#  Phases of $\mathcal{N}=2$ Sachdev-Ye-Kitaev models

###### Abstract

We study $\mathcal{N}=2$ supersymmetric Sachdev-Ye-Kitaev (SYK) models with
complex fermions at non-zero background charge. Motivated by multi-charge
supersymmetric black holes, we propose a new $\mathcal{N}=2$ SYK model with
multiple $U(1)$ symmetries, integer charges, and a non-vanishing
supersymmetric index, realizing features not present in known SYK models. In
both models, a conformal solution with a super-Schwarzian mode emerges at low
temperatures, signalling the appearance of nearly AdS2/BPS physics. However,
in contrast to complex SYK, the fermion scaling dimension depends on the
background charge in the conformal limit. For a critical charge, we find a
high to low entropy phase transition in which the conformal solution ceases to
be valid. This transition has a simple interpretation– the fermion scaling
dimension violates the unitarity bound. We offer some comments on a
holographic interpretation for supersymmetric black holes.

###### Contents

  1. 1 Introduction
  2. 2 $\mathcal{N}=2$ SYK with fractional charge
    1. 2.1 Mean field and the conformal solution
    2. 2.2 Breakdown of conformal ansatz
    3. 2.3 Operator spectrum
    4. 2.4 Comments on the holographic interpretation
  3. 3 $\mathcal{N}=2$ SYK with multiple fermions
    1. 3.1 Mean field and the conformal solution
    2. 3.2 Breakdown of conformal ansatz
    3. 3.3 Operator spectrum
  4. 4 Discussion and Open Questions
  5. A Luttinger-Ward relation
  6. B Kernels with multiple fermions in components

##  1 Introduction

The Sachdev-Ye-Kitaev (SYK) models [1, 2] of randomly interacting fermions are
the simplest known quantum mechanical systems without quasi-particles that
display maximal chaos. In a seemingly unrelated arena of gravitational
physics, a charged black hole near the extremal “mass = charge” limit develops
an approximate two-dimensional anti-de Sitter (AdS2) region. The physics of
AdS2 displays chaos, scrambling, and an approximate conformal symmetry. In
light of the AdS/CFT correspondence, these features (among others) make SYK
models candidate toy descriptions of quantum black holes, following some
earlier insight given by [3, 4]. For applications to charged black holes, one
may consider of SYK models with $N$ complex fermions and at least one
conserved $U(1)$ symmetry. A model of this kind is the complex SYK model
(cSYK), where a set of four or more fermions are coupled through a random
interaction drawn from a fixed Gaussian ensemble. See [5] for a detailed
exposition of this model.

In this paper we will focus on a simple (but ultimately drastic) modification
of the complex SYK model, where the Gaussian random interaction is roughly
replaced by its covariance matrix. A precise version of this modification,
which we will review below, was introduced by Fu, Gaiotto, Maldacena, and
Sachdev in [6]. The seemingly small modification of the random Hamiltonian
allows one to write it as the square of a complex (Gaussian random) operator–
the combination of the conserved supercharges $\mathcal{Q}+\bar{\mathcal{Q}}$,
which leads to very different physics. In particular, the model in fact
possesses $\mathcal{N}=2$ supersymmetry; the $U(1)$ phase rotation of the
fermions does not commute with supersymmetry and thus becomes an R-symmetry.
Supersymmetry typically relates bosons and fermions via the action of
$\mathcal{Q}$, however, as noted in [6], a supersymmetric model consisting
only of fermions may be achieved as long as supersymmetry is non-linearly
realized.

While the proposal in [6] seems to be the simplest generalization of SYK to
incorporate $\mathcal{N}=2$ supersymmetry, for reasons that will be explained
shortly, we will sometimes refer to this model as $\mathcal{N}=2$ SYK with
fractional charges. Unlike cSYK and even $\mathcal{N}=1$ versions, the model
with extended supersymmetry possesses states annihilated by one complex
supercharge; these BPS states have a large exact degeneracy which survives at
strong coupling (this feature of the strong coupling partition function was
observed in [7, 8]). It is natural to conjecture there is some relationship
between a large BPS degeneracy in supersymmetric SYK and the degeneracy of
supersymmetric AdS2 black holes [9, 10]. However, the $\mathcal{N}=2$ SYK
model with fractional charges does not have all of the expected features of
these near-BPS black holes– a point we will come back to later.

The first goal of this paper will be to perform a more exhaustive study of
this $\mathcal{N}=2$ SYK model, in particular at non-zero R-charge, or
equivalently a non-zero chemical potential. For example, in [6] and subsequent
work, the focus was placed on analyzing the model in an ensemble of zero
$U(1)$ R-charge [7, 8, 11, 12, 13, 14, 15, 16]. Because the background
R-charge does not commute with supersymmetry, this choice of zero charge
ensemble preserves supersymmetry and dramatically simplifies the solution of
the model. We generalize this analysis and will uncover an interesting phase
structure as a function of the charge when supersymmetry is broken.

We will contrast the results obtained in a non-zero R-charge sector of
$\mathcal{N}=2$ SYK with the solution of complex SYK. In particular we are
interested in the strong coupling limit, or equivalently temperatures smaller
than the scale set by the variance of the couplings. In this limit we expect
an emergent conformal symmetry at the level of correlation functions between
the fermions; this is the infrared reparametrization symmetry. It is
spontaneously broken to $SL(2,\mathbb{R})$ by the associated “conformal
solution” of the Schwinger-Dyson equations which determines the two-point
functions. In complex SYK the scaling dimension of the fermions is simply
obtained by dimensional analysis arguments and has a universal value
independent of the charge density.

In $\mathcal{N}=2$ SYK, the model possesses not just time translation and
$U(1)_{R}$ symmetries, but also a complex supercharge which is charged under
the $U(1)_{R}$. In the infrared, we will show that there is an emergent super-
reparametrization symmetry, that is spontaneously broken to $SU(1,1|1)$
superconformal symmetry by the conformal solution. The superconformal group
$SU(1,1|1)$ extends the $SL(2,\mathbb{R})$ with bosonic subgroups
$SL(2,\mathbb{R})\times U(1)_{R}\subset SU(1,1|1)$. This superconformal
symmetry is further broken spontaneously by the background R-charge, and
because of this, the scaling dimension of the fermion is no longer fixed by
dimensional analysis. Instead, one needs to look more carefully at the
solution of the model to derive a constraint that uniquely determines the
scaling dimension $\Delta$. The result gives a non-trivial function of the
charge which can be found in equation (2.27) (this expression was
independently derived by S. Sachdev [17]). Another feature is that the
coefficient of the fermion two-point function in the conformal limit is
undetermined from the IR at non-zero charge (when the charge is zero it is
fully determined by supersymmetry). We will verify numerically that this
parameter of the solution is fixed once the UV behavior of the model is
incorporated, consistent with the idea that there is a unique two-point
function at strong coupling. We also derive a Luttinger-Ward relation given in
equation (2.50) and the grand potential in (2.90) (by proposing a “spooky
fermion” picture for $\mathcal{N}=2$ SYK similar to [5]), and verify
numerically its validity. Finally, we analyze the spectrum of bilinear
operators within the conformal solution at any charge.

This takes us to the second departure between complex and $\mathcal{N}=2$ SYK.
While the conformal solution of complex SYK is well-behaved for any charge, we
find a critical charge in $\mathcal{N}=2$ SYK for which the scaling dimension
of the fermion and the overall coefficient of the two-point function vanish.
At higher values of the charge, the scaling dimension becomes either negative
or complex, signaling the fact that this solution is no longer physical (see
for example figure 2). Instead, for these charges the model develops a gap and
behaves as a set of massive complex free fermions. We note that the phenomenon
is distinct from the complex modes already observed in the literature such as
in [18, 19, 20], where some bilinear operator becomes complex. In our case,
the fundamental fermion directly gets a dimension that violates unitarity.

This phase transition we found in $\mathcal{N}=2$ SYK is analogous to a
similar phenomena observed in [21, 22, 23] for complex SYK. Those references
pointed out that even in complex SYK there is a phase transition at non-zero
charge where the conformal solution stops dominating and a gap develops. This
leads to a new phase referred to as a “low entropy phase”. This is in contrast
to the conformal phase which is “high entropy”, in the sense that there is a
large order $N$ zero temperature entropy (at least when the temperature is
taken to zero after taking the large $N$ limit, in this order). Even though
there is a phase transition, it is an open problem to understand analytically
the origin of this phase transition since nothing seems to go wrong with the
conformal solution. Some observations on this direction were made in [5] which
show some unusual features in the four-point function kernel when the charge
is too large, but the analysis is not conclusive and does not determine the
precise value of the critical charge. In contrast we can understand this
transition much better in $\mathcal{N}=2$ SYK. Now there is a sharp issue with
the conformal solution at finite charge since the fermion itself develops an
instability. We give some numerical evidence that the charge at which the
fermion becomes unstable corresponds to the same charge at which there is a
high- to low-entropy phase transition.

There is a nice analogy between this phase transition and black hole physics.
The phase diagram described above can be compared with the one for a near
extremal charged black hole, which develops an AdS2 throat close to its
horizon. This black hole is unstable towards discharging when a charged
fermion is present with a mass (in AdS2 units) smaller than the electric
field, even though the black hole is perfectly stable in the vacuum. This
happens through Schwinger pair production, see for example [24]. A similar
phenomenon can also occur with scalar fields when their effective mass is
below their Breitenlohner-Freedman (BF) bound close to the horizon [25]. The
supersymmetric model seems to have more in common with the candidate bulk
description than complex SYK, since we find an electric field dependent
scaling dimension and an instability. Thus, we can interpret the appearance of
a new boundary phase at a critical charge as the endpoint of the black hole
stability. Since conformal symmetry is lost, the bulk spacetime will be
drastically changed and we can no longer use a dual description as a disc of
Euclidean AdS2 with $SL(2,\mathbb{R})$ isometry. The fact that the entropy is
low in the new phase suggests that the horizon completely disappears. A
schematic picture of the proposal for the bulk is in figure 1.

Figure 1: A depiction of the transition of $\mathcal{N}=2$ SYK, along with a
schematic dual bulk geometries in each phase. At background charge $Q=0$, it
was shown in [6] that the model possesses a superconformal solution. We show
that for $Q<Q_{crit}$ supersymmetry may be lost (depending on which model is
under consideration), but a conformal solution exists. This system is dual to
a (possibly supersymmetric) AdS2 geometry with a Euclidean horizon and a large
extremal entropy. Schematically, a single boundary fermion two point function
is computed by a charged fermion in AdS in which the semi-classical scaling
dimension depends on the background charge. However, we find above a critical
background charge, the dimension of the fermion violates the unitarity bound
and the conformal solution ceases to be valid. In SYK, we find a new massive
phase with vanishing large $N$ entropy. We conjecture the bulk dual is a
horizon-less geometry.

We can now move to the second part of the paper. So far, we have discussed the
conformal solution (and when it becomes unstable). However, even though the
solution of the SYK model develops a conformal symmetry in the IR, this
symmetry is both spontaneously and explicitly broken. The breaking of the
conformal symmetry dominates the dynamics of the model at these scales,
explaining the maximal chaos for example [2]. Similarly, in $\mathcal{N}=2$
SYK there is a supersymmetric Schwarzian mode controlling the breaking of
superconformal symmetry. We verify a similar statement may be made at finite
charge before the phase transition.

However, there is an essential restriction on the $\mathcal{N}=2$ SYK model
thusfar considered in the literature, and this restriction continues to the
$\mathcal{N}=2$ Schwarzian theory one derives from this model. As we mentioned
previously, the model of [6] has the particular feature that the fermionic
charge is fractional compared to the R-charge of the supercharge111In
$\mathcal{N}=2$ SYK, $q$ labels the number of fermions appearing in the
supercharge, and not the number of fermions appearing in the Hamiltonian. If
we normalize the supercharge to have unit R-charge, the fundamental fermion
has charge $1/q$. This is consistent with charge quantization for abelian
groups.. This leads to a Schwarzian theory with fractional charges. An
interesting feature of these models of fractional charge fermions is that the
BPS states come with a range of charges. This may be seen directly in the
grand partition function $Z(\beta,\mu)=\textrm{Tr}[e^{-\beta H+\beta\mu Q}]$,
and in particular can lead to cancellations in the supersymmetric index
$\mathcal{I}=\textrm{Tr}[(-1)^{F}]$ which counts BPS states weighted by their
fermion number. For example, in the large $N$ limit one obtains the super-
Schwarzian theory which may be solved exactly– [7, 8] gives a density of
states of the BPS sector for even $N$ given by

$Z(\beta,\mu)=\sum_{Q\in\mathbb{Z},~{}|Q|<\frac{q}{2}}e^{\beta\mu
Q}~{}\frac{2e^{Ns_{0}}\cos(\frac{\pi Q}{q})}{q}+\ldots$ (1.1)

where $s_{0}=\log(2\cos\frac{\pi}{2q})$. (For odd $N$ the same expression is
valid but now the charge ranges over half-integers instead of integers.) The
dots denote the non-BPS contributions to the spectrum. For even $N$ they are
separated on average by a gap given by $E_{\rm gap}=0.412J/N$ (obtained for
the $q=3$ model using the Schwarzian coupling computed in section 2.2). For
odd $N$ the gap is exponentially small in $N$ instead. This is written with
respect to the charge $Q$ that assigns unit charge to the fermions. The
R-charge $Q_{R}$ assigning unit charge to the supercharge is related to it by
$Q_{R}=Q/q$. The formula (1.1) happens to be exact for $q=3$ and $N$ even or
$N=3~{}{\rm mod}~{}4$, but for $q>3$ is only approximate in the large $N$
limit. We would like to emphasize that even though we are describing the
density of BPS states, their R-charge distribution (1.1) is not protected by
supersymmetry. For example, because of the spread in charges the Witten index
of this theory vanishes, something that can be reproduced by neglecting the
interaction between the fermions (the index being independent of the
temperature and the couplings).

As has been touched upon already, the Schwarzian theory also appears in the
context of near extremal black holes in higher dimensions, describing both the
classical dynamics of excitations above extremality [4, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39] and quantum effects that become large as
we take the extremal limit [40, 41, 42, 9, 10]. In particular, it was recently
shown that a black hole in AdS${}_{5}\times S^{5}$ develops an emergent
$SU(1,1|1)$ symmetry in the BPS limit in [10]. However, the Schwarzian theory
describing its near-BPS dynamics is not the same as the ones constructed in
[6] (Some interesting questions are being raised regarding how to understand
the BPS black hole microstates in $\mathcal{N}=2$ JT gravity in [43, 44]).
Instead, the charge of the fundamental fermion is equal to the charge of the
supercharge. One can then use the same formula for the large $N$ spectrum
(1.1) with $q=1$, so that the BPS states have only zero R-charge and the index
and degeneracy match.

Naively setting $q=1$ does not make sense from the SYK perspective– with only
fundamental fermions and a supercharge linear in them, the model would become
trivial and not display any emergent conformal symmetry. To circumvent this we
will instead construct models with multiple fermions222We thank E. Witten for
this suggestion. and show one can then define interacting theories described
at low energies by unit fundamental R-charge333In three spacetime dimensions,
related models with and without supersymmetry and multiple fields were
constructed in [45, 46]. The presence of dynamical bosons in higher dimensions
can lead to different conclusions about the IR effective solution compared to
what is discussed in this work.. We study in detail a prototypical model with
these features, but more general multi-fermion theories are possible. One
interesting feature of $\mathcal{N}=2$ SYK models with multiple fermions is
the presence of flavor symmetries that commute with the supercharge. In this
case we obtain the Schwarzian with integer R-charge and a non-zero index, in a
sector of fixed flavor charge (in the simplest realization we can even gauge
the flavor charge). The partition function is $Z(\beta,\mu)=e^{Ns_{0}}+Z_{\rm
non-BPS}(\beta,\mu)$ for some order one number $s_{0}$, in the Schwarzian
approximation. Our new $\mathcal{N}=2$ SYK model is thus the first of its kind
to realize the $q=1$ Schwarzian, and passing to a sector of fixed flavor
charge is somewhat analogous to passing to a particular charge sector of the
AdS5 black holes in [10].

For completeness we study these models with multiple fermions at non-zero
flavor and R-charges. We find a similar phase structure as in the models of
[6] with the fundamental fermion becoming unstable at large charges. When
charges are too large the conformal solution breaks down and the system is
driven to a low-entropy phase, although we leave a more exhaustive analysis
for future work of the phase structure of these models. We also find a
potential instability as we raise the flavor charge, without breaking
supersymmetry (see figure 13). Using the Luttinger-Ward relation we show that
the conformal ansatz breaks down precisely where the charge of one of the
fermions becomes maximal saturation value of $\pm N/2$.

The rest of the paper is organized as follows. In section 2 we analyze the
model of [6] at non-zero charge. After reviewing the definition of the model
and the derivation of the mean field action we solve the IR Schwinger-Dyson
equations in section 2.1. We discuss the breakdown of the conformal ansatz in
section 2.2. We also solve the full Schwinger-Dyson equations numerically to
verify our claims, derive a Luttinger-Ward relation for $\mathcal{N}=2$ SYK
relating the charge to the spectral asymmetry present in the IR limit of the
two-point functions, find the grand potential, and compute the Schwarzian
coupling numerically. In section 2.3 we analyze the operator spectrum by
looking at the four-point function. In section 2.4 we elaborate on the
holographic interpretation. In section 3 we carry out most of the same
analysis for a model with multiple fermions. We analyze the behavior of the
Witten index, solve the IR Schwinger-Dyson equations, study the phase
structure at non-zero charge, and analyze the operator spectrum. We conclude
in section 4 pointing out some future directions.

##  2 $\mathcal{N}=2$ SYK with fractional charge

We begin with the $\mathcal{N}=2$ supersymmetric SYK model of [6]. We will
study this model at non-zero charge (some of the results were independently
derived in [17]). This model consists of $N$ complex fermions which obey the
standard Dirac algebra:

$\\{\psi^{i},\bar{\psi}_{j}\\}=\delta^{i}_{j}\,,\qquad\\{\psi^{i},\psi^{j}\\}=0\,,\qquad\\{\bar{\psi}_{i},\bar{\psi}_{j}\\}=0\,.$
(2.1)

In terms of these fermions, the model is defined by a complex supercharge
$\mathcal{Q}$ with the property that it involves $q$-fermion interactions444In
contrast to [6], we employed Einstein summation rather than an explicit
ordered sum $\mathcal{Q}=i^{\frac{q-1}{2}}\\!\sum_{1\leq
i_{1}<\ldots<i_{q}\leq N}C_{i_{1}i_{2}\ldots
i_{q}}~{}\psi^{i_{1}}\psi^{i_{2}}\ldots\psi^{i_{q}}$. This leads to some
different numerical factors.:

$\mathcal{Q}=i^{\frac{q-1}{2}}C_{i_{1}i_{2}\dots
i_{q}}\psi^{i_{1}}\psi^{i_{2}}\ldots\psi^{i_{q}},$ (2.2)

where we take the couplings $C_{i_{1}i_{2}\ldots i_{q}}$ to be totally
antisymmetric. We introduce disorder by integrating out these couplings with
Gaussian statistics and we will fix the normalization of the random variables
$\langle C_{i_{1}i_{2}\ldots i_{q}}\bar{C}^{i_{1}i_{2}\ldots i_{q}}\rangle\sim
J/N^{q-1}$, with a $q$-dependent constant we will fix later to match the mean
field action and equations of motion (2.19). Additionally, we take $q$ to be
odd so $\mathcal{Q}$ has Fermi statistics. We see that (2.2) has
$\mathcal{Q}^{2}=\bar{\mathcal{Q}}^{2}=0$, and the Hamiltonian is

$\displaystyle H=\\{\mathcal{Q},\bar{\mathcal{Q}}\\}\,,$ (2.3)

as dictated by the supersymmetry algebra. When the Hamiltonian is expanded as
a sum of interactions between $2q-2$ fermions, it has the same form as complex
SYK [47, 48, 3, 4, 49, 5, 50, 51], but the complex SYK couplings are not
Gaussian independent variables, given instead in terms of quadratic
combinations of $C_{i_{1}\ldots i_{q}}$.

Supersymmetry relates bosons and fermions, but with supersymmetric SYK models
we realize the supersymmetry nonlinearly. For example take

$\displaystyle\\{\mathcal{Q},\psi^{i}\\}=0\,,\quad\\{\mathcal{Q},\bar{\psi}_{i}\\}=i^{\frac{q-1}{2}}q\,C_{i,i_{1},\ldots,i_{q-1}}\psi^{i_{1}}\ldots\psi^{i_{q-1}}\sim\bar{b}_{i}\,,$
(2.4)

where the second equation defines the bosonic composite complex field $b^{i}$,
up to a choice of normalization. This means we do not need to include
fundamental bosons in the theory in order for it to be supersymmetric. It is
convenient nevertheless to do a Hubbard-Stratonovich transformation
integrating-in a fundamental boson $b^{i}$ such that on-shell its given by the
expression above. This has two advantages. The first is that now supersymmetry
transformations (2.4) are linearly realized. The second is that the theory
with only fermions is not melonic and is not described by a mean field action,
but the equivalent theory with the boson integrated-in is melonic and solvable
at low temperatures [6].

The Hamiltonian constructed above has an additional symmetry that corresponds
to a $U(1)_{R}$ phase rotation of the fundamental fermions. The complex
fermions transform in conjugate representations, and therefore the supercharge
is purely (anti)holomorphic with respect to the rotation, which guarantees for
example that $\mathcal{Q}^{2}=0$. This implies that the $U(1)_{R}$ symmetry is
the $R$-symmetry of the one-dimensional $\mathcal{N}=2$ Poincare algebra. The
explicit generator is:

$Q=\sum_{j}\bar{\psi}_{j}\psi^{j}-\frac{N}{2},~{}~{}~{}~{}Q_{R}=\frac{1}{q}~{}Q.$
(2.5)

The first expression defines the fermion charge $Q$, with a shift of $N/2$ to
enforce charge conjugation symmetry. This is the standard definition with a
fermion having unit charge. On the right we give the natural definition of the
$R$-charge $Q_{R}$, such that the R-charge of the supercharge is one. In these
units the fundamental fermion has a fractional R-charge $1/q$. For this reason
we will refer to these modes as fractional charge $\mathcal{N}=2$ SYK, as
opposed to the models we will study in the next section.

#### Refined Witten Index

The $\mathcal{N}=2$ SYK model is a particular instance of supersymmetric
quantum mechanics, and therefore one may compute the Witten index [52] which
counts the number of bosonic and fermionic ground states with a
$(-1)^{F}=e^{-i\pi Q_{R}}$, with $F$ the fermion number. This is a protected
quantity which may be computed in the free field limit with all $C_{i_{1}\dots
i_{q}}$ set to zero. However, due to the presence of fractionally charged
ground states with opposite bose-fermi statistics, the Witten index of this
$\mathcal{N}=2$ SYK model vanishes. As explained in [6], this model possesses
a $\mathbb{Z}_{q}$ global symmetry under which the fermions transform with a
$q$-th root of unity, and in particular commutes with the supercharge. Turning
on a (discrete) chemical potential for this symmetry allows one to define a
non-vanishing refined Witten index. This refined Witten index
$\mathcal{I}(r)\equiv{\rm Tr}\left[(-1)^{F}e^{2\pi irQ_{R}}\right]$ for this
theory is given by

$\mathcal{I}(r)=e^{\pi iN(\frac{1}{2}-\frac{r}{q})}(1-e^{\frac{2\pi
ir}{q}})^{N}=\left(2\sin\frac{\pi r}{q}\right)^{N},$ (2.6)

which vanishes for $r=0$. In general, the index and the number of ground
states do not agree because there may be cancellation in the index. A bound on
the number of ground states by computing the maximum absolute value of the
index. The answer is

${\rm max}_{r}\log|\mathcal{I}(r)|=N\log\left(2\cos\frac{\pi}{2q}\right)$
(2.7)

The maximum value is given for $r_{\rm max}=(q\pm 1)/2$. We will match this
later at large $N$ to the zero temperature entropy using the low temperature
solution of the model.

###  2.1 Mean field and the conformal solution

We now review the mean field formulation of this theory. The supersymmetry can
be made manifest by introducing superspace coordinates
$Z\equiv(\tau,\theta,\bar{\theta})$. In terms of these coordinates, the
infinitesimal supersymmetry and R-symmetry transformations are realized
respectively as:

$\displaystyle\tau\rightarrow\tau+\theta\bar{\eta}+\bar{\theta}\eta\,,\qquad\theta\rightarrow\theta+\eta\,,\qquad\bar{\theta}\rightarrow\bar{\theta}+\bar{\eta}\,$
(2.8)

for an infinitesimal complex Grassmann parameter $\eta$ and,

$\displaystyle\theta\rightarrow e^{ia}\theta\,,\qquad\bar{\theta}\rightarrow
e^{-ia}\bar{\theta}\,,$ (2.9)

for a phase $a$. The fermion and boson (2.4) are encoded in a chiral fermionic
superfield $\Psi^{i}(\tau,\theta,\bar{\theta})$, where the chirality is
defined by a suitable supercovariant derivative,

$\displaystyle
D_{\bar{\theta}}\equiv(\partial_{\bar{\theta}}+\theta\partial_{\tau})\,,\quad
D_{\bar{\theta}}\Psi^{i}(\tau,\theta,\bar{\theta})=0\,\,\Longrightarrow\,\,\Psi^{i}(\tau,\theta,\bar{\theta})=\psi^{i}(\tau+\theta\bar{\theta})+\sqrt{2}~{}\theta~{}b^{i}(\tau)\,.$
(2.10)

A similar set of expressions hold for the anti-chiral superfield
$\bar{\Psi}_{i}(\tau,\theta,\bar{\theta})$ annihilated by $D_{\theta}$ defined
by conjugation555In our conventions, conjugation always reverses the order of
the fermions..

The supersymmetric Lagrangian density corresponding to the Hamiltonian (2.3)
may be written in terms of the chiral superfields as

$\mathcal{L}=\frac{1}{2}\\!\int\\!d^{2}\theta\,\bar{\Psi}_{i}\Psi^{i}+i^{\frac{q-1}{2}}\int\\!d\theta\,C_{i_{1}i_{2}\ldots
i_{q}}\Psi^{i_{1}}\Psi^{i_{2}}\ldots\Psi^{i_{q}}+i^{\frac{q-1}{2}}\int\\!d\bar{\theta}\,\bar{C}^{i_{1}i_{2}\ldots
i_{q}}\bar{\Psi}_{i_{1}}\bar{\Psi}_{i_{2}}\ldots\bar{\Psi}_{i_{q}}.$ (2.11)

In component form, the Lagrangian becomes after integrating by parts

$\mathcal{L}=\bar{\psi}_{i}\partial_{\tau}\psi^{i}-\bar{b}_{i}b^{i}+i^{\frac{q-1}{2}}\sqrt{2}qC_{i_{1}i_{2}\ldots
i_{q}}b^{i_{1}}\psi^{i_{2}}\dots\psi^{i_{q}}+i^{\frac{q-1}{2}}\sqrt{2}q\bar{C}^{i_{1}i_{2}\ldots
i_{q}}\bar{b}_{i_{1}}\bar{\psi}_{i_{2}}\dots\bar{\psi}_{i_{q}}$ (2.12)

Following [6] we introduce the superspace two-point function
$\mathcal{G}(Z_{1},Z_{2})=\frac{1}{N}\langle\bar{\Psi}_{i}(Z_{1})\Psi^{i}(Z_{2})\rangle$.
The components of this anti-chiral-chiral superfield contains the fermion and
boson two point functions (noting the sum over $i$ is implicit):

$G_{\psi\psi}(\tau_{1},\tau_{2})\equiv\frac{1}{N}\langle\bar{\psi}_{i}(\tau_{1})\psi^{i}(\tau_{2})\rangle,~{}~{}~{}~{}G_{bb}(\tau_{1},\tau_{2})\equiv\frac{1}{N}\langle\bar{b}_{i}(\tau_{1})b^{i}(\tau_{2})\rangle.$
(2.13)

The full two-point function in superspace also includes fermionic correlators
mixing fermions and bosons. The complete expansion is

$\displaystyle\mathcal{G}(Z_{1},Z_{2})$ $\displaystyle=$ $\displaystyle
G_{\psi\psi}(\tau_{1}-\theta_{1}\bar{\theta}_{1},\tau_{2}+\theta_{2}\bar{\theta}_{2})+2\bar{\theta}_{1}\theta_{2}G_{bb}(\tau_{1},\tau_{2})$
(2.14)
$\displaystyle+\sqrt{2}\bar{\theta}_{1}G_{b\psi}(\tau_{1},\tau_{2}+\theta_{2}\bar{\theta}_{2})-\sqrt{2}\theta_{2}G_{\psi
b}(\tau_{1}-\theta_{1}\bar{\theta}_{1},\tau_{2}).$

In the large $N$ limit we will see that the classical solution of the mean
field equations we derive satisfies $G_{\psi b}=0$ and we can consistently set
the mixed correlators to zero for now. It will be necessary to include them
later when considering the kernel giving quadratic fluctuations around the
mean field classical solution.

Starting with the action as in (2.11), our next goal is to produce the
aforementioned mean-field equations of motion which are valid at large $N$. We
will follow a series of now standard steps for SYK models, working largely in
superspace:

  * •

Integrate out the random couplings to generate a bi-local Lagrangian:

$\mathcal{L}=\frac{1}{2}\\!\int\\!d^{2}\theta\,\bar{\Psi}_{i}\Psi^{i}+\int\\!d\bar{\theta}_{1}d\theta_{2}\,\bar{\Psi}_{i_{1}}\bar{\Psi}_{i_{2}}\ldots\bar{\Psi}_{i_{q}}\langle\bar{C}^{i_{1}i_{2}\ldots
i_{q}}C_{j_{1}j_{2}\ldots
j_{q}}\rangle\Psi^{j_{1}}\Psi^{j_{2}}\ldots\Psi^{j_{q}}.$ (2.15)

  * •

Integrate in the field $\mathcal{G}(Z_{1},Z_{2})$ which is then fixed to be
the superspace two-point function by further integrating in an anti-chiral-
chiral bilocal field $\Sigma(Z_{1},Z_{2})$. The components of
$\Sigma(Z_{1},Z_{2})$ contain the fermionic and bosonic self energies,
$\Sigma_{\psi\psi}(\tau_{1},\tau_{2})$ and $\Sigma_{bb}(\tau_{1},\tau_{2})$,
respectively. This amounts to inserting a superspace identity

$1=\int D\mathcal{G}D\Sigma\exp{\left(-N\int
d\bar{Z}_{1}dZ_{2}\,\Sigma(Z_{1},Z_{2})\left(\mathcal{G}(Z_{1},Z_{2})-\frac{1}{N}\bar{\Psi}_{i}(Z_{1})\Psi^{i}(Z_{2})\right)\right)}\,,$
(2.16)

with the order of indices chosen to match the fact that $\mathcal{G}$ and
$\Sigma$ have the same chirality.

  * •

Integrate out the fundamental fermions to obtain an action in terms of the
collective variables only. We will not write the action explicitly, but
instead focus on the equations of motion.

  * •

Vary the action with respect to the bilocal fields. To write the equations of
motion in superspace, we will introduce chiral and anti-chiral integrations

$\displaystyle\int dZ\equiv\int d\tau d\theta\,,\qquad\int d\bar{Z}\equiv\int
d\tau d\bar{\theta}\,,$ (2.17)

which manifest supersymmetry when acting on superfields of the appropriate
chirality. Because the mean field action is bi-local, we will sometimes
encounter several such integrations over distinct superspace coordinates.

The equations of motion resulting from this procedure can be written as

$\displaystyle\Sigma(Z_{2},Z_{3})$ $\displaystyle=$
$\displaystyle\frac{1}{2}J~{}\mathcal{G}(Z_{2},Z_{3})^{q-1},$
$\displaystyle\frac{1}{2}D_{\theta_{3}}\mathcal{G}(Z_{1},Z_{3})+\int
dZ_{2}\,\mathcal{G}(Z_{1},Z_{2})~{}\Sigma(Z_{3},Z_{2})$ $\displaystyle=$
$\displaystyle\delta(Z_{1}-Z_{3}),$ (2.18)

where we define a supersymmetric delta function
$\delta(Z_{1}-Z_{2})\equiv(\bar{\theta}_{1}-\bar{\theta}_{2})\delta(\tau_{1}-\theta_{1}\bar{\theta}_{1}-\tau_{2}+\theta_{2}\bar{\theta}_{2})$.
This equation defines the normalization we chose for $J$ the coupling constant
which leads to particularly simple component equations (2.19). Note also that
the second equation is properly anti-chiral-anti-chiral in $Z_{1}$ and
$Z_{3}$; in particular the superspace integration over $Z_{2}$ as well as the
ordering of the indices inside $\mathcal{G}$ and $\Sigma$ means we do not need
to introduce explicit conjugate bilocals $\bar{\mathcal{G}}$ and
$\bar{\Sigma}$.

The superspace presentation of the mean field action for the $\mathcal{N}=2$
SYK model is well known, but as already discussed, we will also introduce and
analyze the consequences of a chemical potential ($\mu$) for the $U(1)_{R}$
symmetry. In our conventions the chemical potential $\mu$ appears as $\mu
G_{\psi\psi}$ in the action which means it couples to $qQ_{R}$. Because the
R-symmetry does not commute with supersymmetry, or equivalently because the
Grassmann variables $(\theta,\bar{\theta})$ are charged under the R-symmetry
(2.9), the inclusion of a chemical potential generically corresponds to
turning on background fields which break supersymmetry. Therefore, we will
study the mean field equations in component form from now on and study the
supersymmetric point as a special case.

In the fermion-number conserving saddle-point we are interested in, the mixed
$G_{b\psi}$ correlators vanish and we can focus on $G_{\psi\psi}$, $G_{bb}$,
and their self energies. Moreover we will also assume the solution is time
translation invariant. Under these assumptions, the Schwinger-Dyson equations
derived from the mean field action with chemical potential $\mu$ are given by

$\displaystyle\Sigma_{\psi\psi}(\tau)$ $\displaystyle=$ $\displaystyle
J(q-1)G_{\psi\psi}(\tau)^{q-2}G_{bb}(\tau),~{}~{}~{}~{}~{}\Sigma_{bb}(\tau)=JG_{\psi\psi}(\tau)^{q-1},$
(2.19) $\displaystyle G_{\psi\psi}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{-i\omega+\mu+\Sigma_{\psi\psi}(-\omega)},~{}~{}~{}~{}~{}G_{bb}(\omega)=\frac{1}{-1-\Sigma_{bb}(-\omega)}.$
(2.20)

The $\tau=\tau_{1}-\tau_{2}$ is the time difference and the second line is
written in Fourier space. Also, one may note that while the boson $b$ carries
$U(1)_{R}$ charge, there is no chemical potential present in the $G_{bb}$
equation because this is only an auxiliary field.

The full Schwinger-Dyson equations are complicated to solve and are typically
studied numerically, as we do in section 2.2. However, they are tractable in
the IR limit, meaning long time separations $|J\tau|\gg 1$ compared to the
scale set by $J$. In this regime we will assume the theory is approximately
conformal, and importantly we will see below that this assumption may be
violated depending on the background charge. This conformal ansatz determines
the fermion and boson two point functions at zero temperature to be

$G_{\psi\psi}(\tau)=\frac{g_{\psi\psi}}{|\tau|^{2\Delta}}\left(e^{\pi\mathcal{E}}\Theta(\tau)-e^{-\pi\mathcal{E}}\Theta(-\tau)\right),~{}~{}G_{bb}(\tau)=\frac{g_{bb}}{|\tau|^{2\Delta_{b}}}\left(e^{\pi\mathcal{E}_{b}}\Theta(\tau)+e^{-\pi\mathcal{E}_{b}}\Theta(-\tau)\right),$
(2.21)

where we allow for the possibility of a spectral asymmetry in order to
incorporate states with non-zero charge, parametrized by $\mathcal{E}$ for the
fermion and $\mathcal{E}_{b}$ for the boson. The spectral asymmetry defined in
the IR should be thought of as related to the chemical potential or the charge
through the UV behavior of the correlators. The prefactors $g_{\psi\psi}$ and
$g_{bb}$ are coefficients to be determined by the equations of motion. Finally
we also introduce the IR scaling dimensions for fermions $\Delta$ and bosons
$\Delta_{b}$. After solving the zero temperature equations, the ansatz can
also be put at finite temperature by a reparametrization

$G_{\psi\psi}(\tau)=g_{\psi\psi}\left(\frac{\beta}{\pi}\sin{\frac{\pi\tau}{\beta}}\right)^{-2\Delta}e^{2\pi\mathcal{E}\left(\frac{1}{2}-\frac{\tau}{\beta}\right)}\,,\,\,\,\,G_{bb}(\tau)=g_{bb}\left(\frac{\beta}{\pi}\sin{\frac{\pi\tau}{\beta}}\right)^{-2\Delta_{b}}e^{2\pi\mathcal{E}_{b}\left(\frac{1}{2}-\frac{\tau}{\beta}\right)},$
(2.22)

valid for $0<\tau<\beta$, together with $J\tau\gg 1$ and $J(\beta-\tau)\gg 1$
so that the IR solution we are going to find is accurate.

Using the two-point functions outlined above, the next step is to solve the
equations (2.20) in the IR strong coupling limit in which the self-energies
dominate over bare propagators. This leads to the simple form
$\Sigma_{\psi\psi}(-\omega)G_{\psi\psi}(\omega)=-\Sigma_{bb}(-\omega)G_{bb}(\omega)=1$
(after a shift of self-energies, see [5]). The solution is obtained by
transforming the two point functions to Fourier space, taking the inverse, and
then transforming back to time. This determines the self-energies (which for
simplicity we write at zero temperature) to be

$\displaystyle\Sigma_{\psi\psi}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{g_{\psi\psi}}~{}\frac{(1-2\Delta)\sin
2\pi\Delta}{2\pi(\cosh 2\pi\mathcal{E}+\cos
2\pi\Delta)}~{}\frac{1}{|\tau|^{2(1-\Delta)}}\left(-e^{\pi\mathcal{E}}\Theta(-\tau)+e^{-\pi\mathcal{E}}\Theta(\tau)\right),$
(2.23) $\displaystyle\Sigma_{bb}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{1}{g_{bb}}~{}\frac{(1-2\Delta_{b})\sin
2\pi\Delta_{b}}{2\pi(\cosh 2\pi\mathcal{E}_{b}-\cos
2\pi\Delta_{b})}~{}\frac{1}{|\tau|^{2(1-\Delta_{b})}}\left(e^{\pi\mathcal{E}_{b}}\Theta(-\tau)+e^{-\pi\mathcal{E}_{b}}\Theta(\tau)\right),~{}~{}~{}$

Having both the two point functions and self-energies, we can now plug these
back into the first set of Schwinger-Dyson equations (2.19). This places
constraints on the dimensions, prefactors and spectral asymmetries. In
contrast to the special case studied in [6] with no spectral asymmetry, we
find a different set of constraints.

The equations (2.19) can be easily solved in steps. We can separately match
the spectral asymmetry, overall scaling dimension, and prefactors in the left
and right-hand sides. Matching the spectral asymmetry in both equations gives
a single constraint

$\mathcal{E}_{b}=-(q-1)\mathcal{E}$ (2.24)

which is consistent with the on-shell expression for the boson in term of
fermions $b\sim\bar{\psi}^{q-1}$ (2.4). Matching scaling dimensions in both
equations gives again a single constraint,

$(q-1)\Delta+\Delta_{b}=1,$ (2.25)

which can be interpreted as demanding the interaction term $C_{i_{1}\ldots
i_{q}}b^{i_{1}}\psi^{i_{2}}\ldots\psi^{i_{q}}$ in the component action to be
marginal. For consistency of the IR ansatz, namely to make sure we can neglect
the $i\omega$ and $+1$ terms in the equations (2.20), we need $\Delta>0$ and
$\Delta_{b}>1/2$ which, upon using the constraint on dimensions (2.25),
implies $0<\Delta<1/(2(q-1))$. This means that a solution outside of this
range should be considered inconsistent in the sense that the IR ansatz is
invalid.

The final step to solving the equations is to match the prefactors in the left
and right hand side of equations (2.19). The two equations give the following
constraint:

$\frac{(1-2\Delta)\sin 2\pi\Delta}{2\pi(\cosh 2\pi\mathcal{E}+\cos
2\pi\Delta)}=(q-1)Jg_{\psi\psi}^{q-1}g_{bb},~{}~{}~{}\frac{(1-2\Delta_{b})\sin
2\pi\Delta_{b}}{2\pi(\cosh 2\pi\mathcal{E}_{b}-\cos
2\pi\Delta_{b})}=Jg_{\psi\psi}^{q-1}g_{bb}$ (2.26)

By matching these two expressions for $g_{\psi\psi}^{q-1}g_{bb}$, we can
completely determine the scaling dimensions as a function of $q$ and the
spectral asymmetry $\mathcal{E}$, after using the expression for $\Delta_{b}$
and $\mathcal{E}_{b}$. The solution can only be found implicitly through the
equation666This expression was independently derived in [17].

$\frac{(1-2\Delta)\sin 2\pi\Delta}{\cosh 2\pi\mathcal{E}+\cos
2\pi\Delta}=(q-1)\frac{(1-2(q-1)\Delta)\sin 2\pi(q-1)\Delta}{\cosh
2\pi(q-1)\mathcal{E}-\cos 2\pi(q-1)\Delta}$ (2.27)

This determines $\Delta$, $\Delta_{b}$ and $\mathcal{E}_{b}$, all as a
function of $q$ and $\mathcal{E}$. Finally, since now the two equations (2.26)
are identical we can only determine the combination
$g_{\psi\psi}^{q-1}g_{bb}$, but not each prefactor separately. As explained in
[6] this can be traced back to an emergent symmetry in the IR Schwinger-Dyson
equations under $G_{\psi\psi}(\tau)\to\lambda G_{\psi\psi}(\tau)$ and
$G_{bb}(\tau)\to\lambda^{1-q}G_{bb}(\tau)$. It is expected that this does not
generate a relevant mode in the IR and simply the UV boundary conditions
determine a precise value of $g_{\psi\psi}$ and $g_{bb}$. As reviewed for
example in [49] unitarity requires both coefficients to be positive, and this
is consistent with the constraint (2.26). We will verify that the full
solution determines a consistent coefficient using a numerical solution of the
Schwinger-Dyson equations below in section 2.2.

#### Emergent $SU(1,1|1)$ symmetry

The equation above that determines $\Delta$ cannot be solved explicitly in
general. There are special values of parameters where we expect the solution
to preserve supersymmetry. In these cases the equations can be exactly solved
and we reproduce the results of [6] (We will see other examples in the next
section where even supersymmetry is not powerful enough to fully fix the IR
solution). The first observation we can make is that for the following values
of the spectral asymmetry the scaling dimensions have a simple expression

$\mathcal{E}_{\rm
susy}=\frac{ir}{q},~{}~{}~{}r\in\mathbb{Z}_{q},~{}~{}~{}~{}\Rightarrow~{}~{}~{}\Delta=\frac{1}{2q},~{}~{}~{}\Delta_{b}=\frac{1}{2}+\frac{1}{2q}.$
(2.28)

The interpretation for this family is that it corresponds to turning on the
discrete chemical potential conjugate to the $\mathbb{Z}_{q}$ global symmetry
that commutes with the supercharge. A simple way of deriving constraints from
global supersymmetry is to construct the unique superspace two-point function
that is anti-chiral in the first variable, chiral in the second, and
manifestly invariant under global super-translation
$(\tau,\theta,\bar{\theta})\to(\tau+\epsilon+\bar{\theta}\eta+\theta\bar{\eta},\theta+\eta,\bar{\theta}+\bar{\eta})$,
with parameters $(\epsilon,\eta,\bar{\eta})$. The answer is given by

$\mathcal{G}_{\rm
susy}(Z_{1},Z_{2})=f(\tau_{1}-\tau_{2}-\theta_{1}\bar{\theta}_{1}-\theta_{2}\bar{\theta}_{2}-2\bar{\theta}_{1}\theta_{2}),$
(2.29)

where the right hand side involves an arbitrary function $f$. Using (2.14) we
can see that this implies
$G_{\psi\psi}(\tau_{1},\tau_{2})=f(\tau_{1}-\tau_{2})$ and
$G_{bb}(\tau_{1},\tau_{2})=-f^{\prime}(\tau_{1}-\tau_{2})$. Combining these
facts we obtain the constraint

$G_{bb}(\tau_{1},\tau_{2})=-\partial_{\tau_{1}}G_{\psi\psi}(\tau_{1}-\tau_{2}),$
(2.30)

which together with the conformal ansatz implies $\Delta_{b}=\Delta+1/2$, and
replacing this in (2.19) and (2.20) gives $\Delta=1/2q$, as obtained in the
equation above (2.28). Moreover, supersymmetry also implies a relation between
coefficients $g_{bb}=2\Delta g_{\psi\psi}$, allowing one to fully find the
solution for the prefactors as well:

$g_{\psi\psi}^{\rm susy}=\left(\frac{1}{2\pi
J}\frac{\sin\frac{\pi}{q}}{\cos\frac{\pi}{q}+\cos\frac{2\pi
r}{q}}\right)^{1/q},~{}~{}~{}~{}g_{bb}^{\rm
susy}=\frac{1}{q}\left(\frac{1}{2\pi
J}\frac{\sin\frac{\pi}{q}}{\cos\frac{\pi}{q}+\cos\frac{2\pi
r}{q}}\right)^{1/q}.$ (2.31)

These solutions correspond to a finite two-point functions except when
$r=(q\pm 1)/2$. For those two values the denominators vanish and
$g_{\psi\psi}^{\rm susy}$ and $g_{bb}^{\rm susy}$ both diverge. It is
interesting to note those are precisely the values at which the refined index
has maximal absolute value.

So far we have imposed only the global $\mathcal{N}=2$ supersymmetric
conditions but the model at low temperatures enjoys a bigger group of
symmetries. In the IR limit we solved the Schwinger-Dyson equations (2.18) in
a regime where we can neglect the first term and get

$\int
dZ_{2}~{}\mathcal{G}(Z_{1},Z_{2})~{}\left[\frac{J}{2}\mathcal{G}(Z_{3},Z_{2})^{q-1}\right]=\delta(\bar{Z}_{1}-\bar{Z}_{3}).$
(2.32)

It was pointed out in [6] that these equations have a symmetry under
$\mathcal{N}=2$ supersymmetric reparametrizations which we denote by ${\rm
Diff}(S^{1|2})$. This consists of super-reparametrizations
$Z=(\tau,\theta,\bar{\theta})\to
Z^{\prime}=(\tau^{\prime},\theta^{\prime},\bar{\theta}^{\prime})$ that satisfy
the constraints

$\displaystyle
D_{\theta}\bar{\theta}^{\prime}=0~{}~{}~{}~{}D_{\theta}\tau^{\prime}=\bar{\theta}^{\prime}D_{\theta}\theta^{\prime},$
(2.33) $\displaystyle
D_{\bar{\theta}}\theta^{\prime}=0~{}~{}~{}~{}D_{\bar{\theta}}\tau^{\prime}=\theta^{\prime}D_{\bar{\theta}}\bar{\theta}^{\prime}.$
(2.34)

As shown in [6] the solutions of these constraints can be parametrized by a
bosonic reparametrization mode $f(\tau)$, a local $U(1)$ transformation
$e^{ia(\tau)}$ and a complex fermionic mode $\eta(\tau)$ which may be grouped
into the bosonic and fermionic reparametrizations as:

$\displaystyle{\rm Bosonic:}~{}~{}~{}\tau^{\prime}$
$\displaystyle=f(\tau)\,,\quad\theta^{\prime}=e^{ia(\tau)}\sqrt{\partial_{\tau}f(\tau)}\theta\,\quad\bar{\theta}^{\prime}=e^{-ia(\tau)}\sqrt{\partial_{\tau}f(\tau)}\bar{\theta}\,,$
(2.35) $\displaystyle{\rm Fermionic:}~{}~{}~{}\tau^{\prime}$
$\displaystyle=\tau+\theta\bar{\eta}(\tau)+\bar{\theta}\eta(\tau)\,,\quad\theta^{\prime}=\theta+\eta(\tau+\theta\bar{\theta})\,\quad\bar{\theta}^{\prime}=\bar{\theta}+\bar{\eta}(\tau-\theta\bar{\theta})\,.$
(2.36)

Then its easy to check that the IR Schwinger-Dyson equation is invariant under

$\mathcal{G}(Z_{1},Z_{2})\to(D_{\theta_{1}}\theta^{\prime}_{1})^{\frac{1}{q}}~{}(D_{\bar{\theta}_{2}}\bar{\theta}^{\prime}_{2})^{\frac{1}{q}}~{}\mathcal{G}(Z^{\prime}_{1},Z^{\prime}_{2}).$
(2.37)

Even though this is a symmetry of the equations, this is not a symmetry of the
solutions. The correlators found above at the supersymmetric point with
$\Delta=1/(2q)$ are only invariant under a subgroup $SU(1,1|1)$ of ${\rm
Diff}(S^{1|2})$. Therefore we see that the emergent super-reparametrization
symmetry is spontaneously broken to the $\mathcal{N}=2$ superconformal group.
Of course both of these symmetries are broken by the UV term, giving rise to
the $\mathcal{N}=2$ Schwarzian mode.

In a state with non-zero charge, we saw that $\Delta$ is not given by the
supersymmetric solution. It is interesting to notice that the IR equations
preserve supersymmetry even at finite charge, when the UV term is ignored.
From this point of view, the breaking of supersymmetry that happens from the
inclusion of a chemical potential is spontaneous: the solution breaks the
symmetry.

From (2.37) one can deduce that the fermion $\psi$ is a superconformal
primary. In particular that transformation rule, applied to the solution with
zero charge density only, implies the R-charge $1/q$ is twice the scaling
dimension $\Delta=1/(2q)$. This will be important to keep in mind in the next
section.

It is instructive to see how a super-reparametrization explicitly acts on the
two-point function. Its enough to do this at the linearized level. Bosonic
reparametrizations and $U(1)_{R}$ transformations look the same as
reparametrizations and local gauge transformations in complex SYK, and we will
not repeat it here. The new generators are the infinitesimal fermion super-
reparametrization, and under them the two-point function changes as

$\displaystyle\delta G_{\psi\psi}$ $\displaystyle=$ $\displaystyle-
G_{b\psi}(\tau_{1},\tau_{2})\bar{\eta}(\tau_{1})$ (2.38) $\displaystyle\delta
G_{b\psi}$ $\displaystyle=$ $\displaystyle
G_{bb}(\tau_{1},\tau_{2})\bar{\eta}(\tau_{1})-\partial_{\tau_{2}}G_{\psi\psi}(\tau_{1},\tau_{2})\bar{\eta}(\tau_{2})-\frac{1}{q}G_{\psi\psi}(\tau_{1},\tau_{2})\bar{\eta}^{\prime}(\tau_{2})$
(2.39) $\displaystyle\delta G_{bb}$ $\displaystyle=$
$\displaystyle\partial_{\tau_{2}}G_{b\psi}(\tau_{1},\tau_{2})\bar{\eta}(\tau_{2})-\frac{1}{q}G_{b\psi}(\tau_{1},\tau_{2})\bar{\eta}^{\prime}(\tau_{2}).$
(2.40)

and a similar transformation for $\eta$. To simplify the expressions we
rescale $\eta\to\eta/\sqrt{2}$ compared to (2.36). Its clear the first and
third equation vanishes on-shell since the fermionic correlators are zero
$G_{b\psi}=0$. The second line is more interesting. On a superconformal
solution of the type discussed above, the variation of $G_{b\psi}$ also
vanishes whenever $\eta=\eta_{0}+\eta_{1}\tau$ and
$\bar{\eta}=\bar{\eta}_{0}+\bar{\eta}_{1}\tau$. These four real parameters are
the fermionic generators of $SU(1,1|1)$. For $\eta\sim\tau^{n}$ with $n\neq
0,1$ the transformation acts non-trivially on the two-point function, and all
these modes have $SL(2,\mathbb{R})$ Casimir given by $h=3/2$ at
$\mathcal{E}=\mathcal{E}_{\rm susy}$777The Casimir operator $C_{1+2}$ acting
on $\delta G_{b\psi}(\tau_{1},\tau_{2})$ is defined in the same way as
equations (3.54) of [53], with the difference that the scaling dimension
appearing in the generators of $SL(2,\mathbb{R})$ are different for the first
and second insertion. Writing the eigenvalue of the Casimir as $h(h-1)$
defines the parameter $h$.. We will see these modes later again when we
compute fluctuations of the action around the classical Schwinger-Dyson
solution. At the same time we see that for $\mathcal{E}\neq\mathcal{E}_{\rm
susy}$ the four fermion zero-modes are broken spontaneously since the on-shell
correlators no long satisfy any supersymmetry relation.

#### The Zero Temperature Entropy

We can compute the zero temperature entropy of the model and compare with the
index, using the IR asymptotic of the solution of the mean field action. In
order to do this it is convenient to take a derivative with respect to $q$ in
the action. This simplifies when evaluating on a solution of the equations of
motion and gives

$\displaystyle\partial_{q}\frac{\log Z}{N}$ $\displaystyle=$ $\displaystyle
J\beta\int G_{\psi\psi}(\tau)^{q-1}G_{bb}(\tau)\log G_{\psi\psi}(\tau),$
(2.41) $\displaystyle=$
$\displaystyle\\#\beta+2\pi^{2}\Delta~{}Jg_{\psi\psi}^{q-1}g_{bb}+\mathcal{O}(\beta^{-1})$
(2.42)

In the second line we inserted the conformal solution, keeping terms
insensitive to the UV behavior. If we denote the temperature independent term
in the partition function by $\log Z=\beta\\#+G+\mathcal{O}(\beta^{-1})$,
where we introduce the grand potential $G$, then it is given by 888This
expression was independently derived in [17].

$\frac{dG}{dq}=N\frac{\pi\Delta(1-2\Delta)\sin 2\pi\Delta}{(q-1)(\cos
2\pi\Delta+\cosh 2\pi\mathcal{E})}.$ (2.43)

In [49] it is explained how to go from this expression to computing
$S_{0}(Q)$. Unfortunately this cannot be done in this model since it is not
clear what boundary conditions to use when integrating over $q$. Instead, for
now we will focus on the particle-hole symmetric point with $Q_{R}=0$. In this
case the solution is the supersymmetric one with $\mathcal{E}=0$ which implies
$\Delta=1/(2q)$. In this case $G=S_{0}$ and we obtain
$\frac{dS_{0}}{dq}=N\frac{\pi\tan\frac{\pi}{2q}}{2q^{2}}$. This can be easily
integrated, using the free fermion limit to fix the integration constant, and
gives

$S_{0}(Q_{R}=0)=N\log\big{(}2\cos\frac{\pi}{2q}\big{)},$ (2.44)

which precisely matches with the maximization of the index in equation (2.7).
We will see a similar phenomenon in the models we study in the next section.

We will later propose a closed formula for the zero-temperature entropy and
grand potential at non-zero charge based on an extension of the spooky
propagator of [5], and verify these relations numerically.

###  2.2 Breakdown of conformal ansatz

Figure 2: Plot showing the conformal solution for the fermion scaling
dimension $\Delta$ as a function of the spectral asymmetry $\mathcal{E}$ for
$q=3$. The behavior is similar for other values of $q$. We see the scaling
dimension reaches zero at the critical spectral asymmetry, and for larger
$\mathcal{E}$, the conformal ansatz is invalid, to the right of the dashed red
line.

We will now analyze what happens to the conformal solution of the Schwinger-
Dyson equations as we turn on the spectral asymmetry, and therefore the
background R-charge. We start with the $\mathcal{E}=0$ case for which we
obtain a supersymmetric solution with $\Delta=\frac{1}{2q}$. As we turn on
$\mathcal{E}$ we find $\Delta$ is a solution of the transcendental equation
given above. The scaling dimension smoothly goes from
$\Delta(\mathcal{E}=0)=\frac{1}{2q}$ to $\Delta(|\mathcal{E}|=\mathcal{E}_{\rm
critical})=0$. The critical asymmetry $\mathcal{E}_{\rm critical}>0$ is
determined by the following equation

$\frac{\sinh\pi(q-1)\mathcal{E}_{\rm critical}}{\cosh\pi\mathcal{E}_{\rm
critical}}=(q-1)$ (2.45)

For $|\mathcal{E}|>\mathcal{E}_{\rm critical}$ there are solution for the
scaling dimension in the complex $\Delta$-plane indicating the conformal
ansatz breaks down999The point $\Delta=0$ is always a solution but one with
$g_{\psi\psi}^{q-1}g_{bb}=0$. Therefore at least one of the two prefactors has
to vanish which is a non-physical solution.. The value for $q=3$ is given
numerically by

$\mathcal{E}_{\rm critical}=\frac{\log(1+\sqrt{2})}{\pi}=0.28055..$ (2.46)

We show the behavior of $\Delta$ as a function of $\mathcal{E}$ in figure 2.
After $\mathcal{E}>\mathcal{E}_{\text{critical}},$ there is no valid conformal
solution. In such a region, the Dyson Schwinger equations Eq.(2.19) can be
studied numerically. We will in fact show that the solution exponentially
decays once $\mathcal{E}>\mathcal{E}_{\text{critical}}.$ In contrast to known
transitions [21, 22], in this case the fundamental fermion $\psi^{i}$ itself
develops an either negative or complex scaling, making the physical
interpretation of this transition more transparent.

#### Numerical Solution to Schwinger Dyson equations

Figure 3: Numerical solutions to large $N$ Dyson Schwinger equations, where
from left to right $\beta J=5,20,50$ and up to down $\mathcal{E}=0,0.15.$ We
plot against dimensionless quantity $\frac{2\pi\tau}{\beta}$. We also plot the
best fit conformal solutions $G^{c}_{\psi\psi}$ and $G_{b_{\psi}b_{\psi}}^{c}$
and we observe good agreement up when $\tau\sim\mathcal{O}(1/J).$

We can solve equations (2.19) numerically by iterations. We work in Euclidean
time with finite temperature, $\tau\sim\tau+\beta,$ and for numerical purpose
we consider a discretized time $\tau_{i}=\frac{i\beta}{N_{\rm step}}.$ To
describe continuous physics we require $N_{\rm step}\gg\beta J.$ We work with
grand canonical ensemble and fix $\mu$. For a given $\mu$, the expectation
value of the charge can be computed by

$\frac{\langle
Q\rangle}{N}=\frac{1}{2N}\langle[\bar{\psi}^{i},\psi^{i}]\rangle=\frac{1}{2}\left(G(0)-G(\beta)\right).$
(2.47)

We fit numerical solutions against the Ansatz given in equation (2.22) where
we leave $g_{\psi\psi}/g_{bb}$ and $\mathcal{E}_{\psi}$ unfixed. Other
parameters are fixed entirely by IR Schwinger Dyson equations. We take the
best fitted $\mathcal{E}$ as the value $\mathcal{E}(\mu).$ Similar to the case
of complex SYK, $Q(\mu)$ and $\mathcal{E}(\mu)$ both non-trivially depend on
$\mu,$ and by tuning $\mu$ we may understand $Q(\mathcal{E})$. Some examples
showing the result of this procedure are presented in figure 3. It is evident
that the conformal solution is a good approximation in the IR when $\beta J$
is large. Moreover, the UV boundary conditions also fixes the undetermined
ratio $g_{bb}/g_{\psi\psi}$ as shown in figure 4.

Figure 4: Plot of $g_{bb}/g_{\psi\psi}$ computed from the numerical solution
for $q=3$ at different values of $\mathcal{E}.$ Note the infrared Schwinger-
Dyson equations can only determine the combination $g_{\psi\psi}^{q-1}g_{bb},$
but the individual values are determined by the full solution. At
$\mathcal{E}=0,$ it agrees with the supersymmetric answer $g_{bb}=2\Delta
g_{\psi\psi}=\frac{1}{3}g_{\psi\psi}.$

We now turn to the determination of the charge. Similar to complex SYK, it is
possible to derive an analytic relation between the charge and the spectral
asymmetry. We do this in Appendix A. First we define the fermionic and bosonic
contribution to the total charge

$\displaystyle\mathfrak{q}_{f}(\Delta,\mathcal{E})$ $\displaystyle=$
$\displaystyle\frac{(\frac{1}{2}-\Delta)\sinh 2\pi\mathcal{E}}{\cosh
2\pi\mathcal{E}+\cos
2\pi\Delta}+\frac{i}{2\pi}\log\left(\frac{\cos\pi(\Delta+i\mathcal{E})}{\cos\pi(\Delta-i\mathcal{E})}\right),$
(2.48) $\displaystyle\mathfrak{q}_{b}(\Delta_{b},\mathcal{E}_{b})$
$\displaystyle=$ $\displaystyle\frac{(\frac{1}{2}-\Delta_{b})\sinh
2\pi\mathcal{E}_{b}}{\cosh 2\pi\mathcal{E}_{b}-\cos
2\pi\Delta_{b}}+\frac{i}{2\pi}\log\left(\frac{\sin\pi(\Delta_{b}+i\mathcal{E}_{b})}{\sin\pi(\Delta_{b}-i\mathcal{E}_{b})}\right).$
(2.49)

The total fermion charge is given in terms of these functions by

$\frac{Q}{N}=\mathfrak{q}_{f}(\Delta,\mathcal{E})+(q-1)\mathfrak{q}_{b}(\Delta_{b},\mathcal{E}_{b}).$
(2.50)

The right hand side is a function of the spectral asymmetry $\mathcal{E}$
explicitly and through the scaling dimensions $\Delta$ and $\Delta_{b}$. This
expression is the sum of two terms. The first is the contribution to the
charge from the fermion itself, while the second term is a contribution to the
fermion charge from the auxiliary boson. This weird behavior is due to the
fact that this zeroth-order boson is an auxiliary field that has been
integrated-in to simplify the interactions. This bosonic contribution is
similar to the one derived in [54, 55] for first-order bosons. We have
verified numerically that this relation is correct, and the result is shown in
figure 5. In particular we can compute the charge that corresponds to the
critical spectral asymmetry, and for $q=3$ it is

$Q_{\rm critical}\equiv Q(\mathcal{E}_{\rm critical})=(\sqrt{2}-1)N,$ (2.51)

which is smaller than the maximal allowed value $N/2$. Therefore the phase
transition we find is physical.

Figure 5: Plot of $Q(\mathcal{E})$ computed with the numerical solution of the
Schwinger-Dyson equations for $q=3$ (blue dots). Shown in solid black line is
the analytic formula (2.50), analogous to the Luttinger-Ward relation.

At low energy and for charges small compared to $N$, the $\mathcal{N}=2$
Schwarzian description is reliable. We can numerically determine the Super-
Schwarzian coupling. Instead of directly fitting the free energy, we find it
more accurate and less computationally intensive to compute the
compressibility. Due to the $\mathcal{N}=2$ supersymmetry, the compressibility
of the $U(1)_{R}$ symmetry is directly related to the super-Schwarzian
coupling, as the relative coefficient between the reparametrization modes
$f(\tau)$ and the $U(1)$ sigma model is fixed by supersymmetry. The bosonic
part of the $\mathcal{N}=2$ Super-Schwarzian is given by [7]

$S_{b}=\frac{2\pi N\alpha_{S}}{\beta
J}\int_{0}^{2\pi}d\tau\left(-\text{Sch}\left(\tan\frac{f}{2},\tau\right)+2q^{2}\left(\partial_{\tau}a\right)^{2}\right)$
(2.52)

where $f$ is the reparametrization mode on the circle and $a(\tau)$ is the
generator of the $U(1)_{R}$ symmetry. We have rescaled time to have period
$2\pi$ for convenience. The additional factor of $q^{2}$ comes from
normalizing the fundamental fermion to have $R$ charge 1 instead of $1/q$. On
the other hand, the compressibility can be found through $U(1)_{R}$ sigma
model action as

$S_{b}\supset
2\pi\int_{0}^{2\pi}\frac{K}{2\beta}\left(\partial_{\tau}a\right)^{2}.$ (2.53)

We can therefore compute compressibility $K$ from the numerical Dyson-
Schwinger equations and extrapolate the $\mathcal{N}=2$ Schwarzian coupling as

$\alpha_{S}=\frac{1}{4q^{2}}\frac{KJ}{N}.$ (2.54)

To compute the compressibility, we turn on a small chemical potential,
numerically solve the Schwinger-Dyson equations on a grid of size $2^{19}$ and
specialize to $q=3$; the result is

$K=\left(\frac{dQ}{d\mu}\right)_{T=0}=\lim_{\mu\to 0}\frac{Q}{\mu}\approx
0.303N/J.$ (2.55)

This in term determines the Schwarzian coupling to be $\alpha_{S}\approx
0.00842.$ This is different from the numerical coupling of the $\mathcal{N}=0$
Schwarzian for complex SYK which is given by $\alpha_{S}^{\rm cSYK}=0.01418$
[53, 5]. As we vary $\mathcal{E},$ we can also compute the free energy and
entropy. At low temperature and zero charge, the Schwarzian theory implies a
free energy which admits an expansion

$-\beta F=-\beta
E_{0}+S_{0}+\frac{c}{2\beta},~{}~{}~{}c=\frac{4\pi^{2}\alpha_{S}N}{J}\approx
0.332\frac{N}{J}.$ (2.56)

where $E_{0}$ corresponds to the ground state energy and takes a non-zero
value due to normal ordering when going from the Hamiltonian to the mean field
action.101010We thank Yingfei Gu and Pengfei Zhang for a very useful
discussion on this point. The $S_{0}$ is the zero temperature entropy and the
$1/\beta$ term is due to the Super-Schwarzian. In figure 6 we checked that the
answer is consistent with a direct linear fitting against the entropy $S(\beta
J).$

Figure 6: A demonstration that the coupling obtained via the compressibility
is consistent with a direct linear fitting of the entropy $S(\beta J).$ The
dots are the numerical entropy computed through numerical Dyson Schwinger
solutions on a grid of size $2^{20}.$ Due to finite size effects, we can not
take $T$ to be too small. We show a best fit with slope fixed to be the answer
determined through compressibility. The intercept, $S_{0}$, is close to its
analytical answer. We can also fit both the slope and the intercept, and we
obtain $S(\beta J)=0.5493+0.3233/(\beta J).$ The difference is about $2.7\%$
and we conclude the answer determined both ways are consistent with each
other.

We now explain how to use the numerical solution to compute the thermodynamic
potentials. Since in the numerical procedure we fix $\mu,$ the grand potential
can be computed by the on-shell action as

$-\beta\Omega(\mu,T)/N=\log\left(2\cosh\left(\frac{\beta\mu}{2}\right)\right)+\sum_{\omega}\log\left(\frac{1+\frac{\Sigma_{\psi\psi}}{i\omega+\mu}}{1+\Sigma_{bb}}\right)-\beta\int_{0}^{\beta}d\tau~{}G_{\psi\psi}(\tau)\Sigma_{\psi\psi}(\tau),$
(2.57)

where we used the Schwinger-Dyson equations and the time translation of the
solution to simplify the last term. To compute the free energy $F$, we can
change to the canonical ensemble by

$\beta F=\beta\Omega+\beta\mu Q.$ (2.58)

Figure 7: Left: Numerical solutions of $G_{\psi\psi}$ in the region where
$\mathcal{E}>\mathcal{E}_{\text{critical}}.$ We observe exponential decay
solutions. Since the solution ceases to be conformal, the infrared parameter
$\mathcal{E}$ is no longer meaningful. The solutions depend on both $\beta J$
and $\mu/J.$ Right: Log plot of $G_{\psi\psi}$ at various values of $\beta J,$
where dashed lines are linear fits. We observe that the exponent is linear in
$\mu$.

For general $\mu$ we compute the free energy and entropy numerically from
(2.58). We show the plot of entropy in the grand canonical ensemble against
chemical potential in figure 8. We observe a sharp transition near $\mu=J,$
where a new decaying exponential solution to the Schwinger-Dyson equations(as
shown in figure. 7) starts to exist. The discontinuous jump in entropy is
consistent with a first order transition. The transition goes from a high
entropy phase to a low entropy phase, where $S_{0}/N$ is approximately zero.
At zero temperature, such exponentially decay solutions can be found
analytically to be

Figure 8: The entropy computed numerically as a function of the chemical
potential $\frac{\mu}{J}.$ For each value of $\mu,$ we compute the free energy
and entropy on a numerical grid of size $2^{25}$ at small temperatures and
extrapolate the zero temperature entropy. Note at $\mu=0,$ we obtain
$S_{0}(0)\approx 0.5484$ which is close to the value predicted by the index
$\log(2\cos\frac{\pi}{6})\approx 0.5493.$ The transition happens slightly
above $\mu=\mu_{c}=J,$ where the exponential decaying solution (2.59) starts
to appear. Extrapolating the precise zero temperature entropy becomes involved
near the transition point.

$G_{\psi\psi}\left(\tau\right)=e^{-(\mu-\mu_{c})\tau}\Theta\left(\tau\right),~{}~{}G_{bb}\left(\tau\right)=-\delta\left(\tau\right)+e^{((q-1)(\mu-\mu_{c})+J)\tau}\Theta\left(-\tau\right),$
(2.59)

where the critical chemical potential is determined to be

$\mu_{c}=\frac{J}{2}(q-1).$ (2.60)

To verify that this is a solution of the Schwinger-Dyson equations, begin by
writing the ansatz
$G_{\psi\psi}\left(\tau\right)=e^{-(\mu-\mu_{c})\tau}\Theta\left(\tau\right),$
and use $\Sigma_{bb}(\tau)=JG_{\psi\psi}(\tau)^{q-1}$ to obtain the boson
self-energy, in both position and Fourier space as

$\Sigma_{bb}\left(\tau\right)=Je^{-(q-1)(\mu-\mu_{c})\tau}\Theta\left(\tau\right),~{}\Rightarrow~{}\Sigma_{bb}\left(\omega\right)=\frac{J}{-i\omega+(q-1)(\mu-\mu_{c})}.$
(2.61)

We can use the momentum space equation to determine $G_{bb}(\omega)$ and
Fourier transform back to position space to obtain
$G_{bb}\left(\tau\right)=-\delta\left(\tau\right)+e^{((q-1)(\mu-\mu_{c})+J)\tau}\Theta\left(-\tau\right)$.
The position space equation $\Sigma_{\psi\psi}(\tau)$ determines the value of
$\mu_{c}$ since

$\Sigma_{\psi\psi}\left(\tau\right)=J(q-1)e^{-(q-2)(\mu-\mu_{c})\tau}\Theta\left(\tau\right)\left(-\delta(\tau)+e^{((q-1)(\mu-\mu_{c})+J)\tau}\Theta(-\tau)\right)=-\frac{J(q-1)}{2}\delta(\tau).$
(2.62)

where we used $\Theta(\tau)\delta(\tau)=\frac{1}{2}\delta(\tau).$ We note that
when $J=0$, $\mu_{c}$ is zero and the solution reduces to the retarded
propagator of free massive fermion. The non-zero value $\mu_{c}$ can be
thought as the minimal value required for the chemical potential to overtake
the random interaction and creates a gap in the spectrum. For $q=3,$we note
$\mu_{c}=J.$ Numerically a sharp transition occurs around this value.

###  2.3 Operator spectrum

In this section we will determine the scaling dimensions of bilinear operators
at the nearly conformal fixed point. The operator spectrum provides important
insights of the IR physics, in directions orthogonal to the reparametrization
modes. The spectrum of the theory (2.3) at zero charge has been worked out and
extensively discussed in [6, 11, 15]. We generalize the analysis to non-zero
chemical potential, which explicitly breaks supersymmetry. However, in the
infrared, supersymmetry is only spontaneously broken by the nearly conformal
solution. Since the infrared equations of motions remain supersymmetric, the
superspace formalism provides a powerful tool to analyze the bilinear
spectrum. The scaling dimensions of operators become continuous function of
the chemical potential $\mu,$ or equivalently the spectral asymmetry parameter
$\mathcal{E}.$

In particular, we will check that the spectrum does not contain an operator
with a complex scaling dimension. Such a complex mode corresponds (for
instance) to a bulk field below its BF bound, and thus suggests an
instability. Such modes are observed in various examples in nearly conformal
theory [18, 19, 20]. In the presence of a complex mode, the infrared physics
is not correctly described by the conformal saddle. In our case we verify that
there are no complex modes in the range $0\leq\mathcal{E}<\mathcal{E}_{\rm
critical},$ and thus the conformal solution exhibits a consistent spectrum. As
$\mathcal{E}\rightarrow\mathcal{E}_{\rm critical}$ (the point of the phase
transition), we observe the spectrum exhibits unusual features such as a
continuum of states.

To determine the spectrum of bilinear operators, we find it convenient to look
at the variation of large $N$ Dyson Schwinger equation, and set it to zero. To
see that such a procedure determines the bilinear spectrum, we note that Dyson
Schwinger equations are operator equations, and we may insert additional
operators at infinity while the equations still hold. Such an insertion
corresponds to a deformation

$\delta
G_{\psi\psi}=\langle\bar{\psi}_{i}(\tau)\psi^{i}(0)\mathcal{O}_{h}(\infty)\rangle\,,\quad\delta
G_{bb}=\langle\bar{b}_{i}(\tau)b^{i}(0)\mathcal{O}_{h}(\infty)\rangle$ (2.63)

To make such deformation non-zero, $\mathcal{O}$ must be a bilinear operator
consisting of fundamental fermions and bosons. Without losing of generality,
we may restrict $\mathcal{O}$ to be primary.

Imposing $\delta G_{\psi\psi}$ satisfies the Dyson Schwinger equation provides
a necessary consistency condition for the deformation to correspond to an
operator insertion, and it ‘bootstraps’ the operator spectrum. Note that this
is only a necessary condition, and it is not guaranteed that all such
deformation corresponds to non-trivial operators. Nontrivial operators are
distinguished by their $SL(2)$ Casimir (or the supersymmetric extension) and
respect the $h\rightarrow(1-h)$ symmetry which may relate different $\delta
G$.

To write the infrared Dyson Schwinger equations (2.32) in a more compact form,
we let $\mathcal{G}(Z_{1},Z_{2})^{T}=\mathcal{G}(Z_{2},Z_{1})$ and let
$(\star$, $\bar{\star})$ be chiral and anti-chiral convolutions, respectively.
In this notation, the infrared equations of motion are:

$\frac{J}{2}\mathcal{G}\star(\mathcal{G}^{q-1})^{T}\equiv\int
dZ_{2}~{}\mathcal{G}(Z_{1},Z_{2})~{}(\frac{1}{2}J\mathcal{G}(Z_{3};Z_{2})^{q-1})=(\bar{\theta}_{1}-\bar{\theta}_{3})\delta(\tau_{1}-\tau_{3}-\theta_{1}\bar{\theta}_{1}+\theta_{3}\bar{\theta}_{3}).$
(2.64) $\frac{J}{2}(\mathcal{G}^{q-1})^{T}\bar{\star}\mathcal{G}\equiv\int
d\bar{Z}_{2}~{}\mathcal{G}(Z_{2},Z_{1})~{}(\frac{1}{2}J\mathcal{G}(Z_{2};Z_{3})^{q-1})=(\theta_{1}-\theta_{3})\delta(\tau_{1}-\tau_{3}+\theta_{1}\bar{\theta}_{1}-\theta_{3}\bar{\theta}_{3}).$
(2.65)

where $\mathcal{G}=\langle\bar{\Psi}_{i}(Z_{1})\Psi^{i}(Z_{2})\rangle$ is the
anti-chiral-chiral propagator, and
$D_{\theta_{1}}\mathcal{G}=\bar{D}_{\theta_{2}}\mathcal{G}=0.$ These equations
are manifestively invariant under the $SU(1,1|1)$ transformation (2.37).
However, the infrared solutions with $\mathcal{E}\neq 0$ are not invariant
under such transformations. Thus supersymmetry is only spontaneously broken in
the infrared. In such a case, the fermionic transformations in $SU(1,1|1)$
transform the solution with zero $G_{\psi b}$ to solutions with non zero
values of $G_{\psi b}.$

We vary Eq.(2.64) on both sides and take the convolution against $\mathcal{G}$
from the right. Noting $(A\star B)\bar{\star}C=-A\star(B\bar{\star}C),$ we
obtain

$\begin{split}\delta\mathcal{G}(Z_{1},Z_{2})&=\frac{J(q-1)}{2}\left(\mathcal{G}\star\left(\left(\mathcal{G}^{q-2}\right)^{T}\delta\mathcal{G}^{T}\right)\right)\bar{\star}\mathcal{G}\\\
&=\frac{J(q-1)}{2}\int
d\bar{Z}_{3}dZ_{4}\mathcal{G}(Z_{1},Z_{4})\mathcal{G}(Z_{3},Z_{4})^{q-2}\delta\mathcal{G}(Z_{3},Z_{4})\mathcal{G}(Z_{3},Z_{2})\\\
&=\int
d\bar{Z}_{3}dZ_{4}K^{\mathcal{N}=2}(Z_{1},Z_{2};Z_{3},Z_{4})\delta\mathcal{G}(Z_{3},Z_{4})\end{split}$
(2.66)

In other words, $\delta\mathcal{G}$ must be an eigenfunction of the kernel

$K^{\mathcal{N}=2}(Z_{1},Z_{2};Z_{3},Z_{4})=\frac{J(q-1)}{2}\mathcal{G}(Z_{1},Z_{4})\mathcal{G}(Z_{3},Z_{2})\mathcal{G}(Z_{3},Z_{4})^{q-2},$
(2.67)

with eigenvalue 1. The eigenvalue equation determines the spectrum. Note the
kernel can always be written in $\mathcal{N}=2$ superspace. If we assume full
$SU(1,1|1)$ invariance on the super-correlator $\mathcal{G}$, the kernel can
also be diagonalized directly in the superspace [14]. In that case, the
spectrum is guaranteed to organize into $SU(1,1|1)$ supermultiplets with
relative scaling dimensions fixed by symmetry. However, our conformal solution
spontaneously breaks SUSY. Within a supermultiplet, the SUSY constraints on
scaling dimensions no longer hold. Thus it is necessary to work in components
when we diagonalize the super-kernel.

To write in components, we assume translational invariance and the
supercorrelator can be expanded as:

$\displaystyle\mathcal{G}(Z_{1},Z_{2})$ $\displaystyle=$ $\displaystyle
G_{\psi\psi}(\tau_{1}-\tau_{2}-\theta_{1}\bar{\theta}_{1}-\theta_{2}\bar{\theta}_{2})+\sqrt{2}\bar{\theta}_{1}G_{b\psi}(\tau_{1}-\tau_{2}-\theta_{2}\bar{\theta}_{2})$
(2.68) $\displaystyle-\sqrt{2}\theta_{2}G_{\psi
b}(\tau_{1}-\tau_{2}-\theta_{1}\bar{\theta}_{1})+2\bar{\theta}_{1}\theta_{2}G_{bb}(\tau_{1}-\tau_{2}).$

Now we evaluate the spectrum of the kernel Eq.(2.67) over the conformal
saddle. In such case the vacuum carries a definite fermionic number, and

$\mathcal{G}(Z_{1},Z_{2})=G_{\psi\psi}(\tau_{1}-\tau_{2}-\theta_{1}\bar{\theta}_{1}-\theta_{2}\bar{\theta}_{2})+2\bar{\theta}_{1}\theta_{2}G_{bb}(\tau_{1}-\tau_{2}),$
(2.69)

but the fluctuation does not:

$\displaystyle\delta\mathcal{G}(Z_{1},Z_{2})$ $\displaystyle=$
$\displaystyle\delta
G_{\psi\psi}(\tau_{1}-\tau_{2}-\theta_{1}\bar{\theta}_{1}-\theta_{2}\bar{\theta}_{2})+\sqrt{2}\bar{\theta}_{1}\delta
G_{b\psi}(\tau_{1}-\tau_{2}-\theta_{2}\bar{\theta}_{2})$ (2.70)
$\displaystyle-\sqrt{2}\theta_{2}\delta G_{\psi
b}(\tau_{1}-\tau_{2}-\theta_{1}\bar{\theta}_{1})+2\bar{\theta}_{1}\theta_{2}\delta
G_{bb}(\tau_{1}-\tau_{2}).$

At this point we specialize to the case of $q=3$ for simplicity. Evaluating
the super kernel integral in components, and matching the corresponding
components in $\delta\bar{\mathcal{G}}$ we obtain

$\delta G_{\psi\psi}(\tau_{12})=-2J\int
d\tau_{3}d\tau_{4}\left(G_{\psi\psi}(\tau_{14})G_{\psi\psi}(\tau_{32})G_{\psi\psi}(\tau_{34})\delta
G_{bb}(\tau_{34})+G_{\psi\psi}(\tau_{14})G_{\psi\psi}(\tau_{32})G_{bb}(\tau_{34})\delta
G_{\psi\psi}(\tau_{34})\right)$ (2.71) $\delta G_{\psi b}(\tau_{12})=2J\int
d\tau_{3}d\tau_{4}G_{bb}(\tau_{32})G_{\psi\psi}(\tau_{14})G_{\psi\psi}(\tau_{34})\delta
G_{\psi b}(\tau_{34})$ (2.72) $\delta G_{bb}(\tau_{12})=2J\int
d\tau_{3}d\tau_{4}G_{bb}(\tau_{14})G_{bb}(\tau_{32})G_{\psi\psi}(\tau_{34})\delta
G_{\psi\psi}(\tau_{34}).$ (2.73)

We can organize the fluctuations in the bosonic and fermionic sectors as

$\displaystyle K_{b}$
$\displaystyle=2J\begin{pmatrix}-G_{\psi\psi}(\tau_{14})G_{\psi\psi}(\tau_{32})G_{bb}(\tau_{34})&-G_{\psi\psi}(\tau_{14})G_{\psi\psi}(\tau_{32})G_{\psi\psi}(\tau_{34})\\\
G_{bb}(\tau_{14})G_{bb}(\tau_{32})G_{\psi\psi}(\tau_{34})&0\end{pmatrix}\,$
(2.74) $\displaystyle K_{b}$ $\displaystyle*\begin{pmatrix}\delta
G_{\psi\psi}\\\ -\delta G_{bb}\end{pmatrix}=\begin{pmatrix}\delta
G_{\psi\psi}\\\ \delta G_{bb}\end{pmatrix}\,,$ (2.75) $\displaystyle K_{f}$
$\displaystyle=2JG_{bb}(\tau_{32})G_{\psi\psi}(\tau_{14})G_{\psi\psi}(\tau_{34})\,,$
(2.76) $\displaystyle\bar{K}_{f}$
$\displaystyle=2JG_{\psi\psi}(\tau_{32})G_{bb}(\tau_{14})G_{\psi\psi}(\tau_{34})\,.$
(2.77)

We note that the spectrum only depends on the combination
$g_{\psi\psi}^{2}g_{bb},$ although individual matrix element can depend on the
precise value of $g_{\psi\psi}$ and $g_{bb}.$

To proceed we note that with non zero spectral asymmetry, the three point
function of (2.63) can be an arbitrary linear combination of the symmetric and
antisymmetric three point functions:

$\delta
G_{\psi\psi}=\frac{A}{|\tau|^{2\Delta-h}}+\frac{B~{}\text{sgn}(\tau)}{|\tau|^{2\Delta-h}}\,,\quad\delta
G_{bb}=\frac{a}{|\tau|^{2\Delta_{b}-h}}+\frac{b~{}\text{sgn}(\tau)}{|\tau|^{2\Delta_{b}-h}}.$
(2.78)

Due to such a mixing, each matrix element in the bosonic and fermionic kernel
should be thought as a 2 by 2 matrix. It is enough to work out the kernel with
general spectral asymmetry, defined by

$\begin{split}&K(\\{\tau,\mathcal{E},\Delta\\})=\\\
&\frac{\left(e^{\pi\mathcal{E}_{1}}\Theta(\tau_{14})-e^{-\pi\mathcal{E}_{1}}\Theta(\tau_{41})\right)}{|\tau_{14}|^{2\Delta_{1}}}\frac{\left(e^{\pi\mathcal{E}_{2}}\Theta(\tau_{32})-e^{-\pi\mathcal{E}_{2}}\Theta(\tau_{23})\right)}{|\tau_{23}|^{2\Delta_{2}}}\frac{\left(e^{\pi\mathcal{E}_{3}}\Theta(\tau_{34})-e^{-\pi\mathcal{E}_{3}}\Theta(\tau_{43})\right)}{|\tau_{34}|^{2\Delta_{3}}}\end{split}$
(2.79)

Explicitly, by evaluating

$\int
d\tau_{3}d\tau_{4}K\left(\\{\tau,\mathcal{E},\Delta\\}\right)\begin{pmatrix}\frac{\text{sgn}(\tau_{34})}{|\tau_{34}|^{2\Delta-h}}\\\
\frac{1}{|\tau_{34}|^{2\Delta-h}}\end{pmatrix}=K(\\{\mathcal{E},\Delta\\})\begin{pmatrix}\frac{\text{sgn}(\tau_{34})}{|\tau_{34}|^{2\Delta-h}}\\\
\frac{1}{|\tau_{34}|^{2\Delta-h}}\end{pmatrix}$ (2.80)

we can find its matrix representation in the basis
$\left(\frac{\text{sgn}(\tau)}{\tau^{2\Delta-h}},\frac{1}{\tau^{2\Delta-h}}\right).$

$K(\\{\mathcal{E},\Delta\\})=\frac{1}{4\pi}\begin{pmatrix}c_{f}(\frac{3}{2}-\sum\Delta_{i},0)(\bar{Q}-Q)&c_{f}(\frac{3}{2}-\sum\Delta_{i},0)(P+\bar{P})\\\
c_{b}(\frac{3}{2}-\sum\Delta_{i},0)(Q+\bar{Q})&-c_{b}(\frac{3}{2}-\sum\Delta_{i},0)(P-\bar{P})\end{pmatrix}$
(2.81)

where

$c_{f}(\Delta,\mathcal{E})=2i\cos(\pi(\Delta+i\mathcal{E}))\Gamma(1-2\Delta)\,,c_{b}=2\sin(\pi(\Delta+i\mathcal{E}))\Gamma(1-2\Delta),$
(2.82)

and we introduced the parameters

$P=c_{f}(\Delta_{i}=1,\mathcal{E}_{1})c_{f}(\Delta_{2},\mathcal{E}_{2})c_{f}(\Delta_{3},-\mathcal{E}_{3}),\bar{P}=c_{f}(\Delta_{1},-\mathcal{E}_{1})c_{f}(\Delta_{2},-\mathcal{E}_{2})c_{f}(\Delta_{3},\mathcal{E}_{3}),$
(2.83)
$Q=c_{f}(\Delta_{1},\mathcal{E}_{1})c_{f}(\Delta_{2},-\mathcal{E}_{2})c_{b}(\Delta_{3},\mathcal{E}_{3}),\bar{Q}=c_{f}(\Delta_{1},-\mathcal{E}_{1})c_{f}(\Delta_{2},-\mathcal{E}_{2})c_{b}(\Delta_{3},\mathcal{E}_{3}).$
(2.84)

Figure 9:  The bilinear spectrum for $\mathcal{E}=0$ and $0.15$ where each
intersection with horizontal axis signifies an operator with dimension
corresponding to the location of the intersection. The blue curve is bosonic
and the black curve is fermionic. Note $K_{f}$ and $\bar{K}_{f}$ have
identical spectra, thus all fermionic curves have multiplicity 2. In addition,
the presence of pairs of lines comes from the spurious doubling of the
spectrum due to unphysical local symmetries. Accounting for the unphysical
modes, the $\mathcal{E}=0$ spectrum possesses the $\mathcal{N}=2$ Schwarzian
multiplet with 2 $h=3/2$ modes. Turning on the chemical potential leads to an
IR theory with spontaneously broken supersymmetry. While the spectrum still
organizes into multiplets, the $3/2$ modes are no longer protected.

We can look at the spectrum of the theory by evaluating (2.74) and (2.76) in
terms of (2.81). The bosonic and fermionic spectrum are presented in Figure 9,
where a line crossing the axis indicates the possible presence of a operator
with that dimension. The fermionic operators always carry a double degeneracy
as the fermionic kernels $K_{f}$ and $\bar{K}_{f}$ have identical spectra. As
explained in [6], a general feature of supersymmetric SYK models is the
presence of spurious modes due to new local scaling or reparametrization
symmetries which act independently on each argument of the bilocals but are
incompatible with the UV boundary conditions. This includes for instance a
local version of the scaling discussed below (2.27).

At $\mathcal{E}=0,$ we observe the supermultiplet
$\left(\mathbf{1},2\times\mathbf{\frac{3}{2}},\mathbf{2}\right)$ associated
with the $\mathcal{N}=2$ super Schwarzian. This contains a $h=1$ mode
corresponding to the R-symmetry, a pair of $h=3/2$ modes indicating the
presence of a complex supercharge, and a $h=2$ mode corresponding to the
Hamiltonian, or the reparametrization mode. We also see another copy of this
spectrum corresponding to the spurious non-diagonal super-reparametrizations
which act differently on the two arguments of the two-point function. As
explained in [6], the spurious reparametrization mode is not expected to
produce a soft mode in the IR, since this transformation affects the UV
behavior of the correlators.

At $0<\mathcal{E}<\mathcal{E}_{\rm critical}$, we still observe two sets of
reparametrizations modes together with the two $h=1$ partners and two
fermionic partners. Even though the field contents follow the structure of
$\mathcal{N}=2$ multiplet, the scaling dimensions do not obey the
superconformal Ward identities. We find the smallest fermionic modes have
$h\neq 3/2$ for $\mathcal{E}\neq 0$ due to the breaking of supersymmetry. For
example at $\mathcal{E}=0.15,$ we find the four $h=\frac{3}{2}$ modes split
and have scaling dimensions $h_{\pm}\approx\frac{1}{2}\pm 0.09178.$ Moreover,
the first supermultiplet after the Schwarizian multiplet has dimensions
$\approx(\mathbf{2.78478},2\times\mathbf{3.32804},\mathbf{3.77582})$ no longer
obeying the superconformal constraints. While we do not pursue it further in
this work, it would be interesting to understand the dynamics of supersymmetry
breaking in theories with an emergent super-Schwarzian mode, perhaps along the
lines of [56].

Figure 10: We plot $\text{Det}(K_{b}-I_{4\times 4})$ with $h=\frac{1}{2}+is,$
where a complex mode is indicated by location of zeros. We check that there is
no complex modes until $\mathcal{E}=\mathcal{E}_{c},$ at which point there is
also a continuous spectrum of operators in the form of $\frac{1}{2}+is.$

In figure 10 we check for the possible complex modes, whose scaling dimension
takes the form $h=\frac{1}{2}+is$ determined by $SL(2,\mathbb{R}).$ In all
admissible range of $\mathcal{E}$ we find no complex modes. In addition, we
find no bosonic modes between $1<h<\frac{3}{2},$ which if present could
dominate over the Schwarzian in the infrared [57, 58]. Therefore the bilinear
spectrum is free of any known problems.

Now we turn to the behavior close to the critical asymmetry $\mathcal{E}_{\rm
critical}$. At the critical charge, although the conformal two point
coefficients (2.26) become zero, the kernel remains finite, and in fact equals
the identity matrix. This gives rises to unusual feature that instead of
having a discrete set of modes, a continuum of modes emerges for any real
value of $h$. In figure 11 we show the situation close but below of the
transition, showing how the eigenvalue curve flattens out. Noticeably, as
$\mathcal{E}\to\mathcal{E}_{\rm crticial},$ a continuum of complex modes also
emerges as shown in figure 10.

Figure 11: Some bosonic correlators near transition, with
$\mathcal{E}=0.2780,0.2801$. Fermions do not flatten out, but the bosonic
eigenvalues start to flatten. At $\mathcal{E}=\mathcal{E}_{\rm critical}$,
there is a continuous bosonic spectrum.

###  2.4 Comments on the holographic interpretation

In this section we make a proposal for a holographic interpretation of the
features we found in $\mathcal{N}=2$ SYK at finite charge; we also derive a
formula for the charge dependence of the zero-temperature entropy. While we do
not have an exact bulk dual of the model, our interpretation has some
qualitative and some quantitative features which can be compared.

We begin with the most conservative point of view. Since there is a slightly
broken emergent $SU(1,1|1)$ symmetry in $\mathcal{N}=2$ SYK, and given that
the dynamics is described by the $\mathcal{N}=2$ Schwarzian theory, the dual
two dimensional black hole is described by $\mathcal{N}=2$ JT gravity (see for
example [59]). This theory of gravity includes a $U(1)$ gauge field dual to
the SYK electric charge. The two dimensional black hole background also
includes $N$ complex fermions $\psi_{\rm bulk}$ which are dual to the $N$ SYK
fermions $\psi^{i}$, and their bosonic partners. Following the notation in
section 5 of [5] we take the fermions to have mass $M$ in the bulk, charge
one, and they move on an electric field $\mathcal{E}$ equal to the spectral
asymmetry introduced above. When this fermion is quantized with Neumann
boundary conditions, the boundary two point function has the conformal form
with scaling dimension $\Delta=\frac{1}{2}-\sqrt{M^{2}-\mathcal{E}^{2}}$. We
cannot use Dirichlet boundary conditions since those would have $\Delta>1/2$
which is not observed in SYK.

Assume first that the masses of the fermions do not depend on the electric
field. Then the two dimensional black hole is stable as long as the electric
field is not too large $\mathcal{E}<\mathcal{E}_{\rm critical}=M$. Beyond the
critical electric field the fermion develops a complex scaling dimension. When
this happens there is a non-zero amplitude for Schwinger pair production that
can screen the electric field and induce an instability of the two dimensional
black hole. This is qualitatively similar to what we find in the analysis of
the $\mathcal{N}=2$ SYK fermion as a function of the spectral asymmetry, with
the only difference that $M$ has a dependence on $\mathcal{E}$111111Black
holes in four dimension can also have electric field dependent dimensions when
placed in AdS, which plays an important role in the recent work [60].. The
numerical analysis done for $\mathcal{N}=2$ SYK indicates that the new phase
after the instability is not a black hole, since it is a low entropy phase not
possessing a nearly conformal symmetry.

So far the fermion coupled to JT gravity is intended to be thought of as a
sector of the dual black hole to SYK. There is a different type of duality
proposed in [5], which we will refer to as the ‘spooky fermion’, between some
exact results in SYK and a set of fermions in the two dimensional hyperbolic
disk. This proposal implies for example that the zero-temperature entropy of
the complex SYK model is given by

$G_{F}(\mathcal{E})=N[\log Z^{F}_{D}-\log Z^{F}_{N}].$ (2.85)

The right hand side involves the one-loop partition function of $N$ complex
fermions with Dirichlet conditions in the boundary of the hyperbolic disk,
denoted by $Z^{F}_{D}$ and $N$ ghosts fermions with Neumann boundary
conditions, denoted by $Z^{F}_{N}$ and contributes with an extra minus sign.
$G(\Delta,\mathcal{E})$ is the grand canonical partition function. It was
shown in [5] that

$\frac{\partial G_{F}}{\partial\Delta}=-N\frac{\pi(1-2\Delta)\sin
2\pi\Delta}{\cosh 2\pi\mathcal{E}+\cos 2\pi\Delta}.$ (2.86)

Then one can compute the grand canonical partition function by integrating
$\Delta$ from $1/2$ to $\Delta=1/q$ which is the answer for complex SYK,
independent of the spectral asymmetry. This identification fails for other
topologies [61].

For $\mathcal{N}=2$ SYK we propose a similar duality at the level of the disk
topology. The first difference is the presence of a bulk boson with Dirichlet
boundary conditions and a bulk ghost boson with Neumann boundary conditions.
Its contribution is $G_{B}(\Delta_{b},\mathcal{E}_{b})=N[\log Z^{B}_{D}-\log
Z^{B}_{N}]$. The total grand potential is then

$G(\Delta,\mathcal{E})=G_{F}(\Delta,\mathcal{E})+G_{B}(\Delta_{b},\mathcal{E}_{b}).$
(2.87)

This boson is required by supersymmetry. For example at $\mathcal{E}=0$ it
should satisfy $\Delta_{b}=\Delta+1/2$. Moreover, the charge of the boson has
to be $(q-1)$ times the charge of the fermion by susy implying
$\mathcal{E}_{b}=(q-1)\mathcal{E}$. For $\mathcal{E}\neq 0$ susy is broken by
based on the dynamics of SYK we found here we impose that
$\Delta_{b}=1-(q-1)\Delta$. It is possible to show that

$\frac{\partial G_{B}}{\partial\Delta_{b}}=-N\frac{\pi(1-2\Delta_{b})\sin
2\pi\Delta_{b}}{\cosh 2\pi\mathcal{E}_{b}-\cos 2\pi\Delta_{b}}.$ (2.88)

As opposed to complex SYK, for the $\mathcal{N}=2$ theory the fermion
dimension is not fixed. The spooky propagator gives a way of understanding the
formula (2.27). The grand potential depends on $\Delta$ and $\mathcal{E}$. It
is reasonable to expect that the value of $\Delta$ should be such that the
grand potential is extremized with respect to changes in the scaling
dimension. Using that $\partial_{\Delta}\Delta_{b}=-(q-1)$ gives

$\partial_{\Delta}G(\Delta,\mathcal{E})=0,~{}\Rightarrow\frac{(1-2\Delta)\sin
2\pi\Delta}{\cosh 2\pi\mathcal{E}+\cos
2\pi\Delta}=(q-1)\frac{(1-2\Delta_{b})\sin 2\pi\Delta_{b}}{\cosh
2\pi\mathcal{E}_{b}-\cos 2\pi\Delta_{b}}$ (2.89)

This is exactly the same equation as (2.27).

We can use this picture to also derive the Luttinger-Ward relation. In order
to do this start from the expression for the grand potential and apply the
thermodynamic relation $Q=\frac{1}{2\pi}\partial_{\mathcal{E}}G$ [49]. This
gets contributions both from the spooky fermion and boson propagators. It is
easy to check that this relation implies the Luttinger-Ward formula (2.50)
which we verified by a numerical solution of the Schwinger-Dyson equations.

Figure 12: Plot showing the zero temperature entropy for $q=3$ $\mathcal{N}=2$
SYK, computed using the spooky fermion and boson propagators in equation
(2.90) (solid black line) as a function of charge from $Q=0$ up to the
critical value $Q=0.4142N$. We also show the result from a numerical solution
of the Schwinger-Dyson equations (blue dots) showing perfect agreement, at
least far enough from the critical charge where the numerical procedure
becomes more involved.

Finally, we can verify the grand potential itself matches with the one
obtained from the Schwinger-Dyson equation. The explicit expression is given
by

$G(\mathcal{E})/N=\int_{\Delta}^{1/2}dx~{}\frac{\pi(1-2x)\sin 2\pi x}{\cosh
2\pi\mathcal{E}+\cos 2\pi
x}+\int_{1-(q-1)\Delta}^{1/2}dx~{}\frac{\pi(1-2x)\sin 2\pi x}{\cosh
2\pi(q-1)\mathcal{E}-\cos 2\pi x},$ (2.90)

where $\Delta$ is the fermion scaling dimension derived from (2.89). Using
this expression we can also compute the entropy $G=S_{0}(Q)-2\pi\mathcal{E}Q$.
In figure 12 we verify this expression coincides with the result from
numerical solution of the mean field equations. We would like to stress that
$G$ cannot be simply obtained from integrating (2.43) over $q$ since it is not
clear what boundary conditions to use for that integral, so this provides a
non-trivial check. We can verify analytically this works at $Q=0$, giving

$\displaystyle G(\mathcal{E}=0)/N$ $\displaystyle=$
$\displaystyle\int_{\frac{1}{2q}}^{1/2}dx~{}\pi(1-2x)\tan\pi
x+\int_{\frac{1}{2q}+\frac{1}{2}}^{1/2}dx~{}\pi(1-2x)\cot\pi x,$ (2.91)
$\displaystyle=$ $\displaystyle\log\left(2\cos\frac{\pi}{2q}\right),$ (2.92)

which coincides with the expectation from the index. This simple final answer
come from non-trivial cancellations between the bosons and fermions
contribution. It is possible to write down an analytic answer as a function of
$\mathcal{E}$ using results of [49] but it is not very illuminating.

##  3 $\mathcal{N}=2$ SYK with multiple fermions

In this section we will study models of $\mathcal{N}=2$ SYK with multiple
complex fermions. In contrast to the model introduced in [6], these new models
have flavor symmetries that preserve supersymmetry. At low temperature, the IR
physics is still dominated by the $\mathcal{N}=2$ Schwarzian theory with
fundamental $U(1)_{R}$ charge equal to the charge of the supercharge.

We will consider theories with two sets of complex fermions $\psi^{i}$ and
$\chi^{i}$ with $i=1,\ldots,N$. Due to $\mathcal{N}=2$ supersymmetry, the
Hamiltonian is completely specified by giving the supercharge. With more
fermions, there is some arbitrariness in what we choose to be the supercharge
which defines the model; in the simplest realization we will take it to be

$\mathcal{Q}=iC_{ijk}~{}\psi^{i}\psi^{j}\chi^{k},$ (3.1)

where we take $C_{ijk}$ to be totally antisymmetric (this can be generalized
since we only need antisymmetry in the first two indices). Then
$\mathcal{Q}^{2}=\bar{\mathcal{Q}}^{2}=0$ and
$H=\\{\mathcal{Q},\bar{\mathcal{Q}}\\}$. We take the complex coupling matrix
to be a gaussian random distributed with variance $\langle C_{i_{1}i_{2}\ldots
i_{q}}\bar{C}^{i_{1}i_{2}\ldots i_{q}}\rangle\sim J/N^{q-1}$. We will show
these models also have an emergent $SU(1,1|1)$ symmetry at low temperatures,
with additional features due to the existence of a new global symmetry.

The supersymmetric Lagrangian following from (3.1) has the global symmetries
$U(1)_{\psi}\times U(1)_{\chi}$ which act independently on the two types of
fermions $\psi$ and $\chi$. Their generators are
$Q_{\psi}=\sum_{i}\bar{\psi}_{i}\psi^{i}-N/2$ and
$Q_{\chi}=\sum_{i}\bar{\chi}_{i}\chi^{i}-N/2$. Both of these generators
commute with the Hamiltonian but not with the supercharge. There is a linear
combination that defines a supersymmetric flavor symmetry:

$Q_{F}=Q_{\psi}-2Q_{\chi},~{}~{}~{}~{}~{}~{}[Q_{F},\mathcal{Q}]=0.$ (3.2)

The existence of this flavor symmetry implies that deciding exactly what is
the superconformal R-symmetry $U(1)_{R}$ inside the infrared superconformal
group is a non-trivial problem. The simplest possibility is to pick $Q_{\chi}$
itself, but in general it could be any linear combination

$Q_{R}=Q_{\chi}+\alpha~{}Q_{F},~{}~{}~{}~{}~{}~{}[Q_{R},\mathcal{Q}]=\mathcal{Q}$
(3.3)

where $\alpha$ is an arbitrary real parameter. We will determine $\alpha$ from
the solution of the model at low temperatures.

It will be useful for some calculations below to generalize the model to
involve an arbitrary number of $\psi$ fermions in the supercharge

$\mathcal{Q}=i^{\frac{q-1}{2}}C_{i_{1}\ldots
i_{q}}~{}\psi^{i_{1}}\ldots\psi^{i_{q-1}}\chi^{i_{q}}.$ (3.4)

This model also has two conserved currents $Q_{\psi}$ and $Q_{\chi}$, and a
generalization of the flavor charge (3.2) is

$Q_{F}=Q_{\psi}-(q-1)Q_{\chi}\,.$ (3.5)

Again, we can propose a trial low temperature R-symmetry
$Q_{R}=Q_{\chi}+\alpha Q_{F}$, for a parameter $\alpha$ we will determine
later.

Finally, all these models again have a global discrete $\mathbb{Z}_{q}$
symmetry. This is generated by $\psi\to e^{\frac{2\pi ir}{q}}\psi$ and
$\chi\to e^{\frac{2\pi ir}{q}}\chi$ and it clearly commutes with the
supercharge.

#### Calculation of the Index

We will now compute the refined Witten index for this theory. The calculation
can be done in the free fermion limit, but it will be useful to compare to the
result we will find using the mean field description below. We should first
define a notion of fermion number that produces cancellations between states
when they do not preserve supersymmetry. We take the following definition:

$(-1)^{F}\equiv e^{i\pi Q_{\chi}}$ (3.6)

Since this only counts the number of $\chi$-oscillators, it is a-priori not
the ‘true’ fermion number. Nevertheless, since the supercharge involves an
even number of $\psi$ fields, all states in a supermultiplet have the same
$\psi$-fermion number modulo 2. Furthermore, the supercharge has fermion
number one with this definition and the contribution from states in the same
supermultiplet will vanish. More formally, we can say that the Hilbert space
breaks up into a $\mathbb{Z}$ graded complex
$\dots\overset{\mathcal{Q}}{\rightarrow}\mathcal{H}^{q_{\chi}-1}\overset{\mathcal{Q}}{\rightarrow}\mathcal{H}^{q_{\chi}}\overset{\mathcal{Q}}{\rightarrow}\mathcal{H}^{q_{\chi}+1}\overset{\mathcal{Q}}{\rightarrow}\dots$.
This reduces mod 2 to $\mathbb{Z}_{2}$, the fermion number grading. There is a
separate $\mathcal{Q}$-cohomology for each $q_{\chi}\in\mathbb{Z}$ charge.

The presence of the flavor $U(1)_{F}$ allows us to define the Witten index
refined by the chemical potential $y$ conjugate to this charge. The index is
then given by

$\displaystyle\mathcal{I}(y)$ $\displaystyle\equiv$ $\displaystyle{\rm
Tr}\Big{[}(-1)^{F}e^{-\beta H}e^{iyQ_{F}}\Big{]},$ (3.7) $\displaystyle=$
$\displaystyle\Big{(}2\cos\Big{(}\frac{y}{2}\Big{)}~{}2\sin\Big{(}\frac{q-1}{2}y\Big{)}\Big{)}^{N},$
(3.8)

where the second line was computed in the free theory. Just like the models of
[6], the index vanishes with no potential; ${\rm Tr}(-1)^{F}=0$.

Already at the level of the index, we may find interesting features of the
model defined by (3.4) by passing from the grand canonical (fixed $y$) to the
canonical (fixed $Q_{F}$) ensemble. This amounts to picking a charge sector of
the theory with $Q_{F}\to q_{F}$ labeling the sector (for example $q_{F}=0$ if
we gauge the flavor symmetry). Now the index in this sector does not vanish
and is given by

$\displaystyle\mathcal{I}(q_{F})$ $\displaystyle=$ $\displaystyle{\rm
Tr}_{q_{F}}\Big{[}(-1)^{F}e^{-\beta H}\Big{]},$ (3.9) $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\frac{dy}{2\pi}~{}e^{-iyq_{F}}~{}e^{NI(y)},~{}~{}~{}I(y)\equiv\log{\left(2\cos\Big{(}\frac{y}{2}\Big{)}~{}2\sin\Big{(}\frac{q-1}{2}y\Big{)}\right)}$
(3.10)

Now we will take the large $N$ limit while keeping $q_{F}$ of order one.
Therefore we can ignore the dependence on the flavor charge and focus on the
$I(y)$ term to find the saddle point of the integral. In this limit the index
can be approximated by

$\mathcal{I}\sim e^{NI(y_{c})}~{}~{}~{}~{}{\rm
where}~{}~{}\partial_{y}I(y_{c})=0.$ (3.11)

The saddle point equation $\partial_{y}I(y_{c})=0$ cannot be explicitly
solved, but can be rewritten in a form that will be useful for a later
comparison

$\tan\left(\frac{y_{c}}{2}\right)=(q-1)\cot\left((q-1)\frac{y_{c}}{2}\right),~{}~{}~{}~{}\Rightarrow~{}~{}~{}\frac{d}{dq}\frac{\log\mathcal{I}}{N}=\frac{y_{c}}{2}\cot\left((q-1)\frac{y_{c}}{2}\right).$
(3.12)

To simplify the expression for the index we took a derivative of $I(y)$ with
respect to $q$, which due to the saddle point equation only acts on explicit
$q$ dependence. We will see below that this quantity can be identified with
the zero temperature entropy $S_{0}$ computed from the mean field action at
the particle-hole symmetry point.

We can also turn both a chemical potential for $Q_{F}$ and for the
$\mathbb{Z}_{q}$ symmetry that commutes with the supercharge. The answer is
given by

$\displaystyle\mathcal{I}(y,r)$ $\displaystyle=$ $\displaystyle{\rm
Tr}\Big{[}(-1)^{F}e^{-\beta
H}e^{iyQ_{F}}e^{i2\pi\frac{r}{q}(Q_{\psi}+Q_{\chi})}\Big{]},$ (3.13)
$\displaystyle=$ $\displaystyle\Big{(}2\cos\Big{(}\frac{y}{2}+\frac{\pi
r}{q}\Big{)}~{}2\sin\Big{(}\frac{q-1}{2}y-\frac{\pi r}{q}\Big{)}\Big{)}^{N}$
(3.14)

The reason that this refinement does not add much information is that it can
be rewritten as

$\mathcal{I}(y,r)=(-1)^{rN}~{}\mathcal{I}\Big{(}y-\frac{2\pi r}{q},0\Big{)},$
(3.15)

and therefore its given up to a sign by the previous index. For example the
fixed charge refined index is $\mathcal{I}(q_{F},r)=e^{i2\pi
r(\frac{N}{2}-\frac{q_{F}}{q})}\mathcal{I}(q_{F},0)$. We can use this index
then to determine the $\mathbb{Z}_{q}$ charge of the BPS states in each fix
flavor charge sector.

###  3.1 Mean field and the conformal solution

The derivation of the mean field action is largely identical to the one in
section 2 so we will be more brief. We will again work in superspace, first
introducing introduce two chiral superfields defined analogously to (2.10):

$\Psi^{i}=\psi^{i}(\tau+\theta\bar{\theta})+\sqrt{2}\,\theta\,b_{\psi}^{i}\,,\quad
X^{i}=\chi^{i}(\tau+\theta\bar{\theta})+\sqrt{2}\,\theta\,b_{\chi}^{i}\,,$
(3.16)

where we have again introduced auxiliary bosons $b_{\psi}^{i}$,
$b_{\chi}^{i}$, and their conjugates with an explicit form obtained from the
analog of (2.4). The action after integrating in the bosons can be written in
superspace in terms of these chiral fields and the interaction term is

$\mathcal{L}\supset i^{\frac{q-1}{2}}\\!\\!\int
d\theta\left(C_{i_{1}i_{2}\dots
i_{q}}\Psi^{i_{1}}\dots\Psi^{i_{q-1}}X^{i_{q}}\right)+h.c.$ (3.17)

More explicitly, the $\theta$ component of $C_{i_{1}i_{2}\dots
i_{q}}\Psi^{i_{1}}\dots\Psi^{i_{q-1}}X^{i_{a}}$ gives the combination
$\psi^{q-1}b_{\chi}$ and $(q-1)b_{\psi}\psi^{q-2}\chi$. We introduce now the
fermions two point function

$\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})=\frac{1}{N}\langle\bar{\Psi}_{i}(Z_{1})\Psi^{i}(Z_{2})\rangle,~{}~{}~{}\mathcal{G}_{XX}(Z_{1},Z_{2})=\frac{1}{N}\langle\bar{X}_{i}(Z_{1})X^{i}(Z_{2})\rangle.$
(3.18)

The expansion of these superfields includes $G_{\psi\psi}(\tau_{1},\tau_{2})$,
$G_{b_{\psi}b_{\psi}}(\tau_{1},\tau_{2})$, etc. To derive the mean field
action we introduce a Lagrange multiplier $\Sigma_{\psi}(Z_{1},Z_{2})$ and
$\Sigma_{\chi}(Z_{1},Z_{2})$, which in components are
$\Sigma_{\psi}(Z_{1},Z_{2})=\frac{1}{2}\Sigma_{b_{\psi}b_{\psi}}(\tau_{1}-\theta_{1}\bar{\theta}_{1},\tau_{2}+\theta_{2}\bar{\theta}_{2})+\ldots$
and similarly for $\chi$. Next we integrate out both the couplings and the
fundamental superfields. The interaction term in the mean field action
involves

$S\supset N\frac{J}{2}\int
d\bar{Z}_{1}dZ_{2}~{}\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})^{q-1}\mathcal{G}_{XX}(Z_{1},Z_{2}).$
(3.19)

Since the procedure to find the action and its equations of motion (the
Schwinger-Dyson equations) is similar to what we reviewed in section 2.1, we
will move straight to the equations and solution. We consider time translation
invariant solutions with vanishing mixed correlators between bosons and
fermions.

$\displaystyle\frac{1}{2}D_{\theta_{3}}\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{3})\\!+\\!\int\\!\\!dZ_{2}~{}\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})\left[\frac{J}{2}(q-1)\mathcal{G}_{\Psi\Psi}(Z_{3},Z_{2})^{q-2}\mathcal{G}_{XX}(Z_{3},Z_{2})\right]=\delta(\bar{Z}_{1}-\bar{Z}_{3}),$
(3.20)
$\displaystyle\frac{1}{2}D_{\theta_{3}}\mathcal{G}_{XX}(Z_{1},Z_{3})\\!+\\!\int\\!\\!dZ_{2}~{}\mathcal{G}_{XX}(Z_{1},Z_{2})\left[\frac{J}{2}\mathcal{G}_{\Psi\Psi}(Z_{3},Z_{2})^{q-1}\right]=\delta(\bar{Z}_{1}-\bar{Z}_{3}),$
(3.21)

where the quantities in brackets are equal to $\Sigma_{\psi}$ and
$\Sigma_{\chi}$, the superspace self-energies.

#### Solutions of Schwinger-Dyson equations

In this section, we solve the Schwinger-Dyson equations in the IR limit using
the conformal ansatz, derived from the mean field action above. In position
space, the component equations that determine the self energies (when off-
diagonal bilinears are set to zero) are

$\displaystyle\Sigma_{\psi\psi}(\tau)$ $\displaystyle=$ $\displaystyle
J(q-1)\left(G_{\psi\psi}(\tau)^{q-2}G_{b_{\chi}b_{\chi}}(\tau)+(q-2)G_{b_{\psi}b_{\psi}}(\tau)G_{\chi\chi}(\tau)G_{\psi\psi}(\tau)^{q-3}\right),$
(3.22) $\displaystyle\Sigma_{\chi\chi}(\tau)$ $\displaystyle=$ $\displaystyle
J(q-1)G_{b_{\psi}b_{\psi}}(\tau)G_{\psi\psi}(\tau)^{q-2},$ (3.23)
$\displaystyle\Sigma_{b_{\psi}b_{\psi}}(\tau)$ $\displaystyle=$ $\displaystyle
J(q-1)G_{\psi\psi}(\tau)^{q-2}G_{\chi\chi}(\tau),$ (3.24)
$\displaystyle\Sigma_{b_{\chi}b_{\chi}}(\tau)$ $\displaystyle=$ $\displaystyle
JG_{\psi\psi}(\tau)^{q-1}.$ (3.25)

We can begin by picking a similar ansatz as in the previous section for all
correlators

$\displaystyle G_{AA}(\tau)$ $\displaystyle=$
$\displaystyle\frac{g_{AA}}{|\tau|^{2\Delta_{A}}}\left(e^{\pi\mathcal{E}_{A}}\Theta(\tau)-e^{-\pi\mathcal{E}_{A}}\Theta(-\tau)\right),$
(3.26) $\displaystyle G_{b_{A}b_{A}}(\tau)$ $\displaystyle=$
$\displaystyle\frac{g_{b_{A}b_{A}}}{|\tau|^{2\Delta_{b_{A}}}}\left(e^{\pi\mathcal{E}_{b_{A}}}\Theta(\tau)+e^{-\pi\mathcal{E}_{b_{A}}}\Theta(-\tau)\right),$
(3.27)

where $A=\psi$ or $\chi$. We now have several different scaling dimensions
$\Delta_{A}$, $\Delta_{b_{A}}$ and spectral asymmetries $\mathcal{E}_{A}$,
$\mathcal{E}_{b_{A}}$ for both sets of bosons and fermions.

The self-energies can be found by solving
$\Sigma_{AA}(\omega)G_{AA}(-\omega)=1$ and
$\Sigma_{b_{A}b_{A}}(\omega)G_{b_{A}b_{A}}(-\omega)=-1$, and the answer is the
same as before for each correlator:

$\displaystyle\Sigma_{AA}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{g_{AA}}\frac{(1-2\Delta_{A})\sin
2\pi\Delta_{A}}{4\pi\prod_{\pm}\cos\pi(\Delta_{A}\pm
i\mathcal{E}_{A})}\frac{1}{|\tau|^{2(1-\Delta_{A})}}\left(-e^{\pi\mathcal{E}_{A}}\Theta(-\tau)+e^{-\pi\mathcal{E}_{A}}\Theta(\tau)\right),$
$\displaystyle\Sigma_{b_{A}b_{A}}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{g_{b_{A}b_{A}}}\frac{(1-2\Delta_{b_{A}})\sin
2\pi\Delta_{b_{A}}}{4\pi\prod_{\pm}\sin\pi(\Delta_{b_{A}}\pm
i\mathcal{E}_{b_{A}})}\frac{1}{|\tau|^{2(1-\Delta_{b_{A}})}}\left(e^{\pi\mathcal{E}_{b_{A}}}\Theta(-\tau)+e^{-\pi\mathcal{E}_{b_{A}}}\Theta(\tau)\right).$

We now follow the same steps as in the previous section to solve the rest of
the equations. We begin by matching the left- and right-hand sides of
equations (3.22), (3.23), (3.24), and (3.25). This again can be done in steps.
We first match the spectral asymmetry of all equations. This gives the
following two independent constraints

$\mathcal{E}_{b_{\psi}}=-(q-2)\mathcal{E}_{\psi}-\mathcal{E}_{\chi},~{}~{}~{}~{}\mathcal{E}_{b_{\chi}}=-(q-1)\mathcal{E}_{\psi}.$
(3.28)

The interpretation of these constraints is clear since integrating out
auxiliary bosons gives by $b_{\psi}\sim\bar{\psi}^{q-2}\bar{\chi}$ and
$b_{\chi}\sim\bar{\psi}^{q-1}$. Therefore their spectral asymmetry should
respect these relations. Moreover, there are only two $U(1)$ charges we are
free to chose in our theory and therefore there should be only two independent
spectral asymmetries which we take to be $\mathcal{E}_{\psi}$ and
$\mathcal{E}_{\chi}$. Next, we can match the time dependence on both sides of
the four equations. This gives the following two constraints on the scaling
dimensions

$\Delta_{b_{\chi}}+(q-1)\Delta_{\psi}=1,~{}~{}~{}~{}\Delta_{b_{\psi}}+(q-2)\Delta_{\psi}+\Delta_{\chi}=1.$
(3.29)

We then determine the bosonic scaling dimensions from these equations and
consider $\Delta_{\psi}$ and $\Delta_{\chi}$ as independent variables. These
constraints on the dimensions are also reasonable since they imply the
interaction terms are marginal in the low energy effective action. We will see
below that as opposed to Yukawa-like interactions without supersymmetry
(examples of Yukawa interactions with first order bosons are [55, 62]), the
scaling dimensions will be determined from the spectral asymmetries uniquely.

Before moving on to matching the prefactors $g_{AA}$ and $g_{b_{A}b_{A}}$ of
the equations for the self-energies, we will analyze the range of allowable
scaling dimensions such that the approximations made in the IR solution are
self consistent. Dropping the kinetic terms requires $\Delta_{\rm fermion}>0$
and $\Delta_{\rm boson}>1/2$, which implies the following inequalities

$0<\Delta_{\psi}<\frac{1}{2(q-1)}\,,~{}~{}~{}0<\Delta_{\chi}<\frac{1}{2}-(q-2)\Delta_{\psi}<\frac{1}{2}$
(3.30)

The bound on $\Delta_{\chi}$ is therefore not fixed and goes from $1/2$ to
$1/(2(q-1))$ as we increase $\Delta_{\psi}$ within the allowed range.

Now we can match the prefactors. This will produce self-consistency equations
that uniquely determine the scaling dimensions. For each of the equations
(3.22), (3.23), (3.24) and (3.25), we get:

$\displaystyle\frac{(1-2\Delta_{\psi})\sin
2\pi\Delta_{\psi}}{4\pi\prod_{\pm}\cos\pi(\Delta_{\psi}\pm
i\mathcal{E}_{\psi})}$ $\displaystyle=$ $\displaystyle
2(q-1)g_{\psi\psi}^{q-1}~{}g_{b_{\chi}b_{\chi}}+2(q-1)(q-2)~{}g_{b_{\psi}b_{\psi}}~{}g_{\chi\chi}~{}g_{\psi\psi}^{q-2}$
(3.31) $\displaystyle\frac{(1-2\Delta_{b_{\chi}})\sin
2\pi\Delta_{b_{\chi}}}{4\pi\prod_{\pm}\sin\pi(\Delta_{b_{\chi}}\pm
i\mathcal{E}_{b_{\chi}})}$ $\displaystyle=$ $\displaystyle
2g_{b_{\chi}b_{\chi}}~{}g_{\psi\psi}^{q-1}$ (3.32)
$\displaystyle\frac{(1-2\Delta_{\chi})\sin
2\pi\Delta_{\chi}}{4\pi\prod_{\pm}\cos\pi(\Delta_{\chi}\pm
i\mathcal{E}_{\chi})}$ $\displaystyle=$ $\displaystyle
2(q-1)~{}g_{\chi\chi}~{}g_{b_{\psi}b_{\psi}}~{}g_{\psi\psi}^{q-2}$ (3.33)
$\displaystyle\frac{(1-2\Delta_{b_{\psi}})\sin
2\pi\Delta_{b_{\psi}}}{4\pi\prod_{\pm}\sin\pi(\Delta_{b_{\psi}}\pm
i\mathcal{E}_{b_{\psi}})}$ $\displaystyle=$ $\displaystyle
2(q-1)~{}g_{b_{\psi}b_{\psi}}~{}g_{\psi\psi}^{q-2}~{}g_{\chi\chi}$ (3.34)

These four equations are only consistent if the following two constraints are
solved

$\displaystyle\frac{(1-2\Delta_{\chi})\sin
2\pi\Delta_{\chi}}{4\pi\prod_{\pm}\cos\pi(\Delta_{\chi}\pm
i\mathcal{E}_{\chi})}$ $\displaystyle=$
$\displaystyle\frac{(1-2\Delta_{b_{\psi}})\sin
2\pi\Delta_{b_{\psi}}}{4\pi\prod_{\pm}\sin\pi(\Delta_{b_{\psi}}\pm
i\mathcal{E}_{b_{\psi}})},$ (3.35) $\displaystyle\frac{(1-2\Delta_{\psi})\sin
2\pi\Delta_{\psi}}{4\pi\prod_{\pm}\cos\pi(\Delta_{\psi}\pm
i\mathcal{E}_{\psi})}$ $\displaystyle=$
$\displaystyle\frac{(1-2\Delta_{b_{\chi}})\sin
2\pi\Delta_{b_{\chi}}}{4\pi\prod_{\pm}\sin\pi(\Delta_{b_{\chi}}\pm
i\mathcal{E}_{b_{\chi}})}+\frac{(q-2)(1-2\Delta_{\chi})\sin
2\pi\Delta_{\chi}}{4\pi\prod_{\pm}\cos\pi(\Delta_{\chi}\pm
i\mathcal{E}_{\chi})}$ (3.36)

After replacing the values of scaling dimensions and spectral asymmetries for
the bosons using (3.29) and (3.28), these two equations are sufficient to
determine $\Delta_{\psi}(\mathcal{E}_{\psi},\mathcal{E}_{\chi})$ and
$\Delta_{\chi}(\mathcal{E}_{\psi},\mathcal{E}_{\chi})$. We will analyze the
behavior of the scaling dimensions next. First, let us point out again that
using the IR solution we can only determine the following combinations of
prefactors $g_{\psi\psi}^{q-2}g_{\chi\chi}g_{b_{\psi}b_{\psi}}$ and
$g_{\psi\psi}^{q-1}g_{b_{\chi}b_{\chi}}$. This is due to a set of two emergent
scaling symmetries, which we refer to as $\lambda_{1}$ and $\lambda_{2}$, in
the IR given by

$\displaystyle\lambda_{1}:~{}~{}$ $\displaystyle
G_{\psi\psi}\to\lambda_{1}G_{\psi\psi},~{}~{}G_{b_{\chi}b_{\chi}}\to\lambda_{1}^{1-q}G_{b_{\chi}b_{\chi}},~{}~{}G_{\chi\chi}\to\lambda_{1}^{2-q}G_{\chi\chi},~{}~{}G_{b_{\psi}b_{\psi}}\to
G_{b_{\psi}b_{\psi}},$ (3.37) $\displaystyle\lambda_{2}:~{}~{}$ $\displaystyle
G_{\psi\psi}\to G_{\psi\psi},~{}~{}G_{b_{\chi}b_{\chi}}\to
G_{b_{\chi}b_{\chi}},~{}~{}G_{\chi\chi}\to\lambda_{2}G_{\chi\chi},~{}~{}G_{b_{\psi}b_{\psi}}\to\lambda_{2}^{-1}G_{b_{\psi}b_{\psi}}.$
(3.38)

Is it easy to see that these transformations can be obtained from acting with
$U(1)_{\psi}$ and $U(1)_{\chi}$ independently on the two insertions appearing
in the two-point function. This is necessary since otherwise a time
independent transformation acting diagonally would leave the correlators
invariant. These symmetries are broken in the UV and therefore the prefactors
should be determined if we had access to the full solution. These
transformations act on the UV correlators and we do not expect a low energy
mode coming from them for the same reasons as in [6].

#### Supersymmetric solution

Since we find a non-vanishing index we know the ground states preserve
supersymmetry. This implies for the conformal solution that supersymmetric
solutions satisfy $G_{b_{A}b_{A}}\sim\partial_{\tau}G_{AA}$ for $A=\psi,\chi$,
analogous to (2.30). Then the bosonic scaling dimensions are given in terms of
the fermionic ones

$\Delta_{b_{\psi}}=\Delta_{\psi}+\frac{1}{2},~{}~{}~{}\Delta_{b_{\chi}}=\Delta_{\chi}+\frac{1}{2}.$
(3.39)

As opposed to the model studied in the previous section, this does not
determine the scaling dimensions anymore from a purely dimensional analysis
argument. Instead we are left with a single constraint on the fermion
dimensions

${\rm Susy}:~{}~{}~{}~{}~{}\Delta_{\chi}+(q-1)\Delta_{\psi}=\frac{1}{2}.$
(3.40)

To determine a unique solution we need to look at the spectral asymmetry. The
supersymmetry relating the bosonic and fermionic correlators also imposes a
matching between the spectral asymmetries
$\mathcal{E}_{b_{\psi}}=\mathcal{E}_{\psi}$ and
$\mathcal{E}_{b_{\chi}}=\mathcal{E}_{\chi}$. Combinging this with the
relations (3.28), one finds

${\rm Susy}:~{}~{}~{}~{}~{}\mathcal{E}_{\chi}+(q-1)\mathcal{E}_{\psi}=0.$
(3.41)

This is not the most general solution and there are discrete configurations
with complex spectral asymmetry corresponding to turning on the
$\mathbb{Z}_{q}$ charge, analogous to (2.28). We will not discuss those
solutions here.

Using the relations above, so far supersymmetric solutions can be parametrized
by $\Delta_{\psi}$ and $\mathcal{E}_{\psi}$. Given $\mathcal{E}_{\psi}$, we
now impose the constraints coming from the prefactors in the Schwinger-Dyson
equations. They give the following equation

$\frac{\sin 2\pi\Delta_{\psi}}{\cos 2\pi\Delta_{\psi}+\cosh
2\pi\mathcal{E}_{\psi}}=\frac{(q-1)\sin 2\pi(q-1)\Delta_{\psi}}{\cosh
2\pi(q-1)\mathcal{E}_{\psi}-\cos 2\pi(q-1)\Delta_{\psi}}$ (3.42)

This should be seen as an implicit equation determining
$\Delta_{\psi}(\mathcal{E}_{\psi})$, and all other quantities can be
determined by the relations above. This is consistent with the discussion of
the index, only one continuous chemical potential can be turned on while still
preserving supersymmetry, the one conjugate to the UV flavor charge (3.2).
Explicit solutions are possible for small $q$. For $q=3$ it is given by

$\Delta_{\psi}=\frac{1}{2\pi}\arcsin\Big{(}\frac{1}{3}\sqrt{9-\cosh^{2}2\pi\mathcal{E}_{\psi}}\Big{)}.$
(3.43)

We will comment later on the fact that there is a critical spectral asymmetry
at which the scaling dimension vanishes.

#### Emergent $SU(1,1|1)$ symmetry

We will briefly point out now that the supersymmetric solution has an emergent
$SU(1,1|1)$ symmetry. The full Schwinger-Dyson equations in superspace (3.20),
(3.21) upon dropping the UV term are

$\displaystyle\int\\!\\!dZ_{2}\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})(J(q-1)\mathcal{G}_{\Psi\Psi}(Z_{3},Z_{2})^{q-2}\mathcal{G}_{XX}(Z_{3},Z_{2}))=2\delta(\bar{Z}_{1}-\bar{Z}_{3}),$
(3.44)
$\displaystyle\int\\!\\!dZ_{2}\mathcal{G}_{XX}(Z_{1},Z_{2})(J\mathcal{G}_{\Psi\Psi}(Z_{3},Z_{2})^{q-1})=2\delta(\bar{Z}_{1}-\bar{Z}_{3}),$
(3.45)

At late times when the UV term is ignored the equations are symmetric under
the following super-reparametrizations

$\displaystyle\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})$ $\displaystyle\to$
$\displaystyle(D_{\theta_{1}}\theta^{\prime}_{1})^{2\Delta_{\psi}}(D_{\bar{\theta}_{2}}\bar{\theta}^{\prime}_{2})^{2\Delta_{\psi}}\mathcal{G}_{\Psi\Psi}(Z^{\prime}_{1},Z^{\prime}_{2}),$
(3.46) $\displaystyle\mathcal{G}_{XX}(Z_{1},Z_{2})$ $\displaystyle\to$
$\displaystyle(D_{\theta_{1}}\theta^{\prime}_{1})^{2\Delta_{\chi}}(D_{\bar{\theta}_{2}}\bar{\theta}^{\prime}_{2})^{2\Delta_{\chi}}\mathcal{G}_{XX}(Z^{\prime}_{1},Z^{\prime}_{2})$
(3.47)

as long as

$(q-1)\Delta_{\psi}+\Delta_{\chi}=\frac{1}{2}.$ (3.48)

This is precisely the supersymmetric relation found above (3.40). This shows
that the fermions are superconformal primaries under the $SU(1,1|1)$, which is
a particular case of the more general superreparametrizations above.

Since the fermions are chiral primary operators, their R-charge has to be
twice their scaling dimension. This can be used to determine the parameter
$\alpha$ appearing in $Q_{R}$ in equation (3.3). For example, the $\psi$
fermion has charge $Q_{\psi}=1$ and $Q_{\chi}=0$ and therefore $Q_{R}=\alpha$.
The condition of $\psi$ being a chiral primary then determines the unknown
coefficient:

$\alpha=2\Delta_{\psi}\,.$ (3.49)

A similar analysis for the $\chi$ fermion gives the same $\alpha$ thanks to
(3.48).

We show later the presence of an $\mathcal{N}=2$ Super-Schwarzian mode in the
bilinear spectrum. For simplicity we can work at fixed $Q_{F}$ (we will
consider relaxing this in the next section). Since the R-charge of the theory
is given, up to a constant shift in the fixed $Q_{F}$ sector, by $Q_{\chi}$
this means the R-charge of the supercharge is one. The bosonic sector of the
action is

$S_{b}=\frac{2\pi N\alpha_{S}}{\beta
J}\int_{0}^{2\pi}d\tau\left(-\text{Sch}\left(\tan\frac{f}{2},\tau\right)+2\left(\partial_{\tau}a\right)^{2}\right),$
(3.50)

with $a\sim a+2\pi$. We gave some details of the quantum mechanical spectrum
of this theory in the introduction and emphasized it has the special feature
that all BPS states have the same R-charge. This is consistent with the fact
that the index in a fixed $Q_{F}$ sector is non-vanishing and exponentially
large in $N$ (We have done some preliminary checks on these features using
exact diagonalization of the model). We consider next what happens when
working in an ensemble in which the flavor charge fluctuates.

#### Emergent local flavor symmetry

Just as the UV $U(1)_{R}$ becomes a local IR symmetry in the conformal limit,
but leads to the $U(1)_{R}$ mode of the $\mathcal{N}=2$ super Schwarzian, one
might also expect that the $U(1)_{F}$ flavor symmetry may become a
spontaneously and explicitly broken local symmetry, leading to a new physical
mode (and potentially a spurious mode). In the case of $\mathcal{N}=1$
supersymmetry with a global $SO(q)$ symmetry, this situation was studied in
[63] where such a Kac-Moody like enhancement and corresponding flavor mode
were found. In this section, we present a generalization appropriate to our
model with larger supersymmetry.

While not immediately relevant for the IR Schwinger Dyson equations and the
$U(1)_{F}$ mode, we will briefly discuss vector multiplets and $U(1)$ gauge
transformations in superspace. A vector superfield is a real superfield with
the expansion

$\displaystyle
V(\tau,\theta,\bar{\theta})=C(\tau)+\sqrt{2}\,\theta\,\lambda(\tau)-\sqrt{2}\,\bar{\theta}\,\bar{\lambda}(\tau)-\theta\bar{\theta}A(\tau)\,,\qquad
V^{\dagger}=V\,.$ (3.51)

Reality ensures the $\lambda$ are conjugates of each other and $C,A$ are real;
this condition is chosen so that $e^{iV}$ is unitary. A supergauge
transformation is represented by two bosonic chiral superfields
$\Phi(\tau,\theta,\bar{\theta})$ and $\bar{\Phi}(\tau,\theta,\bar{\theta})$:

$\displaystyle V\rightarrow
V+\bar{\Phi}-\Phi\,,\qquad\Phi(\tau,\theta,\bar{\theta})=\phi(\tau)+\sqrt{2}\,\theta\,\xi(\tau)+\theta\bar{\theta}\partial_{\tau}\phi(\tau)\,,$
(3.52)

The final term in the superfield identifies one real degree of freedom of
$\phi$ with the usual gauge parameter. While we will not explore this
direction further, it is interesting to note that this vector superfield would
be required to introduce a supersymmetric generalization of the background
chemical potential $\mu$ for the flavor symmetry.

A naive way to implement a local flavor symmetry in superspace would be to
perform bi-local transformations of the form
$\mathcal{G}(Z_{1},Z_{2})\rightarrow
e^{iV(Z_{1})}e^{-iV(Z_{2})}\mathcal{G}(Z_{1},Z_{2})$ on the Schwinger Dyson
equations (3.44) and (3.45). However, this transformation does not take chiral
superfields to chiral superfields, so the correct prescription to perform bi-
local independent chiral and anti-chiral gauge transformations with the
correct flavor charges using (3.52):

$\displaystyle\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})$ $\displaystyle\to
e^{-i\bar{\Phi}(Z_{1})}e^{i\Phi(Z_{2})}\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})\,,$
(3.53) $\displaystyle\mathcal{G}_{XX}(Z_{1},Z_{2})$ $\displaystyle\to
e^{i(q-1)\bar{\Phi}(Z_{1})}e^{-i(q-1)\Phi(Z_{2})}\mathcal{G}_{XX}(Z_{1},Z_{2})$
(3.54)

This leaves

$\displaystyle e^{i\left(-\bar{\Phi}(Z_{1})+\bar{\Phi}(Z_{3})\right)}$
$\displaystyle\int\\!\\!dZ_{2}\,\mathcal{G}_{\Psi\Psi}(Z_{1},Z_{2})[J(q-1)\mathcal{G}_{\Psi\Psi}(Z_{3},Z_{2})^{q-2}\mathcal{G}_{XX}(Z_{3},Z_{2})]=2\delta(\bar{Z}_{1}-\bar{Z}_{3})\,,$
(3.55) $\displaystyle
e^{i(q-1)\left(\bar{\Phi}(Z_{1})-\bar{\Phi}(Z_{3})\right)}$
$\displaystyle\int\\!\\!dZ_{2}\,\mathcal{G}_{XX}(Z_{1},Z_{2})[J\mathcal{G}_{\Psi\Psi}(Z_{3},Z_{2})^{q-1}]=2\delta(\bar{Z}_{1}-\bar{Z}_{3})\,,$
(3.56)

which is a symmetry of the equations of motion under the support of the
supersymmetric $\delta$-function on the right hand side. Actually, the
holomorphic dependence on $\Phi$ cancels inside the integral, and the
resulting equations appear to have a full complex mode $\bar{\Phi}$ of
transformations. This was already anticipated from (3.37) which was the global
scaling symmetry of the conformal answer.

The above analysis indicates there are 2 bosonic and 2 fermionic flavor
reparametrizations, but it is only the real part of the field $\phi(\tau)$
which leads to a compact $U(1)$ compatible with the UV picture. The imaginary
part leads to a scale symmetry which we believe is not associated to any
physical IR mode. It is less clear, however, that one fermionic mode should be
physical and the other should be unphysical. This is because even if we take
$\phi(\tau)$ to be purely real, the complex supersymmetry generators produce
both $\xi$ and $\bar{\xi}$ fields, schematically:

$\displaystyle\begin{matrix}\,&\,&\phi\in\mathbb{R}&\,&\,\\\
\,&\overset{\mathcal{Q}}{\swarrow}&\\!\\!\\!\\!\\!&\overset{\bar{\mathcal{Q}}}{\searrow}&\,\\\
\xi&\,&\,&\,&\bar{\xi}\\\
\,&\overset{\bar{\mathcal{Q}}}{\searrow}&\,&\overset{\mathcal{Q}}{\swarrow}&\,\\\
\,&\,&\partial_{\tau}\phi&\,&\,\\\ \end{matrix}$ (3.57)

We will see in the bilinear spectrum which modes actually appear in the
infrared, where the global supersymmetry transformations in the diagram would
be replaced with the local super-reparametrizations.

Understanding now that there is one physical local $U(1)$ symmetry of the
Schwinger-Dyson equations, we expect the breaking of this $U(1)$ leads to a
new dynamical IR mode in addition to the $\mathcal{N}=2$ super Schwarzian. We
already reviewed the analogous IR action in Eq. (2.52) in which we wrote down
the bosonic part of the $\mathcal{N}=2$ Schwarzian; the $U(1)_{R}$ breaking
boson becomes a particle moving on the $U(1)$ group manifold (with radius
given in terms of the $q$ parameter). A similar effective action appears for
the $U(1)_{F}$ mode, but there are some important differences. First, because
the $U(1)_{R}$ is part of the super-reparametrizations, it appears in the same
multiplet as the Schwarzian. In contrast, the global flavor symmetry commutes
with supersymmetry, so we expect there to be a multiplet of modes in
correspondence with (3.52). Because the group element can be written in
superspace, the corresponding multiplet should describe a superparticle moving
on the $U(1)_{F}$ group manifold. While we will not derive the effective
action, the situation here is similar to that in [63]. One might further guess
that the collective AdS2 description would include $\mathcal{N}=2$ JT gravity
coupled to $U(1)_{F}$ $\mathcal{N}=2$ BF theory, but we leave this question
for future work.

#### The Zero Temperature Entropy

We now repeat the procedure in the previous section to compute the temperature
independent contribution to the partition function

$\partial_{q}\frac{\log Z}{N}=\frac{J\beta}{2}\int
G_{\psi\psi}(\tau)^{q-1}G_{b_{\chi}b_{\chi}}(\tau)\log
G_{\psi\psi}(\tau)+G_{\psi\psi}(\tau)^{q-2}G_{b_{\psi}b_{\psi}}(\tau)G_{\chi\chi}(\tau)(1+(q-1)\log
G_{\psi\psi}(\tau)).$

We can evaluate the right hand side using the conformal solution. This is
divergent but only the temperature independent piece is independent of the UV
behavior. The answer is given by

$\partial_{q}\frac{\log
Z}{N}=\\#\beta+\Delta_{\psi}\pi^{2}J(g_{\psi\psi}^{q-1}g_{b_{\chi}b_{\chi}}+(q-1)g_{\psi\psi}^{q-2}g_{\chi\chi}g_{b_{\psi}b_{\psi}})+\mathcal{O}(\beta^{-1}),$
(3.58)

where again only the temperature independence piece is insensitive to the UV
behavior. Since this calculation involves only the IR behavior of the action
and the solution, the expression depends only on the combination that we can
determine without information about the UV. Using equations (3.31) to (3.34),
we obtain for the temperature independent piece

$\partial_{q}\frac{\log
Z}{N}=\pi\Delta_{\psi}\left(\frac{(1-2\Delta_{\chi})\sin
2\pi\Delta_{\chi}}{\cos 2\pi\Delta_{\chi}+\cosh
2\pi\mathcal{E}_{\chi}}+\frac{(2(q-1)\Delta_{\psi}-1)\sin
2\pi(q-1)\Delta_{\psi}}{\cos 2\pi(q-1)\Delta_{\psi}-\cosh
2\pi(q-1)\mathcal{E}_{\psi}}\right)+\ldots,$ (3.59)

If we restrict to supersymmetric configurations this simplifies to

$\partial_{q}\frac{\log Z}{N}=\frac{\pi\Delta_{\psi}\sin
2\pi(q-1)\Delta_{\psi}}{\cosh 2\pi(q-1)\mathcal{E}_{\psi}-\cos
2\pi(q-1)\Delta_{\psi}}+\ldots,$ (3.60)

Finally we note that if we define the zero temperature entropy as the
contribution for $\mathcal{E}_{\psi},\mathcal{E}_{\chi}\to 0$ then we obtain

$\frac{dS_{0}}{dq}=N\pi\Delta_{\psi}\cot\left(\pi(q-1)\Delta_{\psi}\right).$
(3.61)

When evaluated at $\mathcal{E}_{\psi}=\mathcal{E}_{\chi}=0$, the equation for
$\Delta_{\psi}$ obtained from the Schwinger-Dyson equation becomes

$\tan\pi\Delta_{\psi}=(q-1)\cot\pi(q-1)\Delta_{\psi}$ (3.62)

As anticipated above, this is exactly the same as what we found from the
index. The equation for $dS_{0}/dq$ and the equation for $\Delta_{\psi}$
matches precisely with the expression (3.12) for the index at fixed charge,
after identifying $y_{c}\to 2\pi\Delta_{\psi}$. This match can be understood
as a manifestation of I-maximization in the context of quantum mechanical
systems with approximate superconformal symmetry [64].

###  3.2 Breakdown of conformal ansatz

In this section we analyze what happens when we turn on the charge by
increasing the spectral asymmetry. As we found in the original $\mathcal{N}=2$
SYK model, for large enough spectral asymmetry the conformal ansatz breaks
down. We first analyze the supersymmetric solutions parametrized by a single
variable, and then move on to the general case.

##### Supersymmetric Solution

The physics of the supersymmetric solution is very different from the case
studied in section 2. To begin with, there is a one-parameter family of
solutions with varying scaling dimensions that are supersymmetric. Second, the
coefficients in front of the Greens functions are also not completely
determined. Using $G_{b_{A}b_{A}}\sim\partial_{\tau}G_{AA}$ we can deduce that
$g_{b_{A}b_{A}}=2\Delta_{A}g_{AA}$ for $A=\psi,\chi$ but now the Schwinger-
Dyson equations in the IR only fix the combination
$g_{\psi\psi}^{q-1}~{}g_{\chi\chi}$, while $g_{\psi\psi}$ and $g_{\chi\chi}$
cannot be independently determined without incorporating the UV behavior.

Figure 13: Fermion scaling dimensions of the supersymmetric solutions, as a
function of the spectral asymmetry of $\psi$ fermion $\mathcal{E}_{\psi}$, for
(a) $q=3$ and (b) $q=5$. We see at a finite critical value of
$\mathcal{E}_{\psi}$ the solution becomes singular with $\Delta_{\psi}\to 0$
while $\Delta_{\chi}\to 1/2$. This point corresponds to maximal $\chi$ charge.
The two curves meet at the dashed line $\Delta=1/2q$ for a special value of
$\mathcal{E}_{\psi}$.

Third, we find a potential breakdown of the conformal ansatz even within the
supersymmetric solution. Solving the equation above numerically we find the
result shown in figure 13. There is again a critical spectral asymmetry
$|\mathcal{E}_{\psi}|=\mathcal{E}_{\rm critical}$ such that for
$|\mathcal{E}_{\psi}|>\mathcal{E}_{\rm critical}$ the fermion scaling
dimensions are either outside the unitarity bound or complex (the other
potential is determined by supersymmetry from $\mathcal{E}_{\rm critical}$).
The critical value is implicitly given by

$\frac{\sinh\pi(q-1)\mathcal{E}_{\rm critical}}{\cosh\pi\mathcal{E}_{\rm
critical}}=(q-1),$ (3.63)

which is exactly the same as the value found in the previous section for the
model defined in [6]. As we see in the figure above, this value decreases with
increasing $q$.

To determine whether this phase transition is physical or not we need to
compute the charges and verify they are not maximal. We can use the Luttinger-
Ward relation derived in the previous section to compute both $Q_{\psi}/N$ and
$Q_{\chi}/N$. Each fermion charge includes a contribution from their
respective bosonic partner, as explained in the previous section. The answer
is given by

$\displaystyle Q_{\psi}/N$ $\displaystyle=$
$\displaystyle\mathfrak{q}_{f}(\Delta_{\psi},\mathcal{E}_{\psi})+(q-2)\mathfrak{q}_{b}(\Delta_{b_{\psi}},\mathcal{E}_{b_{\psi}})+(q-1)\mathfrak{q}_{b}(\Delta_{b_{\chi}},\mathcal{E}_{b_{\chi}}),$
(3.64) $\displaystyle Q_{\chi}/N$ $\displaystyle=$
$\displaystyle\mathfrak{q}_{f}(\Delta_{\chi},\mathcal{E}_{\chi})+\mathfrak{q}_{b}(\Delta_{b_{\psi}},\mathcal{E}_{b_{\psi}}),$
(3.65)

in terms of the functions defined in (2.48) and (2.49). Using these
expressions for the charges we can ask now whether the critical spectral
asymmetry corresponds to an instability or not. The first observation is that
$\mathfrak{q}_{f}(\Delta=1/2,\mathcal{E})=-1/2$ and
$\mathfrak{q}_{b}(\Delta=1/2,\mathcal{E})=0$ regardless of $\mathcal{E}$. At
the critical $\mathcal{E}=\mathcal{E}_{\rm critical}$, the scaling dimension
of $\chi$ and $b_{\psi}$ take precisely these values
$\Delta_{\chi}(\mathcal{E}_{\rm critical})=1/2$ and
$\Delta_{b_{\psi}}(\mathcal{E}_{\rm critical})$. For these reasons we obtain
$|Q_{\chi}|(\mathcal{E}_{\rm critical})=N/2$. Since both fermion charges are
bounded $0\leq|Q_{\psi}|,|Q_{\chi}|\leq N/2$ this means that there is no
instability in the range. The previous analysis shows that in the canonical
ensemble there is no phase transition since we can turn on $Q_{\chi}$ and
$Q_{\psi}$ in a supersymmetric way satisfying the constraint imposed by the
conformal solution. Nevertheless there can be other gapped solutions that have
support in different curves in the $(Q_{\psi},Q_{\chi})$ plane. Then, in the
grand canonical ensemble there might be phase transitions between these two
sets of solutions. We expect this to be so, since at
$\mathcal{E}_{\text{critical}}$, while the $Q_{\chi}$ is saturated, the flavor
charge $Q_{F}$ is not. For example, for $q=3,$ at critical spectral asymmetry,
$Q_{F}/N=\sqrt{2}<\frac{3}{2}.$ We leave a more detailed study of these
possible supersymmetric phase transitions for future work.