††institutetext: Department of Physics, University of California, Santa
Barbara, CA 93106, USA

#  Entanglement Negativity Transitions in Chaotic Eigenstates

Sean McBride and  Fernando Iniguez<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It was recently noted that the entanglement entropy for a subsystem of a
chaotic eigenstate exhibits an enhanced correction when the subsystem
approaches a phase transition at half the total system size. This enhanced
correction was derived for general subsystems by Dong and Wang by summing over
noncrossing permutations, which can be thought of as “saddles” either in a sum
emerging from averaging over Wick contractions or in an analogous
gravitational calculation. We extend these results to the case of entanglement
negativity, an entanglement measure defined on a bipartite density matrix. We
focus on a particular transition previously studied in a toy model of JT
gravity, one for which the sum over permutations was found to give similar (or
even stronger) enhanced corrections. We derive and resum the relevant
permutations to give a form for the averaged negativity spectrum, reproducing
the gravitational answer for some quantities and finding tension with other
quantities, namely the partially transposed entropy. Along the way, we extend
the results of Dong and Wang to the case of $n<1$ Rényi entropy, showing that
it always receives volume law corrections.

##  1 Introduction

The application of ideas from quantum chaos to gravitational settings has been
particularly fruitful. Gravitational observables have been shown to be well
approximated by observables obeying the eigenstate thermalization hypothesis
(ETH) PhysRevA.43.2046 ; PhysRevE.50.888 ; Srednicki:1995pt ; Srednicki_1999 .
This is due to the fact that a holographic quantum field theory with a
semiclassical Einstein gravity dual is expected to be maximally chaotic, i.e.
it saturates the bound of Maldacena:2015waa , up to higher derivative/stringy
corrections which take one away from this regime. The power of ETH is that it
allows us to approximate observables in the microcanonical ensemble by an
observable’s long-time quantum expectation value. The resulting microcanonical
expectation value should resemble that of the canonical ensemble, up to
corrections expected to be suppressed in the system size $V$ by the
thermodynamic ensembles’ equivalence at large $N$. This gives a quantitative
idea of the process of thermalization in isolated quantum many-body systems.

From the perspective of subsystem ETH Garrison:2015lva ; Dymarsky:2016ntg ;
Dymarsky:2017zoc ; Lashkari:2017hwq ; Lu:2017tbo , for a subsystem with volume
fraction $f<1/2$, the corrections to ETH are suppressed in system size.
Formally, this means the trace-norm distance between the canonical and
microcanonical density matrices vanishes in the large volume/thermodynamic
limit, implying that off-diagonal matrix elements of operators vanishes and
expectation values are roughly thermal. This line of thinking is expected to
apply to Rényi entropies.

When $f=1/2$ exactly, the usual wisdom would say there’s an $\mathcal{O}(1)$
correction to the Rényi entropies. One way of seeing this is that there exists
a phase transition in the Rényi entropy at $f=1/2$, and the correction at this
phase transition should be given by the uncertainty in choosing between an
$\mathcal{O}(1)$ number of equivalent dominant phases. However, in a model-
specific result, a correction to the entanglement entropy of
$\mathcal{O}(\sqrt{V})$ was observed in VidmarRigol , a correction derived in
Murthy:2019qvb  and explicated in Dong:2020iod . In particular, the von
Neumann entropy of a subregion $A$ with volume fraction $f=1/2$ takes the form

$S_{A}=\frac{S(E)}{2}-\sqrt{\frac{C_{V}}{2\pi}}+\mathcal{O}(a)$ (1)

where $S(E)$ is the thermodynamic entropy at energy $E$ and $C_{V}$ is the
heat capacity at constant volume. As $C_{V}$ is extensive in the system size,
the correction is “enhanced” to $\mathcal{O}(\sqrt{V})$. This formula is valid
in the large volume limit, where $\sqrt{V}\gg a$, $a$ being the area of the
splitting surface.

In a parallel story, the attempt to match results from tensor networks with
the gravitational path integral led to the understanding of “fixed area
states” Dong:2018seb ; Akers:2018fow . These states, which are eigenstates of
the area operator in semiclassical gravity, have a flat entanglement spectrum,
up to fluctuations about a fixed saddle point which can naïvely be at most
$\mathcal{O}(1)$ in units of $G_{N}$, where $G_{N}\ll 1$. One can think of
these fluctuations as the difference in the “canonical” ensemble where one
fixes the canonical conjugate to the area operator, namely the relative boost
between the entanglement wedges of the two sides Dong:2022ilf , and a
“microcanonical” ensemble where the eigenvalue of the area operator is fixed
at its most probable value.

It was noted in Marolf:2020vsi ; Dong:2020iod  that the universal enhanced
correction to the entanglement entropy also appears in fixed area states near
transition, where the “transition” in this context occurs due to a competition
between two extremal surfaces. One way of understanding this correction is
that near transition we no longer care about fluctuations about a fixed
saddle, but instead we care about resumming an infinite number of saddles
which appear in the sum over topologies in the replicated geometries. Both of
these results match with a more detailed calculation of the same quantity in
Penington:2019kki , where in a model of Jackiw-Teitelboim (JT) gravity + end-
of-the-world (EOW) branes a particular subsystem entropy $S(\rho_{R})$ had the
form

$S(\rho_{R})=\log k-\sqrt{\frac{2\pi}{\beta}}+\mathcal{O}(\log\beta).$ (2)

This $\sqrt{1/\beta}$ correction is analogous to the $\sqrt{C_{V}}$ correction
in chaotic eigenstates. Here we’ve set Newton’s constant (which is analagous
to $N$) to one, but it can be restored via $\beta\rightarrow G_{N}\beta$.

Recently, it was shown by Dong:2021oad  that similar enhanced corrections
exist near transitions in entanglement negativity, a tripartite entanglement
measure defined on a bipartite density matrix $\rho_{A_{1}A_{2}}$. In
particular, the logarithmic negativity was shown to have the following form at
transition:

$\mathcal{E}(\rho_{R_{1}R_{2}})=\log
k_{2}-\frac{\pi^{2}}{8\beta}+\mathcal{O}(\log\beta),$ (3)

for two subsystems $R_{1}$ and $R_{2}$ with $R_{1}\cup R_{2}=R$. Further
corrections were derived for measures descending from a Rényi version of
negativity.

There exists a rich phase diagram for entanglement negativity in holographic
states, and we show that a similar phase diagram exists for a generic chaotic
eigenstate. Our aim is to systematically derive the corrections at transitions
in this phase space. There are two possible transitions, but as was explored
in Dong:2021oad  we only expect interesting behavior near one of the
transitions, for reasons we’ll recapitulate in the main text.

The outline of the paper is as follows. In section 2 we review the derivation
of (1), in particular the resolvent formalism of Dong:2020iod . In section 3
we review the various negativity measures discussed in Dong:2021oad  and the
sum over relevant permutations for the phase transition of interest. In
section 4 we compute corrections to the entanglement measures of interest. We
conclude with some discussion and future directions.

##  2 Diagrammatics for Chaotic Subsystems

We first review the formalism of Murthy:2019qvb ; Dong:2020iod , which was
used to compute the universal form of corrections to the entanglement entropy
of a subsystem at transition. We focus on Dong:2020iod , as their formalism
more easily generalizes to our future calculations. Readers familiar with
their formalism may skip this section, whose only purpose is to make this work
self-contained.

A generic eigenstate $\ket{E}$ of a Hamiltonian $H$ defined on a bipartite
system such that $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ can be
Schmidt decomposed via

$\ket{E}=\sum_{iJ}M_{iJ}\ket{E_{i}}_{A}\otimes\ket{E_{J}}_{B},$ (4)

where $\ket{E_{i}}$ and $\ket{E_{J}}$ denote eigenstates of the subsystem
Hamiltonians $H_{A}$ and $H_{B}$, respectively. As is convention, we use
lowercase indices for states of $A$ and uppercase indices for states of $B$.
ETH instructs us to think of $M_{iJ}$ as a Gaussian random variable with zero
mean and energy banded with width $\Delta$ Deutsch2 ; LuGrover . In
particular, for a system with spatial dimension $d\geq 2$, we have the ansatz

$M_{iJ}=e^{-S(E_{Ai}+E_{BJ})/2}\left(\frac{e^{-\epsilon^{2}/2\Delta^{2}}}{\sqrt{2\pi}\Delta}\right)^{1/2}C_{iJ},$
(5)

where $\epsilon=E_{i}+E_{J}-E$ is the deviation from the total microcanonical
energy. When averaged over a small energy band in $E_{A}$ and $E_{B}$, the
random coefficients $C_{iJ}$ satisfy

$\overline{C_{iJ}}=0,\quad\overline{C_{iJ}C_{i^{\prime}J^{\prime}}}=\delta_{ii^{\prime}}\delta_{JJ^{\prime}}.$
(6)

The effects of finite $\Delta$ will not affect the current and future
computation, so we work in the limit $\Delta\rightarrow 0$, where we
approximate

$M_{iJ}\approx e^{-S(E)/2}C_{iJ}.$ (7)

This approximation assumes the true density of states in a narrow energy band
is well approximated by the thermodynamic entropy in the canonical ensemble.
To leading order in the system volume, we can further approximate the density
of states of the total system as the product of the density of states of the
subsystems $A$ and $B$, evaluated at the subsystem energy $E_{A}$. In other
words,

$S(E)\approx S_{A}(E_{A})+S_{B}(E-E_{A}).$ (8)

This leads to the following form for a subsystem density matrix $\rho_{A}$

$\rho_{A}=\frac{1}{\mathcal{N}}\sum_{E_{i}-2\Delta<E_{j}<E_{i}+2\Delta}\sum_{E-E_{i}-\Delta<E_{J}<E-E_{i}+\Delta}C_{iJ}C_{jJ}\ket{E_{i}}_{A}\bra{E_{j}}_{A}.$
(9)

The double sum takes into account energies in a region of width $2\Delta$.
Averaging over the $C_{ij}$’s gives the averaged subsystem density matrix

$\overline{\rho_{A}}=\frac{1}{\mathcal{N}}\sum_{i}d_{B}(E-E_{i})\ket{E_{i}}_{A}\bra{E_{i}}_{A},$
(10)

where the normalization is given by

$\mathcal{N}=\sum_{i}d_{A}(E_{i})d_{B}(E-E_{i}).$ (11)

Here $d_{A}$ and $d_{B}$ are the degeneracies at a given energy. As a
shorthand and as a motivation for our future computation, we can instead write

$d_{A}(E_{i})=e^{S_{A}(E_{i})}\equiv e^{S_{A}};\quad
d_{B}(E-E_{i})=e^{S_{B}(E-E_{i})}\equiv e^{S_{B}}.$ (12)

As our goal is to compute subsystem von Neumann entropy, we should proceed by
generalizing this procedure to compute $\Tr\overline{\rho_{A}^{n}}$. Before
averaging, from (9) we have

$\rho_{A}^{n}=\frac{1}{\mathcal{N}^{n}}\sum_{E_{i_{1}}}\sum_{i_{2},\cdots,i_{n+1};J_{1},\cdots,J_{n}}\prod_{m=1}^{n}C_{i_{m}J_{m}}C_{i_{m+1}J_{m}}\ket{E_{i_{1}}}_{A}\bra{E_{i_{n+1}}}_{A},$
(13)

where the second sum is understood to be over a strip of width $2n\Delta$, but
we assume $\Delta$ vanishes quickly enough at finite $n$ that this isn’t a
significant effect.

The difference between $\overline{\log\Tr\left(\rho_{A}\right)^{n}}$ and
$\log\Tr\overline{\left(\rho_{A}\right)^{n}}$ is exponentially suppressed in
the system volume Lu:2017tbo , so our goal will be to compute
$\Tr\overline{\left(\rho_{A}\right)^{n}}$, as it is a more tractable
calculation. This involves a sum over Wick contractions, as we assume higher
point connected correlations of the $C_{iJ}$’s vanish. The result is

$\Tr\overline{(\rho_{A})^{n}}=\begin{cases}\frac{1}{\mathcal{N}^{n}}e^{S_{A}+nS_{B}}{}_{2}F_{1}\left(1-n,-n;2;e^{S_{A}-S_{B}}\right),S_{A}<S_{B}\\\
\frac{1}{\mathcal{N}^{n}}e^{nS_{A}+S_{B}}{}_{2}F_{1}\left(1-n,-n;2;e^{S_{B}-S_{A}}\right),S_{A}>S_{B}.\end{cases}$
(14)

As a sanity check, we recover $S_{n}(\rho_{A})=S/2+\mathcal{O}(1)$ where
$S_{A}=S_{B}=S/2$ at $f=1/2$. The derivation of this expression from the
resolvent sum over noncrossing permutations is given in Appendix B.

To study the corrections to this quantity at transition, we upgrade the
putative constant density of states to an integral over an energy dependent
density of states. In other words, we send

$S_{A}\rightarrow S_{A}(E_{A}),\quad S_{B}\rightarrow S_{B}(E-E_{A})$ (15)

and integrate over $E_{A}$. The new averaged trace is given by

$\Tr\overline{(\rho_{A})^{n}}=\frac{1}{\mathcal{N}^{n}}\int
dE_{A}e^{S_{A}(E_{A})+S_{B}(E-E_{A})}G_{n}(E_{A}),$ (16)

where $G_{n}(f,E_{A})$ encompasses the $n$-dependent piece of the trace:

$G_{n}(E_{A})=\begin{cases}e^{(n-1)S_{B}(E-E_{A})}{}_{2}F_{1}\left(1-n,-n;2;e^{S_{A}(E_{A})-S_{B}(E-E_{A})}\right),S_{A}(E_{A})<S_{B}(E-E_{A})\\\
e^{(n-1)S_{A}(E_{A})}{}_{2}F_{1}\left(1-n,-n;2;e^{S_{B}(E-E_{A})-S_{A}(E_{A})}\right),S_{A}(E_{A})>S_{B}(E-E_{A})\end{cases}$
(17)

and the normalization is now

$\mathcal{N}=\int dE_{A}e^{S_{A}(E_{A})+S_{B}(E-E_{A})}.$ (18)

From this, we can directly calculate the ensemble averaged Rényi entropies
$S_{n}(\rho_{A})$:

$\overline{S_{n}}=\frac{1}{1-n}\log\left(\frac{1}{\mathcal{N}^{n}}\int
dE_{A}e^{S_{A}(E_{A})+S_{B}(E-E_{A})}G_{n}(E_{A})\right).$ (19)

###  2.1 Saddle Point Analysis

We make the ansatz that the entropy is extensive in the subsystem size, that
is

$S_{A}(E_{A})=fVs\left(\frac{E_{A}}{fV}\right),\quad
S_{B}(E-E_{A})=(1-f)Vs\left(\frac{E-E_{A}}{(1-f)V}\right),$ (20)

where $f\equiv V_{A}/V$ is the volume fraction, $s(e)$ is the entropy density
as a function of the energy density $e$, and the other factors come from
dimensional analysis. We’re mainly interested in what happens at the
transition $f=1/2$. The “featureless” or infinite temperature case is when
$s(e)=1$ such that all subsystem entropies are proportional to subsystem
volume. We’re only interested in the corrections from finite temperature,
which can be thought of as the difference between the answer in the canonical
ensemble and the microcanonical ensemble. The microcanonical Rényi entropy is
the contribution of the global “unaveraged” microcanonical state
$\rho=\sum_{E-\Delta<E_{i}<E+\Delta}\ket{E_{i}}\bra{E_{i}}:$

$S^{MC}_{n}=\frac{1}{1-n}\log\left(\frac{1}{\mathcal{N}^{n}}\int
dE_{A}e^{S_{A}(E_{A})+nS_{B}(E-E_{A})}\right).$ (21)

We are interested in the correction away from the dominant microcanonical
saddle, so we are interested in computing the following quantity:

$\overline{S_{n}}-S^{MC}_{n}=\frac{1}{1-n}\ln\left(\frac{\int
dE_{A}\exp(F_{1}(E_{A}))}{\int dE_{A}\exp(F_{2}(E_{A}))}\right),$ (22)

where $F_{1}(E_{A})$ and $F_{2}(E_{A})$ are functions defined by

$\displaystyle F_{1}(E_{A})$
$\displaystyle=fVs\left(\frac{E_{A}}{fV}\right)+(1-f)Vs\left(\frac{E-E_{A}}{(1-f)V}\right)+\ln
G_{n}(E_{A})$ $\displaystyle F_{2}(E_{A})$
$\displaystyle=fVs\left(\frac{E_{A}}{fV}\right)+n(1-f)Vs\left(\frac{E-E_{A}}{(1-f)V}\right).$
(23)

As both functions scale with volume, we can perform a saddle point analysis.
The saddle point equations for these functions are

$\displaystyle s^{\prime}\left(\frac{E_{1}}{fV}\right)$
$\displaystyle=s^{\prime}\left(\frac{E-E_{1}}{(1-f)V}\right)-\frac{G^{\prime}_{n}(f,E_{1})}{G_{n}(f,E_{1})}$
$\displaystyle s^{\prime}\left(\frac{E_{2}}{fV}\right)$
$\displaystyle=ns^{\prime}\left(\frac{E-E_{2}}{(1-f)V}\right),$ (24)

where $E_{1}$ and $E_{2}$ are the saddle point energies of $F_{1}(E_{A})$ and
$F_{2}(E_{A})$, respectively. The analysis of these saddle point equations was
done in totality for $n>1$ in Dong:2020iod .111See also Kudler-Flam:2021alo
for a similar study of relative entropy with the same ansatz. Here we fill in
a small gap and study the case of $n<1$. This will be useful later when we are
computing analytic continuations of Rényi negativities below $n=1$.

###  2.2 Corrections at Transition for $n<1$

$s(x)$ is a monotonically increasing function of $x$ with a monotonically
decreasing first derivative (take $s(x)=\sqrt{x}$ as a concrete example). For
the case $n<1$, we can therefore write the iequality

$\frac{E_{2}}{f}>\frac{E-E_{2}}{1-f},$ (25)

which immediately implies

$S_{A}(E_{2})>\frac{f}{1-f}S_{B}(E-E_{2}),$ (26)

and therefore $S_{A}(E_{1})>S_{B}(E-E_{1})$ for $f>1/2$.

The first thing to notice is that there is only one saddle point for both
$F_{1}(E_{A})$ and $F_{2}(E_{A})$, as $G_{n}(f,E_{A})$ is now a strictly
concave function. The single saddle for $F_{1}(E_{A})$ depends sensitively on
the saddle point of $G_{n}(f,E_{A})$, which itself only depends on the
crossover point between the two hypergeometrics. As the crossover point is
completely determined by the $n$-independent quantity

$S_{A}(E_{A})-S_{B}(E-E_{A})$ (27)

and the rest of the $E_{1}$ saddle point equation is independent of $n$, the
full saddle similarly becomes completely independent of $n$. This should be
contrasted with the obviously $n$-dependent saddle point of $F_{2}(E_{A})$.
This difference will generically cause the two saddles to differ by an
$\mathcal{O}(1)$ factor, so for all volume fractions we expect the different
in Rényi entropies to be volume law:

$\overline{S_{n}}-S_{n}^{MC}=\mathcal{O}(V),\quad n<1.$ (28)

Note that this applies for all volume fractions, implying that the $n<1$ Rényi
entropies do not obey the principle of canonical typicality.

This clarifies a conceptual point. For $n\rightarrow 1^{+}$, the $\sqrt{V}$
correction lies in between an exponentially suppressed $\mathcal{O}(e^{-cV})$
region ($f<1/2$) and a strongly enhanced $\mathcal{O}(V)$ region ($f>1/2$).
Why, then, do we not get a similar enhancement for $n\rightarrow 1^{-}$? The
answer is that the dominant behavior in $F_{1}(E_{A})$, which previously
supplied the emergent “soft mode” for the flat interval between two saddles,
becomes independent of the Rényi index. This nonanalyticity might be worrying
if one is used to a Rényi entropy analytic in $n$, but the thermodynamic limit
breaks this assumption. The form of these corrections agrees with the analysis
of a gravitational model in Appendix C of Dong:2021oad .

##  3 Entanglement Negativity

In this section we compute similar quantities as Dong:2020iod  for
entanglement negativity measures. We begin by reviewing some salient
properties of entanglement negativity and its utility as a tripartite measure
of entanglement before diving into the calculation.

###  3.1 Review of Negativity

Entanglement negativity refers to an entanglement measure based on properties
of the partial transpose operation applied to a bipartite density matrix
$\rho_{A_{1}A_{2}}$, defined via

$\expectationvalue{a_{1},a_{2}|\rho_{A_{1}A_{2}}^{T_{A_{2}}}|a_{1}^{\prime},a_{2}^{\prime}}=\expectationvalue{a_{1},a_{2}^{\prime}|\rho_{A_{1}A_{2}}|a_{1}^{\prime},a_{2}}$
(29)

for basis states $\\{\ket{a_{1}}\\}$ in $A_{1}$ and $\\{\ket{a_{2}}\\}$ in
$A_{2}$ VW ; Plenio:2005cwa ; Audenaert:2003 . The partial transpose is a
positive but not completely positive map, which means some of the eigenvalues
of $\rho_{A_{1}A_{2}}^{T_{A_{2}}}$ (hereafter $\rho_{A_{1}A_{2}}^{T_{2}}$) can
be negative. Entanglement negativity quantifies the different between the
eigenvalues of the partially transposed density matrix and the original
density matrix via

$\mathcal{N}(\rho_{A_{1}A_{2}})=\sum_{i}\frac{\absolutevalue{\lambda_{i}}-\lambda_{i}}{2}=\sum_{i:\lambda_{i}<0}\absolutevalue{\lambda_{i}}.$
(30)

As with the von Neumann entropy, there exist Rényi generalizations of
entanglement negativity:

$\mathcal{N}_{n}=\Tr\left(\rho_{A_{1}A_{2}}^{T_{2}}\right)^{n}.$ (31)

Due to the absolute value, one needs to define two different analytic
continuations for even and odd Rényi index $n$, so there are in fact two Rényi
negativities given by

$\displaystyle\mathcal{N}_{2k}^{\textrm{(even)}}$
$\displaystyle=\sum_{i}\absolutevalue{\lambda_{i}}^{2k}$
$\displaystyle\mathcal{N}_{2k-1}^{\textrm{(odd)}}$
$\displaystyle=\sum_{i}\textrm{sgn}{\lambda_{i}}\absolutevalue{\lambda_{i}}^{2k-1}$
(32)

for integer $k$. We define relevant entanglement measures via analytic
continuation from these quantities. The most common quantity to talk about is
the logarithmic negativity, given via a $k\rightarrow 1/2$ analytic
continuation of the even Rényi negativity

$\mathcal{E}(\rho_{A_{1}A_{2}})=\lim_{k\rightarrow
1/2}\log\mathcal{N}_{2k}^{\textrm{(even)}}(\rho_{A_{1}A_{2}})=\log\sum_{i}\absolutevalue{\lambda_{i}}.$
(33)

One other quantity of interest is the partially transposed entropy, also known
as the odd entropy, which is related to the $k\rightarrow 1$ analytic
continuation of the odd Rényi negativity and is explicitly given by

$S^{T_{2}}\equiv\lim_{k\rightarrow
1}\frac{1}{2k-2}\log\mathcal{N}_{2k-1}=-\sum_{i}\lambda_{i}\log\absolutevalue{\lambda_{i}}.$
(34)

We need to include the Rényi entropy-like singular term out front as
$\mathcal{N}^{\textrm{(odd)}}_{1}=\Tr\rho_{A_{1}A_{2}}^{T_{2}}=1$.

###  3.2 Disorder Averaged Negativity

Now we can discuss the disorder average222In the condensed matter literature,
disorder averaging has a different meaning and what we’re doing should more
properly be called “ensemble averaging”. Ensemble averaging, however, already
has a meaning in the high energy literature, so we keep with the terminology
of Dong:2020iod . in the Gaussian approximation described in the previous
section. The Schmidt decomposition of the energy eigenstate $\ket{E}$ is now

$\ket{E}=\sum_{i_{1}j_{1}J}M_{ijJ}\ket{E_{i}}_{A_{1}}\otimes\ket{E_{j}}_{A_{2}}\otimes\ket{E_{J}}_{B}.$
(35)

Once again we’ll consider $M_{ijJ}$ as a Gaussian random variable, in
particular with

$M_{ijJ}\approx e^{-S(E)/2}C_{ijJ}$ (36)
$\overline{C_{ijJ}}=0,\quad\overline{C_{ijJ}C_{i^{\prime}j^{\prime}J^{\prime}}}=\delta_{ii^{\prime}}\delta_{jj^{\prime}}\delta_{JJ^{\prime}}.$
(37)

The partially transposed density matrix is

$\rho_{A_{1}A_{2}}^{T_{2}}=\frac{1}{\mathcal{N}}\sum_{E_{i}E_{j}E_{J}}C_{i_{1}j_{1}J}C_{i_{2}j_{2}J}\ket{E_{i_{1}},E_{j_{2}}}\bra{E_{i_{2}}E_{j_{1}}},$
(38)

or, by replacing dummy variables

$\rho_{A_{1}A_{2}}^{T_{2}}=\frac{1}{\mathcal{N}}\sum_{E_{i}E_{j}E_{J}}C_{i_{1}j_{2}J}C_{i_{2}j_{1}J}\ket{E_{i_{1}},E_{j_{1}}}\bra{E_{i_{2}}E_{j_{2}}},$
(39)

where the sum over energies is understood to be in a window of width
3$\Delta$, though again we take this width to vanish. We also have

$\left(\rho_{A_{1}A_{2}}^{T_{2}}\right)^{n}=\frac{1}{\mathcal{N}^{n}}\sum_{E_{i_{1}},E_{j_{1}}}\sum_{i_{1},\cdots,i_{n},j_{1},\cdots,j_{n},J_{1},\cdots
J_{n}}\prod_{m=1}^{n}C_{i_{m}j_{m+1}J_{m}}C_{i_{m+1}j_{m}J_{m}}\ket{E_{i_{1}},E_{j_{1}}}\bra{E_{i_{n+1}}E_{j_{n+1}}}.$
(40)

Note that the partial transpose has made it so the $i$ ($A_{1}$) indices are
contracted cyclically, while the $j$ ($A_{2}$) indices are contracted anti-
cyclically. The resolvent equation for these Wick contractions is the same as
derived in Dong:2021oad , and a more detailed explanation is given in Appendix
B. We quote the result here:

$\lambda
R(\lambda)=e^{S_{A_{1}}+S_{A_{2}}}+\frac{e^{S_{B}}}{e^{S_{A_{2}}}}\frac{R(\lambda)(1+R(\lambda))}{1-e^{2S_{A_{2}}}R(\lambda)^{2}}.$
(41)

Figure 1: Phase diagram of Rényi negativity for various subsystem densities of
state. The $g$’s label the dominant permutation which appears in the sum over
Wick contractions; their exact forms are given in Appendix A. The resolvent
equation (41) is valid in the regime $S_{A_{2}}<<S_{A_{1}}+S_{B}$; we’ve
indicated the forbidden region $g=X^{-1}$ in a lighter shade. Reproduced with
minor alterations from Dong:2021oad

This resolvent equation furnishes a negativity spectrum described by the phase
diagram in Figure 1. There are two transitions to consider. The first is when
the $A$ and $B$ subsystems are the same size, i.e.
$S_{A_{1}}+S_{A_{2}}=S_{B}$, corresponding to the transition from $g=\text{1}$
to $g=X$ in the phase diagram. From the calculation in Dong:2021oad , we don’t
expect any enhanced corrections at this transition, so we don’t study it in
any detail, though the calculation would presumably follow the same steps. The
second transition of interest is when the $A_{1}$ subsystem is the same size
as the combined $A_{2}B$ subsystem, $S_{A_{1}}=S_{A_{2}}+S_{B}$, corresponding
to the transition from $g=\tau$ to $g=X$ in the phase diagram. In this regime
the sum over diagrams is known explicitly, and the disorder averaged partially
transposed density matrices are

$\Tr\overline{(\rho^{T_{2}}_{A_{1}A_{2}})^{2k}}=\begin{cases}\frac{1}{\mathcal{N}^{2k}}e^{2k(S_{A_{2}}+S_{B})+S_{A_{1}}}e^{S_{A_{2}}}{}_{2}F_{1}\left(1-k,-2k;2;e^{S_{A_{1}}-S_{A_{2}}-S_{B}}\right),S_{A_{1}}<S_{A_{2}}+S_{B}\\\
\frac{1}{\mathcal{N}^{2k}}e^{2kS_{A_{1}}+S_{A_{2}}+S_{B}}e^{S_{A_{2}}}{}_{2}F_{1}\left(1-k,-2k;2;e^{S_{A_{2}}+S_{B}-S_{A_{1}}}\right),S_{A_{1}}>S_{A_{2}}+S_{B}\end{cases}$
(42)

for even $n=2k$ and

$\Tr\overline{(\rho^{T_{2}}_{A_{1}A_{2}})^{2k-1}}=\begin{cases}\frac{1}{\mathcal{N}^{2k-1}}e^{(2k-1)(S_{A_{2}}+S_{B})+S_{A_{1}}}{}_{2}F_{1}\left(1-2k,1-k;1;e^{S_{A_{1}}-S_{A_{2}}-S_{B}}\right),S_{A_{1}}<S_{A_{2}}+S_{B}\\\
\frac{1}{\mathcal{N}^{2k-1}}e^{(2k-1)S_{A_{1}}+S_{A_{2}}+S_{B}}{}_{2}F_{1}\left(1-2k,1-k;1;e^{S_{A_{2}}+S_{B}-S_{A_{1}}}\right),S_{A_{1}}>S_{A_{2}}+S_{B}\end{cases}$
(43)

for odd $n=2k-1$. We give derivations for these formulae in Appendix A; the
gist is that we sum over all permutations which lie on a geodesic between two
dominant regions in phase space. The permutations on this geodesic can be
enumerated, and the previous formulae are functions whose moments reproduce
the combinatoric factors for these permutations.

##  4 Negativity Phase Transitions

We can use (42) and (43) to understand the difference between the
microcanonical and canonical Rényi negativities in a chaotic eigenstate, using
much the same techniques as were used in Dong:2020iod . We denote by
$f_{A_{1}}$ the volume fraction of $A_{1}$ such that the naïve phase
transition happens at $f_{A_{1}}=1/2$. We also denote the volume fraction of
$A_{2}$ by $f_{A_{2}}$ and use $f_{A}=f_{A_{1}}+f_{A_{2}}$ to denote the total
volume fraction of system $A$.

We impose energy conservation in all three subsystems, such that our ansatz is
for subsystem entropies is

$\displaystyle S_{A_{1}}(E_{A_{1}})$
$\displaystyle=f_{A_{1}}Vs\left(\frac{E_{A_{1}}}{f_{A_{1}}V}\right)$
$\displaystyle S_{A_{2}}(E_{A_{2}})$
$\displaystyle=f_{A_{2}}Vs\left(\frac{E_{A_{2}}}{f_{A_{2}}V}\right)$
$\displaystyle S_{B}(E-E_{A_{1}}-E_{A_{2}})$
$\displaystyle=(1-f_{A})Vs\left(\frac{E-E_{A_{1}}-E_{A_{2}}}{(1-f_{A})V}\right).$
(44)

These again follow from ergodicity and imposing that the subsystem entropy is
only a function of the subsystem energy density.

###  4.1 Even Rényi Negativity

We’ll start with studying the even Rényi negativities, from which the
logarithmic negativity descends. Our expressions for the logarithms of the
canonical ensemble and microcanonical ensemble Rényi negativities using our
previous ansatzes are as follows:

$\displaystyle\overline{\mathcal{N}_{2k}}$
$\displaystyle=\frac{1}{\mathcal{N}^{2k}}\int
dE_{A_{1}}dE_{A_{2}}e^{S_{A_{1}}(E_{A_{1}})+2S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}})}G_{k}(f_{A_{1}},f_{A_{2}},E_{A_{1}},E_{A_{2}})$
$\displaystyle\mathcal{N}_{2k}^{MC}$
$\displaystyle=\frac{1}{\mathcal{N}^{2k}}\int
dE_{A_{1}}dE_{A_{2}}e^{S_{A_{1}}(E_{A_{1}})+S_{A_{2}}(E_{A_{2}})+2k(S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}}))},$
(45)

where the function $G_{k}(f_{A_{1}},f_{A_{2}},E_{A_{1}},E_{A_{2}})$ is defined
as the $k$-dependent part of (42)

$G_{k}(f_{A_{1}},f_{A_{2}},E_{A_{1}},E_{A_{2}})=\begin{cases}e^{(2k-1)(S_{A_{2}}+S_{B})}{}_{2}F_{1}\left(1-k,-2k;2;e^{S_{A_{1}}-S_{A_{2}}-S_{B}}\right),S_{A_{1}}<S_{A_{2}}+S_{B}\\\
e^{(2k-1)S_{A_{1}}}{}_{2}F_{1}\left(1-k,-2k;2;e^{S_{A_{2}}+S_{B}-S_{A_{1}}}\right),S_{A_{1}}>S_{A_{2}}+S_{B},\end{cases}$
(46)

and $\mathcal{N}$ (with no other sub/superscripts) is an overall normalization
given by

$\mathcal{N}=\int
dE_{A_{1}}dE_{A_{2}}e^{S_{A_{1}}(E_{A_{1}})+S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}})}.$
(47)

Whenever unspecified, the subsystem entropies should now be understood to be
valued at the subsystem energies, which we only omit for notational clarity.
We write the difference between the logarithms of these quantities as

$\log\overline{\mathcal{N}_{2k}}-\log\mathcal{N}^{MC}_{2k}\equiv\log\left(\frac{\int
dE_{A_{1}}dE_{A_{2}}\exp(F_{1}(E_{A_{1}},E_{A_{2}}))}{\int
dE_{A_{1}}dE_{A_{2}}\exp(F_{2}(E_{A_{1}},E_{A_{2}}))}\right),$ (48)

where the functions $F_{1}(E_{A_{1}},E_{A_{2}})$ and
$F_{2}(E_{A_{1}},E_{A_{2}})$ are defined via the corresponding integrands in
(45). The strategy will be to find saddle points for
$F_{1}(E_{A_{1}},E_{A_{2}})$ and $F_{2}(E_{A_{1}},E_{A_{2}})$ and use the
relative behavior of those saddle points to determine the scaling of the
correction at transition.

We have two coupled saddle point equations for each both functions, which are
given by

$\displaystyle s^{\prime}\left(\frac{E_{1}^{(1)}}{f_{A_{1}}V}\right)$
$\displaystyle=s^{\prime}\left(\frac{E-E_{1}^{(1)}-E_{1}^{(2)}}{(1-f_{A})V}\right)-\frac{\partial_{E_{A_{1}}}G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}{G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}$
$\displaystyle 2s^{\prime}\left(\frac{E_{1}^{(2)}}{f_{A_{2}}V}\right)$
$\displaystyle=s^{\prime}\left(\frac{E-E_{1}^{(1)}-E_{1}^{(2)}}{(1-f_{A})V}\right)-\frac{\partial_{E_{A_{2}}}G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}{G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}$
$\displaystyle s^{\prime}\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)$
$\displaystyle=2ks^{\prime}\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right)$
$\displaystyle(2k+1)s^{\prime}\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)$
$\displaystyle=2ks^{\prime}\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right),$
(49)

where the pair $\mathcal{E}_{1}=(E_{1}^{(1)},E_{1}^{(2)})$ denotes a saddle
point for $F_{1}(E_{A_{1}},E_{A_{2}})$, while
$\mathcal{E}_{2}=(E_{2}^{(1)},E_{2}^{(2)})$ denotes the saddle point for
$F_{2}(E_{A_{1}},E_{A_{2}})$. As $F_{2}(E_{A_{1}},E_{A_{2}})$ is a strictly
concave function, there is only one global maximum.
$F_{1}(E_{A_{1}},E_{A_{2}})$ on the other hand can have two maxima, as
$G_{k}(f_{A_{1}},f_{A_{2}},E_{A_{1}},E_{A_{2}})$ is strictly nonmonotonic.

Figure 2: A schematic plot of regions (shaded in red) in the
$E_{A_{1}}-E_{A_{2}}$ plane where our ansatz for the dominant sum over
permutations does not hold. The lines separating the regions will depend
sensitively on the form of $s(e)$ and the volume fractions of the subsystems.

The first thing we have to be careful about is whether we are still within our
regime of validity for probing the transition of interest. In the case of
Rényi entropy, the fact that the dominant contribution comes from noncrossing
partitions was assumed to hold for all of parameter space, that is for all
values of subsystem entropy. This can be traced back to the fact that the
dominant permutations all lie on a single geodesic $G(\text{1},X)$. In our
case, we’re trying to probe the transition on one geodesic $G(\tau,X)$ while
suppressing diagrams from other geodesics, which imposes some natural
constraints on the size of our subsystems.

We are justified in only considering the diagrams from Appendix A only if the
saddle point energies satisfy the conditions:

$\displaystyle
S_{A_{2}}(E_{1,2}^{(2)})<S_{A_{1}}(E_{1,2}^{(1)})+S_{B}(E-E_{1,2}^{(1)}-E_{1,2}^{(2)})$
$\displaystyle
S_{B}(E-E_{1,2}^{(1)}-E_{1,2}^{(2)})<S_{A_{1}}(E_{1,2}^{(1)})+S_{A_{2}}(E_{1,2}^{(2)}),$
(50)

such that all contributions from subleading permutations remain subleading. We
include a rough phase diagram of the allowed region to explore in Figure 2. If
the saddle point lies outside the allowed region, our answer for the dominant
sum over permutations no longer holds, so we shouldn’t try to explore those
regions of phase space.

This means before attempting to compute corrections at transition for all
subsystem volume fractions, we should derive some bounds on the regime of
validity of our approximation. We’ll make use of the following inequality:

$S_{A_{1}}(E_{A_{1}})+S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}})\leq
Vs\left(\frac{E}{V}\right),$ (51)

which follows from the fact that our subsystem entropy function $s(e)$ is
concave. Plugging in the saddle points and using the first constraint in (50)
we can write

$\displaystyle 2S_{A_{2}}(E_{2}^{(2)})$
$\displaystyle<S_{A_{1}}(E_{2}^{(1)})+S_{A_{2}}(E_{2}^{(2)})+S_{B}(E-E_{2}^{(1)}-E_{2}^{(2)})<Vs\left(\frac{E}{V}\right)$
$\displaystyle\Rightarrow
S_{A_{2}}(E_{2}^{(2)})<\frac{V}{2}s\left(\frac{E}{V}\right).$ (52)

We can use this relation to find

$\displaystyle
S_{A_{2}}(E_{2}^{(2)})=f_{A_{2}}s\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)>f_{A_{2}}s\left(\frac{E_{2}^{(2)}}{V}\right)$
$\displaystyle\Rightarrow
f_{A_{2}}s\left(\frac{E_{2}^{(2)}}{V}\right)<\frac{V}{2}s\left(\frac{E}{V}\right),$
(53)

as $E_{2}^{(2)}<E$, a constraint on $f_{A_{2}}$ which makes this true for all
subsystem entropy densities is

$f_{A_{2}}<1/2.$ (54)

Therefore our calculations are only valid when subsystem $A_{2}$ is less than
half of the total system size. We can find a similar inequality on $S_{B}$
using the second constraint in (50). We have

$\displaystyle 2S_{B}(E-E_{2}^{(1)}-E_{2}^{(2)})$
$\displaystyle<S_{A_{1}}(E_{2}^{(1)})+S_{A_{2}}(E_{2}^{(2)})+S_{B}(E-E_{2}^{(1)}-E_{2}^{(2)})\leq
Vs\left(\frac{E}{V}\right)$ $\displaystyle\Rightarrow
S_{B}(E-E_{2}^{(1)}-E_{2}^{(2)})$
$\displaystyle<\frac{V}{2}s\left(\frac{E}{V}\right).$ (55)

We can therefore write

$\displaystyle
S_{B}(E-E_{2}^{(1)}-E_{2}^{(2)})=(1-f_{A})Vs\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right)$
$\displaystyle>(1-f_{A})Vs\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{V}\right)$
$\displaystyle\Rightarrow(1-f_{A})Vs\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{V}\right)$
$\displaystyle<\frac{V}{2}s\left(\frac{E}{V}\right).$ (56)

Again, a result that makes this inequality true for all saddle point energies
is

$f_{A}>1/2.$ (57)

This ties together a nice family of restrictions: both subsystems $A_{2}$ and
$B$ have to have volume fraction less then half of the system. We illustrate
these constraints in Figure 3. This makes some sense, as we want to probe
transitions dominated by the behavior of $A_{1}$ relative to the rest of the
system.

Figure 3: Excluded volume fractions from our analysis of the cyclic to
pairwise phase transition. Describing the colored “forbidden” regions would
require a sum over permutations we assert to be subdominant.

Another way of seeing there should be a restricted regime for our procedure is
as follows: entanglement negativity is agnostic as to which subsystem $A_{1}$
or $A_{2}$ one applies the partial transpose to. This would of course result
in an averaged density matrix trace symmetric under exchange of $S_{A_{1}}$
and $S_{A_{2}}$, which our expressions (42) and (43) are not. However, by
writing a resolvent equation valid only in a certain parameter regime, we can
no longer comfortably integrate over all energies. This is an important point
because the deviations from the featureless case can in principle be of order
the system size, and so corrections are not necessarily perturbative as they
were assumed to be in Dong:2021oad .

We can, however, be comfortable in the validity of our calculation if the
saddle points for $F_{1}(E_{A_{1}},E_{A_{2}})$ and
$F_{2}(E_{A_{1}},E_{A_{2}})$ obey the conditions above, so restricting to the
set of entropy functions which satisfy (50), let’s first look at the saddle
point equations for $\mathcal{E}_{2}$. Setting the third and fourth equations
equal yields

$s^{\prime}\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)=(2k+1)s^{\prime}\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right).$
(58)

As $s^{\prime}(e)$ is a monotonically decreasing function, for all $k>0$ we
have the inequality

$E_{2}^{(2)}>\frac{f_{A_{2}}}{f_{A_{1}}}E_{2}^{(1)}.$ (59)

We can use this inequality to write a simple inequality on $E_{2}^{(1)}$ by
rewriting the $E_{2}^{(1)}$ saddle point equation as

$s^{\prime}\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)>2ks^{\prime}\left(\frac{E-\frac{f_{A}}{f_{A_{1}}}E_{2}^{(1)}}{(1-f_{A})V}\right).$
(60)

Now we have an inequality which depends on $k$, as we can write

$E_{2}^{(1)}<f_{A_{1}}E,\quad k\geq 1/2.$ (61)

Note that this result is also valid for $k=1/2$, as the relation (59) is a
strict inequality which is never saturated for positive $k$. We can use a
similar strategy to write an inequality for $E_{2}^{(2)}$. Rewriting the
$E_{2}^{(2)}$ equation with (59) yields

$(2k+1)s^{\prime}\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)<2ks^{\prime}\left(\frac{E-\frac{f_{A}}{f_{A_{2}}}E_{2}^{(2)}}{1-f_{A}}\right).$
(62)

The resulting inequality has a slightly different $k$ dependence:

$E_{2}^{(2)}>f_{A_{2}}E,\quad k>0.$ (63)

The last inequalities we can write are those for the saddle point values of
$S_{A_{1}}$ and $S_{A_{2}}$:

$\displaystyle S_{A_{1}}(E_{2}^{(1)})$
$\displaystyle=f_{A_{1}}Vs\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)<f_{A_{1}}Vs\left(\frac{E}{V}\right)$
$\displaystyle S_{A_{2}}(E_{2}^{(2)})$
$\displaystyle=f_{A_{2}}Vs\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)>f_{A_{2}}Vs\left(\frac{E}{V}\right).$
(64)

We’d like to find conditions on the hypergeometric being stuck on the first
branch, i.e. $S_{A_{1}}<S_{A_{2}}+S_{B}$. This is guaranteed to happen if the
weaker inequality $S_{A_{1}}<S_{A_{2}}$ is satisfied, which from (64) is
necessarily true when

$f_{A_{2}}>f_{A_{1}}.$ (65)

If we assume $S_{A_{1}}<S_{A_{2}}$ for the $\mathcal{E}_{1}$ saddle point as
well, the argument of the hypergeometric is exponentially suppressed and we
can approximate it by

${}_{2}F_{1}(1-k,-2k;2;x)\approx 1+k(k-1)x,$ (66)

where the small parameter $x$ is now

$x\equiv
e^{S_{A_{1}}(E_{1}^{(1)})-S_{A_{2}}(E_{1}^{(2)})-S_{B}(E-E_{1}^{(1)}-E_{1}^{(2)})}.$
(67)

Under this assumption the saddle point equations for $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ are the same up to exponentially suppressed terms, and
therefore the saddle points $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are
exponentially close. This leads to the following form of corrections to ETH:

$\log\overline{\mathcal{N}_{2k}}-\log\mathcal{N}_{2k}^{MC}\propto\mathcal{O}(e^{-cV}),\quad
k\geq 1/2,f_{A_{2}}>f_{A_{1}}$ (68)

We can write a similar inequality for which $S_{A_{1}}<S_{B}$ is always
satisfied. We recall the $E_{2}^{(1)}$ saddle point equation:

$s^{\prime}\left(\frac{E_{2}^{(1)}}{f_{A_{1}}}\right)=2ks^{\prime}\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right).$
(69)

At $k=1/2$ there’s clearly an equality between the arguments of the functions
on the right and left, so for $k\geq 1/2$ we have the inequality

$\frac{E_{2}^{(1)}}{fV}\leq\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V},\quad
k\geq 1/2.$ (70)

We’d like to satisfy the inequality $S_{A_{1}}<S_{B}$, or

$f_{A_{1}}Vs\left(\frac{E_{2}^{(1)}}{fV}\right)<(1-f_{A})Vs\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right).$
(71)

This is always satisfied if

$f_{A_{1}}<1-f_{A}.$ (72)

So far we have two constraints which carve out a corner of the phase space for
all $k\geq 1/2$.

Now let’s try to find a condition such that $S_{A_{1}}>S_{A_{2}}+S_{B}$. Using
our previous ansatz this condition is written as

$f_{A_{1}}s\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)>f_{A_{2}}s\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)+(1-f_{A})s\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right).$
(73)

For all $k>0$ we can use (59) to rewrite this as

$(f_{A_{1}}-f_{A_{2}})s\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)>(1-f_{A})s\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right).$
(74)

Using the $E_{2}^{(2)}$ saddle point equation, there exists for $k>0$:

$\frac{E_{2}^{(2)}}{f_{A_{2}}}>\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{1-f_{A}}.$
(75)

Therefore, $S_{A_{1}}>S_{A_{2}}+S_{B}$ is always satisfied if

$f_{A_{1}}-f_{A_{2}}>1-f_{A}\Rightarrow f_{A_{1}}>1/2,\quad k>0.$ (76)

For these volume fractions the corrections to the Rényi negativity are
extensive in the system size, as $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ have
no relation:

$\log\mathcal{N}_{2k}-\log\mathcal{N}_{2k}\propto\mathcal{O}(V),\quad
k>0,f_{A_{1}}>1/2.$ (77)

Figure 4: Phase diagram for corrections to even Rényi negativities. The
concave region with $\mathcal{O}(e^{-cV})$ corrections comes from requiring
$S_{A_{1}}<S_{A_{2}}$ and/or $S_{A_{1}}<S_{B}$. The $\mathcal{O}(V)$ region
requires $S_{A_{1}}>S_{A_{2}}+S_{B}$. The interpolation between these regions
will lie somewhere with $f_{A_{1}}<1/2$ and is outlined by the dashed lines.
The yellow curve represents a system specific boundary which will depend on
$k$ and potentially on the specifics of $s(e)$.

We summarize the results so far in Figure 4. In that phase diagram, none of
the boundaries should be though of as sharp, that is as (59) is never
saturated for $k>0$, neither are any constraints that depend on it. The
interpolation between $\mathcal{O}(e^{-cV})$ corrections and $\mathcal{O}(V)$
corrections will happen somewhere in this “unknown region”, though the only
relevant point is that at $f_{A_{1}}=1/2$ we should still be in a region with
extensive corrections. In particular this implies the logarithmic negativity
receives $\mathcal{O}(V)$ corrections, as was noted in Dong:2021oad .333At
$k=1$, the even Rényi negativity is equal to the second Rényi entropy
$S_{2}(\rho_{A})$, which for $f_{A_{1}}+f_{A_{2}}>1/2$ is expected to always
receive volume law corrections, which we don’t see for all volume fractions.
This is an important consequence of the restriction to a particular phase
transition; we require the full sum over diagrams to reproduce the partially
transposed density matrix exactly.

We won’t comment on the case $k<1/2$ for $f_{A_{1}}<1/2$, though the
expectation is that, like the $n<1$ Rényi entropy, these measures always
receive volume law corrections. It’s also entirely possible the interpolating
line continues moving towards the point $(0,1/2)$, meaning there’s some set of
volume fractions for which arbitrarily small but positive $k$ are well-
approximated by ETH.

###  4.2 Odd Rényi Negativity

We can repeat the previous analysis for odd $n$. We have different expressions
for the canonical and microcanonical Rényi negativities:

$\displaystyle\log\overline{\mathcal{N}_{2k-1}}$
$\displaystyle=\frac{1}{\mathcal{N}_{2k-1}}\int
dE_{A_{1}}dE_{A_{2}}e^{S_{A_{1}}(E_{A_{1}})+S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}})}G_{k}(f_{A_{1}},f_{A_{2}},E_{A_{1}},E_{A_{2}})$
$\displaystyle\log\mathcal{N}_{2k-1}^{MC}$
$\displaystyle=\frac{1}{\mathcal{N}_{2k-1}}\int
dE_{A_{1}}dE_{A_{2}}e^{S_{A_{1}}(E_{A_{1}})+(2k-1)(S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}}))},$
(78)

where $G_{k}(f_{A_{1}},f_{A_{2}},E_{A_{1}},E_{A_{2}})$ is now defined by (43)
as:

$G_{k}(f_{A_{1}},f_{A_{2}},E_{A_{1}},E_{A_{2}})=\begin{cases}e^{(2k-2)(S_{A_{2}}+S_{B})}{}_{2}F_{1}\left(1-k,1-2k;1;e^{S_{A_{1}}-S_{A_{2}}-S_{B}}\right),S_{A_{1}}<S_{A_{2}}+S_{B}\\\
e^{(2k-2)S_{A_{1}}}{}_{2}F_{1}\left(1-2k,1-k;1;e^{S_{A_{2}}+S_{B}-S_{A_{1}}}\right),S_{A_{1}}>S_{A_{2}}+S_{B}.\end{cases}$
(79)

Again the subsystem entropies should be valued at their respective subsystem
energies. Notably $\log\overline{\mathcal{N}_{2k-1}}$ enjoys a symmetry under
$S_{A_{1}}\leftrightarrow S_{A_{2}}+S_{B}$. We again write the difference
between the canonical and microcanonical answers as

$\log\overline{\mathcal{N}_{2k-1}}-\log\mathcal{N}^{MC}_{2k-1}=\log\left(\frac{\int
dE_{A_{1}}dE_{A_{2}}\exp(F_{1}(E_{A_{1}},E_{A_{2}}))}{\int
dE_{A_{1}}dE_{A_{2}}\exp(F_{2}(E_{A_{1}},E_{A_{2}}))}\right)$ (80)

and use the same ansatz (44) to write the saddle point equations for $F_{1}$
and $F_{2}$ as

$\displaystyle s^{\prime}\left(\frac{E_{1}^{(1)}}{f_{A_{1}}V}\right)$
$\displaystyle=s^{\prime}\left(\frac{E-E_{1}^{(1)}-E_{1}^{(2)}}{(1-f_{A})V}\right)-\frac{\partial_{E_{A_{1}}}G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}{G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}$
$\displaystyle s^{\prime}\left(\frac{E_{1}^{(2)}}{f_{A_{2}}V}\right)$
$\displaystyle=s^{\prime}\left(\frac{E-E_{1}^{(1)}-E_{1}^{(2)}}{(1-f_{A})V}\right)-\frac{\partial_{E_{A_{2}}}G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}{G_{k}(f_{A_{1}},f_{A_{2}},E_{1}^{(1)},E_{1}^{(2)})}$
$\displaystyle s^{\prime}\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)$
$\displaystyle=(2k-1)s^{\prime}\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right)$
$\displaystyle s^{\prime}\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)$
$\displaystyle=s^{\prime}\left(\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{(1-f_{A})V}\right).$
(81)

Let’s again investigate the saddle point for $F_{2}$. We immediately see

$\frac{E_{2}^{(2)}}{f_{A_{2}}}=\frac{E-E_{2}^{(1)}-E_{2}^{(2)}}{1-f_{A}}$ (82)

for all $k$! This is a striking result, as it means we can write the sum of
subsystem entropies in $A_{2}$ and $B$ as

$S_{A_{2}}(E_{2}^{(2)})+S_{B}(E-E_{2}^{(1)}-E_{2}^{(2)})\equiv
S_{\overline{A_{1}}}(E_{2}^{(2)})=(1-f_{A_{1}})s\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)$
(83)

This is important as for the odd Rényi negativity, $S_{A_{2}}$ and $S_{B}$
always appear summed, so if we’re only interested in the leading saddle point
approximation we can treat them as one subsystem entropy
$S_{\overline{A_{1}}}$. As such we can rewrite the single saddle point
equation as

$s^{\prime}\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)=(2k-1)s^{\prime}\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right).$
(84)

We recognize this as similar to the saddle point equation (24) for $F_{2}(E)$
in the Rényi entropy, but we’ll go through the discussion nonetheless. At
$k=1$ we can exactly solve for the subsystem energies and they are,
unsurprisingly, proportional to the volume fractions of their respective
subsystems:

$\displaystyle E_{2}^{(1)}$ $\displaystyle=f_{A_{1}}E,\quad k=1$
$\displaystyle E_{2}^{(2)}$ $\displaystyle=f_{A_{2}}E,\quad k=1.$ (85)

When $k>1$, we again have

$E_{2}^{(2)}>\frac{f_{A_{2}}}{f_{A_{1}}}E_{2}^{(1)},$ (86)

which was true for general $k$ in the even case. Similar inequalities on
volume fraction hold in the odd case; we still have

$\displaystyle E_{2}^{(1)}$ $\displaystyle<f_{A_{1}}E,\quad k>1$
$\displaystyle E_{2}^{(2)}$ $\displaystyle>f_{A_{2}}E,\quad k>1.$ (87)

From this the inequality $S_{A_{1}}<S_{\overline{A_{1}}}$ is clearly satisfied
when

$f_{A_{1}}<1/2,$ (88)

and corrections are exponentially suppressed. For $f_{A_{1}}>1/2$, this won’t
be true generically and the corrections are extensive.

We can also say interesting things about $k<1$. In this case the inequalities
are flipped:

$\displaystyle E_{2}^{(1)}$ $\displaystyle>f_{A_{1}}E,\quad k<1$
$\displaystyle E_{2}^{(2)}$ $\displaystyle<f_{A_{2}}E,\quad k<1$
$\displaystyle E_{2}^{(2)}$
$\displaystyle<\frac{f_{A_{2}}}{f_{A_{1}}}E_{2}^{(1)}.$ (89)

We can check where $S_{A_{1}}>S_{\overline{A_{1}}}$. From (89) we have

$\displaystyle S_{A_{1}}(E_{2}^{(1)})$
$\displaystyle=f_{A_{1}}Vs\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)>f_{A_{1}}Vs\left(\frac{E}{V}\right)$
$\displaystyle S_{\overline{A_{1}}}(E_{2}^{(2)})$
$\displaystyle=(1-f_{A_{1}})Vs\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)<(1-f_{A_{1}})Vs\left(\frac{E}{V}\right).$
(90)

We see that $S_{A_{1}}>S_{\overline{A_{1}}}$ is guaranteed to be satisfied if
$f_{A_{1}}>1/2$, and indeed there is no generic behavior for $f_{A_{1}}<1/2$.
Thus the corrections are extensive for all volume fractions for $k<1$.

###  4.3 Odd Rényi Negativity at Transition

We would like to study this case in analogy with the entanglement entropy, for
reasons that will be clear shortly. Let’s follow the same procedure of
dividing $F_{1}$ into two pieces, $F_{\textrm{dom}}$ and $F_{\Delta}$, defined
as

$\displaystyle F_{\textrm{dom}}$
$\displaystyle=S_{A_{1}}(E_{A_{1}})+S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}})$
$\displaystyle+(2k-2)\textrm{max}\\{S_{A_{1}}(E_{A_{1}}),S_{A_{2}}(E_{A_{2}})+S_{B}(E-E_{A_{1}}-E_{A_{2}})\\}$
$\displaystyle F_{\Delta}$
$\displaystyle=\log{}_{2}F_{1}\left(1-2k,1-k;1;e^{-|S_{A_{1}}(E_{A_{1}})-S_{A_{2}}(E_{A_{2}})-S_{B}(E-E_{A_{1}}-E_{A_{2}})|}\right).$
(91)

That is, we take the dominant contribution and relegate the subleading
contributions to a term bounded by $\mathcal{O}(1)$ in volume factors:

$1\leq e^{F_{\Delta}}\leq a_{k},\quad a_{k}\equiv{3k-2\choose
k-1}=\frac{\Gamma(3k-1)}{\Gamma(k)\Gamma(2k)}=1+(k-1)+\mathcal{O}(k-1)^{2}.$
(92)

The averaged Rényi negativity, with a $\frac{1}{2k-2}$ factor which will be
important later, can be rewritten as

$\frac{1}{2k-2}\log\overline{\mathcal{N}_{2k-1}}=\frac{1}{2k-2}\log\left(\frac{1}{\mathcal{N}_{2k-1}}\int
dE_{A_{1}}dE_{A_{2}}e^{F_{\textrm{dom}}+F_{\Delta}}\right),$ (93)

and we can bound $\log\overline{\mathcal{N}_{2k-1}}$ via

$\log\overline{\mathcal{N}_{2k-1}}-\log\mathcal{N}_{2k-1}^{\textrm{dom}}\leq\frac{1}{2}+\mathcal{O}(k-1).$
(94)

As such $\mathcal{N}^{\textrm{dom}}_{2k-1}$ is enough to look for corrections
larger than $\mathcal{O}(1)$.

Unlike the Rényi entropy, at $f=1/2$ there’s no obvious reflection symmetry of
the energies in $F_{\textrm{dom}}$, and indeed we don’t find one numerically.
There is, however, a symmetry in the saddle points, which we’ll argue for as
follows. Call the two saddle points for $F_{\textrm{dom}}$ (or $F_{1}$, it
makes no difference here)
$\mathcal{E}_{1}^{(a)}=(E_{1}^{(1,a)},E_{1}^{(2,a)})$ and
$\mathcal{E}_{1}^{(b)}=(E_{1}^{(1,b)},E_{1}^{(2,b)})$. Under the exchange
$S_{A_{1}}\leftrightarrow S_{\overline{A}_{1}}$, the saddles are swapped due
to the symmetry of the odd Rényi negativity. It’s clear then at
$f_{A_{1}}=1/2$ there exists the equivalence

$\displaystyle\frac{E_{1}^{(1,a)}}{f_{A_{1}}}$
$\displaystyle=\frac{E_{1}^{(2,b)}}{f_{A_{2}}}$
$\displaystyle\frac{E_{1}^{(1,b)}}{f_{A_{2}}}$
$\displaystyle=\frac{E_{1}^{(2,a)}}{f_{A_{2}}}.$ (95)

This means that the two saddle points contribute with equal magnitude, which
contributes an $\mathcal{O}(1)$ factor to the difference between the canonical
and microcanonical negativities:

$\frac{1}{2k-2}\left(\log\mathcal{N}^{\textrm{dom}}_{2k-1}-\log\mathcal{N}^{MC}_{2k-1}\right)=\frac{\log
2}{2-2k}\sim\mathcal{O}(1)$ (96)

However, as in the case of von Neumann entropy, there is a subtletly related
to the fact that the two saddles collide in the limit $k\rightarrow 1$, i.e.
the partially transposed entropy. As they collide, there is an emergent region
between the saddles which contributes to the integral, so we can’t treat the
presence of multiple equivalent saddles at leading order, we must integrate
over the interpolating region. We show a plot of this phenomenon in Figure 5.

Figure 5: Plots of $F_{1}(E_{A_{1}},E_{A_{2}})$ at phase transition. We’ve set
$E=V=1,f_{A_{1}}=1/2$, and $f_{A_{2}}=3/10$. For large $k$ (upper left), the
two saddle points are well-separated and can be treated separately. As we
decrease $k$ (upper right) the saddle points approach one another and produce
an emergent flat region. At exactly $k=1$ (bottom) the saddle points coincide
at $(f_{A_{1}}E,f_{A_{2}}E)$. The dotted line connecting the saddle points is
given by $E_{A_{2}}=-2f_{A_{2}}(E_{A_{1}}-E)$; all saddles at $f_{A_{1}}=1/2$
lie along this line.

Let’s solve the $F_{2}$ saddle point equations perturbatively in $\delta\equiv
2k-2$. The $E_{2}^{(1)}$ saddle point equation (84) becomes

$s^{\prime}\left(\frac{E_{2}^{(1)}}{f_{A_{1}}V}\right)=(1+\delta)s^{\prime}\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}\right)\approx
s^{\prime}\left(\frac{E_{2}^{(2)}}{f_{A_{2}}V}+\delta\frac{s^{\prime}\left(E/V\right)}{s^{\prime\prime}\left(E/V\right)}\right),$
(97)

where we’ve again used that $E_{2}^{(2)}=f_{A_{2}}E$. Combining this with the
unchanged (82) and plugging in $f_{A_{1}}=1/2$ yields

$\displaystyle E_{2}^{(1)}$
$\displaystyle=\frac{E}{2}+\frac{V\delta}{4}\frac{s^{\prime}(E/V)}{s^{\prime\prime}(E/V)}$
$\displaystyle E_{2}^{(2)}$
$\displaystyle=f_{A_{2}}E-\frac{f_{A_{2}}V\delta}{2}\frac{s^{\prime}(E/V)}{s^{\prime\prime}(E/V)}$
(98)

From this we can write our subsystem entropies $S_{A_{1}}$ and
$S_{\overline{A_{1}}}$ in the familiar form

$\displaystyle S_{A_{1}}(E_{2}^{(1)})$
$\displaystyle=\frac{1}{2}s\left(E+\frac{V\delta}{2}\frac{s^{\prime}(E/V)}{s^{\prime\prime}(E/V)}\right)$
$\displaystyle S_{\overline{A_{1}}}(E_{2}^{(2)})$
$\displaystyle=\frac{1}{2}s\left(E-\frac{V\delta}{2}\frac{s^{\prime}(E/V)}{s^{\prime\prime}(E/V)}\right)$
(99)

What happens as $k\rightarrow 1$ for the odd Rényi negativity is precisely the
same as what happens for the $n\rightarrow 1$ von Neumann entropy, namely that
the $F_{\Delta}$ term “fills in” the space between the two saddles. The only
difference is that this flat direction runs between two saddles separated
along a line in the $E_{A_{1}}-E_{A_{2}}$ plane specified by $f_{A_{2}}$. The
rest of the calculation is completely unchanged from that of the von Neumann
entropy, and there is an enhanced correction exactly of the same form:

$\overline{S^{T_{2}}}-S^{T_{2}}_{MC}=-\sqrt{\frac{C_{V}}{2\pi}}+\mathcal{O}(\delta)\sim\mathcal{O}(\sqrt{V})$
(100)

In Dong:2021oad , it was noted that a naïve calculation shows the partially
transposed entropy receives $\mathcal{O}(\sqrt{V})$ corrections, but a more
accurate analysis shows it receives $\mathcal{O}(V)$ corrections. It would be
interesting to understand the difference between our calculation and
theirs.444A possible resolution is that our calculation was done at fixed
$f_{A_{2}}$, roughly the same as fixing $k_{2}$ in Dong:2021oad . Only when
$k=k_{1}k_{2}$ was fixed, similar to fixing $f_{A}$, do they see
$\mathcal{O}(V)$ corrections.

##  5 Discussion and Future Work

In this work we’ve studied a class of tripartite entanglement measures, the
Rényi negativities, in a toy model of a chaotic eigenstate. We’ve resummed the
relevant noncrossing permutations obtained via Wick contractions relevant at
the transition of interest and studied the corrections to the dominant
microcanonical saddle.

The main takeaway is as follows: logarithmic negativity and its Rényi
generalizations thereof are not always “good” chaotic observables in the sense
that their fluctuations (the difference between the canonical and
microcanincal expectation values) are often of the same order as the
quantities themselves, implying they are not self-averaging for all volume
fractions. We’ve shown this is the case for the even Rényi negativity at
transition, as well as for both even and odd Rényi negativities for
$f_{A_{1}}>1/2$. In particular we’ve shown that odd Rényi negativity behaves
mostly the same as Rényi entropy at the $\tau$ to $X$ transition, exhibiting a
$\mathcal{O}(\sqrt{V})$ enhanced correction at exactly $k=1$. One surprising
outcome is that, for both Rényi negativities, canonical typicality holds in
some cases where the partially transposed density matrix is defined on a
subsystem $A_{1}A_{2}$ larger than half of the total system.

One interesting question is what bearing these volumetric corrections have on
the validity of the cosmic brane prescription. It’s expected that the
holographic dual of subregion entanglement measures is given by the action of
a geometric solution with a massive cosmic brane (or branes) inserted
Lewkowycz:2013nqa ; Dong:2016fnf . Away from transition, it’s expected that
there is a single dominant saddle, or at the very least an $\mathcal{O}(1)$
number of equivalent saddles, all of which have have small enough fluctuations
that we can treat the calculation of the brane area perturbatively. What
happens if this saddle doesn’t exist?555We thank Pratik Rath for discussions
on this point. For the $n<1$ Rényi entropy, for example, the dual
gravitational description is expected to be a cosmic brane with negative
tension Dong:2016fnf , so the minimal energy configuration would be a brane
that falls towards the boundary. This is roughly the “holographic dual” of the
$\mathcal{O}(V)$ corrections to ETH; it represents a failure of a single
approximately geometric state to describe the dual system.

We now discuss some extensions to our work. A necessary restriction in our
analysis is only summing over a subset of all relevant permutations near a
particular phase transition. It would be useful to find a closed form
expression for the moments of a block transposed Wishart matrix without these
assumptions, which would involve finding a closed form solution to the
recursion relation in banica2013asymptotic . This would be especially nice as
we could probe the region $f_{A}<1/2$, which is where one could expect ETH to
hold as the partially transposed density matrix is defined on less than half
of the total system.

A technical point in our analysis was the use of 2-Dyck paths and 2-Narayana
numbers, as opposed to (1-)Dyck paths which appear in the calculation of
entanglement entropy. It’s possible some further generalization of Narayana
numbers (as in e.g. Novelli ) will be relevant for calculating transitions in
higher party entanglement measures in a similar model.

So far, we’ve only discussed Rényi negativity, but there exists a family of
holographically inspired measures termed refined Rényi negativities, which are
given by

$\displaystyle
S^{T_{2}(n)}(\rho_{A_{1}A_{2}})=-n^{2}\partial_{n}\left(\frac{1}{n}\log\mathcal{N}^{\textrm{(odd/even)}}_{n}(\rho_{A_{1}A_{2}})\right)$
(101)

We have not touched on the structure of transitions in these measures, but
they could presumably be treated in the same way we’ve presented. Of
particular interest is the refined Rényi 2-negativity $S^{T_{2}(2)}$, the
$n\rightarrow 2$ limit of the even refined Rényi entropy. This quantity is
explicitly given by

$S^{T_{2}(2)}=-\lim_{m\rightarrow
1}m^{2}\partial_{m}\left(\frac{1}{m}\log\mathcal{N}^{\textrm{(even)}}_{2m}\right)=-\sum_{i}\frac{\lambda_{i}^{2}}{\sum_{j}\lambda_{j}^{2}}\log\left(\frac{\lambda_{i}^{2}}{\sum_{j}\lambda_{j}^{2}}\right)$
(102)

which is the von Neumann entropy of the normalized density matrix
$\left(\rho^{T_{2}}_{A_{1}A_{2}}\right)^{2}$. Consequently, the expectation is
that the corrections will be $\mathcal{O}(\sqrt{V})$, which is indeed what is
seen in the gravitational setting. It would be nice to derive this relation
from our formalism.

Additionally, this formalism could be applied to study the reflected entropy
Dutta:2019gen  and its Rényi generalizations thereof Akers:2021pvd ;
Akers:2022max ; Akers:2022zxr . Reflected entropy has been studied in a
similar gravitational system Akers:2022max  and was shown to have
$\mathcal{O}(\sqrt{V})$ corrections at transition, as in the case of the von
Neumann entropy, derived via a resolvent calculation. Presumably the relevant
permutations could be enumerated and the corrections calculated as we’ve done
in this work.

We only considered the case where energy is conserved in all three subsystems.
The authors of Vardhan:2021mdy  consider some cases in a similar model where
some subystems are fixed at infinite temperature, which would correspond to
freezing the density of states in those subsystems; it would be interested to
understand to what extent this changes our results.

##  6 Acknowledgements

We thank Xi Dong, David Grabovsky, Jesse Held, Adolfo Holguin, Jonah Kudler-
Flam, Ion Nechita, Pratik Rath, Mark Srednicki, and Wayne Weng for useful
discussions. The work of SAM was supported by the Air Force Office of
Scientific Research under award number FA9550-19-1-0360, the National Science
Foundation under Grant No. PHY-1820908, and funds from the University of
California. The work of FI was supported by an NSF Graduate Research
Fellowship under Grant No. 2139319 and funds from the University of
California.

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##  Appendix A Deriving the Relevant Sum Over Permutations

Let’s recall a few facts about the permutation group. For an element $g\in
S_{n}$, we denote the number of swaps from the identity permutation
$\text{1}=(1)(2)\cdots(n)$ to $g$ by $\ell(g)$ and the number of distinct
cycles in $g$ by $\chi(g)$. These quantities satisfy the relation

$\ell(g)+\chi(g)=n$ (103)

The number of swaps between two permutations $\ell(g^{-1}h)\equiv d(g,h)$
introduces a natural distance measure between two permutations. In particular,
there exists the traingle inequality

$d(g,g_{1})+d(g_{1},h)\geq d(g,h)$ (104)

A geodesic between two permutations $G(g,h)$ is the set of $g_{1}^{\prime}s$
which saturate this inequality.

The sum over permutations we’re interested in takes the form
Shapourian:2020mkc ; Dong:2021oad ; Vardhan:2021mdy

$\texttt{SUM}=\sum_{g\in
S_{n}}(e^{S_{B}})^{m}(e^{S_{A_{1}}})^{p}(e^{S_{A_{2}}})^{q}$ (105)

where we’ve made the substitutions

$m=\chi(g),\quad p=\chi(g^{-1}X),\quad q=\chi(g^{-1}X^{-1})$ (106)

Here $X$ is the cyclic permutation $(12\cdots n)$ and $X^{-1}$ is the anti-
cyclic permutation $(n\;n-1\cdots 1)$. This is the sum relevant for
calculating the moments of a block transposed Wishart matrix
banica2013asymptotic , i.e. the weighting of Wick contractions when averaging
over a random density matrix with Gaussian correlations. In our work, we’re
interested in the permutations which live on the geodesic $G(\text{1},X)$ and
the geodesic $G(X,X^{-1})$ but not necessarily on the geodesic
$G(\text{1},X^{-1})$. These permutations satisfy the following three
equations:

$\displaystyle m+p$ $\displaystyle=n+1$ $\displaystyle m+q$ $\displaystyle\leq
n+1$ $\displaystyle p+q$ $\displaystyle=n+f(n)$ (107)

where the function $f(n)=1$ if $n$ is odd and $2$ if $n$ is even. When the
second inequality is saturated, we’re talking about the set of noncrossing
pairings $\tau$. There are $C_{n}$ of these permutations, where $C_{n}$ are
the Catalan numbers

$C_{n}=\frac{1}{n+1}\binom{2n}{n}$ (108)

An example of a $\tau$ permutation on an even number of elements is
$(12)(34)\cdots(n-1\;n)$. A noncrossing pairing on an odd number of elements
will have a single cycle of length 1 and all other cycles of length 2.
Permutations which live on $G(\text{1},X)$ and $G(X,X^{-1})$ but not
$G(\text{1},X^{-1})$ are precisely those which live on the single geodesic
$G(\tau,X)$, which is the phase transition we’re interested in. How do we
enumerate these permutations? From the two equalities, we have:

$\displaystyle p=n-1-mq=m+f(n)-1$ (109)

From this, we can derive an upper bound on $m$:

$m\leq\frac{n+2-f(n)}{2}$ (110)

So we’ve reduced the sum over all permutations to a sum over a single
parameter $m=\chi(g)$. We now have

$\texttt{SUM}=\sum_{m=1}^{\frac{n-f(n)+2}{2}}T^{\prime}(n,m)\left(e^{S_{A_{2}}}\right)^{f(n)-1}\left(e^{S_{A_{1}}}\right)^{n+1}\left(\frac{e^{S_{B}}e^{S_{A_{2}}}}{e^{S_{A_{1}}}}\right)^{m}$
(111)

for some counting function $T^{\prime}(n,m)$ which denotes the multiplicity at
every $\chi(g)$. What is this function? Let’s consider it for both even and
odd $n$. For even $n=2k$, the sum is

$\texttt{EVEN
SUM}=\sum_{m=1}^{k}T_{e}(k,m)e^{S_{A_{2}}}\left(e^{S_{A_{1}}}\right)^{2k+1}\left(\frac{e^{S_{B}}e^{S_{A_{2}}}}{e^{S_{A_{1}}}}\right)^{m}$
(112)

This is a sum over permutations starting with the cyclic permutation $X$ at
$m=1$ and ending with the pairwise connected permutations $\tau$ at $m=k$.
Each $m$ corresponds to a permutation with $m$ cycles of even length. In this
case, the numbers $T_{e}(k,m)$ are equivalent to the number of 2-Dyck paths of
order $k$ with $m$ peaks and are given by

$T_{e}(k,m)=\frac{1}{k}\binom{k}{m}\binom{2k}{m-1}$ (113)

The $T_{e}(k,m)$ that appear here are analagous to the Narayana numbers which
appear in the sum over non-crossing permutations. They are sometimes referred
to as 2-Narayana numbers and appeared in various contexts elsewhere
Brezin:1977sv ; Novelli ; Cachazo:2022voc . We therefore have

EVEN SUM
$\displaystyle=e^{S_{A_{2}}}\left(e^{S_{A_{1}}}\right)^{2k+1}\sum_{m=1}^{k}T_{e}(k,m)\left(\frac{e^{S_{B}}e^{S_{A_{2}}}}{e^{S_{A_{1}}}}\right)^{m}$
$\displaystyle=e^{2kS_{A_{1}}}e^{2S_{A_{2}}}e^{S_{B}}{}_{2}F_{1}\left(1-k,-2k,2;\frac{e^{S_{B}}e^{S_{A_{2}}}}{e^{S_{A_{1}}}}\right)$
(114)

Now let’s look at the odd case. When $n=2k-1$, the sum over permutations is

$\texttt{ODD
SUM}=\sum_{m=1}^{k}T_{o}(k,m)\left(e^{S_{A_{1}}}\right)^{2k}\left(\frac{e^{S_{B}}e^{S_{A_{2}}}}{e^{S_{A_{1}}}}\right)^{m}$
(115)

Now the counting function is slightly different. We can derive it as follows:
consider a permutation allowed in the even sum (112) with $m$ cycles. The
second binomial factor in (113) can morally be thought of as choosing $m-1$
distinct elements to belong to different cycles, while the rest is a symmetry
factor that controls the number of noncrossing permutations modulo that
choice. Therefore, one can think of each noncrossing permutation as living in
a “labelled” superselection sector of size $\binom{2k}{m-1}$. By ignoring this
choice and dividing by this factor, we can find a degenerate set of
“unlabelled” noncrossing permutations. From this set we can remove an element
from each cycle, so one unlabelled permutation in the even sum generates $m$
distinct unlabelled permutations in the odd sum, which then have to be
relabelled to give the correct counting. This strategy of unlabelling,
removing an element, and relabelling gives us

$T_{o}(k,m)=m\frac{\binom{2k-1}{m-1}}{\binom{2k}{m-1}}T_{e}(k,m)=\binom{2k-1}{m-1}\binom{k-1}{m-1}$
(116)

Figure 6: The procedure of generating permutations on $G(\tau,X)$ for odd $n$
from even $n$. We identify even permutations on $G(\tau,X)$ which are the same
up to the choice of $m-1$ elements. Each of these pieces produces $m$
unlabelled odd pieces by removing an element, and identifying $m-1$ elements
again gives us all odd permutations on $G(\tau,X)$.

If that was a bit too abstract, we illustrate this procedure in Figure 6. The
sum over permutations for odd $n$ is now

ODD SUM
$\displaystyle=\left(e^{S_{A_{1}}}\right)^{2k}\sum_{m=1}^{k}T_{e}(k,m)\left(\frac{e^{S_{B}}e^{S_{A_{2}}}}{e^{S_{A_{1}}}}\right)^{m}$
$\displaystyle=\left(e^{S_{A_{1}}}\right)^{2k-1}e^{S_{A_{2}}}e^{S_{B}}{}_{2}F_{1}\left(1-2k,1-k,1;\frac{e^{S_{B}}e^{S_{A_{2}}}}{e^{S_{A_{1}}}}\right)$
(117)

As a sanity check for our counting functions $T_{e}(k,m)$ and $T_{o}(k,m)$,
both $T_{e}(k,k)$ and $T_{o}(k,k)$ are equal to the symmetry factors for the
pairwise connected geometries:

$T_{e}(k,k)=C_{k},\quad T_{o}(k,k)=(2k-1)C_{k-1}$ (118)

and $T_{e}(k,1)=T_{o}(k,1)=1$.

##  Appendix B Resolvent for Disorder Averaging

The resolvent matrix $R_{ij}(\lambda)$ encodes the eigenvalue spectrum given
by

$R(\lambda)_{ij}=\frac{1}{\lambda}\delta_{ij}+\sum_{n=1}^{\infty}\frac{1}{\lambda^{n+1}}\left(\rho_{A}^{n}\right)_{ij}$
(119)

or in traced version

$R(\lambda)=\frac{e^{S_{A}}}{\lambda}+\sum_{n=1}^{\infty}\frac{1}{\lambda^{n+1}}\Tr\left(\rho_{A}^{n}\right)$
(120)

There is a resolvent equation for the Wick contractions

$\displaystyle
R(\lambda)_{ij}=\frac{\delta_{ij}}{\lambda}+\frac{e^{S_{B}}}{\lambda}\sum_{n=1}^{\infty}R(\lambda)^{n}R(\lambda)_{ij}$
(121)

We can take the trace and resum this equation:

$\lambda R(\lambda)=e^{S_{A}}+\frac{e^{S_{B}}R(\lambda)}{1-R(\lambda)}$ (122)

This is the resolvent equation for the eigenvalues of a Wishart matrix. This
equation can be solved exactly for $R(\lambda)$. It admits an expansion

$R(\lambda)=\frac{e^{S_{A}}}{\lambda}+\frac{e^{S_{B}}}{\lambda}\sum_{n=1}^{\infty}\sum_{k=1}^{n}N(n,k)\left(\frac{e^{S_{A}}}{\lambda}\right)^{n}e^{(k-1)(S_{B}-S_{A})}$
(123)

where $N(n,k)$ are the Narayana numbers. From this we can read off
$\Tr\overline{(\rho_{A})^{n}}$:

$\Tr\overline{(\rho_{A})^{n}}=e^{nS_{A}+S_{B}}\sum_{k=1}^{n}N(n,k)e^{(k-1)(S_{B}-S_{A})}$
(124)

This sum has a nice closed form expression

$\Tr\overline{(\rho_{A})^{n}}=\begin{cases}e^{S_{A}+nS_{B}}{}_{2}F_{1}\left(1-n,-n;2;e^{S_{A}-S_{B}}\right),S_{A}<S_{B}\\\
e^{nS_{A}+S_{B}}{}_{2}F_{1}\left(1-n,-n;2;e^{S_{B}-S_{A}}\right),S_{A}>S_{B}\end{cases}$
(125)

where there are two branches such that the final argument of the
hypergeometric always always lies within the unit circle on the complex plane.
A similar resolvent exists for the disorder averaging over Wick contractions
for the partially transposed density matrix $\rho_{A_{1}A_{2}}^{T_{2}}$. We
work in the regime where $S_{A_{2}}\ll S_{A_{1}}+S_{B}$. The resolvent
equation is Dong:2021oad

$\lambda
R(\lambda)_{j_{1}j_{2}}^{i_{1}i_{2}}=\delta^{i_{1}i_{2}}_{j_{1}j_{2}}+e^{S_{B}}\left(\sum_{m=1}^{\infty}\frac{R(\lambda)^{2m-2}}{e^{(2m-2)S_{A_{2}}}}R(\lambda)^{i_{1}i_{2}}_{j_{1}j_{2}}+\sum_{m=1}^{\infty}\frac{R(\lambda)^{2m-1}}{e^{(2m-2)S_{A_{2}}}}R(\lambda)^{i_{1}i_{2}}_{j_{1}j_{2}}\right)$
(126)

Taking the trace:

$\lambda
R(\lambda)=e^{S_{A_{1}}+S_{A_{2}}}+e^{S_{B}}\left(\sum_{m=1}^{\infty}\frac{R(\lambda)^{2m-1}(1+R(\lambda))}{e^{(2m-2)S_{A_{2}}}}\right)$
(127)

and resumming gives the final resolvent equation

$\lambda
R(\lambda)=e^{S_{A_{1}}+S_{A_{2}}}+\frac{e^{S_{B}}}{e^{S_{A_{2}}}}\frac{R(\lambda)(1+R(\lambda))}{1-e^{2S_{A_{2}}}R(\lambda)^{2}}$
(128)

We recognize this as the resolvent equation for the moments of a block
transposed Wishart matrix. In the case $S_{A_{2}}=0$ this reduces to the
resolvent equation for the untransposed density matrix. This is a cubic
equation and can be solved exactly, but the solution is not enlightening.

Previously, we saw that the moments of a Wishart matrix are given in closed
form by a sum of Narayana numbers. An equivalent statement is that the inverse
Stieltjes transform of the the resolvent defined by (122), a generating
function for the Narayana numbers, gives the eigenvalue spectrum of a Wishart
matrix. The inverse Stieltjes transform of the solution to (128), a generating
function for the moments of a block transposed Wishart matrix (“block
transposed Narayana numbers”) will produce the eigenvalue spectrum of a block
transposed Wishart matrix (the “negativity spectrum”). The block transposed
Narayana numbers are not known in closed form; see banica2013asymptotic  for a
recursive definition.