# Conductance and thermopower fluctuations
in interacting quantum dots

Henry Shackleton  Laurel E. Anderson  Philip Kim  Subir Sachdev  Department of
Physics, Harvard University, Cambridge MA 02138, USA

###### Abstract

We model an interacting quantum dot of electrons by a Hamiltonian with random
and all-to-all single particle hopping (of r.m.s. strength $t$) and two-
particle interactions (of r.m.s. strength $J$). For $t\ll J$, such a model has
a regime exhibiting the non-quasiparticle physics of the Sachdev-Ye-Kitaev
model at temperatures $E_{\rm coh}\ll T\ll J$, and that of a renormalized
Fermi liquid at $T\ll E_{\rm coh}$, where $E_{\rm coh}=t^{2}/J$. Extending
earlier work has computed the mean thermoelectric properties of such a dot
weakly coupled to two external leads, we compute the sample-to-sample
fluctuations in the conductance and thermopower of such a dot, and describe
several distinct regimes. In all cases, the effect of the SYK interactions is
to reduce the strength of the sample-to-sample fluctuations. We also find that
in the regime where the mean transport co-efficients are determined only by
the value of $J$ at leading order, the sample-to-sample fluctuations can be
controlled by the influence of the smaller $t$.

###### Contents

  1. I Introduction
  2. II Setup
    1. II.1 Hamiltonian and transport coefficients
      1. II.1.1 Single site lead coupling
      2. II.1.2 All-to-all couplings
      3. II.1.3 Disordered all-to-all couplings
    2. II.2 Comparison to other analyses
  3. III Free fermion analysis
    1. III.1 Conductance statistics
    2. III.2 Thermopower statistics
  4. IV Pure SYK analysis
    1. IV.1 Conductance statistics
    2. IV.2 Thermopower statistics
  5. V Interplay between hoppings and interactions
    1. V.1 Fermi liquid regime
    2. V.2 SYK Regime
    3. V.3 Thermopower statistics
  6. VI Conclusion
  7. A Path integral calculation of fluctuations
  8. B Replica off-diagonal fluctuations in the $(G,\Sigma)$ action
  9. C Statistics of ratio distributions
  10. D Conductance fluctuations for single-lead coupling

##  I Introduction

The Sachdev-Ye-Kitaev (SYK) model [1, 2] is a strongly interacting quantum
many-body system without quasiparticle excitations, whose exact solvability in
the large-$N$ limit - with $N$ the number of sites - has led to significant
interest in it both as a toy model for non-Fermi liquid behavior as well as an
analytically tractable example of holographic duality [3, 4].

In contrast to its analytic solvability, experimental realizations of the SYK
model have proved to be challenging. The SYK model is defined microscopically
as a system of fermions with random all-to-all quartic interactions, and is
unstable at low temperatures to single-particle hopping. As such, any
experimental proposal must generate strongly-disordered interactions with a
high degree of connectivity, while simultaneously quenching any single-
particle hopping terms. Several promising proposals have been made to this
extent, involving Majorana zero modes [5, 6], quantum processors [7, 8],
ultracold gases [9, 10, 11] and disordered graphene flakes [12, 13].
Simulations of the SYK model have been achieved on quantum processors [14] and
controllable nuclear-spin-chain simulators [15]. Our study here was motivated
by experiments on disordered graphene flakes [16], results of which will be
reported in a separate paper [17].

Each experimental realization of the SYK model will have a different set of
observables that it is best suited to study. Our focus will be on proposals
for realizing the SYK model with complex fermions in a disordered graphene
flake, for which the measurable quantities are thermoelectric transport
observables, such as conductance and thermopower. Theoretical predictions for
the average values of these quantities have been calculated [18] for realistic
models that include both SYK terms and experimentally-relevant perturbations.
The conclusion of this analysis is that thermoelectric quantities display a
crossover from Fermi liquid-like behavior at temperatures below a coherence
energy $E_{\text{coh}}=t^{2}/J$, where small single-particle hopping terms,
with r.m.s. value $t$, produce coherent quasiparticle excitations, to SYK-like
behavior at temperatures $E_{\rm coh}\ll T\ll J$, where $T$ is the
temperature, and $J$ is the r.m.s. value of the SYK interactions.

In experimental realizations of these mesoscopic systems, transport quantities
such as the conductance and thermopower will display sample-to-sample
fluctuations, or alternatively fluctuations as a function of tuning external
parameters such as chemical potential or magnetic field. For weakly-
interacting disordered quantum dots at zero temperature coupled to broad
multi-channel leads, this results in the well-studied phenomenon of universal
conductance fluctuations (UCF) at zero temperature, where the conductance
displays $\order{1}$ fluctuations (in units of the conductance quanta,
$e^{2}/h$) whose magnitude is independent of the disorder strength [19, 20,
21, 22]. An analogous treatment of disorder fluctuations in strongly-
correlated quantum dots has not been explored previously. In this work, we
analyze the fluctuations in transport properties in quantum dots with strong
SYK interactions, and study the behavior of these fluctuations as their
average values crossover from Fermi liquid-like for $T\ll E_{\text{coh}}$ to
SYK-like for $T\gg E_{\text{coh}}$. We contrast analysis of these properties
in the SYK regime, which involve statistical fluctuations of the single-
particle Green’s function, with the large body of work analyzing statistical
properties of the many-body spectrum [23, 24, 25, 26].

Our analysis is able to recover UCF behavior for zero temperature, while the
variance of the conductance in the Fermi liquid regime displays a $T^{-1}$
falloff at higher temperatures, consistent with prior studies of weakly-
interacting disordered quantum dots [27]. However, we find a surprising
feature of these fluctuations for temperatures larger than the coherence
energy. In contrast to the mean values of transport quantities, whose behavior
for $T\gg E_{\text{coh}}$ is well-described by a pure SYK model ($t=0$), the
same is not true for the variance - at leading order in $N^{-1}$, the variance
of the conductance for a pure SYK model is distinct from the variance in a
model with SYK interactions and random hopping with r.m.s. value
$t>\sqrt{TJ}/N$. The self-averaging properties of the pure SYK model are so
strong that, to leading-order in $N^{-1}$, fluctuations of the physical
transport properties remain driven by fluctuations of random hopping terms,
even if their mean values are well-described by the pure SYK solution.
Distinct predictions are still found for the two temperature regimes, arising
from the different form of the average spectral function in the two limits,
and we find a $T^{-2}$ suppression of the conductance variance in the SYK
regime in contrast with the $T^{-1}$ Fermi liquid prediction.

These aspects of our results are illustrated by the following summary of our
prediction for the mean $(\overline{\sigma})$ and variance of the electrical
conductance $(\sigma)$:

$\displaystyle\overline{\sigma}_{FF}\propto\frac{\Gamma
e^{2}}{\hbar}\frac{1}{t}\quad$ $\displaystyle,\quad\text{Var
}\sigma_{FF}\propto\left(\frac{\Gamma e^{2}}{\hbar}\right)^{2}\frac{1}{NtT}$
(1) $\displaystyle\overline{\sigma}_{SYK}\propto\frac{\Gamma
e^{2}}{\hbar}\frac{1}{\sqrt{JT}}\quad$ $\displaystyle,\quad\text{Var
}\sigma_{SYK}\propto\left(\frac{\Gamma
e^{2}}{\hbar}\right)^{2}\frac{1}{N^{3}JT}$ (2)
$\displaystyle\overline{\sigma}_{tSYK}\propto\frac{\Gamma
e^{2}}{\hbar}\frac{1}{t}\quad$ $\displaystyle,\quad\text{Var
}\sigma_{tSYK}\propto\left(\frac{\Gamma
e^{2}}{\hbar}\right)^{2}\frac{1}{NJT},\quad T\ll E_{\rm coh}$ (3)
$\displaystyle\overline{\sigma}_{tSYK}\propto\frac{\Gamma
e^{2}}{\hbar}\frac{1}{\sqrt{JT}}\quad$ $\displaystyle,\quad\text{Var
}\sigma_{tSYK}\propto\left(\frac{\Gamma
e^{2}}{\hbar}\right)^{2}\frac{\mathcal{E}^{2}t^{2}}{NJ^{2}T^{2}},\quad E_{\rm
coh}\ll T\ll J$ (4)

Here (i) $FF$ refers to the free-fermion results in Section III.1 with
$\Gamma$ a measure of the coupling to the leads, and the various of
$\sigma_{FF}$ crosses over to the UCF value when $T<\Gamma^{2}/Nt$; (ii) the
pure SYK results are in Section IV.1; (iii) $tSYK$ refers to the model with
both hopping and interactions with $t\ll J$, $E_{\rm coh}=t^{2}/J$, the
results for $T\ll E_{\rm coh}$ are in Section V.1, and the results for $E_{\rm
coh}\ll T\ll J$ are in Section V.2 ($\mathcal{E}$ is a measure of the
particle-hole asymmetry). All these results are obtained for the case where
the coupling to the leads, $\Gamma$, is the smallest energy scale, and to
leading order in a $1/N$ expansion.

Note that in all cases, the effect of the SYK interactions is to reduce the
strength of the conductance fluctuations:
(i) Eq. 2 is suppressed by a factor of $1/N^{3}$ in contrast to $1/N$ in all
other cases,
(ii) Eq. 3 is smaller than Eq. 1 by a factor of $t/J$, and
(iii) Eq. 4 is smaller than Eq. 3 by a factor of $E_{\rm coh}/T$.

The structure of this paper is as follows. In Section II, we make explicit the
setup of our theoretical model as well as the assumptions used in calculating
thermoelectric quantities. In Section III, we calculate the fluctuations of
transport quantities in the non-interacting limit, where properties are
governed by single-particle random matrix theory (RMT). We emphasize that this
approach is distinct from more standard approaches of modeling UCF phenomena
using RMT [28], where calculations are done at zero temperature and involve
the statistical treatment of transmission eigenvalues. Our treatment is
primarily done at non-zero temperature and in the limit of weak environmental
coupling, although we show that it is possible to extend our results down to
zero temperature and recover $\order{1}$ universal fluctuations in an
appropriate limit. In Section IV, we study transport fluctuations in the SYK
regime, presenting results both for pure SYK as well as more realistic models
with random single-particle hopping. In Section V, we study a model with both
SYK and random hopping terms and demonstrate that the transport fluctuations
for $T\gg E_{\text{coh}}$ are qualitatively different than that of a pure SYK
model.

In each of these sections, we discuss the fluctuations of the thermopower in
addition to the conductance. The statistical properties of the thermopower
require more care; in our formalism, we find that the thermopower is
determined by a ratio of two Gaussian random variables, and hence the variance
is formally not well-defined. An approximation to normality is still
appropriate in certain parameter regimes for small fluctuations around the
mean, and hence we can formally define a variance within this approximation.
We state results given this assumption and find qualitatively similar behavior
as the conductance variance, which is that the presence of strong SYK
interactions serves to reduce the fluctuations around the mean value.

##  II Setup

###  II.1 Hamiltonian and transport coefficients

Our goal is to characterize fluctuations in transport properties of disordered
quantum dots with random all-to-all interactions. We model this quantum dot by
the Hamiltonian

$\displaystyle H_{\text{dot}}$
$\displaystyle=\frac{1}{(2N)^{3/2}}\sum_{ij;kl}J_{ij;kl}c_{i}^{\dagger}c_{j}^{\dagger}c_{k}c_{l}+\frac{1}{N^{1/2}}\sum_{ij}t_{ij}c_{i}^{\dagger}c_{j}-\mu\sum_{i}c_{i}^{\dagger}c_{i}$
(5)

where $J_{ij;kl}$ and $t_{ij}$ are complex random numbers with zero mean and
variances $J^{2}$ and $t^{2}$, respectively. The complex SYK model is given by
the first term, whereas the second term is a random single-particle hopping
which leads to Fermi liquid behavior at low temperatures.

The quantum dot is coupled to two leads. Following the approach of [29], we
model the leads by considering the Hamiltonian

$\displaystyle
H=H_{\text{dot}}+\sum_{\mathbf{q}}\epsilon_{\mathbf{q}}a_{\mathbf{q}}^{\dagger}a_{\mathbf{q}}+\sum_{i,\mathbf{q},\alpha}\left[\lambda_{i\alpha}c_{i}^{\dagger}a_{\mathbf{q}\alpha}+\lambda_{i\alpha}^{*}a_{\mathbf{q}\alpha}^{\dagger}c_{i}\right]\,.$
(6)

where $\alpha=R\,,L$ labels the right and left leads. To parameterize the
coupling to the leads, we define the matrices

$\displaystyle\Gamma_{ij}^{\alpha}=\pi\rho_{\text{lead},\alpha}\lambda_{i\alpha}\lambda_{j\alpha}^{*}\,,$
(7)

where $\rho_{\text{lead},\alpha}$ is the density of states in lead $\alpha$
near the Fermi level. We will assume
$\rho_{\text{lead},L}=\rho_{\text{lead},R}\equiv\rho_{\text{lead}}$.

We will find that the nature of the conductance fluctuations depends
sensitively on how we model the coupling to the leads, $\lambda_{i\alpha}$.
This is in contrast to the mean values of transport quantities, which is not
as sensitive. We first make the assumption that the two couplings are
proportional to each other, i.e. $\lambda_{iR}=\alpha\lambda_{iL}$ for some
constant $\alpha$. With this constraint, it becomes possible to express
transport properties solely in terms of the equilibrium Green’s functions of
the quantum dot. Using expressions derived in [30], we define

$\displaystyle\mathcal{L}_{ab}$
$\displaystyle=-\frac{i}{2\pi\hbar}\int_{-\infty}^{\infty}\differential{\omega}\omega^{a+b-2}f^{\prime}(\omega)\operatorname{Im}\Tr\left[\bm{\Gamma}\bm{G}^{R}\right]\,,$
(8)

with $\bm{G}^{R,A}(\omega)$ the local retarded and advanced Green’s function
of $H$, both $N\times N$ matrices, $f(\omega)$ the Fermi function
$f(\omega)=1/\left(e^{\omega/T}+1\right)$, and
$\bm{\Gamma}\equiv\bm{\Gamma}^{L}\bm{\Gamma}^{R}/\left(\bm{\Gamma}^{L}+\bm{\Gamma}^{R}\right)$.
For cases where the matrix $\bm{\Gamma}^{L}+\bm{\Gamma}^{R}$ is non-
invertable, this equation is modified by omitting the null subspace of the
matrix. The Green’s functions must be solved for the full Hamiltonian,
including the coupling to the leads; however, we will primarily be focused on
the parameter regime where $\Gamma$ is the smallest energy scale and the
Green’s functions of the isolated system $H_{\text{dot}}$ are used.

The electric conductance $\sigma$, thermal conductance $\kappa$, and
thermopower $\Theta$ are given by

$\displaystyle\sigma=e^{2}\mathcal{L}_{11}\,,\quad\kappa=\beta\left(\mathcal{L}_{22}-\frac{\mathcal{L}_{12}^{2}}{\mathcal{L}_{11}}\right)\,,\quad\Theta=\frac{\beta}{e}\frac{\mathcal{L}_{12}}{\mathcal{L}_{11}}\,.$
(9)

where $\beta=1/T$.

Beyond this point, we must make further assumptions on the nature of the
coupling to the leads. For notational simplicity, we will assume
$\lambda_{iR}=\lambda_{iL}\equiv\lambda_{i}$ \- generalization to the case
where the magnitude of the couplings are asymmetric does not qualitatively
affect our results.

####  II.1.1 Single site lead coupling

In this model, we take our two leads to be coupled to a single site, i.e.
$\lambda_{i}\equiv\delta_{i1}\lambda$. Defining
$\Gamma\equiv\pi\rho_{\text{lead}}\absolutevalue{\lambda}^{2}$, we have

$\displaystyle\overline{\mathcal{L}_{ab}}$
$\displaystyle=\frac{2\Gamma}{\pi\hbar}\int_{-\infty}^{\infty}\differential{\omega}\omega^{a+b-2}f^{\prime}(\omega)\overline{\operatorname{Im}G^{R}_{11}(\omega)}$
(10)

Recall that the Green’s functions are dependent on the random variables
$J_{ij;kl},t_{ij}$. Averaging over disorder, we find that
$\overline{\operatorname{Im}G^{R}_{11}(\omega)}=N^{-1}\sum_{ii}\overline{\operatorname{Im}G^{R}_{ii}(\omega)}\equiv\overline{\operatorname{Im}G^{R}(\omega)}$.
Note that this relation relies on neglecting corrections to $\bm{G}^{R}$
arising from the couplings to the leads, as these corrections will be site-
dependent.

Higher moments of these transport coefficients are given by

$\displaystyle\overline{\mathcal{L}_{ab}\mathcal{L}_{ab}}-\overline{\mathcal{L}_{ab}}\,\overline{\mathcal{L}}_{ab}$
$\displaystyle=\left(\frac{2\Gamma}{\pi\hbar}\right)^{2}\int_{-\infty}^{\infty}\differential{\omega}\differential{\epsilon}\omega^{a+b-2}\epsilon^{a+b-2}f^{\prime}(\omega)f^{\prime}(\epsilon)\rho_{d}(\omega,\epsilon)$
(11)

where we define

$\displaystyle\rho_{d}(\omega,\epsilon)$
$\displaystyle\equiv\frac{1}{N^{2}}\sum_{ij}\left[\overline{\operatorname{Im}G^{R}_{ii}(\omega)\operatorname{Im}G^{R}_{jj}(\epsilon)}-\overline{\operatorname{Im}G^{R}_{ii}(\omega)}\,\overline{\operatorname{Im}G^{R}_{jj}(\epsilon)}\right]$
(12)

The subscript $d$ indicates that this quantity describes the covariance of the
diagonal component of the Green’s function, $G^{R}_{ii}$.

####  II.1.2 All-to-all couplings

Here, we take the leads to be coupled to all sites with equal hopping,
$\lambda_{i}\equiv\frac{\lambda}{\sqrt{N}}$. This model is also appropriate
for hoppings that are equal in magnitude but with site-dependent phases, as
the overall phase can be absorbed by a unitary transformation on the quantum
dot operators. Defining
$\Gamma\equiv\pi\rho_{\text{lead}}\absolutevalue{\lambda}^{2}$ as before, we
have

$\displaystyle\overline{\mathcal{L}_{ab}}$
$\displaystyle=\frac{1}{N}\sum_{ij}\frac{2\Gamma}{\pi\hbar}\int_{-\infty}^{\infty}\differential{\omega}\omega^{a+b-2}f^{\prime}(\omega)\overline{\operatorname{Im}G^{R}_{ij}(\omega)}=\frac{2\Gamma}{\pi\hbar}\int_{-\infty}^{\infty}\differential{\omega}\omega^{a+b-2}f^{\prime}(\omega)\overline{\operatorname{Im}G^{R}(\omega)}\,.$
(13)

where we utilize the fact that $\overline{G^{R}_{ij}(\omega)}=0$ for $i\neq
j$. The overall scaling of $N^{-\frac{1}{2}}$ in $\lambda_{i}$ was chosen such
that the mean value of the conductance is consistent with the previous model.
The second moment is given by

$\displaystyle\overline{\mathcal{L}_{ab}\mathcal{L}_{ab}}-\overline{\mathcal{L}_{ab}}\,\overline{\mathcal{L}_{ab}}$
$\displaystyle=\frac{1}{N^{2}}\sum_{ij,kl}\left(\frac{2\Gamma}{\pi\hbar}\right)^{2}\int_{-\infty}^{\infty}\differential{\omega}\differential{\epsilon}\omega^{a+b-2}\epsilon^{a+b-2}f^{\prime}(\omega)f^{\prime}(\epsilon)\left[\rho_{d}(\omega,\epsilon)+\rho_{o}(\omega,\epsilon)\right]\,.$
(14)

where now we define the off-diagonal Green’s function covariance,

$\displaystyle\rho_{o}(\omega,\epsilon)$
$\displaystyle\equiv\frac{1}{N^{2}}\sum_{ij}\left[\overline{\operatorname{Im}G^{R}_{ij}(\omega)\operatorname{Im}G^{R}_{ji}(\epsilon)}-\overline{\operatorname{Im}G^{R}_{ij}(\omega)}\,\overline{\operatorname{Im}G^{R}_{ji}(\epsilon)}\right]\,.$
(15)

####  II.1.3 Disordered all-to-all couplings

If our sites physically correspond to spatially random modes, as is the case
in graphene realizations of strongly interacting quantum dots in the zeroth
Landau level, then it may be appropriate to model the coupling to the leads as
additional random variables. To analyze this case, we treat $\lambda_{i}$ as
Gaussian random variables:

$\displaystyle\overline{\lambda_{i}}$ $\displaystyle=0\,,$ (16)
$\displaystyle\overline{\lambda_{i}^{*}\lambda_{j}}$
$\displaystyle=\delta_{ij}\frac{\lambda^{2}}{N}\,,$

which in turn implies that
$\overline{\Gamma_{ij}^{\alpha}}=\delta_{ij}\frac{\pi\rho_{\text{lead}}\lambda^{2}}{N}\equiv\delta_{ij}\frac{\Gamma}{N}$.
Crucial to the calculation of fluctuations, we note the identity

$\displaystyle\overline{\Gamma_{ij}^{\alpha}\Gamma_{kl}^{\beta}}=\left(\frac{\Gamma}{N}\right)^{2}\left(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{jk}\right)\,.$
(17)

The average values of the transport coefficients are the same as in the
previous models. Using the relation

$\displaystyle\overline{(\Gamma_{ij}^{R}+\Gamma_{ij}^{L})(\Gamma_{kl}^{R}+\Gamma_{kl}^{L})}=\left(\frac{2\Gamma}{N}\right)^{2}\left(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{jk}\right)\,,$
(18)

we can obtain higher moments of the transport coefficients. This leads to a
result for the variance almost identical to the uniform all-to-all couplings
in the previous section. The crucial difference is that in this case, the
disconnected component of $\rho_{o}(\omega,\epsilon)$, defined in Eq. 15, is
not subtracted off in the expression for the variance of $\mathcal{L}_{ab}$.
The consequence of this is a trivial contribution to the variance of
$\mathcal{L}_{ab}$, which is given by $N^{-1}\overline{\mathcal{L}_{ab}}^{2}$
and can be thought of as being driven by the disorder in the leads in contrast
to the intrinsic disorder in the quantum dot. While this is suppressed by a
factor of $N^{-1}$, we will find that fluctuations generically only appear at
the order or higher, so this contribution cannot be disregarded on these
grounds.

We have shown that the variance of transport quantities, such as the
conductance, are determined by the single-particle Green’s function
covariances $\rho_{d}$, $\rho_{o}$. The primary focus of our paper will be an
analysis of these functions, and their implications for conductance
fluctuations. For concreteness, we will give our predictions for conductance
fluctuations in a model with uniform all-to-all couplings, such that both
$\rho_{o}$ and $\rho_{d}$ contribute, and so that there is no trivial
contribution to the variance arising from disordered leads. We summarize
results for single-mode couplings in Appendix D.

###  II.2 Comparison to other analyses

Due to the extensive literature on conductance fluctuations in mesoscopic
systems, we make precise here the connection between our setup and prior work.

The most well-established results for conductance fluctuations pertain to the
$T=0$ conductance of a sufficiently weakly-interacting quantum dot such that a
single-particle picture is appropriate. In this limit, conductance
fluctuations can be understood most directly via a random matrix analysis of
the scattering matrices, which take values in the circular ensemble [31, 32].
An alternative approach, suitable for studying the effects of non-zero
temperature and weak magnetic fields, is to start with a microscopic single-
particle Hamiltonian modeled as a random matrix, much like our Hamiltonian in
the limit $J=0$. In the non-interacting limit, the conductance for a generic
set of lead couplings $\lambda_{jL}$, $\lambda_{jR}$ is given by the Landauer
formula for a single channel,

$\displaystyle\sigma$
$\displaystyle=\frac{e^{2}}{h}\int\differential{\omega}f^{\prime}(\omega)t(\omega)t^{*}(\omega)\,,$
(19) $\displaystyle t(\omega)$ $\displaystyle\equiv
2\pi\rho_{\text{lead}}\sum_{ij}\lambda_{iL}^{*}G^{R}_{ij}(\omega)\lambda_{jR}\,.$

The conductance variance is thus related to the disorder average of four
copies of $G^{R}$, solved in the presence of the leads. This becomes a
tractable problem in the limit where the number of channels in the leads is
large, and can be dealt with rigorously using supersymmetry techniques [33,
34, 35, 27, 36] to give results consistent with random matrix predictions.
This formulation can also be generalized to non-zero temperature [27], where a
$T^{-1}$ falloff of the conductance variance is observed. However, these
supersymmetry techniques cannot be generalized to accommodate strong
interactions. Our results are most appropriately compared to the prior results
on closed quantum dots, where the number of channels is small and are weakly
coupled to the dot. These works have primarily focused on the $T=0$
conductance in either the weakly-interacting limit [37, 38], where non-
Gaussian behavior of the conductance was found, or in the Coulomb blockade-
dominated limit [32] which found a non-Gaussian distribution of the
conductance peaks. To our knowledge, conductance fluctuations in the parameter
regime of closed quantum dots with $T\gg\Gamma$ and for negligible Coulomb
blockade effects has not been studied previously. This is the regime where we
will conduct our analysis, as it is in this limit that the effects of strong
SYK interactions becomes analytically tractable.

##  III Free fermion analysis

###  III.1 Conductance statistics

We begin with an analysis of conductance fluctuations in the non-interacting
limit ($J=0$). In this limit, the conductance is independent of temperature
[18],

$\displaystyle\overline{\sigma}=\frac{e^{2}}{\hbar}\frac{\Gamma\sqrt{4t^{2}-\mu^{2}}}{\pi
t^{2}}\,.$ (20)

In order to understand the behavior of conductance flucutations, we must
calculate the single-particle covariances $\rho_{d,o}(\omega,\epsilon)$. This
may be done diagrammatically, only keeping diagrams to leading order in
$N^{-1}$. We do this by calculating the covariance of the Green’s function in
imaginary time and analytically continuing to the real axis. The calculation
of $\rho_{d}$ involves analytic continuation of the quantity
$\sum_{ij}G_{ii}(i\omega)G_{jj}(i\epsilon)$, and for $\rho_{o}$,
$\sum_{ij}G_{ij}(i\omega)G_{ji}(i\epsilon)$.

Diagrams that contribute to the covariance of the Green’s function consists of
diagrams of pairs of Green’s functions that are only connected along disorder
lines. The structure of these diagrams is shown in Fig. 1. The diagrammatic
structure of both the $\rho_{d}$ and $\rho_{o}$ fluctuations are similar -
both involve an infinite summation over a set of ladder diagrams, given in the
first figure in Fig. 1. The leading order contributions to $\rho_{o}$ are just
given by this set of diagrams. For $\rho_{d}$, two additional classes of
diagrams must be considered and are shown in Fig. 1. The first class yields an
$n$-fold degeneracy of ladders with $n$ rungs, and the second class gives
additional disorder averaging on either side of the ladder rungs.

Putting all this together, we obtain the final form for the Green’s function
covariances,

$\displaystyle g_{d}(i\omega,i\epsilon)$
$\displaystyle\equiv\frac{1}{N^{2}}\sum_{ij}\left(\overline{G_{ii}(i\omega)G_{jj}(i\epsilon)}-\overline{G_{ii}(i\omega)}\times\overline{G_{jj}(i\epsilon)}\right)$
(21)
$\displaystyle=\frac{1}{N^{2}}\frac{t^{2}G(i\omega)^{2}G(i\epsilon)^{2}}{\left[1-t^{2}G(i\omega)G(i\epsilon)\right]^{2}}\frac{1}{1-t^{2}G(i\omega)^{2}}\frac{1}{1-t^{2}G(i\epsilon)^{2}}$
$\displaystyle g_{o}(i\omega,i\epsilon)$
$\displaystyle\equiv\frac{1}{N^{2}}\sum_{ij}\left(\overline{G_{ij}(i\omega)G_{ji}(i\epsilon)}-\overline{G_{ij}(i\omega)}\times\overline{G_{ji}(i\epsilon)}\right)=\frac{1}{N}\frac{t^{2}G(i\omega)^{2}G(i\epsilon)^{2}}{1-t^{2}G(i\omega)G(i\epsilon)}$

where in the RHS, we use the average Green’s function

Figure 1: Ladder diagrams that contribute to the fluctuations of the single-
particle spectral function. The first class of diagrams contributes to both
the covariances $\rho_{d}$ and $\rho_{o}$, with the contribution to $\rho_{d}$
coming from the $i=j$ case. The last two classes only contribute to
$\rho_{d}$. Disorder-averaging of the single-particle hopping (SYK
interactions) is represented in red (blue).

$\displaystyle
G_{0}(i\omega)=\frac{i\omega+\mu}{2t^{2}}-i\frac{\text{sgn}(\omega)}{2t^{2}}\sqrt{4t^{2}+(\omega-i\mu)^{2}}\,.$
(22)

To obtain expressions for $\rho_{o,d}$, we analytically continue these to the
real axis,

$\displaystyle\rho_{\alpha}=-\frac{1}{4}\left[g_{\alpha}(\omega^{+},\epsilon^{+})+g_{\alpha}(\omega^{-},\epsilon^{-})-g_{\alpha}(\omega^{-},\epsilon^{+})-g_{\alpha}(\omega^{+},\epsilon^{-})\right]$
(23)

where $\omega^{\pm}\equiv\omega\pm i\eta$, $\eta\rightarrow 0$. The expression
for $\rho_{d}$ has been derived before using a similar diagrammatic approach
[39], although we are not aware of an analogous calculation for $\rho_{o}$.

From this analysis, we see that fluctuations arising from $\rho_{o}$ are
enhanced relative to the $\rho_{d}$ fluctuations by a factor of $N$, and hence
will be the main focus of our analysis. However, we will show that a more
careful analysis of $\rho_{d}$ will be necessary to recover UCF behavior at
zero temperature.

Due to the form of the average Green’s function, we find a singular behavior
for the Green’s function covariances in Eq. 21 for
$\absolutevalue{\omega-\epsilon}\rightarrow 0$, as

$\displaystyle 1-t^{2}G(\omega^{+})G(\epsilon^{-})$
$\displaystyle=\frac{1}{t}\left(-\eta+\frac{i}{2}(\omega-\epsilon)\right)+\order{(\omega/t)^{2},(\epsilon/t)^{2}}\,,$
(24) $\displaystyle\rho_{d}(\omega,\epsilon)$
$\displaystyle=-\frac{1}{8N^{2}}\operatorname{Re}\left[\frac{1}{(i(\omega-\epsilon)/2-\eta)^{2}}\right]$
$\displaystyle\rho_{o}(\omega,\epsilon)$
$\displaystyle=-\frac{1}{2Nt}\operatorname{Re}\left[\frac{1}{i(\omega-\epsilon)/2-\eta}\right]$

The above divergence holds for arbitrary chemical potential $\mu$. We see that
the $(\omega-\epsilon)^{-2}$ divergence in $\rho_{d}(\omega,\epsilon)$ is
independent of the energy scale $t$. The correlation function $\rho_{d}$
determines fluctuations of the single-particle energy levels - for the non-
interacting system, the distribution of single-particle energy levels is
determined by the Gaussian Unitary Ensemble (GUE) in which fluctuations are
known to take this universal form [40, 41, 39].

For $T\neq 0$, this divergence may be regulated by carefully taking the
$\eta\rightarrow 0$ limit in the analytic continuation to the real axis. We
state the calculation in a general form, for use later. For real-valued
functions $A(\omega)$, $B(\omega)$, and
$\rho(\omega-\epsilon)=\rho(\epsilon-\omega)$, we have the identity

$\int\differential{\omega}\differential{\epsilon}A(\omega)B(\epsilon)\rho(\omega-\epsilon)=\sqrt{2\pi}\int\differential{k}\operatorname{Re}\left[\widetilde{A}(k)\widetilde{B}^{*}(k)\right]\widetilde{\rho}(k)\,.$
(25)

where we define the Fourier transform
$\widetilde{A}(k)\equiv\frac{1}{\sqrt{2\pi}}\int
e^{-ikx}A(x)\differential{x}$.

The Fourier transform of the Green’s function covariances are:

$\displaystyle\widetilde{\rho}_{d}(k)$
$\displaystyle=\frac{1}{8N^{2}}\sqrt{\frac{\pi}{2}}\absolutevalue{k}\,,$ (26)
$\displaystyle\widetilde{\rho}_{o}(k)$
$\displaystyle=\frac{\sqrt{2\pi}}{4Nt}\,.$

This analysis for $\rho_{d}$ recovers the well-known Dyson-Mehta formula for
the variances of linear statistics in RMT [41, 40], and the more general
covariance formula for linear statistics [42]; however, these fluctuations are
a factor of $N^{-1}$ smaller than the contributions from $\rho_{o}$
fluctuations.

This formula yields the result for the conductance variance,

$\displaystyle\text{Var }\sigma=\left(\frac{\Gamma
e^{2}}{\hbar}\right)^{2}\frac{2}{3\pi tTN}\,,$ (27)

which agrees well with a numerical simulation, shown in Fig. 2. This
expression is valid for $\Gamma^{2}\ll NTt$, as suggested by the $T\rightarrow
0$ divergence. In order to obtain results for $T=0$, a more careful treatment
of the coupling to the leads is required.

Figure 2: We plot the mean and variance of both the conductance and
thermopower, calculated in the non-interacting ($J=0$) limit of our disordered
quantum dot and using Eq. LABEL:eq:transportFormula averaged over $100000$
realizations of the hoppings $t_{ij}$. We set the chemical potential
$\mu=0.33$. In this calculation, the Green’s function of the quantum dot is
solved independent of the leads. We set the strength of the leads coupling
$\Gamma=0.1$ in order for the mean thermopower and conductance to have
comparable magnitudes, although we emphasize that this value only appears as
an overall coefficient in the conductance. These numerical results are
compared with the analytic predictions given in Eq. 27 and Eq.
LABEL:eq:syk2ThermopowerVariance, which show good agreement.

To obtain a $T\rightarrow 0$ result, we must include the self-energy arising
from the coupling to the leads. The form of this correction is dependent on
the manner in which we choose the coupling. For uniform all-to-all couplings,
we have

$\displaystyle\Sigma_{ij}(i\omega)=2\int\differential{\epsilon}\frac{\rho(\epsilon)\absolutevalue{\lambda}^{2}}{i\omega-\epsilon}\approx\frac{2i\Gamma}{N}\text{sgn}(\omega)$
(28)

where $\rho$ is the density of states in the leads, which we approximate by
its value at the Fermi level. To leading order in $\Gamma$,

$\displaystyle\overline{G_{ij}(i\omega)}=\delta_{ij}G_{0}(i\omega)+\frac{2i\Gamma
G_{0}(i\omega)^{2}}{N}\text{sgn}(\omega)$ (29)

Using this result for $G$ in Eq. 21, we can directly evaluate the $T=0$
conductance fluctuations

$\displaystyle\Gamma^{2}\rho_{d}(0,0)$
$\displaystyle=-\frac{1}{32}+\order{\Gamma/t}\,,$ (30)
$\displaystyle\Gamma^{2}\rho_{o}(0,0)$ $\displaystyle=\order{\Gamma/t}\,,$
$\displaystyle\text{Var }\sigma(T=0)$
$\displaystyle=\left(\frac{e^{2}}{h}\right)^{2}\left(\frac{1}{8}+\order{\Gamma/t}\right)\,.$

Note that in this case, $\rho_{d}$ and $\rho_{o}$ contribute at the same
order; the $T=0$ divergences are regulated by an $\order{N^{-1}}$ self-energy,
and $\rho_{d}$ is more singular at $T=0$. We stress that this result is not
rigorous - as evident from the above results, this manner of including the
corrections from the leads is not done consistently in an $N^{-1}$ expansion.
A proper extrapolation down to $T=0$ necessitates, for example, the use of
supersymmetric techniques [34].

###  III.2 Thermopower statistics

Although less well-studied than conductance fluctuations, thermopower
fluctuations have been studied analytically for single-mode contacts at $T=0$
[43] and for broad contacts [44]. Experimental measurements [45, 46] have
found good agreement with these predictions. Our analysis will fall in a
distinct parameter regime to these results, where we consider a quantum dot
weakly coupled to its environment, at temperatures much larger than the
coupling strength.

In the free fermion limit, the mean thermopower vanishes linearly with
temperature [18]

$\displaystyle\overline{\Theta}=\frac{\pi^{2}T}{3e}\frac{\mu}{4t^{2}-\mu^{2}}\,.$
(31)

The linear temperature dependence is a consequence of the linear temperature
dependence of the entropy, and hence is generic for systems with quasiparticle
excitations.

In our framework, the statistical properties of the thermopower is determined
by the ratio of two random variables, $\mathcal{L}_{12}$ and
$\mathcal{L}_{11}$. As higher order moments are suppressed by additional
factors of $N^{-1}$, our transport coefficients are Gaussian to leading order
in $N^{-1}$. The thermopower distribution is then determined by the ratio of
two Gaussian statistics, which in general is non-Gaussian. Nevertheless, an
approximation to Gaussian is appropriate [47] for capturing small fluctuations
around the mean value, so long as the width of the Gaussian distribution is
small relative to the mean. We provide more details on this approximation in
Appendix C. A similar approach was used to characterize fluctuations of the
Fano factor in weakly-interacting quantum dots [48]. Such an approximation
requires knowledge of the covariance of the two quantities $\mathcal{L}_{12}$
and $\mathcal{L}_{11}$. This is given by

$\displaystyle\text{Cov}(\mathcal{L}_{11},\mathcal{L}_{12})$
$\displaystyle=-\left(\frac{\Gamma}{\pi\hbar}\right)^{2}\int\differential{\omega}\differential{\epsilon}\omega
f^{\prime}(\omega)f^{\prime}(\epsilon)\left[\rho_{d}(\omega,\epsilon)+\rho_{o}(\omega,\epsilon)\right]$
(32)

which vanishes by an application of Eq. 25. Therefore to leading order in
$N^{-1}$, the random variables $\mathcal{L}_{11}$ and $\mathcal{L}_{12}$ are
both uncorrelated and have a bivariate normal distribution, so we treat them
as independent. With this assumption, the typical fluctuations of $\Theta$
around its mean value
$\frac{\beta}{e}\frac{\overline{\mathcal{L}_{12}}}{\overline{\mathcal{L}_{11}}}$
are Gaussian with variance

$\displaystyle\frac{\text{Var }\Theta}{\overline{\Theta}^{2}}=\frac{\text{Var
}\mathcal{L}_{11}}{\overline{\mathcal{L}_{11}}^{2}}+\frac{\text{Var
}\mathcal{L}_{12}}{\overline{\mathcal{L}_{12}}^{2}}\,,$ (33)

with

$\displaystyle\text{Var }\mathcal{L}_{11}$
$\displaystyle=\left(\frac{\Gamma}{\hbar}\right)^{2}\frac{2}{3\pi
NTt}\quad\overline{\mathcal{L}}_{11}=\frac{\Gamma}{\hbar}\frac{\sqrt{4t^{2}-\mu^{2}}}{2\pi
t^{2}}$ (34) $\displaystyle\text{Var }\mathcal{L}_{12}$
$\displaystyle=\left(\frac{\Gamma}{\hbar}\right)^{2}\frac{(\pi^{2}-6)T}{9\pi
Nt}\quad\overline{\mathcal{L}_{12}}=-\frac{\Gamma}{\hbar}\frac{\pi\mu
T^{2}}{6t^{2}\sqrt{4t^{2}-\mu^{2}}}\,.$

This analytic prediction agrees well with the numerically calculated variance,
shown in Fig. 2. Similar to the conductance variance, the thermopower variance
scales as $T^{-1}$ at low temperature, although the fact that the mean value
scales linearly with temperature means that, in contrast to the conductance,
the variance normalized by the mean squared diverges as $T^{-3}$.

##  IV Pure SYK analysis

###  IV.1 Conductance statistics

We now move to an analysis of conductance fluctuations for a pure SYK model
($t=0$), where the average value takes the form at half-filling

$\displaystyle\overline{\sigma}$
$\displaystyle=\frac{e^{2}}{\hbar}\frac{0.72\Gamma}{\sqrt{JT}}$ (35)

(the exact value of the prefactor is
$2\sqrt{2}\pi^{-1/4}\Gamma(3/4)\Gamma(1/4)\approx 0.72$). Deviations away from
half-filling only constitute a change in the numerical coefficient. For full
generality, we present results for an $\text{SYK}_{q}$ model with $q$-fermion
interactions - the case $q=4$ is the one of experimental relevance. The
diagrammatic prescription for calculating the Green’s function covariances
$\rho_{d}$, $\rho_{o}$ remain the same, and we consider pairs of Green’s
functions that are only connected via disordered lines. The $N$ scaling of
disorder-connected diagrams has been considered in SYK-like models previously
[4, 49, 50, 51], although an explicit evaluation of such diagrams has only
been carried out for the off-diagonal covariance, $\rho_{o}$, in the Majorana
SYK model [50].

The simplest leading-order diagram, which contributes to both $\rho_{d}$ and
$\rho_{o}$, is shown on top in Fig. 3. This contributes to $\rho_{o}$ with
coefficient $N^{1-q}$ and $\rho_{d}$ with coefficient $N^{-q}$. The different
coefficients arise because the $\rho_{d}$ contribution appears with a factor
of $\delta_{ij}$. We find that $\rho_{d}$ contains additional “ladder”
diagrams as shown in the bottom of Fig. 3 that also contribute at
$\order{N^{-q}}$.

Figure 3: a) The leading-order diagram for a pure SYK model that contributes
to the Green’s function covariance. This contributes to $\rho_{o}$ with a
factor of $N^{1-q}$, and the specialized $i=j$ case contributes to $\rho_{d}$
with a factor of $N^{-q}$. b) For the diagonal covariance $\rho_{d}$, the
diagram in a) is the first in an infinite series of diagrams, generated from
the first by attaching ladder rungs to either the top or bottom diagram in the
manner shown here. We have deformed the diagram from a) in order to more
clearly illustrate the structure of the ladder rungs.

Both these covariances can be evaluated analytically in the conformal limit,
when $\beta J\gg 1$. To see this, we examine the first diagram, in the top of
Fig. 3. In the conformal limit, the Green’s functions take the following form:

$g^{R}(\overline{\omega},\overline{T})=-ie^{-i\theta}\left(\frac{\pi}{\cos
2\theta}\right)^{1/4}\left(\frac{1}{2\pi\overline{T}}\right)^{1/2}\frac{\Gamma\left(\frac{1}{4}-\frac{i\overline{\omega}}{2\pi\overline{T}}+i\mathcal{E}\right)}{\Gamma\left(\frac{3}{4}-\frac{i\overline{\omega}}{2\pi\overline{T}}+i\mathcal{E}\right)}$
(36)

where $\theta$ and $\mathcal{E}$ characterize the spectral asymmetry and are
related to the total charge $\mathcal{Q}$ by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{2\pi}\ln\frac{\sin(\pi/4+\theta)}{\sin(\pi/4-\theta)}\,,$
(37) $\displaystyle\mathcal{Q}$
$\displaystyle=\frac{1}{2}-\frac{\theta}{\pi}-\frac{\sin(2\theta)}{4}\,.$

We have the bounds $-\frac{\pi}{4}\leqslant\theta\leqslant\frac{\pi}{4}$ which
implies $0\leqslant\mathcal{Q}\leqslant 1$, and the particle-hole symmetric
point is $\mathcal{Q}=\frac{1}{2}$. Note that in contrast to the free fermion
case, the SYK solution is most easily analyzed in the canonical ensemble with
fixed charge $\mathcal{Q}$. These Green’s functions satisfy the Schwinger-
Dyson (SD) equation $\Sigma(\omega)G(\omega)=-1$,
$\Sigma(\tau_{1},\tau_{2})=J^{2}G\left(\tau_{1},\tau_{2}\right)^{\frac{q}{2}}(-G(\tau_{2},\tau_{1}))^{\frac{q}{2}-1}$.
We can evaluate the $\tau$ integrals of the top and bottom part of the Feynman
diagram independently. Making use of the conformal SD equations, we find that
each of these parts evaluates to

$\displaystyle
J^{2}\int\differential{\tau_{a}}\differential{\tau_{b}}G(\tau_{1},\tau_{a})G(\tau_{a},\tau_{b})^{\frac{q}{2}}(-G(\tau_{b},\tau_{a}))^{\frac{q}{2}-1}G(\tau_{b},\tau_{2})$
(38)
$\displaystyle=\int\differential{\tau_{a}}\differential{\tau_{b}}G(\tau_{1},\tau_{a})\Sigma(\tau_{a},\tau_{b})G(\tau_{b},\tau_{2})=-G(\tau_{1},\tau_{2})\,.$

A careful analysis of combinatoric factors from the disorder lines yields the
result

$\displaystyle\rho_{o}(\omega,\epsilon)\frac{(q/2)!(q/2-1)!}{N^{q-1}}\operatorname{Im}\left[G^{R}(\omega)\right]\operatorname{Im}\left[G^{R}(\epsilon)\right]\,.$
(39)

For calculating $\rho_{d}$, the summation of ladder diagrams in Fig. 3 must be
carried out. These ladder diagrams are well-studied in the SYK literature; in
particular, evaluation in the strict conformal limit often leads to a
divergent summation, with the regularizing near-conformal corrections taking a
universal form that reflect the underlying dual quantum gravity description.
This divergent summation is a consequence of “resonant” eigenfunctions of the
ladder kernel which have eigenvalue unity. Remarkably, there is no such effect
in this class of ladder diagrams - because of the relation in Eq. 38, we find
that the conformal Green’s function is an exact eigenfunction with eigenvalue
$q-1$, and therefore no resonance occurs. Because of this, the ladder diagrams
may be evaluated via a geometric series to obtain the result

$\displaystyle\rho_{d}(\omega,\epsilon)\frac{(q/2)!(q/2-1)!}{q^{2}N^{q}}\operatorname{Im}\left[G^{R}(\omega)\right]\operatorname{Im}\left[G^{R}(\epsilon)\right]\,.$
(40)

To leading order in $N^{-1}$, the conductance fluctuations are driven by
$\rho_{o}$. The fact that $\rho_{o}$ factorizes into two copies of the
spectral function leads to the simple result,

$\displaystyle\frac{\text{Var
}\sigma}{\overline{\sigma}^{2}}=\frac{(q/2)!(q/2-1)!}{N^{q-1}}\,.$ (41)

The statement that the variance divided by the mean squared takes the above
form holds for any linear statistic $A$ of the spectral function,
$A=\int_{=\infty}^{\infty}\differential{\omega}A(\omega)\operatorname{Im}G^{R}(\omega)$.

For the SYK model with four-fermion interactions, this gives

$\displaystyle\text{Var
}\sigma=\left(\frac{e^{2}}{\hbar}\right)^{2}\frac{1.04\Gamma^{2}}{N^{3}JT}\,.$
(42)

We compare this result to numerical calculations of the conductivity variance
using exact diagonalization, shown in Fig. 4. We also plot the mean values,
which show decent agreement with their respective analytic predictions despite
the relatively small system sizes.

Figure 4: We present numerical results for the conductance and thermopower of
a complex SYK model, for even system sizes $6\leqslant N\leqslant 12$. All
results are averaged over $10^{5}$ realizations with $J=1$, $\mu=0.05$. a) The
average thermopower, and conformal prediction. b) The average conductance, and
conformal prediction. c) The system size scaling of the conductance and
thermopower variance, obtained by fitting the variance as a function of $N$ to
a power law at each temperature. d) We Temperature dependence of the
normalized conductance and thermopower variance for $6\leqslant N\leqslant
12$, both rescaled by their appropriate system size - $N^{-3}$ for conductance
and $N^{-2}$ for thermopower. Darker plots indicate larger system sizes.

Recall that Eq. 41 is only valid in the conformal limit, where $\beta J\gg 1$.
For finite size systems, we also require $\beta J\ll N$ due to Schwarzian
fluctuations setting in at lower temperatures [52, 53, 49, 54]. Exact
diagonalization studies are restricted to small system sizes, with $N=12$ the
maximum size studied here. This implies a rather narrow temperature window
where conformal behavior could be expected. We are unable to establish the
constant temperature dependence predicted by Eq. 41; however, the predicted
$N^{-3}$ scaling is expected to hold even at higher temperatures, away from
the conformal limit, and this is validated by our numerical results.

###  IV.2 Thermopower statistics

We now analyze statistics of the thermopower, where the extensive entropy
leads to a constant thermopower

$\displaystyle\overline{\Theta}=\frac{4\pi}{3e}\mathcal{E}\,.$ (43)

Recall that the thermopower is given by the ratio of two random variables,
whose linear covariance vanished in the free fermion limit. Strikingly, the
opposite behavior is true for an SYK model. To quantify this, we examine the
Pearson correlation coefficient of the two random variables $A$ and $B$,

$\displaystyle r_{A,B}\equiv\frac{\text{Cov}(A,B)}{\sqrt{\text{Var
}A\times\text{Var }B}}\,.$ (44)

$r_{A,B}$ lies between $-1$ and $+1$ and measures the degree of correlation
between two random variables. For the SYK model, a particular property of
$\rho_{o}$ is that it that it factorizes into a product
$\alpha(\omega)\alpha(\epsilon)$ to leading order in $N^{-1}$. Therefore, in
sharp contrast to the Fermi liquid regime where the variables
$\mathcal{L}_{11}$ and $\mathcal{L}_{12}$ were uncorrelated, we generically
expect $r_{A,B}=1-\mathcal{O}(N^{-1})$.

Despite this, we may still approximate our distribution as Gaussian. The
approximation to normality of the distribution of two correlated Gaussian
random variables follows along similar lines as the uncorrelated ratio [47],
which we also discuss in more detail in Appendix C. Defining $r$ as the
correlation coefficient between $\mathcal{L}_{11}$ and $\mathcal{L}_{12}$,

$\displaystyle\frac{\text{Var }\Theta}{\overline{\Theta}^{2}}=\frac{\text{Var
}\mathcal{L}_{11}}{\overline{\mathcal{L}_{11}}^{2}}+\frac{\text{Var
}\mathcal{L}_{12}}{\overline{\mathcal{L}_{12}}^{2}}-\frac{2r\sqrt{\text{Var
}\mathcal{L}_{11}\times\text{Var
}\mathcal{L}_{12}}}{\overline{\mathcal{L}_{11}}\overline{\mathcal{L}_{12}}}\,.$
(45)

Both $\mathcal{L}_{11}$ and $\mathcal{L}_{12}$ are linear statistics, so the
conformal prediction of $\frac{\text{Var
}\mathcal{L}_{11}}{\overline{\mathcal{L}_{11}}^{2}}=\frac{\text{Var
}\mathcal{L}_{12}}{\overline{\mathcal{L}_{12}}^{2}}$ leads to a vanishing
thermopower variance. The leading order non-zero result in the conformal limit
is hence suppressed by an additional factor of $N^{-1}$. However, high-
temperature non-conformal corrections will still give an $\order{N^{1-q}}$
contribution.

Surprisingly, we find strong disagreement between this prediction and the
exact diagonalization in Fig. 4. The thermopower variance in the temperature
regime $\beta J\gg N^{-1}$ is well fit by a $N^{-2}$ scaling, rather than the
$N^{-3}$ high-temperature contribution or the $N^{-4}$ conformal contribution.
This arises due to an anomalous $N^{-2}$ scaling in the variance of the
numerator, $\mathcal{L}_{12}$. As this quantity is proportional to the
particle-hole asymmetry, we conjecture that this is related to additional
fluctuations in the asymmetry not captured by our diagrammatic approach.

##  V Interplay between hoppings and interactions

In the previous sections, we have derived results for the conductance variance
for both the limiting cases of non-interacting fermions with random hopping
and a pure SYK model. In this section, we more carefully analyze the
physically-relevant model with includes both random hopping and SYK terms.

Analysis of crossover behavior in these models has been performed previously
[55, 56, 18] for the average values of observables. The conclusion of these
analyses is that there exists a coherence energy scale
$E_{\text{coh}}\equiv\frac{t^{2}}{J}$ such that transport properties closely
resemble the free fermion model for temperatures $T\ll E_{\text{coh}}$, with
SYK behavior emerging for $T\gg E_{\text{coh}}$ (throughout this analysis, we
assume $T\ll t\,,J$). The source of this behavior lies in the solution to the
set of Schwinger-Dyson equations for the average value of the Green’s
function, which is exact in the large-$N$ limit:

$\displaystyle G(i\omega_{n})^{-1}$
$\displaystyle=-i\omega_{n}+\mu-t^{2}G(i\omega_{n})-\Sigma(i\omega_{n})\,,$
(46) $\displaystyle\Sigma(\tau)$ $\displaystyle=-J^{2}G^{2}(\tau)G(-\tau)\,.$

It is this Green’s function that displays a crossover at $T\sim
E_{\text{coh}}$ from the free fermion-like solution to an SYK-like solution,
which in turn leads to a crossover of the average values of transport
properties.

In contrast, we claim that the variance of transport quantities displays a
qualitatively different type of crossover behavior. This is a consequence of
the free fermion variance in Sec III and the SYK variance in Sec IV containing
different powers of $N$. Fluctuations driven by the randomness in SYK
interactions are strongly suppressed relative to fluctuations driven by the
random single-particle hopping. As a result, to leading order in $N^{-1}$, the
free fermion Feynman diagrams in Fig. 1 \- which exist for any arbitrarily
small random hopping - are always the relevant ones for calculating
fluctuation properties so long as the ratio $\frac{t}{J}$ does not scale with
some inverse power of $N$. The effect of SYK interactions is to renormalize
the average Green’s functions, such that the Green’s function that appear in
Eq. 21 are given by the solution to Eq. 46 rather than just the free fermion
result. One can verify that to leading order in $N^{-1}$, the inclusion of SYK
interactions does not modify the diagrammatic structure any further than this,
with the exception of a class of diagrams illustrated in Fig. 5 \- these
diagrams only contribute to $\rho_{d}$ and hence will not be relevant for our
analysis.

Figure 5: Ladder diagrams that contribute to the fluctuations of the single-
particle spectral function to leading order in $N^{-1}$, for a model that
includes both random single-particle hopping and SYK interactions. Disorder-
averaging of the single-particle hopping (SYK interactions) is represented in
red (blue). The structure of the diagrams are largely identical to the free
fermion case illustrated in Fig. 1, with the SYK interactions having the
effect of renormalizing the average Green’s functions. An exception to this is
the additional set of diagrams, illustrated in the last diagram, which are
qualitatively distinct from the free fermion limit. These diagrams only
contribute to the diagonal covariance $\rho_{d}$ and hence will be neglected
as they are suppressed by a factor of $N^{-1}$ relative to the off-diagonal
covariances.

The key difference that results in the average values of thermoelectric
properties being described by pure SYK for $T\gg E_{\text{coh}}$ and not their
variances may be best understood conceptually within the framework of the
$(G,\Sigma)$ action, which is worked out explicitly in Appendix B. The
intuition is as follows. For systems such as $H_{\text{dot}}$ with random all-
to-all couplings, the fermionic degrees of freedom may be integrated out and
the problem reformulated as a path integral over bilocal fields
$G(\tau_{1},\tau_{2})$, $\Sigma(\tau_{1},\tau_{2})$, with an action that
includes an explicit pre-factor of $N$; hence, the large-$N$ solution is
described by the saddle point value of this action, which is precisely Eq. 46.
The disorder-averaged spectral function, and in turn the average values of
thermoelectric quantities, depend solely on this saddle-point solution. This
is not true for fluctuations, which are subleading in $N^{-1}$ and is governed
by replica off-diagonal fluctuations around the large-$N$ saddle point. The
structure of the perturbation theory around the saddle point may be completely
modified by the presence of a hopping term $t$ \- Feynman diagrams
proportional to $t$ may appear at lower orders in $N^{-1}$, and whose
contributions will a priori be dominant even in a parameter regime where the
saddle point is well-described by the $t=0$ solution.

Our approach to studying the behavior of transport fluctuations for an
interacting quantum dot will again involve calculating the single-particle
covariance $\rho_{d,o}(\omega,\epsilon)$. We will work in the regime where
$\omega,T\ll t,J$, and the average Green’s function takes the universal form
[55]

$\displaystyle G(\omega,T)=\frac{1}{t}g\left(\frac{\omega}{E_{\rm
coh}},\frac{T}{E_{\rm
coh}}\right)\equiv\frac{1}{t}g(\overline{\omega},\overline{T})\,,$ (47)

where we define the dimensionless quantities
$\overline{\omega}\equiv\omega/E_{\text{coh}}$, $\overline{T}\equiv
T/E_{\text{coh}}$. We find that the system sizes accessible to exact
diagonalization are inadequate for establishing even the approximate crossover
of the average Green’s function; due to the narrow temperature window
$N^{-1}\ll T\ll J\,,t$ where our analysis is valid, any crossover behavior is
obscured by combination of high temperature or finite size effects. As a
consequence, numerical results in this section will be restricted to self-
consistent solutions of the Schwinger-Dyson equations given in Eq. 46.

###  V.1 Fermi liquid regime

For $\overline{T},\overline{\omega}\ll 1$, it is known [55] that
$g^{R}(\overline{\omega},\overline{T})$ has a Fermi liquid behavior. These
properties can most simply stated at half filling ($\mu=0$), where the Fermi
liquid nature implies

$\displaystyle g^{R}(\overline{\omega}\ll 1,\overline{T}\ll 1)\approx-i\,.$
(48)

This behavior is determined by Luttinger’s theorem, which for a generic charge
$\mathcal{Q}$ says that

$\displaystyle\mu(\mathcal{Q})-\Sigma(i0^{+})=\mu_{0}(\mathcal{Q})$ (49)

where $\mu(\mathcal{Q})$ is the chemical potential necessary to tune to the
charge $\mathcal{Q}$, and $\mu_{0}(\mathcal{Q})$ is that same value for the
non-interacting ($J=0$) system. This fixes $G^{R}(\omega\rightarrow
0,T\rightarrow 0)$ to be that of the non-interacting Green’s function, the
latter of which we know has the property
$\absolutevalue{g^{R}(\omega\rightarrow 0,T\rightarrow 0)}^{2}=1$ for generic
filling. This property is sufficient for recovering the temperature-
independent non-interacting prediction for the mean value of the conductance
at low temperature, given in Eq. 20 and likewise properly recovers the small
$\omega,\epsilon$ divergence of $\rho_{d,o}$ given in Eq.
LABEL:eq:divergences. Although an explicit calculation of the conductance
variance requires knowledge of the small frequency and temperature behavior of
$g^{R}$, which is not fixed by Luttinger’s theorem, the degree of the
$T\rightarrow 0$ divergence and the assumption that small
frequency/temperature corrections appear at linear order in
$\overline{\omega},\overline{T}$ imply from dimensional analysis that

$\displaystyle\text{Var }\sigma(T\ll E_{\text{coh}})$
$\displaystyle\propto\frac{1}{t^{2}N\overline{T}}=\frac{1}{NJT}\,,$ (50)
$\displaystyle\frac{\text{Var }\sigma(T\ll E_{\text{coh}})}{\sigma(T\ll
E_{\text{coh}})^{2}}$ $\displaystyle\propto\frac{1}{N\overline{T}}\,.$

This result is confirmed by calculating the conductance variance using the
Green’s function $G^{R}$ obtained from numerically solving the large-$N$
Schwinger-Dyson equations, shown in Fig. 7.

###  V.2 SYK Regime

We now analyze the conductance fluctuations for $\overline{T}\gg 1$, where the
average Green’s function approaches the conformal SYK result given in Eq. 36.
The mean value of the conductance is then given by the pure SYK result in Eq.
35. Using this form of the Green’s function, we find that the
$(\omega-\epsilon)^{-1}$ divergence of $\rho_{o}(\omega,\epsilon)$ is no
longer present. The infinite sum of ladder diagrams that yields $\rho_{o}$ is
convergent for large $\overline{T}$. Expanding in powers of
$\overline{T}^{-1}$, we obtain the leading-order expression

$\displaystyle\text{Var }\sigma(T\gg E_{\text{coh}})$
$\displaystyle=\left(\frac{e^{2}}{\hbar}\frac{\Gamma}{t\overline{T}}\right)^{2}\frac{1}{N\pi^{5}}\frac{1}{2\cos(2\theta)}\left[\int_{-\infty}^{\infty}\differential{x}\frac{e^{x}}{\left(1+e^{x}\right)^{2}}\text{Im}\left[h(x)^{2}\right]\right]^{2}\,,$
(51) $\displaystyle h(x)$ $\displaystyle\equiv
e^{-i\theta}\frac{\Gamma\left(\frac{1}{4}-\frac{ix}{2\pi}+i\mathcal{E}\right)}{\Gamma\left(\frac{3}{4}-\frac{ix}{2\pi}+i\mathcal{E}\right)}\,.$

This integral must be done numerically; however, one can see that at the
particle-hole symmetric point ($\theta=\mathcal{E}=0$, $\mathcal{Q}=1/2$), the
integrand vanishes. We emphasize that this expression for the conductance
variance is obtained by using the conformal SYK form of the Green’s function
and taking to leading order a large-$\overline{T}$ expansion of the integral
for the conductance variance, the latter of which is not a homogeneous
function of $\overline{T}$. In particular, Eq. 51 does not imply that the
conductance variance vanishes exactly in the conformal limit when
$\mathcal{E}=0$. Rather, the variance for $\mathcal{E}=0$ is given by a
subleading $\overline{T}^{-3}$ term.

Figure 6: We plot the numerical coefficient of the leading-order conductance
variance in the conformal SYK limit, obtained by a numerical evaluation of the
integral in Eq. 51, along with a quadratic approximation
$4.05\mathcal{E}^{2}$.

For general $\mathcal{Q}$, we find that the resulting expression is well-fit,
see Fig. 6 by the function

$\displaystyle\text{Var }\sigma(T\gg E_{\text{coh}})$
$\displaystyle\approx\left(\frac{e^{2}}{\hbar}\frac{\Gamma}{t\overline{T}}\right)^{2}\times\frac{2.02\mathcal{E}^{2}}{N}\,,$
(52) $\displaystyle\frac{\text{Var }\sigma(T\gg E_{\text{coh}})}{\sigma(T\gg
E_{\text{coh}})^{2}}$
$\displaystyle\approx\frac{3.91\mathcal{E}^{2}}{N\overline{T}}\,.$

We see that the conductance variance normalized by the mean squared has a
$T^{-1}$ scaling, identical to the Fermi liquid regime. However, both
quantities individually have distinct behavior, with the conductance variance
scaling as $T^{-2}$ for $T\gg E_{\text{coh}}$ in contrast to the $T^{-1}$
scaling for $T\ll E_{\text{coh}}$. As an aside, we state the generalization to
an $\text{SYK}_{q}$ model with $q$-fermion interactions; using the conformal
Green’s function gives a $\overline{T}^{\frac{8}{q}-4}$ scaling of the
conductance variance, and a $\overline{T}^{\frac{4}{q}-2}$ scaling of the
normalized conductance variance.

This crossover behavior is demonstrated in Fig. 7, where we solve for the
conductance variance given the form of the Green’s function covariance in Eq.
21, where we use the average Green’s function $G^{R}(\omega)$ obtained from a
full self-consistent solution of the Schwinger-Dyson equations in real time.
Details on the numerical implementation for solving the real-time Schwinger-
Dyson equations can be found in [56]. We note a unique difficulty in
calculating the conductance variance not present in the average value, which
comes from the denominator $1-t^{2}G^{R}(\omega)G^{A}(\omega)$ in the Green’s
function covariance. As discussed previously, it is characteristic of a Fermi
liquid that this denominator goes to zero as $T\rightarrow 0$. As a
consequence, the accuracy with which one must numerically solve for
$G^{R}(\omega)$ diverges as $T\rightarrow 0$; small errors at low temperatures
can easily lead to an unphysical divergence in the conductance variance. Our
self-consistent solution for $G^{R}(\omega)$ utilizes a grid of $2^{28}$
frequency points on the real axis, which gives a sufficiently accurate
solution down to $T/E_{\text{coh}}\approx 0.03$ and is enough to recover the
predicted $T^{-1}$ scaling at low temperatures.

Figure 7: For parameters $J=10$, $t=0.1$, $\mathcal{Q}=0.4$, $N=30$, and
$\Gamma=0.1$, we numerically solve for the leading order contribution to the
conductance variance in the large-$N$ limit by solving the Schwinger-Dyson
equations for the average Green’s function over a range of temperatures. We
demonstrate a crossover from $T^{-1}$ behavior at low temperatures, indicative
of Fermi liquid behavior, to a more rapid $T^{-2}$ falloff at higher
temperatures which reflects the average Green’s function approaching the
conformal SYK form.

###  V.3 Thermopower statistics

The mean thermopower in a model with both random hopping and SYK interactions
displays a crossover from the linear temperature scaling characteristic of a
Fermi liquid for $T\ll E_{\text{coh}}$ to the constant SYK value for $T\gg
E_{\text{coh}}$. The coefficient of the mean thermopower in the Fermi liquid
regime receives a renormalization due to the presence of SYK interactions,
from $\overline{\Theta}\sim(et)^{-1}T$ in the free fermion model to
$\overline{\Theta}\sim(eE_{\text{coh}})^{-1}T$. This is not true for the mean
conductance, whose value for $T\rightarrow 0$ is determined by the zero-
frequency spectral density and is fixed by Luttinger’s theorem, Eq. 48.

We now discuss the crossover behavior of the thermopower variance. For $T\ll
E_{\text{coh}}$, the thermopower variance follows from the free fermion
analysis in Section III and diverges as $\overline{T}^{-1}$ for low
temperatures, albeit with a renormalized coefficient. For $T\gg
E_{\text{coh}}$, we find that the Pearson correlation coefficient $r$ between
$\mathcal{L}_{11}$ and $\mathcal{L}_{12}$ is $1$ to leading order in
$\overline{T}$. We apply Eq. 45, which gives the thermopower variance in terms
of $r$ and the statistics of $\mathcal{L}_{11}$ and $\mathcal{L}_{12}$, where
now we have

$\displaystyle\text{Var }\mathcal{L}_{11}$
$\displaystyle=\left(\frac{\Gamma}{\hbar}\right)^{2}\times\frac{2.02\mathcal{E}^{2}}{Nt^{2}\overline{T}^{2}}\,,\quad\overline{\mathcal{L}_{11}}=\frac{\Gamma}{\hbar}\frac{0.72}{t\sqrt{\overline{T}}}\,,$
(53) $\displaystyle\text{Var }\mathcal{L}_{12}$
$\displaystyle=\left(\frac{\Gamma}{\hbar}\right)^{2}\times\frac{0.07}{Nt^{2}}\,,\quad\overline{\mathcal{L}_{12}}=\frac{\Gamma}{\hbar}\frac{3.01\overline{T}^{1/2}\mathcal{E}}{t}\,.$

All of the terms in Eq. 45 decay as $\overline{T}^{-1}$, which implies that in
the limit $r=1$,

$\displaystyle\frac{\text{Var
}\Theta}{\overline{\Theta}^{2}}=\frac{1}{N\overline{T}}\left(1.97\absolutevalue{\mathcal{E}}-0.09\absolutevalue{\mathcal{E}}^{-1}\right)^{2}\,.$
(54)

The coefficient is rather striking, as it predicts a suppression of this
leading-order variance at a critical value of the particle-hole asymmetry
$\absolutevalue{\mathcal{E}_{c}}\approx 0.24$. Recall that this leading-order
suppression happens generically for a pure SYK model - this is a consequence
of expanding around the limit of perfect correlation between
$\mathcal{L}_{11}$ and $\mathcal{L}_{12}$, along with the identity
$\frac{\text{Var}\mathcal{L}_{11}}{\overline{\mathcal{L}_{11}}^{2}}=\frac{\text{Var}\mathcal{L}_{12}}{\overline{\mathcal{L}_{12}}^{2}}$.
The latter identity is not true generically in this model, but only occurs at
the aforementioned fine-tuned value $\mathcal{E}_{c}$. This value of
$\mathcal{E}$ corresponds to a rather large particle-hole asymmetry however,
$\mathcal{Q}_{c}\approx\frac{1}{2}\pm 0.41$, and is hence not easily
accessible.

##  VI Conclusion

We have analyzed the fluctuations of thermoelectric transport properties in
strongly-correlated quantum dots. Despite the apparent simplicity of our
microscopic model due to its exact large-$N$ solution, this saddle point only
describes the mean value of transport quantities; higher-order moments are
controlled by replica off-diagonal fluctuations around this saddle point, and
as such require a more unconventional analysis. We find distinct system size
scalings for these fluctuations in a free fermion model $(N^{-1})$ and an SYK
model $(N^{-3})$. The SYK prediction is qualitatively changed by the inclusion
of a small random hopping, which we find is able to drive conductance
fluctuations at the same order as the free fermion prediction. However, we
still find distinct temperature scalings, with a $T^{-2}$ suppression for
temperatures above the coherence energy in contrast to the $T^{-1}$ scaling at
lower temperatures predicted by the free fermion result.

Our main analytic results for the conductance, $\sigma$ were summarized in
Section I. We also computed the thermopower, $\Theta$. The mean thermopower
vanishes linearly with $T$ in the Fermi liquid regime (see Eq. 31), while the
SYK regime has a $T$-independent thermopower (see Eq. 43). Furthermore, the
finite $N$ Schwarzian corrections are quite small for the mean thermopower in
the SYK regime [18]. These features make the thermopower an ideal probe for
detecting the SYK regime in experiments. However, analytic computations of the
sample-to-sample fluctuations in the thermopower are not straightforward
because the expression for the thermopower involves the ratio of electron
Green’s functions. We made partial analytic progress assuming small Gaussian
fluctuations about the mean of both the numerator and the denominator, and
also obtained numerical exact-diagonalization results for small values of $N$.
Our main results are as follows. For a free fermion model, the thermopower
variance scales as $t\left(NT\right)^{-1}$, in good agreement with numerical
results. For a pure SYK model, we find surprisingly that the leading order
$N^{-3}$ contribution to the thermopower variance vanishes in the conformal
limit ($T\ll J$) due to perfect correlation between the numerator and
denominator. Fluctuations in this regime are hence governed by a combination
of high-temperature and $\order{N^{-4}}$ corrections, although we are unable
to verify this behavior numerically due to anomalous $\order{N^{-2}}$
fluctuations. For a model with both random hopping and SYK interactions, our
predictions once again are qualitatively modified. The scaling of the variance
in the low temperature Fermi liquid regime is suppressed from the free fermion
result $t\left(NT\right)^{-1}$ by an additional factor of $t/J$. In the SYK
regime, the scaling is identical, albeit arising from distinct mechanisms. A
noteworthy feature in the SYK regime is that this leading-order variance
vanishes at a critical value of the particle-hole asymmetry $\mathcal{E}_{c}$,
in which case the first non-zero contribution scales as
$N^{-1}(T/E_{\text{coh}})^{-2}$.

A more careful treatment of the effects of the coupling between the quantum
dot and the leads may reveal richer physics. In this work, we restrict our
parameter regime to a “closed” quantum dot, where the coupling to the leads is
the smallest energy scale in the system and transport quantities follow from
the properties of the isolated quantum dot. A more robust framework for
treating the effects of the leads can be developed by treating both the
single-particle hopping in the leads and the coupling to the quantum dot as
random variables, for which an exact (in the large-$N$ limit) set of
Schwinger-Dyson equations can be obtained for the non-equilibrium Green’s
functions [57]. The mean value of the conductance has been studied using this
framework, although the effects of single-particle hopping within the quantum
dot were not considered. In addition to treating conductance fluctuations
within this framework, an analysis of the effects of inter-dot single-particle
hopping, which was not considered in [57], may lead to new predictions even in
the average value of transport properties.

The nature of conductance fluctuations for a pure SYK model is also deserving
of further analysis. The results we present are confined to the conformal
regime. Deviations from this prediction at higher temperatures can be captured
by an analysis of the large-$N$ numerical solution to the Schwinger-Dyson
equations, and low-temperature deviations may be understood analytically
through Schwarzian fluctuations. This analysis is also expected to give
greater agreement with numerical results for small system sizes, where clear
agreement with the conformal prediction is absent.

### Acknowledgements

We thank Alex Kruchkov for significant discussions at the initial stages of
this work. We also thank Yigal Meir for helpful comments. This research was
supported by the U.S. National Science Foundation grant No. DMR-2245246 and by
the Simons Collaboration on Ultra-Quantum Matter which is a grant from the
Simons Foundation (651440, S.S.). PK and LA acknowledge support from ONR MURI
(N00014-21-1-2537).

##  Appendix A Path integral calculation of fluctuations

In this Appendix, we review the procedure for calculating the fluctuations of
observables in disordered systems using the path integral approach.

Calculating statistical quantities in disordered systems, such as averages and
variances, is in general a non-trivial task. This arises from the fact that
correlation functions such as $G(\tau-\tau^{\prime})$ for a given disorder
realization $J_{ijkl}$ (this notation is specific to an SYK model, which we
will use without loss of generality) are given by functional integrals of the
form

$\displaystyle
G(\tau-\tau^{\prime})=\frac{1}{N}\frac{\int\mathcal{D}c^{\dagger}\mathcal{D}c\,\sum_{i}c_{i}^{\dagger}(\tau)c_{i}(\tau^{\prime})e^{-S[c\,,c^{\dagger}\,,J_{ijkl}]}}{\int\mathcal{D}c^{\dagger}\mathcal{D}c\,e^{-S[c\,,c^{\dagger}\,,J_{ijkl}]}}\,.$
(55)

The mean of this quantity over an ensemble $P(J_{ijkl})$ is given by
integrating it over all realizations of $J_{ijkl}$. This averaging cannot
simply be done, as Eq. 55 is a ratio of two quantities. What can be done
analytically is carry out the average of the numerator and denominator
separately - this constitutes treating the random variables $J_{ijkl}$ on the
same footing as our physical variables $c_{i}^{\dagger}\,,c_{i}$. Treating the
disorder average properly requires techniques such as the replica trick [58],
which we will employ here. Supersymmetric techniques have also been developed
for dealing with these averages [59], which is the primary method used for
calculating conductance fluctuations of free electrons and generally yields
more reliable results than the replica approach, the latter of which requires
a generally-uncontrolled analytical continuation of the number of replicas
$M\rightarrow 0$. However, these supersymmetric techniques are not appropriate
for including the effects of strong interactions. Recent advances have
generalized these supersymmetry techniques to a particular variant of the SYK
model [60], and an interesting direction for future research would be to see
whether such an approach is applicable to our model or a variant thereof that
would allow for more controlled calculations of transport fluctuations.

Here, we make explicit the setup we use to calculate fluctuations of
quantities like $G(i\omega)$. What we are interested in is the covariance of
the Green’s function at different frequencies, such as
$\frac{1}{N^{2}}\sum_{ij}\left[\overline{G_{ii}(i\omega)G_{jj}(i\epsilon)}-\overline{G_{ii}(i\omega)}\,\overline{G_{jj}(i\epsilon)}\right]$.
Using the replica trick, we can rewrite the product of Green’s functions
$G(\tau_{1}-\tau_{2})G(\tau_{3}-\tau_{4})$ as a functional integral taken over
two copies of fermionic variables,$c_{i}^{a}\,,c_{i}^{\dagger
a}\,,\widetilde{c}_{i}^{a^{\prime}}\,,\widetilde{c}_{i}^{\dagger a^{\prime}}$,
with $i$ a site index and $a\,,a^{\prime}$ replica indices,

$\displaystyle\lim_{M\,,M^{\prime}\rightarrow
0}\frac{1}{N^{2}MM^{\prime}}\sum_{\begin{subarray}{c}1<a<M\\\
1<a^{\prime}<M^{\prime}\end{subarray}}\int\sum_{i,j}c_{i}^{\dagger
a}(\tau_{1})c_{i}^{a}(\tau_{2})\widetilde{c}_{j}^{\dagger
a^{\prime}}(\tau_{3})\widetilde{c}_{j}^{a^{\prime}}(\tau_{4})e^{-\sum_{a}S[c_{i}^{\dagger
a}\,,c_{i}^{a},J_{ijkl}]-\sum_{a^{\prime}}S[\widetilde{c}_{i}^{\dagger
a^{\prime}}\,,\widetilde{c}_{i}^{a^{\prime}},J_{ijkl}]}$ (56)

We can dispense of the independent replica summations and the distinction
between $c$ and $\widetilde{c}$ by combining them into an enlarged summation,

$\displaystyle\lim_{M\rightarrow
0}\frac{1}{N^{2}M^{2}}\sum_{\begin{subarray}{c}1<a\,,b<M\\\ a\neq
b\end{subarray}}\int\sum_{i,j}c_{i}^{\dagger
a}(\tau_{1})c_{i}^{a}(\tau_{2})c_{j}^{\dagger
b}(\tau_{3})c_{j}^{b}(\tau_{4})e^{-\sum_{d}S[c_{i}^{\dagger
d}\,,c_{i}^{d},J_{ijkl}]}$ (57)

The action $S$ is a function of the random variables $J_{ijkl}$, and the
disorder average is performed over the above quantity. Doing this induces
interactions between the different replicas. Subtracting off the disconnected
contribution,
$\overline{G(\tau_{1}-\tau_{2})}\,\overline{G(\tau_{3}-\tau_{4})}$ constitutes
disregarding contributions that do not contain any interactions between the
two replica indices. An analogous treatment of the off-diagonal covariance,
$\frac{1}{N^{2}}\sum_{ij}\left[\overline{G_{ij}(i\omega)G_{ji}(i\epsilon)}-\overline{G_{ij}(i\omega)}\,\overline{G_{ji}(i\epsilon)}\right]$
leads to an expectation value of the form $c_{i}^{\dagger
a}(\tau_{1})c_{i}^{b}(\tau_{2})c_{j}^{\dagger
b}(\tau_{3})c_{j}^{a}(\tau_{4})$.

For our calculations, we will proceed perturbatively starting from the
replica-symmetric saddle point. If we use this as our starting point, our
propagators will remain replica-symmetric to all orders in perturbation theory
[61]. It has been shown that for free fermions, this approximation is
sufficient for accurately recovering the leading-order contribution to the
mean value of $G(\tau_{1}-\tau_{2})$, although $N^{-1}$ corrections require
replica-off-diagonal saddles [62]. For four-point functions like Eq. 56, it is
known that a replica-diagonal ansatz is insufficient for reproducing the full
spectral correlations of random matrix theory [63] for small $\order{N^{-1}}$
energy differences, but can be recovered by considering off-diagonal saddle
manifolds [62]. This discrepancy is not relevant for our analysis, as we will
only be interested in spectral correlations over $\order{T}$ energy
differences.

##  Appendix B Replica off-diagonal fluctuations in the $(G,\Sigma)$ action

The calculation of the Green’s function covariances may be performed within
the formalism of the $(G,\Sigma)$ path integral, which we describe here.
Although this perspective does not provide a direct computational advantage
over the fermionic diagram approach in the main text - all non-trivial
integrals are still present - it admits an explicit $N^{-1}$ expansion, in
contrast with the diagrammatic approach in the main text where the task of
writing down all diagrams that contribute at a given order requires careful
analysis of index summations. The approach here is more easily generalizable
to the calculation of higher order moments, and also provides a more general
framework for understanding which observables obey a straightforward crossover
from SYK-like to Fermi liquid-like as a function of temperature and which ones
have more subtle crossover behavior - the former are functions of only the
saddle point solutions of the $(G,\Sigma)$ path integral, whereas the latter
are properties of fluctuations around the saddle point. Here, we rederive the
off-diagional Green’s function covariance, $\rho_{o}$, using this formulation.

We begin with a derivation of the $(G,\Sigma)$ path integral. Recall that our
Hamiltonian is given by

$\displaystyle H$
$\displaystyle=\frac{1}{(2N)^{3/2}}\sum_{ij;kl=1}^{N}J_{ij;kl}c_{i}^{\dagger}c_{j}^{\dagger}c_{k}c_{l}+\frac{1}{N^{1/2}}\sum_{ij=1}^{N}t_{ij}c_{i}^{\dagger}c_{j}-\mu\sum_{i}c_{i}^{\dagger}c_{i}$
(58)

where $J_{ij;kl}$ and $t_{ij}$ are complex random numbers with zero mean and
variances $J^{2}$ and $t^{2}$, respectively. In path integral form, we have
the partition function

$\displaystyle\overline{Z[h]^{M}}$
$\displaystyle=\int\mathcal{D}J\mathcal{D}t\mathcal{D}c\mathcal{D}c^{\dagger}e^{-\sum_{a=1}^{M}S_{a}[J]}$
(59) $\displaystyle S_{a}[J]$
$\displaystyle=\sum_{ij}\int\differential{\tau}c_{i}^{\dagger
a}(\tau)\left[\left(\partial_{\tau}-\mu\right)\delta_{ij}+\frac{t_{ij}}{N^{1/2}}\right]c_{j}^{a}(\tau)$
$\displaystyle+\frac{1}{(2N)^{3/2}}\sum_{ij;kl}\int\differential{\tau}J_{ij;kl}c_{i}^{\dagger
a}(\tau)c_{j}^{\dagger a}(\tau)c^{a}_{k}(\tau)c^{a}_{l}(\tau)$

Integrating over disorder, our path integral becomes

$\displaystyle Z[h]$
$\displaystyle=\int\mathcal{D}c\mathcal{D}c^{\dagger}e^{-S}$ (60)
$\displaystyle S$
$\displaystyle=\sum_{a,i}\int\differential{\tau}c_{i}^{\dagger
a}(\partial_{\tau}-\mu)c_{i}^{a}-\sum_{a\,,b}\int\differential{\tau_{1}}\differential{\tau_{2}}\Bigg{[}\frac{NJ^{2}}{4}\left(\frac{1}{N}\sum_{i}c_{i}^{\dagger
a}(\tau_{1})c_{i}^{b}(\tau_{2})\right)^{2}\left(\frac{1}{N}\sum_{i}c_{i}^{\dagger
b}(\tau_{2})c_{i}^{a}(\tau_{1})\right)^{2}$
$\displaystyle-\frac{Nt^{2}}{2}\left(\frac{1}{N}\sum_{i}c_{i}^{\dagger
a}(\tau_{1})c_{i}^{b}(\tau_{2})\right)\left(\frac{1}{N}\sum_{i}c_{i}^{\dagger
b}(\tau_{2})c_{i}^{a}(\tau_{1})\right)\Bigg{]}$

We now insert the field

$\displaystyle G^{ab}(\tau_{1}\,,\tau_{2})\equiv\frac{1}{N}\sum_{i}c^{\dagger
a}_{i}(\tau_{1})c_{i}^{b}(\tau_{2})$ (61)

where the equivalence is enforced with a Lagrange multiplier
$\Sigma^{ab}(\tau_{1},\tau_{2})$. The $c\,,c^{\dagger}$ fields can be
integrated out to yield the action

$\displaystyle\frac{S[G\,,\Sigma\,,h]}{N}$
$\displaystyle=-\ln\det(-\partial_{\tau}+\mu-\Sigma)-\sum_{a\,,b}\int\differential{\tau_{1,2}}\Bigg{(}\Sigma^{ab}(\tau_{1},\tau_{2})G^{ba}(\tau_{2},\tau_{1})$
(62)
$\displaystyle+\frac{J^{2}}{4}\left(G^{ab}(\tau_{1},\tau_{2})G^{ba}(\tau_{2},\tau_{1})\right)^{2}-\frac{t^{2}}{2}G^{ab}(\tau_{1},\tau_{2})G^{ba}(\tau_{2},\tau_{1})\Bigg{)}\,.$

We take the replica-diagonal saddle point,
$G^{ab}(\tau_{1},\tau_{2})=\delta_{ab}G(\tau_{1}-\tau_{2})$ and likewise for
$\Sigma^{ab}$. The replica-diagonal Schwinger-Dyson equations are given by Eq.
46 in the main text - as emphasized earlier, it is the solution to this set of
equations that displays a crossover from SYK-like for $T\gg E_{\text{coh}}$ to
Fermi liquid-like for $T\ll E_{\text{coh}}$. This saddle-point solution does
not contribute to the Green’s function covaraince; to obtain a non-zero value,
we must consider fluctuations around it,
$G^{ab}(\tau_{1},\tau_{2})\equiv\delta_{ab}G(\tau_{1}-\tau_{2})+\delta
G^{ab}(\tau_{1},\tau_{2})$.

In this representation, our observables of interest are

$\displaystyle g_{o}(\tau_{1,2,3,4})$
$\displaystyle\equiv\frac{1}{N^{2}}\sum_{ij}\left[\overline{G_{ij}(\tau_{1}-\tau_{2})G_{ji}(\tau_{3}-\tau_{4})}-\overline{G_{ij}(\tau_{1}-\tau_{2})}\,\overline{G_{ji}(\tau_{3}-\tau_{4})}\right]$
(63) $\displaystyle=\langle
G^{ab}(\tau_{1}-\tau_{2})G^{ba}(\tau_{3}-\tau_{4})\rangle-\frac{1}{N}\langle
G^{aa}(\tau_{1}-\tau_{2})G^{bb}(\tau_{3}-\tau_{4})\rangle$

for $a\neq b$. Note the subleading correction in $g_{o}$, which arises from
the $i=j$ term in the disconnected contribution (the “standard” disconnected
part of $g_{o}$ vanishes due to the fact that $\langle G^{ab}\rangle=0$ for
fluctuations around the replica-diagonal saddle point).

These replica off-diagonal observables vanish at the replica-diagonal saddle
point. To find the leading order non-zero result, we expand the action around
its saddle-point solution. The expansion of everything other than the
determinant is rather straightforward. For evaluation of the determinant, we
use Jacobi’s formula

$\displaystyle\frac{1}{\det(-\partial_{\tau}+\mu-\Sigma)}\frac{\partial\det(-\partial_{\tau}+\mu-\Sigma)}{\partial\Sigma^{ab}(\tau_{1},\tau_{2})}=-\Tr\left[\left(-\partial_{\tau}+\mu-\Sigma\right)^{-1}\frac{\partial\Sigma}{\partial\Sigma^{ab}(\tau_{1},\tau_{2})}\right]$
(64)
$\displaystyle=-\left[\left(-\partial_{\tau}+\mu-\Sigma\right)^{-1}\right]^{ba}\left(\tau_{2},\tau_{1}\right)=-\delta_{ab}G(\tau_{2}-\tau_{1})$

where in the final line we evaluate the expression at the replica-diagonal
saddle point. To second order, we use

$\displaystyle\frac{1}{\det(-\partial_{\tau}+\mu-\Sigma)}\frac{\partial^{2}\det(-\partial_{\tau}+\mu-\Sigma)}{\partial\Sigma^{ab}(\tau_{1},\tau_{2})\partial\Sigma^{cd}(\tau_{3},\tau_{4})}$
(65)
$\displaystyle=-\frac{1}{\det(-\partial_{\tau}+\mu-\Sigma)}\partialderivative{\Sigma^{cd}(\tau_{3},\tau_{4})}\left[\det(-\partial_{\tau}+\mu-\Sigma)\Tr\left[\left(-\partial_{\tau}+\mu-\Sigma\right)^{-1}\frac{\partial\Sigma}{\partial\Sigma^{ab}(\tau_{1},\tau_{2})}\right]\right]$
$\displaystyle=\delta_{ab}\delta_{cd}G(\tau_{2}-\tau_{1})G(\tau_{4}-\tau_{3})-\Tr\left[\delta\Sigma^{ab}G\delta\Sigma^{ba}G\right]$

This leads to the quadratic action

$\displaystyle\frac{\delta S\left[\delta
G\,,\delta\Sigma\right]}{N}=\sum_{ab}\Big{[}\frac{1}{2}\Tr\left[G\delta\Sigma^{ab}G\delta\Sigma^{ba}\right]-\int\differential{\tau_{1}}\differential{\tau_{2}}\delta
G^{ab}(\tau_{1},\tau_{2})\left[\delta\Sigma^{ba}(\tau_{2},\tau_{1})-\frac{t^{2}}{2}\delta
G^{ba}(\tau_{2},\tau_{1})\right]$ (66)
$\displaystyle-\frac{J^{2}\delta_{ab}}{2}\int\differential{\tau_{1}}\differential{\tau_{2}}\Bigg{(}2G(\tau_{1},\tau_{2})G(\tau_{2},\tau_{1})\delta
G^{aa}(\tau_{1},\tau_{2})\delta G^{aa}(\tau_{2},\tau_{1})$
$\displaystyle+G(\tau_{1},\tau_{2})^{2}\delta G^{aa}(\tau_{2},\tau_{1})\delta
G^{aa}(\tau_{1},\tau_{2})\Bigg{)}$

Figure 8: We illustrate the propagators for use in a diagrammatic expansion in
$N^{-1}$ around the saddle point of the $(G\,,\Sigma)$ action. The fields $G$
and $\Sigma$ are a function of two times and two replica indices, which
necessitates the sheet-like representation above. The colors indicate
different replica indices $a\,,b$, and solid (dotted) lines indicate a $G$
($\Sigma$) field.

Figure 9: Interactions arise in an expansion around the $(G\,,\Sigma)$ saddle
point from expanding the $\ln det(-\partial_{\tau}+\mu-\Sigma)$ term, which
leads to arbitrary order sheets for which $\Sigma$ propagators can be attached
to. Additionally, four $G$ fields can be attached at a “seam.”

The trace notation in the first term is shorthand for four time integrals,
i.e.
$\Tr[G\Sigma]\equiv\int\differential{\tau_{a}}\differential{\tau_{b}}G(\tau_{a},\tau_{b})\Sigma(\tau_{b},\tau_{a})$.
We can invert the quadratic action to obtain a propagator, which we can do
separately for the replica diagonal and replica off-diagonal components. For
the latter, we have

$\displaystyle-\frac{2\delta
S}{N}=\int\differential{\tau_{1,2,3,4}}\begin{pmatrix}\delta
G^{ab}(\tau_{1},\tau_{2})&\delta\Sigma^{ab}(\tau_{1},\tau_{2})\end{pmatrix}\begin{pmatrix}-t^{2}\delta_{\tau_{1},\tau_{3}}\delta_{\tau_{2},\tau_{4}}&\delta_{\tau_{1},\tau_{3}}\delta_{\tau_{2},\tau_{4}}\\\
\delta_{\tau_{1},\tau_{3}}\delta_{\tau_{2},\tau_{4}}&-G(\tau_{1}-\tau_{3})G(\tau_{2}-\tau_{4})\end{pmatrix}\begin{pmatrix}\delta
G^{ba}(\tau_{4},\tau_{3})\\\
\delta\Sigma^{ba}(\tau_{4},\tau_{3})\end{pmatrix}$ (67)

The matrix must be inverted, which can most easily be done in Matsubara
frequency space. This leads to the result

$\displaystyle\langle\delta G^{a\neq b}(\tau_{1},\tau_{2})\delta G^{b\neq
a}(\tau_{4},\tau_{3})\rangle=\frac{1}{N\beta^{2}}\sum_{i\omega_{n},i\epsilon_{n}}e^{-i\omega_{n}(\tau_{1}-\tau_{3})-i\epsilon_{n}(\tau_{4}-\tau_{2})}\frac{G(i\omega_{n})G(i\epsilon_{n})}{1-t^{2}G(i\omega_{n})G(i\epsilon_{n})}\,,$
(68) $\displaystyle\langle\delta\Sigma^{a\neq b}(\tau_{1},\tau_{2})\delta
G^{b\neq
a}(\tau_{4},\tau_{3})\rangle=\frac{1}{N\beta^{2}}\sum_{i\omega_{n},i\epsilon_{n}}e^{-i\omega_{n}(\tau_{1}-\tau_{3})-i\epsilon_{n}(\tau_{4}-\tau_{2})}\frac{1}{1-t^{2}G(i\omega_{n})G(i\epsilon_{n})}\,,$
$\displaystyle\langle\delta\Sigma^{a\neq
b}(\tau_{1},\tau_{2})\delta\Sigma^{b\neq
a}(\tau_{4},\tau_{3})\rangle=\frac{1}{N\beta^{2}}\sum_{i\omega_{n},i\epsilon_{n}}e^{-i\omega_{n}(\tau_{1}-\tau_{3})-i\epsilon_{n}(\tau_{4}-\tau_{2})}\frac{t^{2}}{1-t^{2}G(i\omega_{n})G(i\epsilon_{n})}\,.$

This gives the expected result for $g_{o}$ in Eq. 21 of the main text once the
trivial disconnected piece of $g_{o}$ is subtracted off. Note that for $t=0$,
while $\langle\delta G^{ab}\delta G^{ba}\rangle$ is non-zero, its contribution
to $g_{o}$ is subtracted off exactly by the disconnected piece. Hence, the
leading order contribution to $g_{o}$ when $t=0$ is given by the first
correction to the $G^{ab}$ propagator, illustrated in Fig. 10. This
corresponds to the fermionic Feynman diagram shown in the top of Fig. 3 in the
main text.

Figure 10: We illustrate Feynman diagrams that contribute to the off-diagonal
Green’s function covariance in a pure SYK model. For a model that includes
random hoppings, there exists a non-trivial contribution in the bare $\delta
G^{ab}\delta G^{ba}$ propagator; for a pure SYK model, this contribution is
subtracted off exactly in the covariance and one must include the leading
order correction to obtain a non-zero result.

##  Appendix C Statistics of ratio distributions

Here, we provide a summary of relevant results involving ratio distributions,
which we utilize for calculating statistical properties of the thermopower.

We take $X_{1}$, $X_{2}$ to be two correlated Gaussian random variables, with
means $\mu_{1,2}$, variances $\sigma^{2}_{1,2}$, and correlation coefficient
$r$. Our quantity of interest is the random variable $Z\equiv X_{1}/X_{2}$.
The probability density function $f(z)$ of $Z$ can be obtained from the joint
density $g(x_{1},x_{2})$ of $X_{1,2}$,

$\displaystyle f(z)$
$\displaystyle=\int_{-\infty}^{\infty}\absolutevalue{y}g(zy,y)\differential{y}\,.$
(69)

This function along with the cumulative distribution function
$F(z)\equiv\int_{-\infty}^{z}f(x)\differential{x}$ are known [47]. However,
much like the Cauchy distribution - which is a limiting case of a ratio
distribution when the numerator and denominator have zero mean - the integrals
$\int_{-\infty}^{\infty}z^{a}f(z)\differential{z}$, $\alpha\geqslant 1$ do not
converge and the mean and variance are formally ill-defined.

One can make progress in the limit where
$\absolutevalue{\sigma_{2}/\mu_{2}}\rightarrow 0$; or in other words, when the
probability of the denominator in $Z$ becoming negative is zero. This result
can equivalently be derived from the assumption that $X_{2}>0$ which implies
$F(z)\equiv P(x_{1}/x_{2}<z)=P(x_{1}-zx_{2}<0)$. Since the sum of two
correlated Gaussians is also a Gaussian, this gives the cumulative
distribution function

$\displaystyle
F(z)=\Phi\left(\frac{\mu_{2}z-\mu_{1}}{\sqrt{\sigma_{1}^{2}-2zr\sigma_{1}\sigma_{2}+z^{2}\sigma_{2}^{2}}}\right)$
(70)

where $\Phi(x)$ is the cumulative distribution function of a Gaussian random
variable, $\Phi(x)\equiv\int_{-\infty}^{x}\phi(y)\differential{y}$,
$\phi(x)\equiv\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^{2}}$.

For small fluctuations around the mean value, $\overline{z}=\mu_{1}/\mu_{2}$,
we have

$\displaystyle\Phi\left(\frac{\mu_{2}z-\mu_{1}}{\sqrt{\sigma_{1}^{2}-2zr\sigma_{1}\sigma_{2}+z^{2}\sigma_{2}^{2}}}\right)\approx\Phi\left(\frac{z-\overline{z}}{\overline{z}\sqrt{\frac{\sigma_{1}^{2}}{\mu_{1}^{2}}-\frac{2r\sigma_{1}\sigma_{2}}{\mu_{1}\mu_{2}}+\frac{\sigma_{2}^{2}}{\mu_{2}^{2}}}}\right)$
(71)

which yields the approximation to normality, with variance

$\displaystyle\frac{\text{Var
}z}{\overline{z}^{2}}\approx\frac{\sigma_{1}^{2}}{\mu_{1}^{2}}-\frac{2r\sigma_{1}\sigma_{2}}{\mu_{1}\mu_{2}}+\frac{\sigma_{2}^{2}}{\mu_{2}^{2}}\,.$
(72)

In the main text, we find several situations where the numerator and
denominator are highly correlated such that $r=1-\order{N^{-1}}$, where we use
$N$ as a stand-in for a generic large dimensionless parameter, which depending
on the context may refer to either the actual system size or
$T/E_{\text{coh}}$. To leading order in $N^{-1}$, we therefore have perfect
correlation between the numerator and denominator, leading to

$\displaystyle\frac{\text{Var
}z}{\overline{z}^{2}}\approx\left(\frac{\sigma_{1}}{\mu_{1}}-\frac{\sigma_{2}}{\mu_{2}}\right)^{2}+\order{N^{-1}}\,.$
(73)

Working in the limit of perfect correlation means that we may think of $X_{1}$
and $X_{2}$ as arising from the same normal distribution $X$, i.e.
$X_{1}=\sigma_{1}X+\mu_{1}$ and $X_{2}=\sigma_{2}X+\mu_{2}$. The ratio
distribution is still non-trivial even if both variables arise from the same
probability distribution. However, it does imply a special limit
$\sigma_{1}/\mu_{1}=\sigma_{2}/\mu_{2}$ where the distribution becomes trivial
and the variance vanishes due to the numerator and denominator being directly
proportional to each other. In this limit, the variance incurs an additional
$N^{-1}$ suppression due to the necessity of expanding out $r$ to higher
order. This prediction is confirmed by numerical simulation, see Fig. 11. We
take $10000$ samples of the ratio distribution $Z$ for parameters
$\sigma_{1}^{2}=1.5$, $\sigma_{2}^{2}=1$, $\mu_{y}=10$, $r=1-1/N$, and
variable $\mu_{x}$. We fit the power law scaling of the variance as a function
of $N$ for $500<N<10000$ and plot the exponent while varying $\mu_{x}$. As
expected, an anomalous suppression of the variance appears at the critical
value where $\sigma_{1}/\mu_{1}=\sigma_{2}/\mu_{2}$.

Figure 11: By drawing from a ratio distribution, where the correlation
coefficient between the numerator and denominator is given by $1-1/N$, we fit
the variance to an $N^{\alpha}$ form and plot the exponent $\alpha$. When the
probability distributions are tuned such that
$\sigma_{1}/\mu_{1}=\sigma_{2}/\mu_{2}$, we obtain a $N^{-1}$ suppression of
the variance.

##  Appendix D Conductance fluctuations for single-lead coupling

In the main text, we present results for conductance fluctuations for a model
where we take our leads to be coupled to all sites with equal magnitude. To
leading order in $N^{-1}$, fluctuations are controlled by the off-diagonal
Green’s function covariance $\rho_{o}$. If we instead choose to model our
leads as only being coupled to a single site in the quantum dot, our results
are modified as fluctuations are driven by the diagonal Green’s function
covariance $\rho_{d}$, which is generically suppressed relative to $\rho_{o}$
by an additional factor of $N^{-1}$. Note that this contribution is still
present in our model in the main text, but is ignored in virtue of this
$N^{-1}$ suppression. We present results for both $\rho_{o}$ and $\rho_{d}$ in
the main text but focus on conductance fluctuations for fully-connected leads.
Here, we present results for conductance fluctuations that arise from
$\rho_{d}$, which are subleading in $N^{-1}$ for fully-connected leads but are
the dominant contribution for leads coupled to a single site. We remind the
reader that average conductance is insensitive to this choice and remains the
same as in the main text.

For a free fermion model, we have the physical interpretation that $\rho_{d}$
gives the covariance of the single-particle eigenvalues, the form of which is
universal and well-known from random matrix theory. In particular, the
variance of linear statistics such as the conductance is given by the Dyson-
Mehta formula [40, 41], which yields the conductance variance

$\displaystyle\text{Var
}\sigma_{FF}=\left(\frac{e^{2}}{\hbar}\frac{\Gamma}{TN}\right)^{2}\frac{3\zeta(3)}{\pi^{4}}\,.$
(74)

For a pure SYK model, our expression for $\rho_{d}$ given in Eq. 40 yields

$\displaystyle\text{Var
}\sigma_{SYK}=\left(\frac{e^{2}\Gamma}{\hbar}\right)^{2}\frac{0.07}{N^{4}JT}\,.$
(75)

We now consider the case with both SYK interactions and random hopping terms.
For the low temperature Fermi liquid phase, we predict a scaling similar to
the free fermion result in Eq. 74, but with a renormalization which can be
deduced on dimensional grounds to be

$\displaystyle\text{Var }\sigma_{tSYK}\propto\left(\frac{\Gamma e^{2}}{\hbar
TN}\frac{t}{J}\right)^{2}\,,\quad T\ll E_{\text{coh}}\,.$ (76)

For the SYK regime, $T\gg E_{\text{coh}}$, we find nearly identical to the
case considered to the main text, due to the fact that in this regime,
$\rho_{d}(\omega,\epsilon)=N^{-1}\rho_{o}(\omega,\epsilon)$ to leading order
in $E_{\text{coh}}/T$. Hence,

$\displaystyle\text{Var }\sigma_{tSYK}=2.02\mathcal{E}^{2}\left(\frac{\Gamma
e^{2}}{\hbar NT}\frac{t}{J}\right)^{2}\,,\quad T\gg E_{\text{coh}}\,.$ (77)

Note that this is the same scaling as in the Fermi liquid regime, albeit with
the crucial difference that the overall coefficient is proportional to the
particle-hole asymmetry.

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