi}\phi}=0\;.$
(3.12)

One can check that this is a consistent ansatz. The equations of motion for
the remaining bosonic components are:

$\displaystyle\Sigma_{\bar{\eta}\eta}(\tau_{1},\tau_{2})$
$\displaystyle=J\,\biggl{(}\frac{p-1}{q}\,G_{\bar{\phi}\phi}\,G_{\bar{f}f}+G_{\bar{\eta}\eta}\,G_{\bar{\psi}\psi}\biggr{)}\,G_{\bar{\eta}\eta}^{\phantom{\bar{\eta}\eta}p-2}\,G_{\bar{\phi}\phi}^{\phantom{\bar{\phi}\phi}q-1}\;,$
(3.13) $\displaystyle\Sigma_{\bar{\phi}\phi}(\tau_{1},\tau_{2})$
$\displaystyle=J\,\biggl{(}G_{\bar{\phi}\phi}\,G_{\bar{f}f}+\frac{q-1}{p}\,G_{\bar{\eta}\eta}\,G_{\bar{\psi}\psi}\biggr{)}\,G_{\bar{\eta}\eta}^{\phantom{\bar{\eta}\eta}p-1}\,G_{\bar{\phi}\phi}^{\phantom{\bar{\phi}\phi}q-2}\;,$
$\displaystyle\Sigma_{\bar{f}f}(\tau_{1},\tau_{2})$
$\displaystyle=\frac{J}{q}\,G_{\bar{\eta}\eta}^{\phantom{\bar{\eta}\eta}p-1}\,G_{\bar{\phi}\phi}^{\phantom{\bar{\phi}\phi}q}\;,\qquad\qquad\Sigma_{\bar{\psi}\psi}(\tau_{1},\tau_{2})=\frac{J}{p}\,G_{\bar{\eta}\eta}^{\phantom{\bar{\eta}\eta}p}\,G_{\bar{\phi}\phi}^{\phantom{\bar{\phi}\phi}q-1}\;,$

where all functions are of $(\tau_{1},\tau_{2})$, and

$\displaystyle\int\\!d\tau_{3}\,\Sigma_{\bar{\eta}\eta}(\tau_{2},\tau_{3})\,G_{\bar{\eta}\eta}(\tau_{1},\tau_{3})$
$\displaystyle=\delta(\tau_{1}{-}\tau_{2}),\;\;\int\\!d\tau_{3}\,\Sigma_{\bar{f}f}(\tau_{2},\tau_{3})\,G_{\bar{f}f}(\tau_{1},\tau_{3})=-\delta(\tau_{1}{-}\tau_{2}),$
(3.14)
$\displaystyle\int\\!d\tau_{3}\,\Sigma_{\bar{\phi}\phi}(\tau_{2},\tau_{3})\,G_{\bar{\phi}\phi}(\tau_{1},\tau_{3})$
$\displaystyle=-\alpha\,\delta(\tau_{1}{-}\tau_{2}),\;\;\int\\!d\tau_{3}\,\Sigma_{\bar{\psi}\psi}(\tau_{2},\tau_{3})\,G_{\bar{\psi}\psi}(\tau_{1},\tau_{3})=\alpha\,\delta(\tau_{1}{-}\tau_{2}).$

We consider turning on chemical potentials $\mu$ for various $\mathrm{U}(1)$
global symmetries. This corresponds to turning on a real background gauge
field $A_{t}=-\mu$ in Lorentzian signature, and an imaginary background field
$A_{\tau}=i\mu$ in Euclidean signature. Notice that the super-
reparametrization symmetry (3.9) can be preserved only if
$\mu_{R}=p\mu_{\eta}+q\mu_{\phi}=0$. For generic chemical potentials, one is
left with the standard reparametrization symmetry $\text{Diff}^{+}(S^{1})$,
under which the component fields transform as

$\displaystyle G_{\bar{A}A}(\tau_{1},\tau_{2})$
$\displaystyle\mapsto\;\bigl{(}\partial_{\tau_{1}}\tau^{\prime}_{1}\bigr{)}{}^{\Delta_{A}}\;\bigl{(}\partial_{\tau_{2}}\tau^{\prime}_{2}\bigr{)}{}^{\Delta_{A}}\;G_{\bar{A}A}(\tau_{1},\tau_{2})$
(3.15) $\displaystyle\Sigma_{\bar{A}A}(\tau_{1},\tau_{2})$
$\displaystyle\mapsto\;\bigl{(}\partial_{\tau_{1}}\tau^{\prime}_{1}\bigr{)}{}^{1-\Delta_{A}}\bigl{(}\partial_{\tau_{2}}\tau^{\prime}_{2}\bigr{)}{}^{1-\Delta_{A}}\,\Sigma_{\bar{A}A}(\tau_{1},\tau_{2})$

where the label $A$ runs over all field species. As argued in [19], shifting
the self-energies $\Sigma$ by the chemical potentials, the EOMs in the IR are
unchanged.

In the presence of a chemical potential, we use the following conformal ansatz
for the two-point function at zero temperature ($\beta=\infty$) of a boson or
fermion with charge $Q$ and dimension $\Delta$:

$G(\tau)=g\,G_{s}\left(\tau;\Delta,\mathcal{E}\right)\equiv
g\,\frac{e^{\pi\mathcal{E}}\,\Theta(\tau)+s\,e^{-\pi\mathcal{E}}\,\Theta(-\tau)}{|\tau|^{2\Delta}}\;,\qquad
s=\pm 1\;,\qquad\mathcal{E}\in\mathbb{R}\;,$ (3.16)

where $s=1$ for a boson and $s=-1$ for a fermion. We review its derivation in
Appendix C. This ansatz is invariant under $\mathrm{PSL}(2,\mathbb{R})$
transformations acting as $\tau\mapsto\tau^{\prime}=\frac{a\tau+b}{c\tau+d}$
with $ad-bc=1$, accompanied by a suitable gauge transformation of
$A_{\tau}$.333When the transformed time difference
$\tau^{\prime}(\tau)-\tau^{\prime}(0)$ has the opposite sign with respect to
$\tau$, one needs to keep track of the extra factor $s\,e^{2\pi\mathcal{E}}$
when the operators commute past each other, see Appendix C.2. The constant $g$
is bound to be positive by unitarity. The two-point function at non-zero
temperature is obtained through the reparametrization
$\phi(\tau)=\tan\frac{\pi\tau}{\beta}$ with
$\tau\in\bigl{(}-\frac{\beta}{2},\frac{\beta}{2}\bigr{)}$:

$G(\tau)=g\,G_{s}^{\beta}(\tau;\Delta,\mathcal{E})\equiv
g\,e^{-2\pi\mathcal{E}\frac{\tau}{\beta}}\,\biggl{\lvert}\frac{\pi}{\beta\sin\bigl{(}\frac{\pi\tau}{\beta}\bigr{)}}\biggr{\rvert}^{2\Delta}\,\Bigl{(}e^{\pi\mathcal{E}}\,\Theta(\tau)+s\,e^{-\pi\mathcal{E}}\,\Theta(-\tau)\Bigr{)}\;.$
(3.17)

The parameter $\mathcal{E}$ is called spectral asymmetry and is related to the
chemical potential. Then the extension of $G(\tau)$ beyond
$\tau\in\bigl{(}-\frac{\beta}{2},\frac{\beta}{2}\bigr{)}$ satisfies
$G(\tau+\beta)=s\,G(\tau)$.

We solve the equations of motion using the same strategy as in [26]. The
equations (3.14) can be uniformly written as

$\int\\!d\tau_{3}\,\Sigma(\tau_{2},\tau_{3})\,G(\tau_{1},\tau_{3})=-s\,\delta(\tau_{1}-\tau_{2})\qquad\Rightarrow\qquad\Sigma(\omega)\,G(-\omega)=-s\;,$
(3.18)

where $\Sigma$ is understood to be $\Sigma_{\overline{\eta}\eta}$ or
$\Sigma_{\overline{f}f}$ for the Fermi multiplet components, and
$\alpha^{-1}\Sigma_{\overline{\phi}\phi}$ or
$\alpha^{-1}\Sigma_{\overline{\psi}\psi}$ for the chiral multiplet ones. The
Fourier transform of (3.16) reads444In our conventions
$G(\omega)=\int\\!d\tau\,e^{-i\omega\tau}\,G(\tau)$.

$G(\omega)=-i\,g\operatorname{sgn}(\omega)\,\Gamma(1-2\Delta)\,|\omega|^{2\Delta-1}\Bigl{(}e^{\pi\mathcal{E}+\pi
i\Delta\operatorname{sgn}(\omega)}-s\,e^{-\pi\mathcal{E}-\pi
i\Delta\operatorname{sgn}(\omega)}\Bigr{)}\;.$ (3.19)

The expressions for $\Sigma$ in Fourier and normal space follow from (3.18):

$\displaystyle\Sigma(\omega)$ $\displaystyle=\frac{e^{-\pi\mathcal{E}-\pi
i\Delta\operatorname{sgn}(\omega)}-s\,e^{\pi\mathcal{E}+\pi
i\Delta\operatorname{sgn}(\omega)}}{2ig\operatorname{sign}(\omega)\,\Gamma(1-2\Delta)\,\bigl{(}\cosh
2\pi\mathcal{E}-s\cos 2\pi\Delta\bigr{)}\,|\omega|^{2\Delta-1}}\;,$ (3.20)
$\displaystyle\Sigma(\tau)$ $\displaystyle=\frac{(1-2\Delta)\sin
2\pi\Delta}{2\pi g\bigl{(}\cosh 2\pi\mathcal{E}-s\cos
2\pi\Delta\bigr{)}}\,\Bigl{(}e^{-\pi\mathcal{E}}\,\Theta(\tau)+s\,e^{\pi\mathcal{E}}\,\Theta(-\tau)\Bigr{)}\,|\tau|^{2\Delta-2}\;.$

We denote the parameters appearing in the ansatz for $G_{\bar{\phi}\phi}$ as
$\mathcal{E}_{\phi}$, $\Delta_{\phi}$, $g_{\phi}$, and similarly for the other
fields. Plugging $\Sigma$ from (3.20) into the algebraic equations in (3.13)
and matching spectral asymmetries and dimensions, one obtains four linear
relations:

$\displaystyle\Delta_{f}$
$\displaystyle=1-(p-1)\Delta_{\eta}-q\,\Delta_{\phi}\;,\qquad\qquad$
$\displaystyle\mathcal{E}_{f}$
$\displaystyle=-(p-1)\mathcal{E}_{\eta}-q\,\mathcal{E}_{\phi}\;,$ (3.21)
$\displaystyle\Delta_{\psi}$
$\displaystyle=1-p\,\Delta_{\eta}-(q-1)\Delta_{\phi}\;,\qquad\qquad$
$\displaystyle\mathcal{E}_{\psi}$
$\displaystyle=-p\,\mathcal{E}_{\eta}-(q-1)\mathcal{E}_{\phi}\;.$

Matching the coefficients in (3.13) and making use of (3.21) gives the
equations

$\displaystyle\frac{(1-2\Delta_{\eta})\sin(2\pi\Delta_{\eta})}{\cosh(2\pi\mathcal{E}_{\eta})+\cos(2\pi\Delta_{\eta})}=2\pi
Jg_{\eta}^{p-1}g_{\phi}^{q-1}\biggl{(}\frac{p-1}{q}g_{\phi}g_{f}+g_{\eta}g_{\psi}\biggr{)}\,,$
(3.22)
$\displaystyle\frac{\bigl{[}1-2(p-1)\Delta_{\eta}-2q\Delta_{\phi}\bigr{]}\sin\bigl{[}2\pi(p-1)\Delta_{\eta}+2\pi
q\Delta_{\phi}\bigr{]}}{\cosh\bigl{[}2\pi(p-1)\mathcal{E}_{\eta}+2\pi
q\mathcal{E}_{\phi}\bigr{]}-\cos\bigl{[}2\pi(p-1)\Delta_{\eta}+2\pi
q\Delta_{\phi}\bigr{]}}=\frac{2\pi J}{q}g_{\eta}^{p-1}g_{\phi}^{q}\,g_{f}\,,$
$\displaystyle\frac{\alpha\,(1-2\Delta_{\phi})\sin(2\pi\Delta_{\phi})}{\cosh(2\pi\mathcal{E}_{\phi})-\cos(2\pi\Delta_{\phi})}=2\pi
Jg_{\eta}^{p-1}g_{\phi}^{q-1}\biggl{(}g_{\phi}g_{f}+\frac{q-1}{p}g_{\eta}g_{\psi}\biggr{)}\,,$
$\displaystyle\frac{\alpha\,\bigl{[}1-2p\Delta_{\eta}-2(q-1)\Delta_{\phi}\bigr{]}\sin\bigl{[}2\pi
p\Delta_{\eta}+2\pi(q-1)\Delta_{\phi}\bigr{]}}{\cosh\bigl{[}2\pi
p\mathcal{E}_{\eta}+2\pi(q-1)\mathcal{E}_{\phi}\bigr{]}+\cos\bigl{[}2\pi
p\Delta_{\eta}+2\pi(q-1)\Delta_{\phi}\bigr{]}}=\frac{2\pi
J}{p}g_{\eta}^{p}\,g_{\phi}^{q-1}g_{\psi}\,.$

Substituting the second and fourth equations into the first and third ones one
can eliminate the parameters $g_{\\#}$ and $J$ and obtain two equations that
determine $\Delta_{\eta}$ and $\Delta_{\phi}$ as functions of
$\mathcal{E}_{\eta}$ and $\mathcal{E}_{\phi}$:

$\displaystyle\frac{(1-2\Delta_{\eta})\sin(2\pi\Delta_{\eta})}{\cosh(2\pi\mathcal{E}_{\eta})+\cos(2\pi\Delta_{\eta})}$
$\displaystyle=\frac{(p-1)[1-2(p-1)\Delta_{\eta}-2q\Delta_{\phi}]\sin[2\pi(p-1)\Delta_{\eta}+2\pi
q\Delta_{\phi}]}{\cosh[2\pi(p-1)\mathcal{E}_{\eta}+2\pi
q\mathcal{E}_{\phi}]-\cos[2\pi(p-1)\Delta_{\eta}+2\pi q\Delta_{\phi}]}$
$\displaystyle\quad+\frac{\alpha
p[1-2p\Delta_{\eta}-2(q-1)\Delta_{\phi}]\sin[2\pi
p\Delta_{\eta}+2\pi(q-1)\Delta_{\phi}]}{\cosh[2\pi
p\mathcal{E}_{\eta}+2\pi(q-1)\mathcal{E}_{\phi}]+\cos[2\pi
p\Delta_{\eta}+2\pi(q-1)\Delta_{\phi}]}\,,$
$\displaystyle\frac{\alpha(1-2\Delta_{\phi})\sin(2\pi\Delta_{\phi})}{\cosh(2\pi\mathcal{E}_{\phi})-\cos(2\pi\Delta_{\phi})}$
$\displaystyle=\frac{q[1-2(p-1)\Delta_{\eta}-2q\Delta_{\phi}]\sin[2\pi(p-1)\Delta_{\eta}+2\pi
q\Delta_{\phi}]}{\cosh[2\pi(p-1)\mathcal{E}_{\eta}+2\pi
q\mathcal{E}_{\phi}]-\cos[2\pi(p-1)\Delta_{\eta}+2\pi q\Delta_{\phi}]}$ (3.23)
$\displaystyle\quad+\frac{\alpha(q-1)[1-2p\Delta_{\eta}-2(q-1)\Delta_{\phi}]\sin[2\pi
p\Delta_{\eta}+2\pi(q-1)\Delta_{\phi}]}{\cosh[2\pi
p\mathcal{E}_{\eta}+2\pi(q-1)\mathcal{E}_{\phi}]+\cos[2\pi
p\Delta_{\eta}+2\pi(q-1)\Delta_{\phi}]}\,.$

Substituting the solution back into (3.22) then determines the combinations of
coefficients $g_{\eta}^{p-1}g_{\phi}^{q}g_{f}$ and
$g_{\eta}^{p}g_{\phi}^{q-1}g_{\psi}$. That only those two combinations can be
determined in this way is due to the fact that the IR Schwinger–Dyson
equations have an emergent symmetry [11]. For solutions to be consistent with
the approximation (3.14) in which we dropped the kinetic terms, we need
$\Delta_{\eta,\phi}>0$ and $\Delta_{f,\psi}>\frac{1}{2}$ which, due to (3.21),
imply:

$\Delta_{\eta}>0\,,\qquad\Delta_{\phi}>0\,,\qquad(p-1)\,\Delta_{\eta}+q\,\Delta_{\phi}<\frac{1}{2}\,,\qquad
p\,\Delta_{\eta}+(q-1)\,\Delta_{\phi}<\frac{1}{2}\,.$ (3.24)

###  3.3 Superconformal solutions

Assuming translational invariance and supersymmetry, the bilocal superfields
must be functions of the invariant $T_{12}$ in (A.9). This implies that they
are determined by their lowest components as defined in (B.1):

$\displaystyle\mathcal{G}_{\mathcal{Y}}(T_{1},T_{2})$
$\displaystyle=G_{\bar{\eta}\eta}(T_{12})\;,\qquad\qquad$
$\displaystyle\mathcal{G}_{\Phi}(T_{1},T_{2})$
$\displaystyle=G_{\bar{\phi}\phi}(T_{12})\;,$ (3.25)
$\displaystyle\Sigma_{\mathcal{Y}}(T_{1},T_{2})$
$\displaystyle=\Sigma_{\bar{f}f}(T_{12})\;,\qquad\qquad$
$\displaystyle\Sigma_{\Phi}(T_{1},T_{2})$
$\displaystyle=\Sigma_{\bar{\psi}\psi}(T_{12})\;.$

Expanding the components of $\mathcal{G}_{\mathcal{Y},\Phi}$ one obtains the
constraints

$\Delta_{f}=\Delta_{\eta}+\frac{1}{2}\,,\quad\Delta_{\psi}=\Delta_{\phi}+\frac{1}{2}\,,\quad\mathcal{E}_{f}=\mathcal{E}_{\eta}\,,\quad\mathcal{E}_{\psi}=\mathcal{E}_{\phi}\,,\quad
g_{f}=2\Delta_{\eta}g_{\eta}\,,\quad g_{\psi}=2\Delta_{\phi}g_{\phi}\,.$
(3.26)

Together with (3.21), they imply

$p\,\Delta_{\eta}+q\,\Delta_{\phi}=\frac{1}{2}\;,\qquad\qquad
p\,\mathcal{E}_{\eta}+q\,\mathcal{E}_{\phi}=0\;.$ (3.27)

In accord with (2.13), it will be convenient to parametrize the spectral
asymmetries as

$\mathcal{E}_{\phi}=-p\,\mathcal{E}\;,\qquad\qquad\mathcal{E}_{\eta}=q\,\mathcal{E}\;.$
(3.28)

At low temperature we can identify $2\pi\mathcal{E}=-\beta\mu_{F}$ in terms of
the chemical potential $\mu_{F}$ for $\mathrm{U}(1)_{F}$ (while a chemical
potential for $\mathrm{U}(1)_{R}$ explicitly breaks supersymmetry). Note that
we should restrict to $\mathcal{E}\geq 0$. The consistency conditions in
(3.22) reduce to two equations. The first one is

$\alpha\sin(2\pi\Delta_{\phi})=\frac{q}{p}\,\sin\biggl{(}\frac{\pi}{p}-\frac{2\pi
q}{p}\Delta_{\phi}\biggr{)}\,\frac{\cosh(2\pi
p\mathcal{E})-\cos(2\pi\Delta_{\phi})}{\cosh(2\pi
q\mathcal{E})+\cos\bigl{(}\frac{\pi}{p}-\frac{2\pi
q}{p}\Delta_{\phi}\bigr{)}}\,,$ (3.29)

which can be used to fix $\Delta_{\phi}$ in terms of $\mathcal{E}$. The second
one then determines the following combination of the coefficients:

$g_{\eta}^{p}\,g_{\phi}^{q}=\frac{\alpha\,p}{2\pi
J}\;\frac{\sin(2\pi\Delta_{\phi})}{\cosh(2\pi
p\mathcal{E})-\cos(2\pi\Delta_{\phi})}\;.$ (3.30)

The bounds (3.24) on the consistency of the conformal solutions reduce to

$0<\Delta_{\phi}<\frac{1}{2q}\;.$ (3.31)

At the lower bound one has $\Delta_{\phi}=0$, while at the upper bound one has
$\Delta_{\eta}=0$. An alternative derivation of the equations in this section
is provided in Appendix A.2.1.

###  3.4 Existence of superconformal solutions at fixed $\boldsymbol{p}$ and
$\boldsymbol{q}$

We search for solutions $\Delta_{\phi}(\mathcal{E})$ to (3.29) in the range
$\Delta_{\phi}\in\bigl{(}0,\frac{1}{2q}\bigr{)}$. To that purpose, we define
$R_{p,q,\mathcal{E}}(\Delta_{\phi})$ as the right-hand-side of (3.29) and
study the equation
$\alpha\sin(2\pi\Delta_{\phi})=R_{p,q,\mathcal{E}}(\Delta_{\phi})$. As
stressed above, we shall restrict to $\mathcal{E}\geq 0$.

For $p=q=1$ the model describes free fields with randomly distributed masses.
For $\alpha\neq 1$ eqn. (3.29) has no solutions, while for $\alpha=1$ it is a
tautology. We will not consider this model any further. The models with $p=1$
or $p>1$ have a rather different behavior and we will consider them
separately.

##### Case $\boldsymbol{p=1}$, $\boldsymbol{q\geq 2}$.

In this case the function $R_{1,q,\mathcal{E}}(\Delta_{\phi})$ is zero at the
endpoints $\Delta_{\phi}=0,\frac{1}{2q}$ while $\sin(2\pi\Delta_{\phi})$ is
monotonically increasing from zero, therefore at least one solution exists if
the gradient of $R_{1,q,\mathcal{E}}(\Delta_{\phi})$ at $\Delta_{\phi}=0$ is
greater than that of $\alpha\sin(2\pi\Delta_{\phi})$. This condition is

$\alpha<\frac{q^{2}\sinh^{2}(\pi\mathcal{E})}{\sinh^{2}(\pi q\mathcal{E})}\;.$
(3.32)

It turns out that there is only one solution if (3.32) is satisfied, and no
solutions otherwise. The function of $\mathcal{E}$ in (3.32) has a maximum of
$1$ at $\mathcal{E}=0$ and it decreases to zero for larger $\mathcal{E}$.
Therefore we must have $\alpha<1$ in order for a solution to exist. At fixed
$\alpha<1$, the solution only exists in the range
$\mathcal{E}\in\bigl{[}0,\,\mathcal{E}_{*}(\alpha)\bigr{)}$, where
$\mathcal{E}_{*}$ saturates (3.32). In this case $\Delta_{\phi}(\mathcal{E})$
has a maximum at $\mathcal{E}=0$ and it monotonically decreases to zero as
$\mathcal{E}\to\mathcal{E}_{*}$. At $\mathcal{E}=\mathcal{E}_{*}$ there is a
phase transition signalled by $\Delta_{\phi}=0$ and $\Delta_{\eta}=1/2$ and
beyond that point the conformal solution ceases to exist.

This transition is similar to the transition found in the multi-fermion model
of [26]. There it was observed that one of the fermion species saturates its
charge while the other one does not. In our model, at the transition point,
the charge of the fermion reaches its maximal value $\mathcal{Q}_{\eta}=1/2$
while the boson is still above its minimal value. This is numerically
investigated in Section 4. Furthermore, in Section 3.8 we find gapped
solutions that mirror massive free particles. In [26] it was proposed that in
the multi-fermion model a jump occurs between two distinct sets of solutions
in the grand canonical ensemble. However, in Section 4 we illustrate that in
our model, for finite $\beta\mu$, the phase after the transition seems to
interpolate between the conformal behaviour and an exponential behaviour of
non-conformal solutions. Another possibility, which neither our analytical nor
numerical analysis would capture, is that the breakdown of the conformal
ansatz signals that a replica-breaking solution is preferred by the path
integral. Such a proposal requires further study.

In the special case $p=1$, $q=2$ we can explicitly solve (3.29) and (3.32):

$\Delta_{\phi}(\mathcal{E})=\frac{1}{2\pi}\arccos\biggl{(}\frac{\alpha}{2-\alpha}\cosh(2\pi\mathcal{E})\biggr{)}\;,\qquad\mathcal{E}_{*}(\alpha)=\frac{1}{\pi}\operatorname{arctanh}\bigl{(}\sqrt{1-\alpha}\,\bigr{)}\;.$
(3.33)

The corresponding solution to (3.30) is

$g_{\eta}\,g_{\phi}^{2}=\frac{\alpha}{4\pi
J\,(1-\alpha)}\,\frac{\sqrt{(2-\alpha)^{2}-\alpha^{2}\cosh^{2}(2\pi\mathcal{E})}}{\cosh(2\pi\mathcal{E})}\;.$
(3.34)

For $p=1$, $q=3$ analytic solutions can be found as well, although they are
lengthier:

$\cos(2\pi\Delta_{\phi})=\frac{2\alpha\cosh(2\pi\mathcal{E})+\sqrt{36+12\alpha(4-\alpha)\sinh^{2}(2\pi\mathcal{E})}}{4(3-\alpha)}$
(3.35)

gives the dimension $\Delta_{\phi}(\mathcal{E})$, while

$\mathcal{E}_{*}(\alpha)=\frac{1}{2\pi}\operatorname{arccosh}\biggl{(}\frac{3-\sqrt{\alpha}}{2\sqrt{\alpha}}\,\biggr{)}$
(3.36)

identifies the phase transition point. The solution for $g_{\eta}g_{\phi}^{3}$
can be easily written.

##### Case $\boldsymbol{p>1}$, $\boldsymbol{q\geq 2}$.

In this case the function $R_{p,q,\mathcal{E}}(\Delta_{\phi})$ is zero at the
endpoints $\Delta_{\phi}=0,\frac{1}{2q}$ if $\mathcal{E}=0$, while it is
positive at $\Delta_{\phi}=0$ if $\mathcal{E}>0$. For $\mathcal{E}=0$ eqn.
(3.29) simplifies to

$\frac{\alpha\,p}{q}=\tan(\pi\Delta_{\phi})\,\tan\biggl{(}\frac{\pi}{2p}-\frac{\pi
q\Delta_{\phi}}{p}\biggr{)}\;.$ (3.37)

The right-hand-side is zero at $\Delta_{\phi}=0,\frac{1}{2q}$ and attains a
maximum in between. If $\alpha$ is small enough such that $\frac{\alpha p}{q}$
is less than (or equal to) the maximal value then there are two (or one)
solutions, otherwise there are no solutions. For $\mathcal{E}>0$, since the
right-hand-side of (3.29) is positive at $\Delta_{\phi}=0$ and it vanishes at
$\Delta_{\phi}=\frac{1}{2q}$, there must be at least one solution regardless
of $\alpha$, $\mathcal{E}$. However, there could be more than one solution.
For fixed $p>1$, $q\geq 2$ and small $\alpha$, we empirically find that by
varying $\mathcal{E}$ from above $0$ to $\infty$, we go from having $3$
solutions to $1$ solution through a critical value of $\mathcal{E}$ where
there are $2$ solutions. When $\alpha$ is raised and this scan in
$\mathcal{E}$ is repeated, there is a regime where the number of solutions
goes from 1 to 3 to 1. For even larger $\alpha$ there is only one solution
regardless of $\mathcal{E}$. Note that unlike the case of $p=1$ or the model
of [26], there is no region in $\alpha$, $\mathcal{E}$ where no solutions
exist.

The behaviour above can be seen explicitly in the case $p=q>1$, where analytic
expressions for $\Delta_{\phi}$ can be obtained by solving the following
quartic equation:

$\tfrac{\alpha-1}{2}(x^{4}-1)+\cosh(2\pi
p\mathcal{E})\Bigl{(}e^{\frac{i\pi}{2p}}(\alpha
x^{3}-x)+e^{-\frac{i\pi}{2p}}(x^{3}-\alpha
x)\Bigr{)}+i(\alpha+1)\sin\bigl{(}\tfrac{\pi}{p}\bigr{)}\,x^{2}=0$ (3.38)

in which
$x=\exp\bigl{[}i\bigl{(}2\pi\Delta_{\phi}-\frac{\pi}{2p}\bigr{)}\bigr{]}$. In
Figure 1 we plot the number of acceptable solutions satisfying
$\Delta_{\phi}\in\bigl{(}0,\frac{1}{2p}\bigr{)}$ for $p=q=3$ as $\alpha$ and
$\mathcal{E}$ vary. Excluding the horizontal axis $\mathcal{E}=0$, there are
$3$ solutions in the yellow region and $1$ solution in the blue region. On the
horizontal axis, there are $2$ solutions along the green segment within the
yellow region, and no solution along the black line. On the boundary of the
two regions, where the discriminant of the quartic (3.38) vanishes, there are
two solutions for $\mathcal{E}>0$ and one solution for $\mathcal{E}=0$.

Figure 1: Left: Plot of the number of acceptable solutions to (3.29) for
$p=q=3$ as a function of $\alpha$ and $\mathcal{E}$. For $\mathcal{E}>0$ there
are 3 solutions in the yellow region, 1 solution in the blue region, and 2
solutions along their boundary. For $\mathcal{E}=0$ there are two solutions
along the green segment, no solutions along the black line, and 1 solution at
their boundary. Right: Solutions $\Delta_{\phi}(\mathcal{E})$ for
$\alpha=0.06$, $0.075$, $0.085$.

There are special values of $\mathcal{E}$ and $\alpha$ for which (3.38)
simplifies. For $\alpha=1$, (3.38) reduces to a quadratic equation and the
unique solution with $\Delta_{\phi}\in\bigl{(}0,\frac{1}{2p}\bigr{)}$ is

$\Delta_{\phi}=\frac{1}{4p}-\frac{1}{2\pi}\arcsin\biggl{[}\frac{\sin\bigl{(}\frac{\pi}{2p}\bigr{)}}{\cosh(2\pi
p\mathcal{E})}\biggr{]}\;,$ (3.39)

acceptable whenever $\mathcal{E}>0$. The corresponding solution to (3.30) is

$g_{\eta}^{p}\,g_{\phi}^{p}=\frac{p\sin\bigl{(}\frac{\pi}{2p}\bigr{)}}{2\pi
J\sqrt{\cosh^{2}(2\pi
p\mathcal{E})-\sin^{2}\bigl{(}\frac{\pi}{2p}\bigr{)}}}\;.$ (3.40)

For $\mathcal{E}=0$, (3.38) can be written as a quadratic equation in
$\tan(\pi\Delta_{\phi})$ which gives two solutions:

$\Delta_{\phi}=\frac{1}{\pi}\arctan\biggl{[}\frac{1-\alpha}{2}\,\tan\Bigl{(}\frac{\pi}{2p}\Bigr{)}\pm\frac{1}{2}\sqrt{(1-\alpha)^{2}\tan^{2}\Bigl{(}\frac{\pi}{2p}\Bigr{)}-4\alpha}\,\biggr{]}\;.$
(3.41)

The two solutions are real if and only if
$\alpha\leq\tan^{2}\bigl{(}\frac{\pi}{4p}\bigr{)}$. Indeed
$\alpha=\tan^{2}\big{(}\frac{\pi}{4p}\bigr{)}$ is where the right boundary of
the yellow region intersects the horizontal axis in Figure 1 and the two
solutions merge into one (disappearing for larger values of $\alpha$). It
follows from (3.41) that both solutions satisfy $\Delta_{\phi}>0$, that the
larger solution becomes $\Delta_{\phi}=\frac{1}{2p}$ for $\alpha=0$, and that
$\Delta_{\phi}<\frac{1}{2p}$ for the allowed values $\alpha>0$. The
corresponding solution to (3.30) is

$g_{\eta}^{p}\,g_{\phi}^{p}=\frac{\alpha\,p}{2\pi J\tan(\pi\Delta_{\phi})}\;.$
(3.42)

##### Case $\boldsymbol{p>1}$, $\boldsymbol{q=1}$.

The behaviour of the function $R_{p,1,\mathcal{E}}(\Delta_{\phi})$ is similar
to the one in the previous case, however the function
$\sin(2\pi\Delta_{\phi})$ vanishes at both endpoints
$\Delta_{\phi}=0,\frac{1}{2}$. For very small values of $\alpha$, the number
of solutions $\Delta_{\phi}(\mathcal{E})$ to (3.29) as we increase
$\mathcal{E}$ from above $0$ to $+\infty$ is 2 (of which only 1 is acceptable
at $\mathcal{E}=0$) and then 0. For larger values of $\alpha$ it is 1 (not
acceptable at $\mathcal{E}=0$), then 2, and then 0. For even larger values of
$\alpha$ there is only 1 solution (that becomes $\Delta_{\phi}=0$ at
$\mathcal{E}=0$ and thus is not acceptable) up to $\mathcal{E}_{*}(\alpha)$
and then no solutions for $\mathcal{E}\geq\mathcal{E}_{*}(\alpha)$.

###  3.5 Solutions at large $\boldsymbol{p}$ and fixed $\boldsymbol{q}$

Using the methods in [6, 11] and Appendix C of [12], we search for analytic
solutions to the Schwinger–Dyson equations at large $p$. We truncate to the
bosonic components only, assuming (3.12), but we do not neglect the kinetic
terms. Let the chemical potentials corresponding to $Q_{\eta,\phi}$ be
$\mu_{\eta,\phi}$ respectively. The equations we want to solve are the
algebraic ones in (3.13) and the integro-differential ones in (3.8) that in
components read as in (B.5):

$\displaystyle-\bigl{(}\partial_{\tau_{2}}+\mu_{\eta}\bigr{)}\,G_{\overline{\eta}\eta}(\tau_{1},\tau_{2})+\int\\!d\tau_{3}\,\Sigma_{\overline{\eta}\eta}(\tau_{2},\tau_{3})\,G_{\overline{\eta}\eta}(\tau_{1},\tau_{3})$
$\displaystyle=\delta(\tau_{1}-\tau_{2})\;,$ (3.43) $\displaystyle
G_{\overline{f}f}(\tau_{1},\tau_{2})+\int\\!d\tau_{3}\,\Sigma_{\overline{f}f}(\tau_{2},\tau_{3})\,G_{\overline{f}f}(\tau_{1},\tau_{3})$
$\displaystyle=-\delta(\tau_{1}-\tau_{2})\;,$
$\displaystyle-\bigl{(}\partial_{\tau_{2}}+\mu_{\phi}\bigr{)}\,G_{\overline{\phi}\phi}(\tau_{1},\tau_{2})+\int\\!d\tau_{3}\,\Sigma_{\overline{\phi}\phi}(\tau_{2},\tau_{3})\,G_{\overline{\phi}\phi}(\tau_{1},\tau_{3})$
$\displaystyle=-\alpha\,\delta(\tau_{1}-\tau_{2})\;,$ $\displaystyle
G_{\overline{\psi}\psi}(\tau_{1},\tau_{2})+\int\\!d\tau_{3}\,\Sigma_{\overline{\psi}\psi}(\tau_{2},\tau_{3})\,G_{\overline{\psi}\psi}(\tau_{1},\tau_{3})$
$\displaystyle=\alpha\,\delta(\tau_{1}-\tau_{2})\;.$

Assuming that the theory becomes free in the large $p$ limit, we search for
solutions in a $\frac{1}{p}$ expansion around the free two-point functions.
Generically, we expand the bilocal fields $G$ (assuming translational
invariance) as

$\displaystyle G(\tau)$
$\displaystyle=G_{0}(\tau)+\frac{1}{p^{n}}G_{1}(\tau)+\mathcal{O}(p^{-n-1})\;,$
(3.44) $\displaystyle G(\omega_{k})$
$\displaystyle=G_{0}(\omega_{k})+\frac{1}{p^{n}}G_{1}(\omega_{k})+\mathcal{O}(p^{-n-1})\;.$

Here $G_{0}$ are the free propagators (we review the correlators of free Fermi
and chiral multiplets in Appendix C), $\omega_{k}$ are the appropriate
Matsubara frequencies,555In our conventions
$G(\tau)=\sum_{n\,\in\,\mathbb{Z}+\frac{1-s}{4}}G(n)\,e^{\frac{2\pi
in\tau}{\beta}}$ and
$G(n)=\frac{1}{\beta}\int_{0}^{\beta}d\tau\,e^{-\frac{2\pi
in\tau}{\beta}}\,G(\tau)$, where $s=+1$ for bosons and $s=-1$ for fermions.
$n>0$ are suitable integers that we determine below, and we require

$\bigl{\lvert}G_{0}(\omega_{k})\bigr{\rvert}\,\gg\,\frac{\lvert
G_{1}(\omega_{k})\rvert}{p^{n}}\;,\qquad\forall\,\omega_{k}\text{
fixed}\,,\quad p\gg 1\;.$ (3.45)

More in detail, substituting the free propagators (C.16), (C.20), (C.24),
(C.28) into the definition (B.2) of the bilocal fields, one obtains:

$\displaystyle G_{\bar{\eta}\eta}^{(0)}(\tau)$ $\displaystyle=\hskip
6.99997pt\frac{e^{\mu_{\eta}\tau}}{2\cosh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\Bigl{[}\Theta(\tau)\,e^{-\frac{\mu_{\eta}\beta}{2}}-\Theta(-\tau)\,e^{\frac{\mu_{\eta}\beta}{2}}\Bigr{]}\;,\quad$
$\displaystyle G_{\bar{f}f}^{(0)}(\tau)$ $\displaystyle=-\delta(\tau)\;,$
(3.46) $\displaystyle G_{\bar{\phi}\phi}^{(0)}(\tau)$
$\displaystyle=\alpha\,\frac{e^{\mu_{\phi}\tau}}{2\sinh\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}}\Bigl{[}\Theta(\tau)\,e^{-\frac{\mu_{\phi}\beta}{2}}+\Theta(-\tau)\,e^{\frac{\mu_{\phi}\beta}{2}}\Bigr{]}\;,\quad$
$\displaystyle G_{\bar{\psi}\psi}^{(0)}(\tau)$
$\displaystyle=\alpha\,\delta(\tau)\;.$

Being them free propagators, they satisfy the free equations of motion,
obtained from (3.43) by setting all $\Sigma$’s to zero. We consider an ansatz
for the bilocal fields that is an expansion in $\frac{1}{p}$ around the free
result:

$\displaystyle G_{\bar{\eta}\eta}(\tau)$
$\displaystyle=G_{\bar{\eta}\eta}^{(0)}(\tau)\biggl{(}1+\frac{\widetilde{G}_{\bar{\eta}\eta}(\tau)}{p}+\mathcal{O}(p^{-2})\biggr{)},$
$\displaystyle G_{\bar{f}f}(\tau)$
$\displaystyle=-\delta(\tau)+\frac{\widetilde{G}_{\bar{f}f}(\tau)}{p^{n_{f}}}+\mathcal{O}\bigl{(}p^{-n_{f}-1}\bigr{)},$
$\displaystyle G_{\bar{\phi}\phi}(\tau)$
$\displaystyle=G_{\bar{\phi}\phi}^{(0)}(\tau)\biggl{(}1+\frac{\widetilde{G}_{\bar{\phi}\phi}(\tau)}{p^{n_{\phi}}}+\mathcal{O}\bigl{(}p^{-n_{\phi}-1}\bigr{)}\biggr{)},$
$\displaystyle G_{\bar{\psi}\psi}(\tau)$
$\displaystyle=\alpha\,\delta(\tau)+\frac{\widetilde{G}_{\bar{\psi}\psi}(\tau)}{p^{n_{\psi}}}+\mathcal{O}\bigl{(}p^{-n_{\psi}-1}\bigr{)}.$
(3.47)

Here we assume that $n_{f},n_{\phi},n_{\psi}>0$ and that
$\widetilde{G}_{\bar{\eta}\eta}$, $\widetilde{G}_{\bar{f}f}$,
$\widetilde{G}_{\bar{\phi}\phi}$, $\widetilde{G}_{\bar{\psi}\psi}$ are
$\mathcal{O}(1)$. We chose the order of the leading correction in
$G_{\bar{\eta}\eta}$ such that various terms in the equations of motion
simplify using

$\bigl{[}G_{\bar{\eta}\eta}\bigr{]}^{p-n}=\bigl{[}G_{\bar{\eta}\eta}^{(0)}\bigr{]}^{p-n}\,e^{\widetilde{G}_{\bar{\eta}\eta}}\,\bigl{(}1+\mathcal{O}(p^{-1})\bigr{)}\;.$
(3.48)

This in turn leads to the same differential equation as in [6, 11], with the
benefit of hindsight. In terms of the chemical potentials $\mu_{F}$ for
$\mathrm{U}(1)_{F}$ and $\mu_{R}$ for any R-charge $\mathrm{U}(1)_{R}$, the
potentials $\mu_{\eta,\phi}$ are decomposed as

$\mu_{\eta}=q\,\mu_{F}+R[\eta]\,\mu_{R}\;,\qquad\mu_{\phi}=-p\,\mu_{F}+R[\phi]\,\mu_{R}\;,\qquad
p\,\mu_{\eta}+q\,\mu_{\phi}=\mu_{R}\;.$ (3.49)

The last equation follows from the constraint that the J-term has R-charge
$1$, i.e., from $p\,R[\eta]+q\,R[\phi]=1$ . We want to be able to impose the
supersymmetry constraint $\mu_{R}=0$ at large $p$. This implies that the
scalings of $\mu_{\eta}$ and $\mu_{\phi}$ must be related as $\mu_{\phi}\sim
p\,\mu_{\eta}$. The simplest scaling is $\mu_{\eta}\sim\mu_{F}\sim
R[\eta]\sim\mathcal{O}(p^{-1})$ and $\mu_{\phi}\sim\mu_{R}\sim
R[\phi]\sim\mathcal{O}(1)$. In this way both $G_{\bar{\eta}\eta}^{(0)}$ and
$G_{\bar{\phi}\phi}^{(0)}$ are $\mathcal{O}(1)$. Besides, the bilocal fields
should agree with the free limits at short distances, which implies the
following boundary conditions: $\widetilde{G}(0)=\widetilde{G}(\beta)=0$ for
all components. Plugging the ansatz (3.47) and the boundary conditions into
(3.43) we get

$\displaystyle 0$
$\displaystyle=\frac{1}{p}\,G_{\bar{\eta}\eta}^{(0)}(\tau_{12})\,\partial_{\tau_{12}}\widetilde{G}_{\bar{\eta}\eta}(\tau_{12})+\int\\!d\tau_{3}\,\Sigma_{\bar{\eta}\eta}(\tau_{23})\,G_{\bar{\eta}\eta}^{(0)}(\tau_{13})+\mathcal{O}(p^{-2})\;,$
(3.50) $\displaystyle 0$
$\displaystyle=\frac{1}{p^{n_{f}}}\widetilde{G}_{\bar{f}f}(\tau_{12})-\Sigma_{\bar{f}f}(\tau_{21})+\mathcal{O}\bigl{(}p^{-n_{f}-1}\bigr{)}\;,$
$\displaystyle 0$
$\displaystyle=\frac{1}{p^{n_{\phi}}}\,G_{\bar{\phi}\phi}^{(0)}(\tau_{12})\,\partial_{\tau_{12}}\widetilde{G}_{\bar{\phi}\phi}(\tau_{12})+\int\\!d\tau_{3}\,\Sigma_{\bar{\phi}\phi}(\tau_{23})\,G_{\bar{\phi}\phi}^{(0)}(\tau_{13})+\mathcal{O}\bigl{(}p^{-n_{\phi}-1}\bigr{)}\;,$
$\displaystyle 0$
$\displaystyle=\frac{1}{p^{n_{\psi}}}\widetilde{G}_{\bar{\psi}\psi}(\tau_{12})+\alpha\,\Sigma_{\bar{\psi}\psi}(\tau_{21})+\mathcal{O}\bigl{(}p^{-n_{\psi}-1}\bigr{)}\;,$

where $\tau_{ij}=\tau_{i}-\tau_{j}$. These equations can be solved to obtain
the $\Sigma$’s in terms of the $G$’s at leading order in $\frac{1}{p}$. In
particular, one acts with $(\partial_{\tau_{1}}-\mu_{\eta})$ on the first
equation and with $(\partial_{\tau_{1}}-\mu_{\phi})$ on the third equation,
obtaining:666We have neglected the terms
$-\frac{1}{p}\,\delta(\tau)\,\partial_{\tau}\widetilde{G}_{\bar{\eta}\eta}(\tau)$
in $\Sigma_{\bar{\eta}\eta}(-\tau)$ and
$-\frac{1}{p^{n_{\phi}}}\,\delta(\tau)\,\partial_{\tau}\widetilde{G}_{\bar{\phi}\phi}(\tau)$
in $\Sigma_{\bar{\phi}\phi}(-\tau)$ since we are interested in the equations
at $\tau\neq 0$.

$\displaystyle\Sigma_{\bar{\eta}\eta}(-\tau)$
$\displaystyle=-\tfrac{1}{p}\,G_{\bar{\eta}\eta}^{(0)}(\tau)\,\partial_{\tau}^{2}\widetilde{G}_{\bar{\eta}\eta}(\tau)+\mathcal{O}(p^{-2}),\hskip
21.00009pt\Sigma_{\bar{f}f}(-\tau)=\tfrac{1}{p^{n_{f}}\rule{0.0pt}{4.68999pt}}\,\widetilde{G}_{\bar{f}f}(\tau)+\mathcal{O}(p^{-n_{f}-1}),$
(3.51) $\displaystyle\Sigma_{\bar{\phi}\phi}(-\tau)$
$\displaystyle=\tfrac{1}{\alpha
p^{n_{\phi}}\rule{0.0pt}{4.68999pt}}\,G_{\bar{\phi}\phi}^{(0)}(\tau)\,\partial_{\tau}^{2}\widetilde{G}_{\bar{\phi}\phi}(\tau)+\mathcal{O}(p^{-n_{\phi}-1}),\;\;\Sigma_{\bar{\psi}\psi}(-\tau)=-\tfrac{1}{\alpha
p^{n_{\psi}}\rule{0.0pt}{4.68999pt}}\,\widetilde{G}_{\bar{\psi}\psi}(\tau)+\mathcal{O}(p^{-n_{\psi}-1}).$

Let us then consider the algebraic equations. Substituting the ansatz (3.47)
into the last two equations of (3.13) we determine the leading behavior of
$\Sigma_{\bar{f}f}$ and $\Sigma_{\bar{\psi}\psi}$. This can then be used in
(3.51) to determine $\widetilde{G}_{\bar{f}f}$ and
$\widetilde{G}_{\bar{\psi}\psi}$. In order to have a well-defined large $p$
limit, the simplest choice is to scale $J$ as $2^{p}p^{-n_{f}}$, so that the
combination

$\mathcal{J}\,\equiv\,\frac{\alpha^{q}\,p^{n_{f}}\,J}{2^{p+q-2}\,q\cosh^{p-2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}\sinh^{q}\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}}$
(3.52)

remains fixed as $p$ becomes large. Recalling that $p$ is odd, we obtain:

$\displaystyle\widetilde{G}_{\bar{f}f}(\tau)$
$\displaystyle=\frac{\mathcal{J}\,e^{\left(\mu_{\eta}-\mu_{R}\right)\tau}}{2\cosh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\,\Bigl{[}\Theta(\tau)\>e^{-\left(\mu_{\eta}-\mu_{R}\right)\frac{\beta}{2}}+\Theta(-\tau)\>e^{\left(\mu_{\eta}-\mu_{R}\right)\frac{\beta}{2}}\Bigr{]}\,e^{\widetilde{G}_{\bar{\eta}\eta}(-\tau)}\;,$
(3.53) $\displaystyle\widetilde{G}_{\bar{\psi}\psi}(\tau)$
$\displaystyle=\frac{q\,\mathcal{J}\,\sinh\bigl{(}\tfrac{\mu_{\phi}\beta}{2}\bigr{)}\,e^{(\mu_{\phi}-\mu_{R})\tau}}{2\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\,\Bigl{[}\Theta(\tau)\>e^{-(\mu_{\phi}-\mu_{R})\frac{\beta}{2}}-\Theta(-\tau)\>e^{(\mu_{\phi}-\mu_{R})\frac{\beta}{2}}\Bigr{]}\,e^{\widetilde{G}_{\bar{\eta}\eta}(-\tau)}\;,$

together with $n_{\psi}=n_{f}+1$. Similarly, substituting these expressions
and the ansatz (3.47) into the first two equations of (3.13), we determine the
leading behavior of $\Sigma_{\bar{\eta}\eta}$ and $\Sigma_{\bar{\phi}\phi}$:

$\displaystyle\Sigma_{\bar{\eta}\eta}(\tau)$
$\displaystyle=-\frac{\mathcal{J}^{2}}{p^{2n_{f}-1}}\;G^{(0)}_{\bar{\eta}\eta}(-\tau)\;e^{\widetilde{G}_{\bar{\eta}\eta}(\tau)+\widetilde{G}_{\bar{\eta}\eta}(-\tau)}+\mathcal{O}(p^{-2n_{f}})\;,$
(3.54) $\displaystyle\Sigma_{\bar{\phi}\phi}(\tau)$
$\displaystyle=\frac{q\,\mathcal{J}^{2}\,\sinh^{2}\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}}{p^{2n_{f}}\,\alpha^{2}\,\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\;G_{\bar{\phi}\phi}^{(0)}(-\tau)\;e^{\widetilde{G}_{\bar{\eta}\eta}(\tau)+\widetilde{G}_{\bar{\eta}\eta}(-\tau)}+\mathcal{O}(p^{-2n_{f}-1})\;.$

Equating these expressions with those in (3.51) to leading order in
$\tfrac{1}{p}$ fixes $n_{f}=1$ and $n_{\phi}=n_{\psi}=2$, and gives the
following differential equations:

$\partial_{\tau}^{2}\,\widetilde{G}_{\bar{\eta}\eta}(\tau)=\mathcal{J}^{2}\;e^{\widetilde{G}_{\bar{\eta}\eta}(\tau)+\widetilde{G}_{\bar{\eta}\eta}(-\tau)}\,,\qquad\partial_{\tau}^{2}\,\widetilde{G}_{\bar{\phi}\phi}(\tau)=\frac{q\,\mathcal{J}^{2}\,\sinh^{2}\bigl{(}\tfrac{\mu_{\phi}\beta}{2}\bigr{)}}{\alpha\,\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\;e^{\widetilde{G}_{\bar{\eta}\eta}(\tau)+\widetilde{G}_{\bar{\eta}\eta}(-\tau)}\,.$
(3.55)

Notice that the free propagators $G_{\bar{\eta}\eta}^{(0)}$ and
$G_{\bar{\phi}\phi}^{(0)}$ have dropped from the equations. The only solution
to the second equation that satisfies the boundary conditions is

$\widetilde{G}_{\bar{\phi}\phi}(\tau)=\frac{q\,\sinh^{2}\bigl{(}\tfrac{\mu_{\phi}\beta}{2}\bigr{)}}{\alpha\,\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\;\widetilde{G}_{\bar{\eta}\eta}\;.$
(3.56)

The differential equation for $\widetilde{G}_{\bar{\eta}\eta}$ is the same as
the one found in [6, 12, 11]. Since the odd part
$\widetilde{G}_{\text{odd}}(\tau)=\frac{1}{2}\bigl{(}\widetilde{G}_{\bar{\eta}\eta}(\tau)-\widetilde{G}_{\bar{\eta}\eta}(-\tau)\bigr{)}$
satisfies $\partial_{\tau}^{2}\,\widetilde{G}_{\text{odd}}(\tau)=0$, the only
solution compatible with the boundary conditions is
$\widetilde{G}_{\text{odd}}(\tau)=0$ and we can take
$\widetilde{G}_{\bar{\eta}\eta}$ to be even. Then the general solution with
integration constants $v,b$ is:777When solving the differential equation, it
is convenient to redefine $x=e^{-\widetilde{G}_{\bar{\eta}\eta}}$.
$\exp\bigl{[}\widetilde{G}_{\bar{\eta}\eta}(\tau)\bigr{]}=v/\bigl{[}\beta\mathcal{J}\sin\bigl{(}\frac{v}{\beta}|\tau|+b\bigr{)}\bigr{]}$
where $v>0$, $\beta$ has been introduced for later convenience, and $b$ is
defined modulo $2\pi$. The boundary conditions
$\widetilde{G}(0)=\widetilde{G}(\beta)=0$ imply $\beta\mathcal{J}=v/\sin(b)$
and $\sin(b)=\sin(v+b)$. One solution to the second equation is $v=2\pi m$ for
$m\in\mathbb{N}$, however this does not lead to a function
$\exp\bigl{[}\widetilde{G}_{\bar{\eta}\eta}(\tau)\bigr{]}$ that is positive
for all values of $\tau$. The other solutions are
$b=\bigl{(}n+\frac{1}{2}\bigr{)}\pi-\frac{v}{2}$ for $n\in\mathbb{Z}$ and
without loss of generality we consider $n=0,1$. The other equation reduces to
$\beta\mathcal{J}=\pm v/\cos\bigl{(}\frac{v}{2}\bigr{)}$, where $\pm$
correspond to $n=0,1$, respectively. Only the case $n=0$, $0<v<\pi$ leads to a
positive function $\exp\bigl{[}\widetilde{G}_{\bar{\eta}\eta}(\tau)\bigr{]}$,
and the second equation has one and only one solution for all values of
$\beta$. The final solution is thus:

$e^{\widetilde{G}_{\bar{\eta}\eta}(\tau)}=\frac{v}{\beta\mathcal{J}\cos\bigl{(}\frac{v}{\beta}|\tau|-\frac{v}{2}\bigr{)}}\;,\qquad\beta\mathcal{J}=\frac{v}{\cos\bigl{(}\frac{v}{2}\bigr{)}}\;,\qquad
0<v<\pi\;.$ (3.57)

In the weak coupling limit $\beta\mathcal{J}\rightarrow 0$ (keeping all
$\mu_{I}\beta$’s fixed) one has $v\to 0$ and
$\frac{v}{\beta\mathcal{J}}\rightarrow 1$ and therefore
$\widetilde{G}_{\bar{\eta}\eta},\,\widetilde{G}_{\bar{\phi}\phi}\to 0$. From
(3.53) we see that also
$\widetilde{G}_{\bar{f}f},\,\widetilde{G}_{\bar{\psi}\psi}\to 0$ and the
solution reduces to the free UV one at leading order in $\tfrac{1}{p}$. In the
strong coupling limit $\beta\mathcal{J}\rightarrow\infty$, instead, one has
$v\rightarrow\pi$ and, to this order of approximation and away from $\tau=0$,
the two-point functions agree with the conformal ones (3.17) with the
following parameters:

$\displaystyle\Delta_{\eta}$
$\displaystyle=\frac{1}{2p}+\mathcal{O}(p^{-2})\;,$ $\displaystyle
2\pi\mathcal{E}_{\eta}$
$\displaystyle=-\beta\mu_{\eta}+\mathcal{O}(p^{-2})\;,$
$\displaystyle\Delta_{f}$ $\displaystyle=\frac{1}{2}+\mathcal{O}(p^{-1})\;,$
$\displaystyle 2\pi\mathcal{E}_{f}$
$\displaystyle=\beta(\mu_{R}-\mu_{\eta})+\mathcal{O}(p^{-1})\;,$ (3.58)
$\displaystyle\Delta_{\phi}$
$\displaystyle=\frac{q\,\sinh^{2}\bigl{(}\tfrac{\mu_{\phi}\beta}{2}\bigr{)}}{2\alpha
p^{2}\,\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\;,\quad$
$\displaystyle 2\pi\mathcal{E}_{\phi}$
$\displaystyle=-\beta\mu_{\phi}+\mathcal{O}(p^{-3})\;,$
$\displaystyle\Delta_{\psi}$
$\displaystyle=\frac{1}{2}+\mathcal{O}(p^{-1})\;,$ $\displaystyle
2\pi\mathcal{E}_{\psi}$
$\displaystyle=\beta(\mu_{R}-\mu_{\phi})+\mathcal{O}(p^{-1})\;,$

as well as

$\displaystyle g_{\eta}$
$\displaystyle=\frac{1}{2\cosh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\biggl{(}1-\frac{1}{p}\log\mathcal{J}\biggr{)}+\mathcal{O}(p^{-2})\,,\hskip
45.00006ptg_{f}=\frac{1}{2p\cosh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-2})\,,$
(3.59) $\displaystyle g_{\phi}$
$\displaystyle=\frac{\alpha}{2\sinh\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}}\biggl{(}1-\frac{q\sinh^{2}\bigl{(}\tfrac{\mu_{\phi}\beta}{2}\bigr{)}}{\alpha
p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\log\mathcal{J}\biggr{)}+\mathcal{O}(p^{-3})\,,\quad
g_{\psi}=\frac{q\sinh\bigl{(}\tfrac{\mu_{\phi}\beta}{2}\bigr{)}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3}).$

The conformal dimensions are compatible with the bounds (3.24) at leading
order in $\tfrac{1}{p}$, in particular $\Delta_{\eta}$ saturates
$p\Delta_{\eta}=\frac{1}{2}+\mathcal{O}(p^{-1})$. A higher order in
$\tfrac{1}{p}$ it would be necessary to verify that all bounds are satisfied.
We see that the condition $\mu_{\phi}>0$ (discussed in Section 2) is necessary
in order to ensure $g_{\phi},g_{\psi}>0$ as required by unitarity. We
tentatively see that the solution at large $p$ interpolates between the free
UV limit and the IR conformal-like behavior. Notice in particular that the
phase transition is not visible in this limit.

It might be tempting to think that the conformal data in (3.58) solves, at
some given order in $\frac{1}{p}$, the self-consistency equations (3.23).
This, in general, does not happen, due to incompatibility between the large
$p$ and low energy limits. Write (3.43) in Fourier space:

$\biggl{[}-\frac{s}{G_{0}(\omega_{k})}+\beta^{2}\,\Sigma(\omega_{-k})\biggr{]}G(\omega_{k})=-s\;,$
(3.60)

where $\Sigma$ should again be understood as $\alpha^{-1}\Sigma$ for $\phi$
and $\psi$, and we used that $G_{0}$ is the free propagator. In the large $p$
limit (3.45), using the expansion (3.44), one further has

$\Sigma(\omega_{-k})=\frac{-s}{\beta^{2}G_{0}(\omega_{k})+\frac{\beta^{2}}{p^{n}}G_{1}(\omega_{k})+\mathcal{O}\bigl{(}\frac{1}{p^{n+1}}\bigr{)}}+\frac{s}{\beta^{2}G_{0}(\omega_{k})}=\frac{s}{p^{n}}\frac{G_{1}(\omega_{k})}{\beta^{2}G_{0}(\omega_{k})^{2}}+\mathcal{O}\bigl{(}\tfrac{1}{p^{n+1}}\bigr{)}.$
(3.61)

In these terms, as one can see from (3.43), the low-energy regime we
previously studied for dynamical and auxiliary fields, respectively,
corresponds to taking

$\bigl{\lvert}\mu_{\phi,\eta}-\beta\,\Sigma_{\phi,\eta}(\omega_{-k})\bigr{\rvert}\gg|\omega_{k}|\;,\qquad\bigl{\lvert}\beta\,\Sigma_{f,\psi}(\omega_{-k})\bigr{\rvert}\gg
1\;,\qquad\omega_{k}\ll J\;.$ (3.62)

Combining with (3.61), for the auxiliary fields we find that

$\frac{1}{p^{n}}\,\bigl{\lvert}G_{1}(\omega_{k})\bigr{\rvert}\gg\bigl{\lvert}G_{0}(\omega_{k})\bigr{\rvert}\;,$
(3.63)

which directly contradicts (3.45) (recall that
$\bigl{\lvert}G_{0}(\omega_{k})\bigr{\rvert}=1/\beta$ for auxiliary fields).
This means that, as long as we must include auxiliary fields in our
considerations, there is no regime of large $p$ and small $\omega_{k}$ in
which the approximation we made in our large $p$ computation and the
approximation of dropping the kinetic terms are both reliable.

This result suggests that, at large $p$, the system at low energies first
enters into a quasi-conformal regime described by (3.58)–(3.59) in which
interactions compete with the kinetic terms, while at extremely low
temperatures the system ends up in a different conformal regime that satisfies
(3.22)–(3.23) and in which the kinetic terms are negligible.

A particular case, in this respect, is the supersymmetric one for which
$\mu_{R}=0$. In this case the IR conformal solution is supersymmetric since
its parameters satisfy the constraints (3.26)–(3.27) within the working
accuracy. Moreover, (3.29) can be independently derived from
$\mathcal{I}$-extremization, as we will do in Section 3.9, without assuming
(3.63). At this point, the zero-temperature limit where the index is computed
can be taken safely by requiring that $p$ goes to infinity faster than
$G_{1}(\omega)$ as $\omega$ goes to zero, in such a way that (3.45) holds.
Because of this, (3.58)–(3.59) in the supersymmetric regime do actually solve
(3.29).

####  3.5.1 Grand potential and entropy at large $\boldsymbol{p}$

It is possible to compute the grand potential $\log Z$ in a large $p$
expansion. To leading order in $1/N$, this is just the averaged action (3.6)
(or (B.3) in components) evaluated on the solution provided by (3.57). Instead
of a direct evaluation, we follow [11, 12] and first compute the derivative of
$\log Z$ with respect to $J$. Only the explicit dependence on $J$ in (3.6)
matters when taking this derivative, and not the dependence through
dimensionful coefficients such as $g_{\eta,\phi}$ in the bilocal fields
$G_{\eta,\phi}$ since the fields solve the equations of motion. We get:

$\displaystyle J\,\partial_{J}\,\frac{\log
Z}{N}=J\int\\!d\tau_{1}\,d\tau_{2}\,\biggl{(}\frac{1}{q}\,G_{\bar{f}f}\,G_{\bar{\eta}\eta}^{p-1}\,G_{\bar{\phi}\phi}^{q}+\frac{1}{p}\,G_{\bar{\eta}\eta}^{p}\,G_{\bar{\psi}\psi}\,G_{\bar{\phi}\phi}^{q-1}\biggr{)}$
(3.64)
$\displaystyle\,\,\qquad\qquad=\frac{\mathcal{J}^{2}\cos^{2}\bigl{(}\frac{v}{2}\bigr{)}}{4p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\int_{-\beta}^{\beta}\\!d\tau\,\frac{\beta-|\tau|}{\cos^{2}\bigl{(}\frac{v}{\beta}|\tau|-\frac{v}{2}\bigr{)}}+\mathcal{O}(p^{-3})=\frac{v\tan\bigl{(}\frac{v}{2}\bigr{)}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\,.$

To the integral on the first line the delta functions in $G_{\bar{f}f}$ and
$G_{\bar{\psi}\psi}$ do not contribute, and hence only the first term
contributes at leading order, since the second term is suppressed by
$\tfrac{1}{p^{2}}$ with respect to the first one. Using the relation between
$\beta\mathcal{J}$ and $v$ in (3.57) as well as the linearity of
$\beta\mathcal{J}$ in $J$ in (3.52), the derivative with respect to $J$ can be
exchanged for a derivative with respect to $v$ using the formula
$J\,\partial_{J}=v/\bigl{[}1+\frac{v}{2}\tan\bigl{(}\frac{v}{2}\bigr{)}\bigr{]}\,\partial_{v}$.
We obtain a differential equation in $v$ for $\log Z$:

$\frac{1}{N}\,\partial_{v}\log
Z=\frac{\tan\bigl{(}\frac{v}{2}\bigr{)}\,\bigl{[}1+\frac{v}{2}\tan\bigl{(}\frac{v}{2}\bigr{)}\bigr{]}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\;.$
(3.65)

The value of $\log Z$ at $v=0$ is known because at that point the theory is
free. We can read off $Z\big{|}_{v=0}$ from (C.18) and (C.26). Integrating
(3.65) from $0$ to $v$ at large $p$ then gives

$\frac{\log
Z}{N}=\log\biggl{[}2\cosh\biggl{(}\frac{\mu_{\eta}\beta}{2}\biggr{)}\biggr{]}-\alpha\log\biggl{[}2\sinh\biggl{(}\frac{\mu_{\phi}\beta}{2}\biggr{)}\biggr{]}+\frac{v\tan\bigl{(}\frac{v}{2}\bigr{)}-\frac{v^{2}}{4}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\;.$
(3.66)

As $p\rightarrow\infty$, $\log Z$ tends to the free value, which is consistent
with our starting ansatz (3.47).

In order to compute the von Neumann entropy $S=(1-\beta\partial_{\beta})\log
Z$, one needs to make the $\beta$ dependence in $v$ explicit. Using
$\beta\partial_{\beta}=\beta\mathcal{J}\cos\bigl{(}\frac{v}{2}\bigr{)}/\bigl{[}1+\frac{v}{2}\tan\bigl{(}\frac{v}{2}\bigr{)}\bigr{]}\cdot\partial_{v}$
we obtain

$\displaystyle\frac{S}{N}$
$\displaystyle=\log\biggl{[}2\cosh\biggl{(}\frac{\mu_{\eta}\beta}{2}\biggr{)}\biggr{]}-\frac{\mu_{\eta}\beta}{2}\tanh\biggl{(}\frac{\mu_{\eta}\beta}{2}\biggr{)}-\alpha\log\biggl{[}2\sinh\biggl{(}\frac{\mu_{\phi}\beta}{2}\biggr{)}\biggr{]}+\frac{\alpha\mu_{\phi}\beta}{2}\coth\biggl{(}\frac{\mu_{\phi}\beta}{2}\biggr{)}$
$\displaystyle\quad-\frac{v^{2}}{8p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\;.$
(3.67)

On the first line is the entropy of the free theory, as in (C.19) and (C.27),
while on the second line is the first correction. In order to obtain the low-
temperature behavior, we take $\beta\mathcal{J}\rightarrow\infty$ while
keeping $\beta\mu_{\eta,\phi}$ fixed and expand $v(\beta\mathcal{J})$ in
(3.57):

$v^{2}=\pi^{2}-\frac{4\pi^{2}}{\beta\mathcal{J}}+\frac{12\pi^{2}}{(\beta\mathcal{J})^{2}}+\mathcal{O}\bigl{(}(\beta\mathcal{J})^{-3}\bigr{)}\;.$
(3.68)

In particular, the zero-temperature specific entropy
$\mathcal{S}_{0}\,\equiv\,S\bigl{(}\beta=\infty\bigr{)}/N$ can be expressed
using the relations $\beta\mu_{\eta,\phi}=-2\pi\mathcal{E}_{\eta,\phi}$
between the chemical potentials and the spectral asymmetries of the IR
conformal solution, which are valid at zero temperature:

$\displaystyle\mathcal{S}_{0}$
$\displaystyle=\log\bigl{[}2\cosh(-\pi\mathcal{E}_{\eta})\bigr{]}-\pi\mathcal{E}_{\eta}\tanh(\pi\mathcal{E}_{\eta})-\alpha\log\bigl{[}2\sinh(-\pi\mathcal{E}_{\phi})\bigr{]}+\alpha\,\pi\mathcal{E}_{\phi}\coth(\pi\mathcal{E}_{\phi})$
$\displaystyle\quad-\frac{\pi^{2}}{8p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\;.$
(3.69)

We shall see in Section 3.9 that this quantity, when evaluated on
supersymmetric spectral asymmetries satisfying
$p\mathcal{E}_{\eta}+q\mathcal{E}_{\phi}=0$, matches the large $p$ expansion
of the entropy extracted from the Witten index.

###  3.6 Solutions at large $\boldsymbol{p}$ and $\boldsymbol{q}$

We can similarly search for analytic solutions to the Schwinger-Dyson
equations at large $p$ and $q$, following the same steps as in Section 3.5. We
consider an ansatz for the bilocal fields similar to the one in (3.47),
however we keep the ratio $p/q$ fixed as we send $p\to\infty$:

$\displaystyle G_{\bar{\eta}\eta}(\tau)$
$\displaystyle=G_{\bar{\eta}\eta}^{(0)}(\tau)\biggl{(}1+\frac{1}{p}\widetilde{G}_{\bar{\eta}\eta}(\tau)+\mathcal{O}(p^{-2})\biggr{)}\,,\;\;$
$\displaystyle G_{\bar{f}f}(\tau)$
$\displaystyle=-\delta(\tau)+\frac{1}{p}\widetilde{G}_{\bar{f}f}(\tau)+\ldots\;,$
(3.70) $\displaystyle G_{\bar{\phi}\phi}(\tau)$
$\displaystyle=G_{\bar{\phi}\phi}^{(0)}(\tau)\biggl{(}1+\frac{1}{q}\widetilde{G}_{\bar{\phi}\phi}(\tau)+\mathcal{O}\bigl{(}p^{-2}\bigr{)}\biggr{)}\,,\;\;$
$\displaystyle G_{\bar{\psi}\psi}(\tau)$
$\displaystyle=\alpha\,\delta(\tau)+\frac{1}{q}\widetilde{G}_{\bar{\psi}\psi}(\tau)+\ldots\;.$

In particular
$\bigl{[}G_{\bar{\phi}\phi}\bigr{]}^{q-n}=\bigl{[}G_{\bar{\phi}\phi}^{(0)}\bigr{]}^{q-n}\,e^{\widetilde{G}_{\bar{\phi}\phi}}\,\bigl{(}1+\mathcal{O}(p^{-1})\bigr{)}$
besides (3.48). We insist that no chemical potential is larger than
$\mathcal{O}(1)$, therefore according to (3.49) we shall choose the scaling
$\mu_{\eta}\sim\mu_{\phi}\sim\mu_{R}\sim\mathcal{O}(1)$ while
$\mu_{F}\sim\mathcal{O}(p^{-1})$, with the possibility of setting $\mu_{R}=0$.
In particular $\mu_{\eta}=-(q/p)\,\mu_{\phi}+\mathcal{O}(p^{-1})$.

Expanding the Schwinger-Dyson equations determines $\widetilde{G}_{\bar{f}f}$
and $\widetilde{G}_{\bar{\psi}\psi}$ algebraically as

$\displaystyle\widetilde{G}_{\bar{f}f}(\tau)$
$\displaystyle=\frac{\mathcal{J}\,e^{(\mu_{\eta}-\mu_{R})\tau}}{2\cosh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\Bigl{[}\Theta(\tau)\,e^{-(\mu_{\eta}-\mu_{R})\frac{\beta}{2}}+\Theta(-\tau)\,e^{(\mu_{\eta}-\mu_{R})\frac{\beta}{2}}\Bigr{]}\,e^{\widetilde{G}_{\bar{\eta}\eta}(-\tau)+\widetilde{G}_{\bar{\phi}\phi}(-\tau)},$
(3.71) $\displaystyle\widetilde{G}_{\bar{\psi}\psi}(\tau)$
$\displaystyle=\frac{q^{2}\,\mathcal{J}\sinh\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}\,e^{(\mu_{\phi}-\mu_{R})\tau}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\Bigl{[}\Theta(\tau)\,e^{-(\mu_{\phi}-\mu_{R})\frac{\beta}{2}}-\Theta(-\tau)\,e^{(\mu_{\phi}-\mu_{R})\frac{\beta}{2}}\Bigr{]}\,e^{\widetilde{G}_{\bar{\eta}\eta}(-\tau)+\widetilde{G}_{\bar{\phi}\phi}(-\tau)}.$

We use the same definition of $\mathcal{J}$ as in (3.52) (with $n_{f}=1$), and
we keep $\mathcal{J}$ fixed as $p\to\infty$. The algebraic equations then also
determine

$\displaystyle\Sigma_{\bar{\eta}\eta}(\tau)$
$\displaystyle=-\frac{\mathcal{J}^{2}}{\gamma\,p}\;G^{(0)}_{\bar{\eta}\eta}(-\tau)\;e^{\widetilde{G}_{\bar{\eta}\eta}(\tau)+\widetilde{G}_{\bar{\eta}\eta}(-\tau)+\widetilde{G}_{\bar{\phi}\phi}(\tau)+\widetilde{G}_{\bar{\phi}\phi}(-\tau)}+\dots$
(3.72) $\displaystyle\Sigma_{\bar{\phi}\phi}(\tau)$
$\displaystyle=\frac{\mathcal{J}^{2}\,(1-\gamma)}{\alpha\,\gamma^{2}\,q}\;G_{\bar{\phi}\phi}^{(0)}(-\tau)\;e^{\widetilde{G}_{\bar{\eta}\eta}(\tau)+\widetilde{G}_{\bar{\eta}\eta}(-\tau)+\widetilde{G}_{\bar{\phi}\phi}(\tau)+\widetilde{G}_{\bar{\phi}\phi}(-\tau)}+\dots$

where we defined

$\frac{1}{\gamma}\,\equiv\,1+\frac{q^{2}\sinh^{2}\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}}{\alpha
p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\;.$ (3.73)

The dynamical equations give two differential equations for
$\widetilde{G}_{\bar{\eta}\eta}$, $\widetilde{G}_{\bar{\phi}\phi}$ that can be
recast as:

$\partial_{\tau}^{2}\,\widetilde{G}(\tau)=\mathcal{J}^{2}\gamma^{-2}\;e^{\widetilde{G}(\tau)+\widetilde{G}(-\tau)}\;,\qquad\widetilde{G}_{\bar{\eta}\eta}(\tau)=\gamma\,\widetilde{G}(\tau)\;,\qquad\widetilde{G}_{\bar{\phi}\phi}(\tau)=(1-\gamma)\,\widetilde{G}(\tau)\;.$
(3.74)

Note that $\gamma\in(0,1)$. The solution for $\widetilde{G}$ is again (3.57),
with the substitution $\mathcal{J}\to\mathcal{J}\gamma^{-1}$. In the weak
coupling limit $\beta\mathcal{J}\rightarrow 0$, the solution reduces to the
free UV solution to first order in $\frac{1}{p}$. In the strong coupling
limit, instead, we reproduce the conformal two-point functions with the
following parameters, at this order of approximation:

$\displaystyle\Delta_{\eta}$
$\displaystyle=\frac{\gamma}{2p}+\mathcal{O}(p^{-2})\,,$ $\displaystyle
2\pi\mathcal{E}_{\eta}$
$\displaystyle=-\beta\mu_{\eta}+\mathcal{O}(p^{-2})\,,$ (3.75)
$\displaystyle\Delta_{f}$ $\displaystyle=\frac{1}{2}+\mathcal{O}(p^{-1})\,,$
$\displaystyle 2\pi\mathcal{E}_{f}$
$\displaystyle=\beta\mu_{R}-\beta\mu_{\eta}+\mathcal{O}(p^{-1})\,,$
$\displaystyle\Delta_{\phi}$
$\displaystyle=\frac{1-\gamma}{2q}+\mathcal{O}(p^{-2})\,,$ $\displaystyle
2\pi\mathcal{E}_{\phi}$
$\displaystyle=-\beta\mu_{\phi}+\mathcal{O}(p^{-2})\,,$
$\displaystyle\Delta_{\psi}$
$\displaystyle=\frac{1}{2}+\mathcal{O}(p^{-1})\,,$ $\displaystyle
2\pi\mathcal{E}_{\psi}$
$\displaystyle=\beta\mu_{R}-\beta\mu_{\phi}+\mathcal{O}(p^{-1})\,,$
$\displaystyle g_{\eta}$
$\displaystyle=\frac{1}{2\cosh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\biggl{(}1+\frac{\gamma}{p}\log\gamma\biggr{)}+\mathcal{O}(p^{-2})\,,$
$\displaystyle
g_{f}=\frac{\gamma}{2p\cosh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-2})\,,$
$\displaystyle g_{\phi}$
$\displaystyle=\frac{\alpha}{2\sinh\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}}\biggl{(}1+\frac{1-\gamma}{q}\log\gamma\biggr{)}+\mathcal{O}(p^{-2})\,,$
$\displaystyle
g_{\psi}=\frac{\alpha\,(1-\gamma)}{2q\sinh\bigl{(}\frac{\mu_{\phi}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-2})\,.$

These values satisfy the consistency bounds (3.24).

As in the large $p$ fixed $q$ case, as long as we must include auxiliary
fields in our considerations, there is no regime of large $p,q$ and small
$\omega_{k}$ for which the large $p,q$ solution and the conformal solution are
both reliable. Again, $\mathcal{I}$-extremization gives us an independent
derivation of (3.29) in the supersymmetric case.

####  3.6.1 Grand potential and entropy at large $\boldsymbol{p}$ and
$\boldsymbol{q}$

We now compute the grand potential $\log Z$ to leading order in $1/N$ in a
large $p$ expansion, at fixed $p/q$, by evaluating the averaged action on the
solution of the large $p$ and $q$ Schwinger-Dyson equations. By
differentiating $\log Z$ with respect to $J$ we get:

$\displaystyle J\,\partial_{J}\,\frac{\log Z}{N}$
$\displaystyle=J\\!\int\\!d\tau_{1}\,d\tau_{2}\,\biggl{(}\frac{1}{q}\,G_{\bar{f}f}\,G_{\bar{\eta}\eta}^{p-1}\,G_{\bar{\phi}\phi}^{q}+\frac{1}{p}\,G_{\bar{\eta}\eta}^{p}\,G_{\bar{\psi}\psi}\,G_{\bar{\phi}\phi}^{q-1}\biggr{)}$
(3.76)
$\displaystyle=\frac{\mathcal{J}^{2}\cos^{2}\bigl{(}\frac{v}{2}\bigr{)}}{4p^{2}\gamma\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\int_{-\beta}^{\beta}\\!d\tau\,\frac{\beta-|\tau|}{\cos^{2}\bigl{(}\frac{v}{\beta}|\tau|-\frac{v}{2}\bigr{)}}+\mathcal{O}(p^{-3})=\frac{\gamma
v\tan\bigl{(}\frac{v}{2}\bigr{)}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\,,$

where $\gamma$ is defined in (3.73). Notice that, this time, both terms in the
action contribute at leading order. We follow the same steps as in Section
3.5.1, we exchange the derivative with respect to $J$ for a derivative with
respect to $v$, we integrate from $0$ to $v$, and obtain

$\frac{\log
Z}{N}=\log\biggl{[}2\cosh\biggl{(}\frac{\mu_{\eta}\beta}{2}\biggr{)}\biggr{]}-\alpha\log\biggl{[}2\sinh\biggl{(}\frac{\mu_{\phi}\beta}{2}\biggr{)}\biggr{]}+\frac{\gamma
v\tan\bigl{(}\frac{v}{2}\bigr{)}-\gamma\frac{v^{2}}{4}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\;.$
(3.77)

In order to compute the von Neumann entropy $S=(1-\beta\partial_{\beta})\log
Z$ we rewrite the $\beta$ dependence through $v$. By using
$\beta\partial_{\beta}=\bigl{(}1-\beta\gamma^{-1}\partial_{\beta}\gamma\bigl{)}\,v\,/\,\bigl{[}1+\frac{v}{2}\tan\bigl{(}\frac{v}{2}\bigr{)}\bigr{]}\cdot\partial_{v}$,
we get

$\displaystyle\frac{S}{N}$
$\displaystyle=\log\biggl{[}2\cosh\biggl{(}\frac{\mu_{\eta}\beta}{2}\biggr{)}\biggr{]}-\frac{\mu_{\eta}\beta}{2}\tanh\biggl{(}\frac{\mu_{\eta}\beta}{2}\biggr{)}-\alpha\log\biggl{[}2\sinh\biggl{(}\frac{\mu_{\phi}\beta}{2}\biggr{)}\biggr{]}+\frac{\alpha\mu_{\phi}\beta}{2}\coth\biggl{(}\frac{\mu_{\phi}\beta}{2}\biggr{)}$
$\displaystyle\quad-\frac{\gamma\,v^{2}}{8p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}\biggl{[}1+(1-\gamma)\,\mu_{\phi}\beta\coth\biggl{(}\frac{\mu_{\phi}\beta}{2}\biggr{)}+\gamma\,\mu_{\eta}\beta\tanh\biggl{(}\frac{\mu_{\eta}\beta}{2}\biggr{)}\biggr{]}$
$\displaystyle\quad+\frac{\gamma\,v\,\mu_{\eta}\beta\,\tan\bigl{(}\frac{v}{2}\bigr{)}\tanh\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}{2p^{2}\cosh^{2}\bigl{(}\frac{\mu_{\eta}\beta}{2}\bigr{)}}+\mathcal{O}(p^{-3})\;.$
(3.78)

We now take $\beta\mathcal{J}\rightarrow\infty$ while keeping $\beta\mu_{I}$
fixed, in order to get the zero-temperature specific entropy. Expressing
everything in terms of the spectral asymmetries of the IR conformal solution,
and keeping only finite terms, we get

$\displaystyle\mathcal{S}_{0}$
$\displaystyle=\log\bigl{[}2\cosh(\pi\mathcal{E}_{\eta})\bigr{]}-\pi\mathcal{E}_{\eta}\tanh(\pi\mathcal{E}_{\eta})-\alpha\log\bigl{[}2\sinh(-\pi\mathcal{E}_{\phi})\bigr{]}+\alpha\,\pi\mathcal{E}_{\phi}\coth(\pi\mathcal{E}_{\phi})$
(3.79)
$\displaystyle\quad-\frac{\gamma\,\pi^{2}}{8p^{2}\cosh^{2}\bigl{(}\pi\mathcal{E}_{\eta}\bigr{)}}\Bigl{[}1+2(1-\gamma)\pi\mathcal{E}_{\phi}\coth(\pi\mathcal{E}_{\phi})+2\gamma\pi\mathcal{E}_{\eta}\tanh(\pi\mathcal{E}_{\eta})\Bigr{]}+\mathcal{O}(p^{-3})\;.$

This result matches the large $p,q$ expansion of the entropy extracted from
the Witten index in Section 3.9 for supersymmetric chemical potentials.

###  3.7 Luttinger–Ward relation

The “Luttinger–Ward” (LW) relation [38] gives the total charge $\langle
Q\rangle$ of a conformal solution in terms of a sum over contributions indexed
by $A$, each associated with a field of abundance $N_{A}$, charge $Q_{A}$,
statistics $s_{A}=\pm 1$, dimension $\Delta_{A}$, and spectral asymmetry
$\mathcal{E}_{A}$. As reviewed in Appendix D, the relation is

$\displaystyle\langle Q\rangle$
$\displaystyle=\sum\nolimits_{A}\,N_{A}\,Q_{A}\,\mathfrak{q}_{s_{A}}(\Delta_{A},\mathcal{E}_{A})+C\;,\qquad\qquad
C=\text{const.}$ (3.80) $\displaystyle\mathfrak{q}_{s}(\Delta,\mathcal{E})$
$\displaystyle=s\Biggl{[}\bigl{(}\Delta-\tfrac{1}{2}\bigr{)}\frac{\sinh(2\pi\mathcal{E})}{\cosh(2\pi\mathcal{E})-\cos\bigl{[}2\pi\bigl{(}\Delta+\frac{1-s}{4}\bigr{)}\bigr{]}}+\frac{1}{2\pi
i}\log\frac{\sin\bigl{[}\pi\bigl{(}\Delta+\frac{1-s}{4}+i\mathcal{E}\bigr{)}\bigr{]}}{\sin\bigl{[}\pi\bigr{(}\Delta+\frac{1-s}{4}-i\mathcal{E}\bigr{)}\bigr{]}}\Biggr{]}.$

The charge is only determined up to a constant $C$, because of ambiguities in
matching the UV limit (see also Appendix A of [26]).

For our model, the charges of fields are in Table 1. For the superconformal
solutions, using (3.26)–(3.27), the charges $\langle Q_{\eta}\rangle$,
$\langle Q_{\phi}\rangle$, $\langle Q_{F}\rangle$ are:

$\displaystyle\frac{\langle Q_{\eta}\rangle}{N}$
$\displaystyle=\frac{q\,\Delta_{\phi}\sinh(2\pi\mathcal{E}_{\eta})}{\cosh(2\pi\mathcal{E}_{\eta})+\cos(2\pi\Delta_{\eta})}+\frac{\alpha\,p\,\Delta_{\phi}\,\sinh(2\pi\mathcal{E}_{\phi})}{\cosh(2\pi\mathcal{E}_{\phi})-\cos(2\pi\Delta_{\phi})}-\frac{p}{2\pi
i}\log\frac{\cos[\pi(\Delta_{\eta}+i\mathcal{E}_{\eta})]}{\cos[\pi(\Delta_{\eta}-i\mathcal{E}_{\eta})]}$
$\displaystyle\quad+\frac{\alpha\,p}{2\pi
i}\log\frac{\sin[\pi(\Delta_{\phi}+i\mathcal{E}_{\phi})]}{\sin[\pi(\Delta_{\phi}-i\mathcal{E}_{\phi})]}+C_{\eta}\;,$
$\displaystyle\frac{\langle Q_{\phi}\rangle}{N}$
$\displaystyle=-\frac{q\,\Delta_{\eta}\sinh(2\pi\mathcal{E}_{\eta})}{\cosh(2\pi\mathcal{E}_{\eta})+\cos(2\pi\Delta_{\eta})}-\frac{\alpha\,p\,\Delta_{\eta}\sinh(2\pi\mathcal{E}_{\phi})}{\cosh(2\pi\mathcal{E}_{\phi})-\cos(2\pi\Delta_{\phi})}-\frac{q}{2\pi
i}\log\frac{\cos[\pi(\Delta_{\eta}+i\mathcal{E}_{\eta})]}{\cos[\pi(\Delta_{\eta}-i\mathcal{E}_{\eta})]}$
$\displaystyle\quad+\frac{\alpha\,q}{2\pi
i}\log\frac{\sin[\pi(\Delta_{\phi}+i\mathcal{E}_{\phi})]}{\sin(\pi(\Delta_{\phi}-i\mathcal{E}_{\phi})]}+C_{\phi}\;,$
$\displaystyle\frac{\langle Q_{F}\rangle}{N}$
$\displaystyle=\frac{q}{2}\,\frac{\sinh(2\pi\mathcal{E}_{\eta})}{\cosh(2\pi\mathcal{E}_{\eta})+\cos(2\pi\Delta_{\eta})}+\frac{\alpha
p}{2}\,\frac{\sinh(2\pi\mathcal{E}_{\phi})}{\cosh(2\pi\mathcal{E}_{\phi})-\cos(2\pi\Delta_{\phi})}+q\,C_{\eta}-p\,C_{\phi}\;.$
(3.81)

The constants $C_{\eta,\phi}$ will be determined by analyzing the Witten index
in Section 3.9.

###  3.8 Non-conformal solutions

In addition to the conformal solutions we discussed so far, which can describe
the system at low energies, we also find non-conformal solutions. This is not
unusual: they have previously been identified in SYK-like models, e.g. in [12,
26]. These solutions can be concurrent with conformal solutions at a given
fixed value of $\mu_{\eta,\phi}$, although they differ in their charge.
However, unlike the conformal solutions, they are exact solutions to the full
Schwinger–Dyson equations at $\beta=\infty$.

We identify two families of solutions that are compatible with the requirement
$\mu_{\phi}>0$.888We further require $G_{\bar{\phi}\phi}>0$, since the sign of
the Euclidean $G_{\bar{\phi}\phi}$ is the sign of the Lorentzian spectral
density which must be non-negative, as we discuss in Appendix C. This rules
out other exponential solutions of the Schwinger–Dyson equations. Furthermore,
we require that these solutions do not diverge for $\tau\to\pm\infty$. Such
solutions only exist when the chemical potentials $\mu_{\eta,\phi}$ are in
certain domains. When both chemical potentials are above some critical value,
i.e., when $\mu_{\eta}>\mu_{\eta}^{\text{c}}$ and $\mu_{\phi}>0$, we find:

$\displaystyle G_{\bar{\eta}\eta}(\tau)$
$\displaystyle=-\Theta(-\tau)\,e^{(\mu_{\eta}-\mu^{\text{c}}_{\eta})\tau},$
$\displaystyle G_{\bar{f}f}(\tau)$
$\displaystyle=-\delta(\tau)+\Theta(\tau)\,a_{f}\,e^{-\left[(p-1)(\mu_{\eta}-\mu^{\text{c}}_{\eta})+q(\mu_{\phi}-\mu^{\text{c}}_{\phi})+a_{f}\right]\tau}$
(3.82) $\displaystyle G_{\bar{\phi}\phi}(\tau)$
$\displaystyle=\alpha\,\Theta(-\tau)\,e^{(\mu_{\phi}-\mu^{\text{c}}_{\phi})\tau},$
$\displaystyle G_{\bar{\psi}\psi}(\tau)$
$\displaystyle=\alpha\,\delta(\tau)+\alpha\,a_{\psi}\,\Theta(\tau)\,e^{-\left[p(\mu_{\eta}-\mu^{\text{c}}_{\eta})+(q-1)(\mu_{\phi}-\mu^{\text{c}}_{\phi})-a_{\psi}\right]\tau}$

where the constants take the following values:999We used the rule
$\Theta(\tau)^{m}\,\Theta(-\tau)^{n}\,\delta(\tau)=2^{-m-n}\,\delta(\tau)$,
namely $\Theta(0)=\frac{1}{2}$, which follows from using the function
$\text{P}\frac{1}{i\omega}+\pi\,\delta(\omega)$ as the Fourier transform of
$\Theta(\tau)$, and will be consistent with our numerical results.

$\mu_{\eta}^{\text{c}}=\frac{(p+q-1)\,\alpha^{q}J}{2^{p+q-2}\,q}\,,\quad\mu_{\phi}^{\text{c}}=-\frac{(p+q-1)\,\alpha^{q-1}J}{2^{p+q-2}\,p}\,,\quad
a_{f}=\frac{J\alpha^{q}}{q}\,,\quad a_{\psi}=\frac{J\alpha^{q-1}}{p}\,.$
(3.83)

Note that the chemical potentials must also satisfy
$p\,(\mu_{\eta}-\mu^{\text{c}}_{\eta})+(q-1)(\mu_{\phi}-\mu^{\text{c}}_{\phi})>a_{\psi}$.
On the other hand, when $\mu_{\eta}$ is below a critical value but
$\mu_{\phi}$ is above, we find:

$\displaystyle G_{\bar{\eta}\eta}$
$\displaystyle=\Theta(\tau)\,e^{-(\mu^{\text{c}}_{\eta}-\mu_{\eta})\tau}\,,$
$\displaystyle G_{\bar{f}f}$
$\displaystyle=-\delta(\tau)+\delta_{p,1}\,\Theta(\tau)\,a_{f}\,e^{-\left[q(\mu_{\phi}-\mu^{\text{c}}_{\phi})+a_{f}\right]\tau}$
(3.84) $\displaystyle G_{\bar{\phi}\phi}$
$\displaystyle=\Theta(-\tau)\,\alpha\,e^{(\mu_{\phi}-\mu^{\text{c}}_{\phi})\tau}\,,\quad$
$\displaystyle G_{\bar{\psi}\psi}$ $\displaystyle=\alpha\,\delta(\tau)$

with

$\mu_{\eta}^{\text{c}}=\frac{(q-p+1)\alpha^{q}J}{2^{p+q-2}q}\;,\qquad\mu_{\phi}^{\text{c}}=\frac{(q-p-1)\alpha^{q-1}J}{2^{p+q-2}p}\;,\qquad
a_{f}=\frac{J\alpha^{q}}{q}\;.$ (3.85)

In particular, for $p=1$ and $q=2$, the critical chemical potentials are
$\mu_{\eta}^{\text{c}}=\frac{1}{2}J\alpha^{2}$ and $\mu_{\phi}^{\text{c}}=0$.
In this solution, the boson behaves like a free boson while the fermion seems
to acquire an effective mass. This makes this solution similar to the non-
conformal solutions found in the two-fermion model of [26]. In the particular
case where the chemical potentials do not break supersymmetry, i.e. when
$\mu_{\eta}=-2\mu_{\phi}$, the solution (3.82) cannot be realized. However,
the solution (3.84) is consistent for any $\mu_{\phi}>0$.

We can also explicitly calculate the entropy of these solutions. We take the
finite-$\beta$ off-shell action and apply $\bigl{(}1-\beta J\partial_{\beta
J}-\beta\mu\partial_{\beta\mu}\bigr{)}$. After simplifying with the equations
of motion, we find

$\displaystyle\frac{S}{N}$
$\displaystyle=\log\Bigl{[}2\cosh\bigl{(}\tfrac{\beta\mu_{\eta}}{2}\bigr{)}\Bigr{]}-\alpha\log\Bigl{[}2\sinh\bigl{(}\tfrac{\beta\mu_{\phi}}{2}\bigr{)}\Bigr{]}+\beta\mu_{\eta}\,Q_{\eta}+\beta\mu_{\phi}\,Q_{\phi}$
(3.86)
$\displaystyle\quad+\sum_{k}\Biggl{[}\log\Biggl{(}\frac{1+\frac{\Sigma_{\bar{\eta}\eta}(\omega_{k})}{i\omega_{k}-\mu_{\eta}}}{1+\Sigma_{\bar{f}f}(\omega_{k})}\Biggr{)}-\alpha\log\Biggl{(}\frac{1+\frac{\Sigma_{\bar{\phi}\phi}(\omega_{k})}{i\omega_{k}-\mu_{\phi}}}{1+\Sigma_{\bar{\psi}\psi}(\omega_{k})}\Biggr{)}\Biggr{]}$
$\displaystyle\quad-\beta\int_{0}^{\beta}\\!d\tau\Bigl{[}\Sigma_{\bar{\eta}\eta}(\tau)\,G_{\bar{\eta}\eta}(\tau)+\Sigma_{\bar{f}f}(\tau)\,G_{\bar{f}f}(\tau)+\Sigma_{\bar{\phi}\phi}(\tau)\,G_{\bar{\phi}\phi}(\tau)+\Sigma_{\bar{\psi}\psi}(\tau)\,G_{\bar{\psi}\psi}(\tau)\Bigr{]}$

where $\omega_{k}$ are either the fermionic or bosonic Matsubara frequencies,
depending on the field. The first two terms are the free action, which appears
from regulating the logarithmic term. Promoting the sum to an integral, we can
plug (3.82) or (3.84) in, so that the sum cancels with the third line. This
leaves the entropy to be solely the “free part”:

$\frac{S}{N}=\lim_{\beta\,\rightarrow\,\infty}\Bigl{\\{}\log\Bigl{[}2\cosh\bigl{(}\tfrac{\beta\mu_{\eta}}{2}\bigr{)}\Bigr{]}-\alpha\log\Bigl{[}2\sinh\bigl{(}\tfrac{\beta\mu_{\phi}}{2}\bigr{)}\Bigr{]}+\tfrac{1}{2}\bigl{(}\beta\mu_{\eta}+\alpha\,\beta\mu_{\phi}\bigr{)}\Bigr{\\}}=0\;,$
(3.87)

where we assumed that $\mu$ does not scale with $\beta$. We obtain this result
for both families, irrespective of $p$ and $q$.

###  3.9 Witten index and $\boldsymbol{\mathcal{I}}$-extremization

In supersymmetric quantum mechanics, a protected and computable quantity is
the Witten index
$\mathcal{I}^{\text{W}}=\operatorname{Tr}_{\mathcal{H}}\,(-1)^{F}e^{-\beta
H}$, which does not depend on $\beta$ [42]. In the presence of a flavour
symmetry $\mathrm{U}(1)_{F}$, the index can be refined by inserting a complex
fugacity $x$ for the flavour charge $Q_{F}$. Besides, with $\mathcal{N}=2$
supersymmetry and in the presence of a $\mathrm{U}(1)$ R-symmetry $R$, one can
construct an alternative index in which $e^{i\pi R}$ is used in place of
$(-1)^{F}$:

$\mathcal{I}(x)=\operatorname{Tr}_{\mathcal{H}}\,e^{i\pi R}\,e^{-\beta
H}\,x^{Q_{F}}\;.$ (3.88)

Let us explain the relation between the two indices, for the models considered
in this paper.

When $\gcd(p,q)=1$ and there is no discrete $\mathbb{Z}_{\gcd(p,q)}$ flavour
symmetry, there exists an assignment of R-charges such that $(-1)^{F}=e^{i\pi
R}$ and hence (3.88) is identical to the ordinary Witten index. Such an
assignment is such that

$R[\eta]=2m+1\;,\qquad R[\phi]=2n\;,\qquad p\,R[\eta]+q\,R[\phi]=1$ (3.89)

for some $m,n\in\mathbb{Z}$ (and recall that $p$ is odd). These equations have
a solution if and only if $\gcd(p,q)=1$. Any other R-charge assignment is then
related to the one in (3.89) by mixing with the flavour symmetry, namely, by
rotating the phase of $x$.

When $d\equiv\gcd(p,q)>1$, although the ordinary Witten index is not equal to
(3.88), one can consider Witten indices refined by an additional twist $g$ for
the $\mathbb{Z}_{d}$ flavour symmetry, where $g$ was defined in (2.11) and
$g^{d}$ is equivalent to a $\mathrm{U}(1)_{F}$ rotation:

$\mathcal{I}^{\text{W}}_{r}(x)=\operatorname{Tr}_{\mathcal{H}}\,(-1)^{F}\,g^{r}\,e^{-\beta
H}\,x^{Q_{F}}\qquad\qquad\text{for}\quad r=0,\ldots,d-1\;.$ (3.90)

The R-charge assignment such that $(-1)^{F}g^{r}=e^{i\pi R}$ and thus
$\mathcal{I}(x)=\mathcal{I}^{\text{W}}_{r}(x)$ should satisfy
$R[\eta]=1+\frac{2r}{p}+2m$, $R[\phi]=2n$ and $p\,R[\eta]+q\,R[\phi]=1$. There
is always one and only one solution for $r$ in the range $0\leq r<d$ which is
$r=\frac{1+d}{2}\text{ mod }d$, with $r\neq 0$ whenever $d>1$. Thus (3.88) is
equivalent to one of the refined Witten indices.

Writing the $\mathrm{U}(1)_{F}$ fugacity as $x=e^{2\pi iu}=e^{2\pi
iy+2\pi\mathcal{E}}$ with $y,\mathcal{E}\in\mathbb{R}$, the insertions in
(3.88) can be recast in the following way:

$\mathcal{I}=\operatorname{Tr}_{\mathcal{H}}\,e^{i\pi(R\,+\,2yQ_{F})}\,e^{-\beta(H\,+\,\mu
Q_{F})}\qquad\text{where}\qquad 2\pi\mathcal{E}=-\mu\beta\;.$ (3.91)

Here we used the symbol $\mu$ in place of $\mu_{F}$ in order to avoid
cluttering, and later on we will identify $\mathcal{E}$ with the same quantity
introduced in (3.28). The $\mathcal{I}$-extremization principle introduced in
[36, 37] states that the values $\hat{y},\hat{\mu}$ that extremize
$\mathcal{I}$ under Laplace transform select the infrared Hamiltonian
$H_{\text{IR}}=H+\hat{\mu}Q_{F}$ and the infrared superconformal R-charge

$R_{\text{sc}}=R+2\hat{y}\,Q_{F}\;.$ (3.92)

Using the relation
$\bigl{\\{}\mathcal{G}_{\frac{1}{2}},\overline{\mathcal{G}}_{-\frac{1}{2}}\bigr{\\}}=L_{0}-R_{\text{sc}}/2$
in the superconformal algebra — see (A.22) — we observe that chiral primaries
annihilated by $\mathcal{G}_{\frac{1}{2}}=Q$ and
$\overline{\mathcal{G}}_{-\frac{1}{2}}=Q^{\dagger}$ must have
$\Delta=\frac{1}{2}R_{\text{sc}}$. Applying this to $\phi$ and $\eta$ we get

$\Delta_{\phi}=-p\,\hat{y}+\frac{R[\phi]}{2}\;,\qquad\qquad\Delta_{\eta}=q\,\hat{y}+\frac{R[\eta]}{2}\;,$
(3.93)

which allow us to trade $\hat{y}$ for the conformal dimensions
$\Delta_{\phi,\eta}$. The fact that $R$ is an R-charge guarantees that the
dimensions satisfy the supersymmetry constraint (3.27). Using (3.93) and Table
1, $R_{sc}$ can also be written as

$R_{\text{sc}}=2\Delta_{\eta}Q_{\eta}+2\Delta_{\phi}Q_{\phi}\,.$ (3.94)

The index has an ambiguity given by overall multiplication by a power of $x$.
In the Hamiltonian formalism this corresponds to the ambiguity in the
assignment of charges to the Fock vacuum (i.e., the normal ordering ambiguity
in the definition of charge operators). In the path-integral formalism it
corresponds to the ambiguity in the regularization of 1-loop
determinants.101010This is similar to the parity anomaly in 3d theories. For a
complex fermion of charge 1, the fermionic Fock space has two states, and if
we insist on assigning integer charges then there is no canonical choice and
one is forced to break charge conjugation. On the other hand, one could assign
charges $\pm\frac{1}{2}$ to the two states in a charge-conjugation invariant
fashion, but then the charges are not integer. We fix the ambiguity by
demanding that the charge operators be written as (anti-)commutators in the
Hamiltonian formalism. Each chiral multiplet contributes
$\bigl{(}e^{-\frac{i\pi R[\phi]}{2}}x^{\frac{p}{2}}-e^{\frac{i\pi
R[\phi]}{2}}x^{-\frac{p}{2}}\bigr{)}{}^{-1}$ while each Fermi multiplet
$e^{-\frac{i\pi R[\eta]}{2}}x^{-\frac{q}{2}}+e^{\frac{i\pi
R[\eta]}{2}}x^{\frac{q}{2}}$. The result is

$\mathcal{I}(x)=\Biggl{[}\frac{e^{-\frac{i\pi
R[\eta]}{2}}x^{-\frac{q}{2}}+e^{\frac{i\pi
R[\eta]}{2}}x^{\frac{q}{2}}}{\bigl{(}e^{-\frac{i\pi
R[\phi]}{2}}x^{\frac{p}{2}}-e^{\frac{i\pi
R[\phi]}{2}}x^{-\frac{p}{2}}\bigr{)}^{\alpha}}\Biggr{]}^{N}\;.$ (3.95)

In order to extract the degeneracy of BPS states $d(Q_{F},R)$ at fixed
charges, which is captured by the index written as

$\mathcal{I}(x)=\sum_{Q_{F},R}d(Q_{F},R)\;e^{i\pi R}\,x^{Q_{F}}\;,$ (3.96)

it is useful to rewrite it as a constrained partition function for BPS states.
Let $x_{\eta}=e^{2\pi iu_{\eta}}$ and $x_{\phi}=e^{2\pi iu_{\phi}}$ be the two
fugacities for $Q_{\eta}$ and $Q_{\phi}$, respectively, and consider

$\operatorname{Tr}_{\text{BPS}}x_{\eta}^{Q_{\eta}}\,x_{\phi}^{Q_{\phi}}=\operatorname{Tr}_{\text{BPS}}\exp\Bigl{\\{}2\pi
i\Bigl{[}\bigl{(}p\,u_{\eta}+q\,u_{\phi}\bigr{)}R+\bigl{(}R[\phi]\,u_{\eta}-R[\eta]\,u_{\phi}\bigr{)}Q_{F}\Bigr{]}\Bigr{\\}}$
(3.97)

where we used (2.10). When the chemical potentials are constrained to
$p\,u_{\eta}+q\,u_{\phi}=\frac{1}{2}$, and we identify $u\equiv
R[\phi]\,u_{\eta}-R[\eta]\,u_{\phi}$ according to (2.13), that quantity
reduces to the index $\mathcal{I}(x)$. Therefore, the Laplace transform of the
index computes:

$\displaystyle\oint\\!\frac{dx}{2\pi
ix}\>\mathcal{I}(x)\,x_{\eta}^{-Q_{\eta}}\,x_{\phi}^{-Q_{\phi}}\Big{|}_{pu_{\eta}+qu_{\phi}=\frac{1}{2}}$
$\displaystyle=\int_{0}^{1}\\!du\,\exp\Bigl{[}\log\mathcal{I}(x)-i\pi R-2\pi
iuQ_{F}\Bigr{]}$ (3.98)
$\displaystyle\equiv\int_{0}^{1}\\!du\;e^{N\,\mathcal{S}(u;\,Q_{F},R)}=\sum\nolimits_{\tilde{R}}d(Q_{F},\tilde{R})\,e^{i\pi(\tilde{R}-R)}\;.$

Here the function $\mathcal{S}(u;Q_{F},R)$, sometimes called the _entropy
function_ , is $\frac{1}{N}$ times the quantity in brackets on the first line.
At large $N$ and $Q_{F}$, one can compute the integral in the saddle-point
approximation extremizing $\mathcal{S}$ with respect to $u$. On the right-
hand-side of (3.98) appears a weighted sum over R-charges of the degeneracy of
states within a sector of fixed $Q_{F}$. Assuming that this sum is dominated
by one value, and comparing with the saddle point computation, one observes
that the R-charge $R$ of each saddle is fixed by requiring that $\mathcal{S}$
is real. Then the value of $N\mathcal{S}$ is the zero-temperature entropy,
$\log d(Q_{F},R)$.

The saddle point in terms of $u$ is given by

$\alpha\,p\cot\Bigl{(}p\pi u-\frac{\pi}{2}R[\phi]\Bigr{)}+q\tan\Bigl{(}q\pi
u+\frac{\pi}{2}R[\eta]\Bigr{)}+2i\mathcal{Q}_{F}=0\;,$ (3.99)

where we introduced $\mathcal{Q}_{F}=Q_{F}/N=\mathcal{O}(1)$. Substituting
(3.93) and (3.27), and using $y=u-i\mathcal{E}$, the real and imaginary parts
of this equation can be written in terms of $\Delta_{\phi}$ and $\mathcal{E}$.
The real part is precisely the relation (3.29) between the conformal dimension
$\Delta_{\phi}$ and the spectral asymmetry $\mathcal{E}$ in superconformal
solutions. The imaginary part reads:

$\frac{\alpha\,p\sinh(2\pi p\mathcal{E})}{\cosh(2\pi
p\mathcal{E})-\cos(2\pi\Delta_{\phi})}-\frac{q\sinh(2\pi
q\mathcal{E})}{\cosh(2\pi q\mathcal{E})+\cos\bigl{(}\frac{\pi}{p}-\frac{2\pi
q}{p}\Delta_{\phi}\bigr{)}}+2\mathcal{Q}_{F}=0\;.$ (3.100)

This coincides with the LW relation that determines $Q_{F}$ in (3.81), if the
constants satisfy $q\,C_{\eta}-p\,C_{\phi}=0$. Setting
$\operatorname{\mathbb{I}m}\mathcal{S}=0$ determines the R-charge to be

$\frac{R}{N}=\frac{1}{2\pi
i}\log\frac{\cos(\pi\Delta_{\eta}-i\pi\mathcal{E}_{\eta})}{\cos(\pi\Delta_{\eta}+i\pi\mathcal{E}_{\eta})}-\frac{\alpha}{2\pi
i}\log\frac{\sin(\pi\Delta_{\phi}-i\pi\mathcal{E}_{\phi})}{\sin(\pi\Delta_{\phi}+i\pi\mathcal{E}_{\phi})}+\frac{\alpha}{2}-2\biggl{(}\\!\frac{R[\phi]}{2p}-\frac{\Delta_{\phi}}{p}\\!\biggr{)}\mathcal{Q}_{F}\,,$
(3.101)

where the principal value is taken in the logarithms. Using (3.92) and (3.93),
we can determine the superconformal R-charge of the BPS states:

$\frac{R_{\text{sc}}}{N}=-\frac{1}{2\pi
i}\log\biggl{[}\frac{\cos(\pi\Delta_{\eta}+i\pi\mathcal{E}_{\eta})}{\cos(\pi\Delta_{\eta}-i\pi\mathcal{E}_{\eta})}\biggr{]}+\frac{\alpha}{2\pi
i}\log\biggl{[}\frac{\sin(\pi\Delta_{\phi}+i\pi\mathcal{E}_{\phi})}{\sin(\pi\Delta_{\phi}-i\pi\mathcal{E}_{\phi})}\biggr{]}+\frac{\alpha}{2}\;.$
(3.102)

On the other hand, substituting the LW relations for $Q_{\eta}$ and $Q_{\phi}$
from (3.81) into (3.94) gives another determination of $R_{\text{sc}}$. The
latter agrees with (3.102) if
$2\Delta_{\eta}C_{\eta}+2\Delta_{\phi}C_{\phi}=\frac{\alpha}{2}$. Solving this
and the previous constraint determines the constants in the Luttinger–Ward
relations to be

$C_{\eta}=\frac{\alpha\,p}{2}\;,\qquad\qquad C_{\phi}=\frac{\alpha\,q}{2}\;.$
(3.103)

We shall see that this is consistent with the numerical result in Figure 7. In
particular, we find out that $R_{\text{sc}}$ takes a common, non-zero value
among the supersymmetric ground states.111111The fact that the ground states
have a non-vanishing but well-defined R-charge (meaning that they are still
eigenvectors of the R-charge operator) means that the R-symmetry is unbroken
in those states. Finally, the real part of the entropy function that gives the
entropy is

$\operatorname{\mathbb{R}e}\mathcal{S}=\frac{1}{2}\log\Biggl{[}2^{1-\alpha}\,\frac{\cosh(2\pi
q\mathcal{E})+\cos\bigl{(}\frac{\pi}{p}-\frac{2\pi
q}{p}\Delta_{\phi}\bigr{)}}{\bigl{[}\cosh(2\pi
p\mathcal{E})-\cos(2\pi\Delta_{\phi})\bigr{]}^{\alpha}}\Biggr{]}-2\pi\mathcal{E}\mathcal{Q}_{F}\;.$
(3.104)

####  3.9.1 Entropy of various solutions

The index $\mathcal{I}(x)$ in (3.95) is the grand canonical partition function
of the theory, while $d(Q_{F},R)$ in (3.98) is the microcanonical degeneracy
of states. It follows that after substituting $\mathcal{E}(\mathcal{Q}_{F})$,
the quantity $\operatorname{\mathbb{R}e}\mathcal{S}$ in (3.104) becomes the
zero-temperature microcanonical entropy $S_{0}(\mathcal{Q}_{F})$. We compute
it in various cases, within the regime of validity of the IR conformal ansatz.

##### Solutions for $\boldsymbol{p=1}$, $\boldsymbol{q=2}$.

Plugging the explicit solution (3.33) for $\Delta_{\phi}(\mathcal{E})$ into
the LW relation (3.100) we can determine the spectral asymmetry $\mathcal{E}$
and $\Delta_{\phi}$ as functions of $\mathcal{Q}_{F}$:

$\mathcal{E}=\frac{1}{2\pi}\operatorname{arctanh}\biggl{(}\frac{2\mathcal{Q}_{F}}{2-\alpha}\biggr{)}\;,\qquad\qquad\Delta_{\phi}=\frac{1}{2\pi}\arccos\biggl{(}\frac{\alpha}{\sqrt{(2-\alpha)^{2}-4\mathcal{Q}_{F}^{2}}}\biggr{)}\;.$
(3.105)

Due to the bound $0\leq\tanh\bigl{(}\pi\mathcal{E}\bigr{)}<\sqrt{1-\alpha}$ in
(3.33) for conformal solutions, the charge is bound by
$0\leq\mathcal{Q}_{F}<\sqrt{1-\alpha}$ at fixed $\alpha<1$. Evaluating the
entropy function (3.104) on the solution, one obtains the entropy
$S_{0}=\operatorname{\mathbb{R}e}\mathcal{S}$:

$S_{0}=\frac{1}{2}\log\Biggl{[}\frac{4^{2-\alpha}\,(1-\alpha)^{1-\alpha}}{\rule{0.0pt}{10.00002pt}\bigl{(}2-\alpha+2\mathcal{Q}_{F}\bigr{)}{}^{1-\frac{\alpha}{2}+\mathcal{Q}_{F}}\,\bigl{(}2-\alpha-2\mathcal{Q}_{F}\bigr{)}{}^{1-\frac{\alpha}{2}-\mathcal{Q}_{F}}}\Biggr{]}\;.$
(3.106)

The quantity $e^{NS_{0}(\mathcal{Q}_{F})}$ is the degeneracy of ground states
at fixed charge $\mathcal{Q}_{F}$. In Figure 2 we plot the entropy
$S_{0}(\mathcal{Q}_{F})$ for various values of $\alpha$ and we observe that it
is always positive, with a maximum at $\mathcal{Q}_{F}=0$. The entropy does
not vanish as $\mathcal{Q}_{F}$ reaches the critical value $\sqrt{1-\alpha}$,
and the dashed line is the envelop of the entropy at those values as $\alpha$
is varied.

Figure 2: Plot of the entropy $S_{0}(\mathcal{Q}_{F})$ in the range
$0\leq\mathcal{Q}_{F}<\sqrt{1-\alpha}$ for different values of $\alpha$, in
the case $p=1$, $q=2$. The dashed line is the value of the entropy at the
upper critical value of $\mathcal{Q}_{F}$ as $\alpha$ is varied.

##### Solutions for $\boldsymbol{p=q>1}$, $\boldsymbol{\alpha=1}$ and
$\boldsymbol{\mathcal{E}\neq 0}$.

Similarly, plugging the explicit solution (3.39) for
$\Delta_{\phi}(\mathcal{E})$ into the LW relation (3.100) we get

$\mathcal{E}=-\frac{1}{2\pi
p}\operatorname{arcsinh}\biggl{(}\frac{B}{\sqrt{2}\,\mathcal{Q}_{F}}\biggr{)}\;,\qquad\Delta_{\phi}=\frac{1}{4p}-\frac{1}{2\pi}\arcsin\biggl{[}\frac{\sqrt{2}\,|\mathcal{Q}_{F}|}{C}\sin\Bigl{(}\frac{\pi}{2p}\Bigr{)}\biggr{]}\;,$
(3.107)

where we defined the following functions of $\mathcal{Q}_{F}$:

$\displaystyle A$
$\displaystyle=\cos\bigl{(}\tfrac{\pi}{2p}\bigr{)}\sqrt{\bigl{(}p^{2}-\mathcal{Q}_{F}^{2}\bigr{)}\rule{0.0pt}{6.99997pt}^{2}\cos^{2}\bigl{(}\tfrac{\pi}{2p}\bigr{)}+4p^{2}\mathcal{Q}_{F}^{2}}$
(3.108) $\displaystyle B$
$\displaystyle=\sqrt{\bigl{(}p^{2}-\mathcal{Q}_{F}^{2}\bigr{)}\cos^{2}\bigl{(}\tfrac{\pi}{2p}\bigr{)}+A}\;,$
$\displaystyle C$
$\displaystyle=\sqrt{p^{2}+\mathcal{Q}_{F}^{2}-\bigl{(}p^{2}-\mathcal{Q}_{F}^{2}\bigr{)}\sin^{2}\bigl{(}\tfrac{\pi}{2p}\bigr{)}+A}\;.$

Note that $C^{2}-B^{2}=2\mathcal{Q}_{F}^{2}$. Evaluating the entropy function
(3.104) on the solution we get

$S_{0}=\log\Biggl{[}\frac{\cos\bigl{(}\frac{\pi}{2p}\bigr{)}\,\bigl{(}pC+\lvert\mathcal{Q}_{F}\rvert
B\bigr{)}\,\bigl{(}C+\operatorname{sgn}(\mathcal{Q}_{F})B)^{\frac{\mathcal{Q}_{F}}{p}}}{2^{\frac{\mathcal{Q}_{F}}{2p}-\frac{1}{2}}\,|\mathcal{Q}_{F}|^{\frac{\mathcal{Q}_{F}}{p}}\,B^{2}}\Biggr{]}\;.$
(3.109)

Figure 3 shows $S_{0}(\mathcal{Q}_{F})$ against $\mathcal{Q}_{F}$ for various
values of $p=q$, and it is always positive.

Figure 3: Plot of $S_{0}(\mathcal{Q}_{F})$ against $\mathcal{Q}_{F}$ for the
solution in the cases $p=q=3,5,7,9$, $\mathcal{E}\neq 0$.

##### Solutions for $\boldsymbol{p=q>1}$ and $\boldsymbol{\mathcal{E}=0}$.

In this case, whenever $\alpha<\tan^{2}\bigl{(}\frac{\pi}{4p}\bigr{)}$ there
are two acceptable conformal solutions (3.41) which coincide at
$\alpha=\tan^{2}\bigl{(}\frac{\pi}{4p}\bigr{)}$. The only solution to the LW
relation (3.100) is $\mathcal{Q}_{F}=0$. Evaluating (3.104) on (3.41) and with
$\mathcal{Q}_{F}=0$ gives

$\\!S_{0}=\log\left[\frac{2\,\Bigl{(}(1-\alpha)\tan\bigl{(}\frac{\pi}{2p}\bigr{)}\pm\sqrt{(1-\alpha)^{2}\tan^{2}\bigl{(}\frac{\pi}{2p}\bigr{)}-4\alpha}\,\Bigr{)}^{\frac{1-\alpha}{2}}}{(1-\alpha)^{\frac{1-\alpha}{2}}\cos^{\alpha}\bigl{(}\frac{\pi}{2p}\bigr{)}\,\Bigl{(}(1+\alpha)\tan\bigl{(}\frac{\pi}{2p}\bigr{)}\pm\sqrt{(1-\alpha)^{2}\tan^{2}\bigl{(}\frac{\pi}{2p}\bigr{)}-4\alpha}\,\Bigr{)}^{\frac{1+\alpha}{2}}}\right]\\!.\\!$
(3.110)

The signs above are correlated with the sign in (3.41). Figure 4 shows $S_{0}$
against $\alpha$ for $p=3$, and it is always positive. In addition, we see
that the solution with the plus sign always has a higher entropy, and is
expected to be the dominant saddle at large $N$.

Figure 4: Plot of $S_{0}$ against
$\alpha\in\bigl{(}0,\tan^{2}\frac{\pi}{4p}\bigr{)}$ for the solutions in the
case $p=q=3$, $\mathcal{E}=0$. The blue/black curves correspond to the
solutions with a plus/minus sign in (3.41).

##### Solution at large $\boldsymbol{p}$.

One can check that the conformal parameters given in (3.58) solve the
Schwinger–Dyson equation (3.29) up to $\mathcal{O}(p^{-3})$. Substituting the
large $p$ values of $\Delta_{\eta}$ and $\Delta_{\phi}$ into (3.104), using
(3.81) and

$\displaystyle\cosh(2\pi\mathcal{E}_{\eta})+\cos(2\pi\Delta_{\eta})$
$\displaystyle=2\cosh^{2}(\pi
q\mathcal{E})-\tfrac{\pi^{2}}{2p^{2}}+\mathcal{O}(p^{-4})\;,$ (3.111)
$\displaystyle\cosh(2\pi\mathcal{E}_{\phi})-\cos(2\pi\Delta_{\phi})$
$\displaystyle=2\sinh^{2}(\pi p\mathcal{E})+\mathcal{O}(p^{-4})\;,$

reproduces the expansion in (3.69). This is a consistency check for the large
$p$ computation.

##  4 Numerical results

In this section we present some results we obtained by numerically solving the
Schwinger–Dyson equations. Our first goal is to check that the analytic
approximations we carried out are sensible. Indeed we verify that the
numerical solutions exhibit the conformal behaviour of equations (3.17) in the
appropriate parameter region, and the non-conformal behavior of (3.84) outside
that region. Second, we extract additional information which is analytically
unreachable: we check that the zero-temperature limit of the entropy matches
the prediction from the supersymmetric index and we estimate the Schwarzian
coupling.

###  4.1 Summary of the numerical method

The SD equations can be solved numerically by adapting the method used in [6].
We discretize the interval $[0,\beta]$ into $M$ points, with $M$ a power of
$2$.121212One can sample the interval at either $\tau_{i}=i/M$ or
$\tau_{i}=\bigl{(}i+\frac{1}{2}\bigr{)}/M$, for $i=0,\dots,M-1$. For
sufficiently high number of points, they give identical solutions within the
numerical precision, however we found the latter choice to be numerically more
stable. Note that since the model depends on $\beta$ only through $\beta J$
and $\beta\mu$, one can always take $\beta=1$ for numerical purposes. In
frequency space we pick the $M$ fermionic and bosonic Matsubara frequencies
with lowest absolute value, namely

$\omega^{\rm
f}_{k}=\frac{2\pi}{\beta}\biggl{(}k-\frac{M}{2}+\frac{1}{2}\biggr{)}\;,\qquad\omega^{\rm
b}_{k}=\frac{2\pi}{\beta}\biggl{(}k-\frac{M}{2}+1\biggr{)}\;,\qquad
k=0,\dots,M-1\;,$ (4.1)

where the discretization is manifest because we take a finite number of
frequencies.131313Since $M$ is even, the truncation is not symmetric around
zero for the bosonic Matsubara frequencies: $\omega^{\rm
b}_{k}=-\frac{\pi(M-2)}{\beta},\dots,0,\dots,\frac{\pi M}{\beta}$. This
creates an apparent difficulty when calculating $\Sigma(-\omega_{k})$ for
$k=M-1$. We fix this by assuming that the bilocal fields are real, so that
$\Sigma(-\omega_{k})=\overline{\Sigma}(\omega_{k})$. We transform between the
two descriptions with Fast Fourier Transforms. We update each iteration with a
weighing factor $x$, as in [6], for which we found $x=2/3$ to be a good
general choice. Schematically,

$G(\omega_{k})_{i}\rightarrow\Sigma(\tau_{k})_{i}=\Sigma_{\text{EOM}}\bigl{[}G(\tau_{k})_{i}\bigr{]}\rightarrow
G(\omega_{k})_{i+1}=(1-x)G(\omega_{k})_{i}+xG_{\text{EOM}}\bigl{[}\Sigma(-\omega_{k})_{i}\bigr{]}.$
(4.2)

We iterate until
$\sum_{k}\bigl{\lvert}G(\omega_{k})_{i}-G(\omega_{k})_{i+1}\bigr{\rvert}^{2}$
is smaller than some precision goal. To begin the iteration, we take the free
solution.

For small $\mu_{\phi}$ and finite $J$, the equation of motion for
$G_{\bar{\phi}\phi}(\omega=0)$ is highly sensitive to numerical errors in
$\Sigma_{\bar{\phi}\phi}(\omega=0)$. One possible workaround, proposed for
example in [43], is to change the equation of motion for $G_{\bar{\phi}\phi}$
by replacing the parameter $\mu_{\phi}$ with $\rho$ so that
$G_{\bar{\phi}\phi}(\omega=0)=\alpha\rho$. We replace the integro-differential
equation for $G_{\bar{\phi}\phi}$ with

$G_{\bar{\phi}\phi}\bigl{[}\Sigma(-\omega^{\text{b}}_{k})_{i}\bigr{]}=-\frac{\alpha\rho}{\rho\bigl{[}-i\omega^{\text{b}}_{k}+\Sigma_{\bar{\phi}\phi}(-\omega^{\text{b}}_{k})_{i}-\Sigma_{\bar{\phi}\phi}(0)_{i}\bigr{]}-1}\;.$
(4.3)

One can then use the original equation of motion at the end to verify which
value of $\mu_{\phi}$ is realized. In principle, one could take
$\mu_{\phi}=\Sigma_{\bar{\phi}\phi}(0)+1/\rho$, however it is best to average
the EOM in a window of frequency space. If one desires a specific value of
$\mu_{\phi}$, this approach incurs a high additional numerical cost, since the
solver must itself be iterated many times to carry out a bisection search or
equivalent technique. Nevertheless, we found this to be a useful approach for
those cases that are more numerically unstable, such as $(p,q)\neq(1,2)$.

###  4.2 Conformal behaviour for $\boldsymbol{(p,q)=(1,2)}$

In this section we focus on the case $(p,q)=(1,2)$. Numerically, this turns
out to be the most accessible case. It is also of particular theoretical
interest, as it is the closest choice to the quantum mechanical model in [29].

We first test that the conformal ansatz is realized at low energies. We take a
solution with large $\beta J$, and we scan the numerical solutions to (3.22)
for the best quadratic fit. These solutions are parameterised by
$\mathcal{E}_{\eta}$, $\mathcal{E}_{\phi}$, $g_{\eta}$, and $g_{\phi}$. In
principle, one could just fit $G_{\bar{\eta}\eta}$ and $G_{\bar{\phi}\phi}$,
which can be beneficial since the auxiliary fields tend to converge slower to
the conformal solution. However, we found that this results in less precise
and less accurate estimates of the $\mathcal{E}$’s, so we fit the four bilocal
fields together. Clearly, we must exclude from the fit the regions with $\tau$
close to $0$ and $\beta$, where the approximation is invalid and singular.
Tipically we exclude
$\bigl{\lvert}\frac{1}{2}-\frac{\tau}{\beta}\bigr{\rvert}>3/8$.

(a) $\bigl{(}\beta\mu_{\eta},\,\beta\mu_{\phi}\bigr{)}=(-2,\,1)$

(b) $\bigl{(}\beta\mu_{\eta},\,\beta\mu_{\phi}\bigr{)}=(1,\,2)$

Figure 5: Two solutions for $\beta J=1000$, $\alpha=1/4$, $(p,q)=(1,2)$. The
colored full lines are the numerical solution of the Schwinger–Dyson equations
with $M=2^{18}$ points. The black dashed lines are a conformal solution for
which the parameters $\mathcal{E}_{\eta,\mu}$ and $g_{\eta,\mu}$ have been
fitted numerically. (a): Supersymmetric chemical potentials. (b): Non-
supersymmetric.

(a) $\bigl{(}\beta\mu_{\eta},\,\beta\mu_{\phi}\bigr{)}=(-2\mu,\,\mu)$

(b) $\bigl{(}\beta\mu_{\eta},\,\beta\mu_{\phi}\bigr{)}=(\mu,\,2\mu)$

Figure 6: Plot of $\mathcal{E}_{\eta}$ (blue points) and $\mathcal{E}_{\phi}$
(red points) as a function of $\mu$. The data was generated with $M=2^{20}$
points, $\beta J=600$, and $\alpha=1/5$. The dashed lines in each plot
correspond to (4.4) (a: SUSY chemical potentials) and (4.5) (b: non-SUSY),
respectively.

We present two illustrative examples in Figure 5. Clearly, the conformal
solution is realized and it matches our proposed ansatz. For $(p,q)=(1,2)$ and
supersymmetric chemical potentials, we verified numerically that, at
sufficiently large $\beta J$, we have

$2\pi\mathcal{E}_{\eta}\,\simeq\,-\beta\mu_{\eta}=2\beta\mu_{\phi}\;,\qquad\qquad
2\pi\mathcal{E}_{\phi}\,\simeq\,-\beta\mu_{\phi}\;,$ (4.4)

which matches the expectation from the supersymmetric index. This suggests
that, even in the deep IR, SUSY is at most lightly broken at small
temperatures. For other choices of chemical potentials, we also find a linear
relation at low temperatures. The ansatz

$2\pi\mathcal{E}_{\eta}\,\simeq\,-\biggl{(}1-\frac{2\alpha}{5}\biggr{)}\beta\mu_{\eta}+\frac{4\alpha}{5}\beta\mu_{\phi}\;,\qquad
2\pi\mathcal{E}_{\phi}\,\simeq\,\frac{1}{2}\biggl{(}1-\frac{2\alpha}{5}\biggr{)}\beta\mu_{\eta}-\frac{2\alpha}{5}\beta\mu_{\phi}$
(4.5)

was numerically successful for large $\beta J$ and finite chemical potentials
well below the critical value, testing with different values of $\alpha$. We
plot a supersymmetric and a non-supersymmetric example in Figure 6. This
ansatz has the curious property of being supersymmetric irrespectively of the
chemical potentials, suggesting an emergent IR supersymmetry. When fitting the
low energy behaviour, $\Delta_{\phi,\eta}$ and $g_{\phi,\eta}$ also satisfy
the respective supersymmetry constraints within $1\%-5\%$ at $\beta J$ between
$200$ and $600$. This ansatz was further confirmed when comparing the
numerical values of the charge and entropy to the Luttinger–Ward relation and
the zero-temperature entropy for non-supersymmetric cases in Section 4.3.
However, we note that one expects, for physical fields at fixed
$\mathcal{Q}_{\eta,\phi}$, to have

$\mu_{I}(\mathcal{Q}_{\eta},\mathcal{Q}_{\phi},T)=\mu_{0,I}(\mathcal{Q}_{\eta},\mathcal{Q}_{\phi})+2\pi\mathcal{E}_{I}(\mathcal{Q}_{\eta},\mathcal{Q}_{\phi})T+\ldots\;,$
(4.6)

see for instance [44, 45, 12]. Here the index $I=\eta,\phi$ labels the
$\mathrm{U}(1)$ symmetries. Since we work at fixed $\mu$ and the function
$\mathcal{Q}_{I}(\mu)$ is a priori complicated, eqn. (4.5) might be the
linearized behaviour of a more intricate function
$\mu_{0,I}\bigl{(}\mathcal{Q}_{\eta}(\mu_{\eta},\mu_{\phi}),\,\mathcal{Q}_{\phi}(\mu_{\eta},\mu_{\phi})\bigr{)}$.

###  4.3 Charge and entropy for $\boldsymbol{(p,q)=(1,2)}$

(a) $\alpha=1/3$

(b) $\alpha=1/5$

Figure 7: We plot $\mathcal{Q}_{\eta}$ (blue points) and $\mathcal{Q}_{\phi}$
(red points) as a function of $\mu_{F}$. The data was generated with
$M=2^{18}$ points, $\beta J=200$ and
$(\beta\mu_{\eta},\beta\mu_{\phi})=(-2\mu,\mu)$. The gray full line is the
Luttinger–Ward formula (4.7), assuming (4.4). The gray dashed lines are the
continuations in (4.8). The vertical dotted line is the value $\mu_{F}$ for
which $\mathcal{E}_{*}$ is reached.

An important check of the conformal ansatz is the Luttinger–Ward relation
between $\mathcal{E}$ and $\mathcal{Q}$, which we presented in Section 3.7. As
previously noted, the Luttinger–Ward calculation determines
$\mathcal{Q}(\mathcal{E})$ up to undetermined constants, which we fixed
exploiting the index in (3.103). Numerically, we can verify these constants to
be:

$\displaystyle\mathcal{Q}_{\eta}$
$\displaystyle=\mathfrak{q}_{-}(\mathcal{E}_{\eta},\Delta_{\eta})+(1-p)\,\mathfrak{q}_{+}(\mathcal{E}_{f},\Delta_{f})-\alpha
p\,\mathfrak{q}_{-}(\mathcal{E}_{\psi},\Delta_{\psi})+\frac{\alpha p}{2}\;,$
(4.7) $\displaystyle\mathcal{Q}_{\phi}$
$\displaystyle=\alpha\,\mathfrak{q}_{+}(\mathcal{E}_{\phi},\Delta_{\phi})+\alpha(1-q)\,\mathfrak{q}_{-}(\mathcal{E}_{\psi},\Delta_{\psi})-q\,\mathfrak{q}_{+}(\mathcal{E}_{f},\Delta_{f})+\frac{\alpha
q}{2}\;,$

where the function $\mathfrak{q}$ is given in (3.80). This matches the index
prediction (3.103). We note that the charges do not vanish when
$\mu_{\eta,\phi}\rightarrow 0$, which is supported by the numerics, as can be
seen in Figure 7. The charges also converge very quickly to their low-
temperature limit, since they depend on $\beta J$ through $\mathcal{E}$. Thus
we can take $\beta J$ to be large and assume (4.4).

In the superconformal solution, as we take $\mathcal{E}$ close to
$\mathcal{E}_{*}(\alpha)$ defined in (3.33), we approach the phase transition
discussed in Section 3.4. In order to avoid working with $\mathcal{E}$ which
is numerically more indirect, we label as
$\beta\mu_{\phi}^{*}=2\operatorname{arccosh}\bigl{(}\alpha^{-\frac{1}{2}}\bigr{)}$
the critical chemical potential, which follows from (4.4).

Figure 8: We plot $\Delta_{\eta}$ and $\Delta_{\phi}$ for $\alpha=1/4$ and
$(\beta\mu_{\eta},\beta\mu_{\phi})=(-2\mu,\mu)$, assuming (4.4). The data was
generated by solving (3.29) numerically. The vertical dotted line is at the
value of $\mu$ for which $\mathcal{E}_{*}$ is reached.

Recall that, as $\mathcal{E}\rightarrow\mathcal{E}_{*}$, the conformal
dimensions converge to $\Delta_{\eta}=1/2$ and $\Delta_{\phi}=0$, see Figure
8. This is a solution — albeit a trivial one — to the consistency equations
(3.23), or equivalently (3.29), for any value of $\mathcal{E}$. Focusing only
on the consistency equations, the absence of other nearby solutions for the
conformal dimensions when $\mathcal{E}>\mathcal{E}_{*}$ suggests that the
would-be conformal solution should have these marginal values. While,
technically, this fact renders the conformal approximation no longer valid,
since the kinetic and interacting terms are of the same order in $\omega$, we
can conjecturally extend the Luttinger–Ward relation by plugging the fixed
dimensions in. We conjecture

$\mathcal{Q}_{\eta}=\frac{1}{2}\;,\qquad\qquad\mathcal{Q}_{\phi}=1-\coth(\beta\mu_{F})+\frac{\alpha}{2}\coth\biggl{(}\frac{\beta\mu_{F}}{2}\biggr{)}\;.$
(4.8)

As we can see in the dashed line in Figure 7, the numerical results do not
deviate significantly from such an analytic continuation.

We can further combine this result with (3.106) in order to obtain a zero-
temperature analytic prediction for the entropy as a function of the flavour
chemical potential, within the conformal phase and, conjecturally, beyond. We
use (3.86) to calculate the entropy numerically at each finite temperature,
and then we take $\beta J\rightarrow\infty$ with $\beta\mu$ fixed.

(a) $\alpha=\frac{1}{4}$, $\mu=0.125$, $r=0$, $M=2^{22}$

(b) $\alpha=\frac{1}{4}$, $\mu=2.58$, $r=3$, $M=2^{24}$

Figure 9: We plot two examples of the extrapolation of the zero-temperature
entropy for $(\beta\mu_{\eta},\beta\mu_{\phi})=(-2\mu,\mu)$. In plot 9(a) we
take a small value of $\mu$ for which it suffices to fit with a line, allowing
us also to estimate the Schwarzian coupling. Meanwhile, in plot 9(b) we take a
value of $\mu$ close to the phase transition. Instead of fitting the points
directly, we fit the third Richardson transform with the ansatz (4.10). Here
$M$ is the number of points in the discretized Schwinger–Dyson equation.

For the extrapolation to zero temperature, the precision strongly depends on
how close to the phase transition we are. For values of the chemical potential
sufficiently below $\beta\mu_{\phi}^{*}$, it suffices to probe values of
$\beta J$ between $50$ and $500$ and use a polynomial fit to extract the
constant term, as can been seen in Figure 9(a). The error is estimated by
dropping points at both ends of the sample. The resulting extrapolation of the
zero-temperature entropy are the red dots in Figure 10. However, as we move
closer to the critical point, this extrapolation becomes harder. The main
limiting factor is that we cannot probe arbitrarily low temperatures due
finite-size effects and numerical instabilities. Furthermore, if we assume
that the entropy behaves as

$S(T)\,\approx\,S_{0}+g_{\text{Sch}}(\beta J)^{-1}+g_{\gamma}(\beta
J)^{-\gamma}+\ldots\;,$ (4.9)

what we observe numerically (as well as in the analysis of the spectrum in
Section 5) is that $\gamma$ becomes close to $1$. Thus, we generate points at
integer values of $\beta J$ between $100$ and $400$ and then take one or more
Richardson transforms, see [46, 47]. After the $r$-th Richardson transform, we
expect to converge to $S_{0}$ as

$\bigl{[}S(T)\bigr{]}_{r}\,\approx\,S_{0}+g_{\gamma,r}(\beta
J)^{-\gamma}+g_{r}(\beta J)^{-(1+r)}+\ldots\;.$ (4.10)

Fitting the form (4.10) for all the constants is more reliable at low
temperature than (4.9) since the exponents are more distinct, and we estimate
the numerical error by varying the points included in the fit and the order
$r$ of the Richardson transform. We used this strategy for the light blue
points in Figure 10, with a step of $10$ for those on the left and a step of
$4$ for those on the right (a smaller step increases the convergence of the
Richardson acceleration). Lastly, for points very close to the critical point
the convergence to $S_{0}$ is particularly slow, so we considered only points
with $\beta J$ between $200$ and $400$ with a step of $2$. These are the dark
blue points in Figure 10. The error is estimated by varying the degree of the
Richardson transform and by comparing with the previous method, taking the
largest estimate. An example of such a fit is given in Figure 9(b). The
exponent $\gamma$ in (4.9) seems to be close to $1$ at the transition point,
but unfortunately the precision attainable with our points is not enough to
extract meaningful estimates, including whether it is smaller or larger than
$1$.

Figure 10: Plot of the zero-temperature entropy $S_{0}$ as a function of the
chemical potentials $(\beta\mu_{\eta},\beta\mu_{\phi})=(-2\mu,\mu)$ at
$\alpha=1/4$. The red dots were generated with $2^{22}$ points, $\beta J$
spanning from $200$ to $1000$, and the entropy was obtained with simple linear
extrapolation. The light blue and dark blue dots were generated with $2^{23}$
and $2^{24}$ points, respectively, with $\beta J$ spanning from $100$ to
$400$, and extrapolated using the Richardson-transform fit described in eqn.
(4.10). The black line is the prediction from combining the index and the
Luttinger–Ward formula (4.7), while the dashed line uses the index and the
continuation (4.8).

Unfortunately, $\beta J\sim 500$ is still slowly converging to the low-
temperature behaviour and the above techniques have limited success in
isolating $S_{0}$. Furthermore, the results are sensitive to small changes in
the extrapolation method, which we attempt to capture with the error estimates
in Figure 10, but which are likely underestimated. With these significant
caveats, what we can at best observe is that, in Figure 10, the kink that
follows from extrapolating $\mathcal{Q}_{F}(\mu)$ with (4.8) seems to be close
to the numerical results. This could suggest some form of second-order phase
transition. However, note that this would-be kink only appears when working at
fixed chemical potential, while at fixed charge we can just apply (3.106)
directly, which is smooth.

$\begin{array}[]{c | cc | cc}\hline\cr\hline\cr&\lx@intercol\hfil\beta\mu_{\phi}=1/8\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\beta\mu_{\phi}=3/2\hfil\lx@intercol\\\ \alpha&\mathcal{Q}_{F}&g_{\text{Sch}}&\mathcal{Q}_{F}&g_{\text{Sch}}\\\ \hline\cr 1/4&0.1087&1.09\pm 0.03&0.7920&1.09\pm 0.01\\\ 1/5&0.1119&0.97\pm 0.03&0.8147&0.96\pm 0.01\\\ 1/6&0.1140&0.90\pm 0.03&0.8298&0.90\pm 0.02\\\ 1/7&0.1155&0.87\pm 0.04&0.8405&0.86\pm 0.02\\\ \hline\cr\hline\cr\end{array}$ Table 2: Estimates of the Schwarzian coupling, for $(\beta\mu_{\eta},\beta\mu_{\phi})=(-2\mu,\mu)$.