# Building Entanglement Entropy out of Correlation Functions for Interacting
Fermions

Saranyo Moitra  0000-0001-7912-1961<EMAIL_ADDRESS>Department of
Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005,
India.  Rajdeep Sensarma  0000-0002-2136-5354<EMAIL_ADDRESS>Department of Theoretical Physics, Tata Institute of Fundamental Research,
Mumbai 400005, India.

(August 28, 2024)

###### Abstract

We provide a prescription to construct Rényi and von Neumann entropy of a
system of interacting fermions from a knowledge of its correlation functions.
We show that Rényi entanglement entropy of interacting fermions in arbitrary
dimensions can be represented by a Schwinger Keldysh free energy on replicated
manifolds with a current between the replicas. The current is local in real
space and is present only in the subsystem which is not integrated out. This
allows us to construct a diagrammatic representation of entanglement entropy
in terms of connected correlators in the _standard field theory with no
replicas_. This construction is agnostic to how the correlators are
calculated, and one can use calculated, simulated or measured values of the
correlators in this formula. Using this diagrammatic representation, one can
decompose entanglement into contributions which depend on the one-particle
correlator, two particle correlator and so on. We provide analytic formula for
the one-particle contribution and a diagrammatic construction for higher order
contributions. We show how this construction can be extended for von-Neumann
entropy through analytic continuation. For a practical implementation of a
quantum state, where one usually has information only about few-particle
correlators, this provides an approximate way of calculating entanglement
commensurate with the limited knowledge about the underlying quantum state.

##  I Introduction

A quantum many body state encodes non-local correlations between degrees of
freedom in one part of the system with those in another part; i.e. the
information about the state is distributed amongst degrees of freedom which
are typically far away from each other. A simple way to see this is to expand
a quantum many body state in a basis which is tensor product of local basis
states. The complex quantum amplitudes of this expansion (or the many-body
wavefunction) store the information about these non-local correlations. The
most obvious example of this is quantum statistics: e.g. Fermions cannot share
a quantum state, which can impose long range non-local constraints on
wavefunctions of many Fermions.

If one is interested in observables which have support only in the Hilbert
space of a subsystem $A$, one can trace over the degrees of freedom in the
complimentary subsystem $B$ and construct a reduced density matrix (RDM)
$\hat{\rho}_{A}$ [1]. This density operator will reproduce all observables in
the subsystem and has the usual interpretation of a “classical” ensemble of
quantum states as specified by its spectral decomposition. If the RDM
represents a pure quantum state in the subsystem, the original state is
separable, otherwise it is entangled [2].

Entanglement and its related measures have played an important role in wide
ranging fields including quantum information and computation [3, 4],
foundations of quantum mechanics [5, *BELL_RMP66_ProblemHiddenVariables,
*Clauser.Holt_PRL69_ProposedExperimentTest,
*Zeilinger_RMP99_ExperimentFoundationsQuantum], black-hole physics and quantum
gravity [9, *hayden_J.HighEnergyPhys.2007_BlackHolesMirrors, *susskind_EREPR,
*penington_J.HighEnerg.Phys.2020_EntanglementWedgeReconstruction,
*Das:2020xoa], and quantum condensed matter systems [14, 15]. An important
measure of entanglement is the bipartite entanglement entropy (EE), which is
the classical entropy corresponding to the probability distribution specified
by the eigenvalues of $\hat{\rho}_{A}$. In quantum many body systems, the
scaling of EE with subsystem size is used to fingerprint states [16, 15, 17,
*Calabrese_2004, *Calabrese_2009, 20, 21], detect presence of topological
order [22, *levin_Phys.Rev.Lett.2006_DetectingTopologicalOrder,
*jiang_NaturePhys2012_IdentifyingTopologicalOrder] or occurrence of quantum
phase transitions [25, *SubirON, *SubirWitczakKrempa,
*ju_Phys.Rev.B2012_EntanglementScalingTwodimensional]. In fact, exotic states
like spin liquids[29] are often best described by the “entanglement patterns”
in the ground state [30,
*isakov_NaturePhys2011_TopologicalEntanglementEntropy,
*pretko_Phys.Rev.B2016_EntanglementEntropyQuantum]. More recently the
entanglement scaling of excited states in the middle of the spectrum [33] has
been used to classify whether a disordered interacting system is in an ergodic
phase, where local observables can be described by usual statistical
mechanics, or in a many body localized phase [34], where laws of statistical
mechanics fail to apply. The question of thermalization [35] in a system has
also been tracked through the time evolution of entanglement under non-
equilibrium dynamics [36, 37, 38]. Thermalizing systems show a linear growth
of entanglement, while many body localized systems have a slower logarithmic
growth of entanglement [39, 40]. The growth of entanglement has also played a
crucial role in the development of ideas related to quantum chaos [41,
*Swingle.Hayden_PRA16_MeasuringScramblingQuantum, 43]. Experimental
measurements of EE in many body systems have been performed on a variety of
platforms including ion traps [44, 45], ultracold Rydberg atoms [46, 47], and
superconducting qubits [48].

There are only a few methods to calculate the EE in interacting many body
systems, either in thermal or in the ground state. Even fewer methods can
tackle non-equilibrium dynamics of EE. The most direct method is to obtain the
quantum state by exact diagonalization, calculate the reduced density matrix
and hence the EE [49, 50]. While this is the most widely used method, it is
limited to small finite size systems in one dimension, since the specification
of a quantum many body state requires knowledge of an exponentially large (in
system size) number of quantum amplitudes. Quantum Monte Carlo based methods
have been extensively used for systems in one and two dimensions [51,
*Grover_PRL13_EntanglementInteractingFermions,
*McMinis.Tubman_PRB13_RenyiEntropyInteracting,
*Drut.Porter_PRB15_HybridMonteCarlo]. In the cases where EE is expected to
have a weak scaling with system size, numerical techniques using Density
Matrix Renormalisation Group ideas [55,
*Schollwock_AoP11_DensitymatrixRenormalizationGroup] in one dimension and
tensor network based methods [57, *Evenbly.Vidal_JSP11_TensorNetworkStates] in
higher dimensions have been used to calculate EE. For non-interacting systems
or integrable systems, progress can be made using specialized numerical
techniques, as one can reduce the complexity of the calculation [59,
*Peschel_2009]. Standard field theoretic techniques for calculation of
entanglement entropy [18, *Calabrese_2009, 21] use replica methods with
complicated boundary conditions. This limits their scope as it is often
impossible to obtain solutions of even simple problems with the complicated
boundary conditions. For 1+1 dimensional CFTs [17, *Calabrese_2004], where
correlators are tightly constrained, one can obtain exact analytical answers
for leading and subleading scaling of EE.

A key problem is calculating EE of a state is the following: we have an
operational prescription to calculate EE if we know the exact quantum state;
however for a generic system, there is no such prescription to calculate the
EE in terms of the correlation functions. There are two aspects to this issue:
(a) A prescription to obtain entanglement from correlators can lead to
efficient estimates of entanglement in numerics, since efficient algorithms to
calculate correlation functions already exist in the literature [61, 56].
There are also a large class of analytic approximation schemes [62, 63] for
calculating interacting correlation functions, which have been developed over
the years. With a prescription connecting correlations to EE, these methods
can increase the scope of study of entanglement in large quantum systems. We
note that in the special cases where this prescription is known, like in non-
interacting systems [60, 20] or 1+1D CFT [19], our knowledge of EE is vastly
more advanced than the cases where such a prescription is not known. (b) In
any realistic situation, it is impossible to know the exact quantum state;
however there are experimental probes to obtain information about the
correlation functions [64]. Even in highly controllable systems like ultracold
atoms [65], experiments can at best have knowledge of few-body correlations
[66, 67, 68]. A prescription to calculate entanglement entropy from knowledge
of correlation functions would thus not only enhance the theoretical space for
such calculations, it will be the only consistent description of realistic
experimental situations. Here we would like to note that if one requires the
knowledge of all $m$-body correlators in a system to calculate EE, the
complexity of the problem is same as knowing the full quantum state. Thus it
would be useful to have estimates of entanglement which involve only few body
correlators, and one should be able to improve these estimates if information
about higher order correlators become available.

In this paper, we take on the task of constructing an operational prescription
for computing EE of a generic interacting Fermionic system in terms of its
correlation functions. We consider the system to be made of mutually exclusive
regions $A$ and $B$, with a Hilbert space which is a tensor product of Hilbert
spaces for degrees of freedom in $A$ and $B$. We are interested in the EE of
the system when the degrees of freedom in $B$ are traced out. We use Schwinger
Keldysh field theory [69, 70], which allows us to consider ground states,
thermal systems, and closed and open quantum systems evolving in time out of
thermal equilibrium on the same footing. Our prescription is agnostic to these
different situations. (i) We show that the $n$th Rényi entropy, $S^{(n)}$, of
a system of interacting Fermions is the Schwinger Keldysh free energy of a
system of $n$ replicas with “inter-replica currents” flowing only in the
subsystem $A$. These currents are local in space (i.e. between same lattice
sites, or same location) and in time (currents are present only at the time of
measurement). The matching of fields across replica-s in standard field theory
technique for calculating EE is replaced by the _inter-replica currents_ in
our formalism. We argue that the doubling of fields in the SK field theory is
a crucial ingredient which allows boundary conditions to be replaced by
quadratic current-like terms, and we do not know of any method which can
achieve this in a single contour Lorentzian or Euclidean field theory. (ii)
Using this identification of EE with a free energy, we provide a prescription
to calculate it in terms of correlations in a single replica theory, which
does not involve complicated boundary conditions (it has the same boundary
conditions as the usual field theory). (iii) We show that if one only has
information of up to $m$-particle connected correlators, one can construct an
estimate of EE, which can be improved if information about $m+1$-particle
correlators become available. More precisely, for the $n$th order Rényi
Entanglement Entropy (REE), $S^{(n)}$, we show

$S^{(n)}=S^{(n)}_{1\mathrm{P}}+S^{(n)}_{2\mathrm{P}}+S^{(n)}_{3\mathrm{P}}+\cdots,$
(1)

where $S^{(n)}_{m\mathrm{P}}$ involves up to $m$-particle connected
correlators, $G^{(m)}_{c}$, and does not involve any correlators of higher
order (particle number). For $m>1$, $G^{(m)}_{c}$ is simply related to the
connected part of the expectation value of a normal ordered string of Fermion
operators ($\hat{c}^{\dagger},\hat{c}^{\vphantom{\dagger}}$) w.r.to the state,

$G^{(m)}_{c}\sim\expectationvalue*{\,\underbrace{\hat{c}^{\dagger}\hat{c}^{\dagger}\cdots\hat{c}^{\dagger}}_{m\rm\
times}\,\underbrace{\hat{c}^{\vphantom{\dagger}}\,\hat{c}^{\vphantom{\dagger}}\,\cdots\hat{c}^{\vphantom{\dagger}}}_{m\rm\
times}\,}_{c}.$ (2)

We calculate $S^{(n)}_{1\mathrm{P}}$ exactly and provide a prescription of
Feynman diagrams for calculating $S^{(n)}_{m\mathrm{P}}$ for $m>1$ in terms of
higher order connected correlators. This decomposition would be most useful
when the state of the system is close to that of a Gaussian theory (this could
be a symmetry broken mean field theory) and higher order connected correlators
are parametrically smaller. (iv) The von-Neumann entropy, $S$, does not have a
simple field theoretic interpretation, but has to be calculated as an analytic
continuation of REE, i.e. $S=\lim_{n\to 1}S^{(n)}$. We posit a similar
$m$-particle decomposition for $S$,

$S=S_{1\mathrm{P}}+S_{2\mathrm{P}}+S_{3\mathrm{P}}+\cdots,$ (3)

where $S_{m\mathrm{P}}=\lim_{n\to 1}S^{(n)}_{m\mathrm{P}}$. We calculate
$S_{1\mathrm{P}}$ exactly, and show that a large class of diagrams for
$S^{(n)}_{m\mathrm{P}}$ vanish when the $n\to 1$ limit is taken. For
$S_{2\mathrm{P}}$, we explicitly calculate the first non-trivial diagram which
involves two 2-particle connected correlators. Note that our prescription is
completely agnostic to how the connected correlation functions are calculated.
Our formulation thus opens up the possibility of computing EE using
approximate analytic techniques, numerically obtained correlators or even
experimentally measured ones. We believe this construction will allow large
scale calculations of EE in higher dimensions ( $d>1$) both in and out of
equilibrium.

Figure 1:  Replica based evaluation of $S^{(4)}$ : (a) Standard technique for
ground states where the reduced density matrix is represented as a Euclidean-
time path integral with boundary conditions in the subsystem $A$. The fields
are matched across consecutive replica as shown. (b) Our formalism expresses
each replica of the density matrix on a _real time_ Keldysh contour with
doubled fields. Fields at each point of the subsystem $A$ in one replica are
coupled to their counterparts in all other replicas. For $S^{(4)}$ this
implies coupling between sheet $1$ and $3$, and between sheet $2$ and $4$, in
addition to the coupling between consecutive ones. The coupling operates only
at the time of observation $t_{o}$, and is “directional” in Replica space,
acting as a “Replica current”. Partial tracing is achieved by setting the
coupling to zero in the complementary subsystem $B$. No field matchings are
required in this new formalism.

The remainder of the text is organised as follows. In Section II we establish
that the $n$th order REE for a generic system of interacting fermions is equal
to the Schwinger Keldysh free energy of $n$ replicas of the system coupled via
inter-replica currents. We use this fact in Section III to show how $S^{(n)}$
can be decomposed into $m$ particle contributions $S^{(n)}_{m\mathrm{P}}$ and
provide explicit diagrammatic rules to construct the same in terms of $k$
particle correlators with $1\leq k\leq m$. Section IV is devoted to the
analytic continuation of these contributions as $n\to 1$. In Section V we
provide an alternate diagrammatic prescription in terms of Green’s functions
of the non-interacting replicas in presence of the quadratic inter-replica
current terms. These correlators depend on replica indices and do not admit
the usual physical interpretations as correlators in standard Keldysh filed
theory. Lastly we conclude in Section VI with a summary of our findings.

##  II  Entanglement Entropy as a Free Energy in presence of Replica Currents

Consider a system of Fermions which is made up of two mutually exclusive
spatial subsystems $A$ and $B$. For our purpose, we consider a lattice with
$V$ sites, where the subsystem $A$ has $V_{A}$ sites. We will explicitly work
with discrete lattice systems, and take appropriate continuum limit at the
end. The Hilbert space of the system is a product of the Hilbert space of
degrees of freedom lying in A and those lying in B, i.e. ${\cal H}_{AB}={\cal
H}_{A}\otimes{\cal H}_{B}$. On tracing over degrees of freedom in $B$, a
quantum state of the full system, $\ket{\psi(t_{o})}$ gives rise to a reduced
density matrix $\hat{\rho}_{A}(t_{o})$ over the subsystem $A$. Here $t_{o}$ is
the time of observation. The $n^{{\text{th}}}$ order Rényi entanglement
entropy (REE) of this state for the partition between $A$ and $B$ is then
given by

$S^{(n)}(t_{o})=-\frac{1}{n-1}\ln\Tr[\left(\hat{\rho}_{A}(t_{o})\right)^{n}],$
(4)

while the von-Neumann entanglement entropy (referred to as EE in the
following) is obtained as the analytic continuation

$S(t_{o})=\lim_{n\to
1}S^{(n)}(t_{o})=-\Tr[\hat{\rho}_{A}(t_{o})\ln\hat{\rho}_{A}(t_{o})].$ (5)

In a steady state or in equilibrium, the RDM and hence the entropies are
independent of $t_{o}$. In the standard field theoretic prescription for
calculation of $S^{(n)}$, one considers $n$ copies(replicas) of the system.
The time evolution of each of these replicas can be written as a functional
integral over fields, giving rise to $n$ fields for each space-time point. The
multiplication of RDMs translates to imposing the boundary condition that the
fields within subsystem $A$ in different replicas have to be matched at the
time of measurement. A schematic representation of this prescription is shown
in Fig. 1(a). This naturally leads to integrals over constrained field
configurations in the replica space [19]. Here, we propose a modified scheme
(schematically shown in Fig. 1(b)) to calculate REE of fermions which uses
Schwinger Keldysh field theory for $n$ replicas [37, 71, 38, 72, 73]. The key
advantage is that we can trade the boundary conditions in favour of “currents”
flowing between replicas in the subsystem A. This allows us to connect REE
with standard correlators of the original non-replicated theory.

To develop our formalism, it is useful to define the Wigner characteristic
$\chi$ of a density matrix, as the expectation value of the fermionic
displacement operator [74, 75, 38],

$\chi({\bm{\bar{\zeta}}},{\bm{\zeta}};t_{o})=\Tr[\hat{\rho}(t_{o})\,\mathrm{e}^{\sum_{i}\hat{c}^{\dagger}_{i}\zeta_{i}-\bar{\zeta}_{i}\hat{c}^{\vphantom{\dagger}}_{i}}],$
(6)

where $\hat{\rho}(t_{o})$ is the density matrix of the full system. The Wigner
characteristic has the useful property that $\chi$ for the RDM is given by

$\chi_{A}({\bm{\bar{\zeta}}},{\bm{\zeta}};t_{o})=\Tr[\hat{\rho}(t_{o})\,\mathrm{e}^{\sum_{i\in
A}\hat{c}^{\dagger}_{i}\zeta_{i}-\bar{\zeta}_{i}\hat{c}^{\vphantom{\dagger}}_{i}}],$
(7)

i.e. one simply needs to restrict the support of the displacement operator to
${\cal H}_{A}$ [37]. From now onwards we will use ${\bm{\zeta}}$ and
${\bm{\eta}}$ to indicate vectors of Grassmann variables with support only in
$A$. Note that $\chi_{A}$ is a Grassmann valued function of the Grassmann
fields $\zeta_{i},\bar{\zeta}_{i}$ and does not have an immediate physical
interpretation. Its usefulness lies in the fact that expectations values of
operators with support in $A$ can be written as integrals over $\chi$,

$\expectationvalue*{\hat{O}}[t_{o}]\equiv\Tr[\hat{\rho}_{A}(t_{o})\hat{O}]=2^{-V_{A}}\int\mathcal{D}\left[{\bm{\bar{\zeta}}},{\bm{\zeta}}\right]\mathcal{D}\left[{\bm{\bar{\eta}}},{\bm{\eta}}\right]\,\chi_{A}\left(\sqrt{2}{\bm{\bar{\eta}}},\sqrt{2}{\bm{\eta}};\,t_{o}\right)\chi_{O}\left(\sqrt{2}{\bm{\bar{\zeta}}},\sqrt{2}{\bm{\zeta}}\right)e^{{\bm{\bar{\zeta}}}\cdot{\bm{\eta}}-{\bm{\bar{\eta}}}\cdot{\bm{\zeta}}},$
(8)

where
$\chi_{O}\left({\bm{\bar{\zeta}}},{\bm{\zeta}}\right)=\Tr[\hat{O}\leavevmode\nobreak\
\,\mathrm{e}^{\sum_{i\in
A}\hat{c}^{\dagger}_{i}\zeta_{i}-\bar{\zeta}_{i}\hat{c}^{\vphantom{\dagger}}_{i}}]$
is the Weyl symbol of the operator $\hat{O}$ and the dot product
${\bm{\bar{\zeta}}}\cdot{\bm{\eta}}=\sum_{i\in A}\bar{\zeta}_{i}\eta_{i}$. It
is then easy to see (substituting $\hat{\rho}_{A}$ for $\hat{O}$) that the
$2^{\text{nd}}$ Rényi entropy is given by

$e^{-S^{(2)}(t_{o})}=2^{-V_{A}}\int\mathcal{D}\left[{\bm{\bar{\zeta}}},{\bm{\zeta}}\right]\mathcal{D}\left[{\bm{\bar{\eta}}},{\bm{\eta}}\right]\chi_{A}\left(\sqrt{2}{\bm{\bar{\eta}}},\sqrt{2}{\bm{\eta}};\,t_{o}\right)\chi_{A}\left(\sqrt{2}{\bm{\bar{\zeta}}},\sqrt{2}{\bm{\zeta}};\,t_{o}\right)e^{{\bm{\bar{\zeta}}}\cdot{\bm{\eta}}-{\bm{\bar{\eta}}}\cdot{\bm{\zeta}}}.$
(9)

Using properties of displacement operators and fermionic coherent states, Ref.
[38] showed that the $n$th order REE corresponding to can be written as
(suppressing the time index),

$\begin{split}\,\mathrm{e}^{-(n-1)S^{(n)}}=2^{-{(n-1)}V_{A}}\int\prod_{\alpha=1}^{n-1}\mathcal{D}\left[{\bm{\bar{\zeta}}}^{(\alpha)},{\bm{\zeta}}^{(\alpha)}\right]\mathcal{D}\left[{\bm{\bar{\eta}}}^{(\alpha)},{\bm{\eta}}^{(\alpha)}\right]\leavevmode\nobreak\
\prod_{\alpha=1}^{n-1}\chi_{A}\left(\sqrt{2}{\bm{\bar{\eta}}}^{(\alpha)},\sqrt{2}{\bm{\eta}}^{(\alpha)}\right)\leavevmode\nobreak\
\chi_{A}\left(\sqrt{2}\sum\nolimits_{\alpha}{\bm{\bar{\zeta}}}^{(\alpha)},\sqrt{2}\sum\nolimits_{\alpha}{\bm{\zeta}}^{(\alpha)}\right)\\\
\exp\left({\sum_{\alpha=1}^{n-1}{\bm{\bar{\zeta}}}^{(\alpha)}\\!\cdot{\bm{\eta}}^{(\alpha)}-{\bm{\bar{\eta}}}^{(\alpha)}\\!\cdot{\bm{\zeta}}^{(\alpha)}+\sum_{\alpha>\beta}{\bm{\bar{\zeta}}}^{(\alpha)}\\!\cdot{\bm{\zeta}}^{(\beta)}-{\bm{\bar{\zeta}}}^{(\beta)}\\!\cdot{\bm{\zeta}}^{(\alpha)}}\right).\end{split}$
(10)

Here $1\leq\alpha,\beta\leq n-1$ are replica indices. Note that while there
are product of $n$ replicas, the integration is over $2(n-1)$ pairs of
Grassmann variables for each site in $A$.

Eq. (10) is the starting point of our attempt to find a field theoretic
construction of EE and hence a general relation between correlators and
entanglement. For this, we work with the Schwinger Keldysh field theory of
Fermions[69], which describes the evolution of a many body density matrix
$\hat{\rho}(t)=\hat{U}(t,0)\hat{\rho}(0)\hat{U}^{\dagger}(t,0)$ in terms of
path/functional integrals over two Fermionic (Grassmann) fields
$\psi_{\pm}(i,t)$ at each space-time point. The $\psi_{+}$ fields are obtained
from the expansion of $\hat{U}$ (i.e. forward propagation of states) while the
$\psi_{-}$ fields come from the expansion of $\hat{U}^{\dagger}$ (i.e.
backward propagation of states), giving ’rise to a field theory on a closed
contour. The partition function $\mathcal{Z}$ on this contour in presence of
sources $J_{\pm},\bar{J}_{\pm}$ is

$\mathcal{Z}[\bar{J}_{\pm},J_{\pm}]=\\!\int\\!\mathcal{D}\left[\bar{\psi}_{\pm},\psi_{\pm}\right]\\!\,\mathrm{e}^{\iota\mathcal{S}_{K}[\bar{\psi}_{\pm},\psi_{\pm}]+\iota\\!\int\differential{t}\bar{J}_{+}\psi_{+}-\bar{J}_{-}\psi_{-}+{\text{h.c.}}},$

where the Keldysh action $\mathcal{S}_{K}$ determines the evolution of the
correlators in the system. Note that although we have used the same notation
$\mathcal{D}\left[\cdot\right]$ to indicate integrals over the Fermion matter
fields $\psi$ as well as integrals over the arguments of the Wigner
characteristic ($\zeta,\eta$), the matter fields fluctuate in space-time,
while the arguments of Wigner characteristic are evaluated at $t_{o}$ and
hence have only spatial variations. In this section, we do not need specific
forms of the action, and will refrain from talking about them. One obvious
advantage of using a Keldysh field theory is that one can treat both
equilibrium and non-equilibrium situations in the same footing.

Figure 2: Schematic representation of the Wigner characteristic function
$\chi_{A}({\bm{\bar{\zeta}}},{\bm{\zeta}};t_{o})$ as a Keldysh path integral
with source insertions. Expanding the evolution operators $\hat{U}(t,0)$ and
$\hat{U}^{\dagger}(t,0)$ leads to a path integral with fields on the
“forward”(red) and “backward”(blue) contours respectively. The fields are
identified at $t=\infty$ and are connected at $t=0$ through the matrix
elements of the initial density matrix $\hat{\rho}_{0}$. The displacement
operator is inserted symmetrically on both forward and backward contours at
the time of observation $t_{o}$ and is equivalent to insertion of sources
$J_{\pm},\bar{J}_{\pm}$. The call-out shows the spatial distribution of these
sources; they take nonzero values only in subsystem $A$.

In earlier works [38, 71, 37], we had shown that the Wigner characteristic of
the reduced density matrix is the Schwinger-Keldysh partition function of the
system in presence of sources which couple to fields in the subsystem $A$ only
at the time when the entropy is measured, i.e.
$\bar{J}_{\pm}(i,t)=\pm\,\iota\,\bar{\zeta}_{i}/2\,\delta(t-t_{o})$ if $i\in
A$ and $0$ otherwise. This is schematically shown in Fig. 2. Note that the
partition function automatically traces over degrees of freedom in $B$, the
sources are inserted in $A$ to compute the Wigner characteristic.

At this point, it will be useful to shift to symmetric
($\psi_{\text{s}},\bar{\psi}_{\text{s}}$) and antisymmetric
($\psi_{\text{a}},\bar{\psi}_{\text{a}}$) combinations of the $\pm$ fields
defined at each spacetime point as,

$\displaystyle\psi_{\text{s}}$
$\displaystyle=(\psi_{+}+\psi_{-})/\sqrt{2},\quad$
$\displaystyle\bar{\psi}_{\text{s}}$
$\displaystyle=(\bar{\psi}_{+}+\bar{\psi}_{-})/\sqrt{2},$
$\displaystyle\psi_{\text{a}}$
$\displaystyle=(\psi_{+}-\psi_{-})/\sqrt{2},\quad$
$\displaystyle\bar{\psi}_{\text{a}}$
$\displaystyle=(\bar{\psi}_{+}-\bar{\psi}_{-})/\sqrt{2}.$

The symmetric source $J_{\text{s}}=(J_{+}+J_{-})/\sqrt{2}$ couples to the
antisymmetric field $\bar{\psi}_{\text{a}}$, while the antisymmetric source
$J_{\text{a}}=(J_{+}-J_{-})/\sqrt{2}$ couples to the symmetric field
$\bar{\psi}_{\text{s}}$. For the Wigner characteristic, the source structure
works out to

$\left.\begin{aligned}
J_{\text{s}}(i,t)&=0,\quad&J_{\text{a}}(i,t)&=-\tfrac{\iota}{\sqrt{2}}\zeta_{i}\delta(t-t_{o}),\\\
\bar{J}_{\text{s}}(i,t)&=0,\quad&\bar{J}_{\text{a}}(i,t)&=\tfrac{\iota}{\sqrt{2}}\bar{\zeta}_{i}\delta(t-t_{o})\end{aligned}\right\\}{\text{
for {\hbox{i \in A}},}}$ (11)

and $0$ otherwise. We can then write

$\displaystyle\chi_{A}(\sqrt{2}{\bm{\bar{\zeta}}},\sqrt{2}{\bm{\zeta}},t_{o})$
(12)
$\displaystyle=\\!\\!\int\mathcal{D}\left[{\bm{\bar{\psi}}}_{\text{s,a}},{\bm{\psi}}_{\text{s,a}}\right]\,\mathrm{e}^{\iota\mathcal{S}_{K}[{\bm{\bar{\psi}}}_{\text{s,a}},{\bm{\psi}}_{\text{s,a}}]+{\bm{\bar{\zeta}}}\cdot\hat{P}_{A}{\bm{\psi}}_{\text{s}}(t_{o})-{\bm{\bar{\psi}}}_{\text{s}}(t_{o})\hat{P}_{A}\cdot{\bm{\zeta}}},$

where $\hat{P}_{A}$ is a projection operator on to the degrees of freedom in
subsystem $A$, added to explicitly account for the sources coupling only to
the degrees of freedom in $A$. Substituting Eq. (12) into Eq. (10), we obtain

$\,\mathrm{e}^{-(n-1)[S^{(n)}-V_{A}\ln
2]}=\int\prod_{\alpha=1}^{n}\mathcal{D}\left[{\bm{\bar{\psi}}}^{(\alpha)}_{\text{s,a}},{\bm{\psi}}^{(\alpha)}_{\text{s,a}}\right]\,\mathrm{e}^{\iota\sum\nolimits_{\alpha=1}^{n}\mathcal{S}_{K}\left[{\bm{\bar{\psi}}}^{(\alpha)}_{\text{s,a}},{\bm{\psi}}^{(\alpha)}_{\text{s,a}}\right]}\,\mathrm{e}^{\iota\mathcal{S}_{\text{ent}}\left[\left\\{\hat{P}_{A}{\bm{\bar{\psi}}}_{\text{s}}^{(\alpha)}(t_{o}),\hat{P}_{A}{\bm{\psi}}_{\text{s}}^{(\beta)}(t_{o})\right\\}\right]},$
(13)

where the _entangling action_ , $\mathcal{S}_{\text{ent}}$, is given by

$\displaystyle\,\mathrm{e}^{\iota\mathcal{S}_{\text{ent}}}$
$\displaystyle=\int\prod_{\alpha=1}^{n-1}\mathcal{D}\left[{\bm{\bar{\zeta}}}^{(\alpha)},{\bm{\zeta}}^{(\alpha)},{\bm{\bar{\eta}}}^{(\alpha)},{\bm{\eta}}^{(\alpha)}\right]\exp\left({\sum_{\alpha=1}^{n-1}{\bm{\bar{\zeta}}}^{(\alpha)}\\!\cdot{\bm{\eta}}^{(\alpha)}-{\bm{\bar{\eta}}}^{(\alpha)}\\!\cdot{\bm{\zeta}}^{(\alpha)}+\sum_{\alpha>\beta}{\bm{\bar{\zeta}}}^{(\alpha)}\\!\cdot{\bm{\zeta}}^{(\beta)}-{\bm{\bar{\zeta}}}^{(\beta)}\\!\cdot{\bm{\zeta}}^{(\alpha)}}\right)$
(14)
$\displaystyle\hphantom{=\int\prod_{\alpha=1}^{n-1}}\exp\left({\sum_{\alpha=1}^{n-1}{\bm{\bar{\eta}}}^{(\alpha)}\\!\cdot\hat{P}_{A}{\bm{\psi}}_{\text{s}}^{(\alpha)}(t_{o})-{\bm{\bar{\psi}}}^{(\alpha)}_{\text{s}}(t_{o})\hat{P}_{A}\\!\cdot{\bm{\eta}}^{(\alpha)}}+{\sum_{\alpha=1}^{n-1}{\bm{\bar{\zeta}}}^{(\alpha)}\\!\cdot\hat{P}_{A}{\bm{\psi}}_{\text{s}}^{(n)}(t_{o})-{\bm{\bar{\psi}}}^{(n)}_{\text{s}}(t_{o})\hat{P}_{A}\\!\cdot{\bm{\zeta}}^{(\alpha)}}\right).$

The first part of Eq. (13) represents the action of $n$ independent replicas,
while the entangling action involves integrals over the source terms in each
of the replicas. Note that in absence of $\mathcal{S}_{\text{ent}}$, the
Keldysh partition function of the independent replicas is $1$, and hence the
entanglement would have vanished, i.e. the finite entanglement comes solely
from the effects of $\mathcal{S}_{\text{ent}}$. This is consistent with the
fact that $\mathcal{S}_{\text{ent}}$ has information about the subsystem $A$,
since the sources are restricted to $A$. While this may seem similar to the
standard path integral over $n$ replicas used in field theoretic treatment of
entanglement [21, 19], we note that we have $2n$ fields here. Beyond the
obvious fact that this allows us to look at non-equilibrium dynamics, we will
see that even for equilibrium systems or ground states, using $2n$ fields
gives us certain advantages in terms of the structure of the resultant theory.

For a non-interacting system, with Gaussian action, both the integrals over
the fields and the sources can be performed exactly (in either order); however
for a generic interacting system, it is not possible to integrate out the
matter fields exactly. Fortunately, even in this case, the integral over the
sources is just a Gaussian integral, which can be done analytically to get

$\iota\mathcal{S}_{{\text{ent}}}=\begin{bmatrix}{\bm{\bar{\psi}_{\text{s}}}}^{(1)},{\bm{\bar{\psi}_{\text{s}}}}^{(2)},\cdots,{\bm{\bar{\psi}_{\text{s}}}}^{(n)}\end{bmatrix}\mathbb{J}\otimes\hat{P}_{A}\begin{bmatrix}{\bm{\psi_{\text{s}}}}^{(1)}\\\
{\bm{\psi_{\text{s}}}}^{(2)}\\\ \vdots\\\
{\bm{\psi_{\text{s}}}}^{(n)}\end{bmatrix},$ (15)

where $\mathbb{J}$ is an $n\times n$ antisymmetric matrix with all entries
below the diagonal equal to one,

$\mathbb{J}=\begin{pmatrix}\hphantom{-}0&-1&\hphantom{-}\cdots&-1\\\
\hphantom{-}1&\hphantom{-}0&\hphantom{-}\ddots&\hphantom{-}\vdots\\\
\hphantom{-}\vdots&\hphantom{-}\ddots&\hphantom{-}\ddots&-1\\\
\hphantom{-}1&\hphantom{-}\cdots&\hphantom{-}1&\hphantom{-}0\end{pmatrix}_{n\times
n}.$ (16)

Note that the integral over the sources couples the fields in different
replicas in $\mathcal{S}_{\text{ent}}$. It is useful to note the following
features of the entangling action:

  * •

The integration of sources produces a replica coupling term which is quadratic
in the fields and only couples fields in the subsystem $A$ across replicas.

  * •

The coupling is local in space and time, i.e. it connects fields on the same
lattice sites, only at the time of measuring the REE.

  * •

The structure of $\mathcal{S}_{\text{ent}}$ has the form of a “current” in the
replica space, flowing between the same sites in the subsystem $A$ across any
two distinct replicas. The entangling action does not couple fields in the
same replica, but couples fields in all distinct replica pairs. In contrast,
the standard replica trick with Euclidean field theory involves fields in one
replica being identified with their counterparts in consecutive replicas. The
schematic difference between the standard approach and our approach is shown
in Fig. 1(a) and (b). The use of the two-contour SK field theory allows the
replacement of the boundary condition by quadratic terms. The boundary
condition in standard replica methods requires one to work with actions
defined on $n$-sheeted Riemann surfaces for generic interacting systems, with
reduction to a single sheet possible in special cases, like in free theories
[21] or in presence of conformal symmetry in $1+1$D [19]. Our formalism on the
other hand, involves “current”-like couplings between replicas without
additional boundary conditions on fields across replicas. This will allow us
to write down a diagrammatic expansion for EE in terms of the local connected
correlators of the single replica for a generic theory. We will take up this
task in the next section.

  * •

It is important to note that the Keldysh indices of the fields making up the
entangling action has the form
$\sim\bar{\psi}^{(\alpha)}_{\text{s}}\psi^{(\beta)}_{\text{s}}$ with
$\alpha\neq\beta$. Such a term involving fields from the same replica is
forbidden in usual Schwinger Keldysh field theory as it would change
normalization of $\hat{\rho}$ and affect the causal structure of the
correlations. Thus there is no obvious analogue of this term in usual single
component field theories, or thermal field theories. The doubling of the
fields in the Keldysh formalism is what allows us to write this term, even if
we are considering the entanglement of a thermal state or a ground state.
Thus, Keldysh field theory is not just a way to access non-equilibrium
dynamics in this case, it is essential to writing down a space-time local
description of entanglement entropies.

  * •

Finally we would like to note that the equations derived till now are agnostic
to the explicit form of the action. For e.g., they do not require conformal
invariance, and can be used for generic interacting systems in any dimension
both in and out of equilibrium.

Thus the $n$th order REE is the Keldysh free energy of the $n$-fold replicated
system in presence of the inter-replica “current” between the same sites in
the subsystem $A$. These currents are active only at the time of measurement
of the entropy of the system. For a system in ground state/ thermal
equilibrium / steady state, the entropy is independent of the time of
measurement, and we can choose the time to simplify calculations. For non-
equilibrium dynamics, which tracks the time evolution of entanglement entropy,
one is interested in calculating the entropy as a function of the measurement
time $t_{o}$. We have thus provided an alternate field theoretic picture of
REE of fermionic systems. We note that a similar construction for Bosonic
systems runs into issues of zero modes, and we will take it up in a future
work [73].

##  III Feynman Diagrams and “m” particle Entanglement

Figure 3:  Diagrammatic representation of (a) the symmetric($\psi_{\text{s}}$)
and antisymmetric($\psi_{\text{a}}$) fields in the theory, (b) the two point
propagators of the free theory in each replica, and (c) the entangling vertex
(for details about vertex factors refer to main text, Eq. (15) & Eq. (16)).
Lattice sites are labelled by $i,j$ and $1\leq\alpha,\beta\leq n$ are replica
indices. (d) A ring diagram with two $\cal J$ vertices. We explicitly evaluate
the diagram with lattice and replica indices marked on each propagator. Since
the latter are same in each replica, the resulting sum factorises into a trace
over replica indices ($\cal R$), and a trace over the subsystem ($A$). (e)
Rényi entropy in a free theory, $S^{(n)}_{0}$, is the sum of all ring
diagrams. Unmarked propagators imply trace over replica and lattice indices.
The term with a single $\cal J$ vertex evaluates to zero, but is still
explicitly mentioned to emphasize the series structure of the ring diagrams.

In the previous section, we have shown that the $n$th order REE of a system of
interacting fermions is the Keldysh free energy of $n$ replicas of the system
with inter-replica currents flowing between the sites in the subsystem $A$.
Here we will try to develop a diagrammatic expansion which will provide a
prescription to construct the REE in terms of the correlation functions of the
interacting system _in a single replica sheet_ restricted to the subsystem
$A$. More precisely, we will show that the $n$th REE can be written as

$\begin{multlined}S^{(n)}=S^{(n)}_{1\mathrm{P}}\left[G^{(1)}\right]\,+\,S^{(n)}_{2\mathrm{P}}\left[G^{(1)},G^{(2)}_{c}\right]\\\
\,+\,S^{(n)}_{3\mathrm{P}}\left[G^{(1)},G^{(2)}_{c},G^{(3)}_{c}\right]\,+\,\cdots,\end{multlined}S^{(n)}=S^{(n)}_{1\mathrm{P}}\left[G^{(1)}\right]\,+\,S^{(n)}_{2\mathrm{P}}\left[G^{(1)},G^{(2)}_{c}\right]\\\
\,+\,S^{(n)}_{3\mathrm{P}}\left[G^{(1)},G^{(2)}_{c},G^{(3)}_{c}\right]\,+\,\cdots,$
(17)

where $G^{(1)}=-\iota\expectationvalue*{\psi_{\text{s}}\bar{\psi}_{\text{s}}}$
is the one-particle (two-point) Keldysh Green’s function at equal times, and
$G^{(m)}_{c}=(-\iota)^{m}\expectationvalue*{\psi_{\text{s}}\psi_{\text{s}}\cdot\cdot\,{m\rm\
times}\leavevmode\nobreak\
\bar{\psi}_{\text{s}}\bar{\psi}_{\text{s}}\cdot\cdot\,{m\rm\ times}}$ is the
equal time $m$-particle (or $2m$-point) connected correlator of the symmetric
fields at the time of observation. The spatial indices of the correlators will
only span the sites in the subsystem $A$. As alluded to in Eq. (2), these
correlators can be expressed in terms of (the connected pieces of) expectation
values of normal ordered strings of Fermion creation and annihilation
operators in the system. We derive the exact form for the same in Appendix A.
In the construction of Eq. (17), $S^{(n)}_{m\mathrm{P}}$ is independent of
$k$-particle connected correlators for $k>m$. Further, if the $m$-particle
correlator $G^{(m)}$ is factorizable, i.e. $G^{(m)}_{c}=0$, then
$S^{(n)}_{m\mathrm{P}}$ vanishes identically. We will provide an explicit
analytic form of $S^{(n)}_{1\mathrm{P}}[G^{(1)}]$ and formulate diagrammatic
rules for constructing $S^{(n)}_{m\mathrm{P}}$ for general $m$. The key point
of this formulation is that it is agnostic to how the correlators are
calculated/measured and provides a recipe to stitch together the exact
correlators of the interacting theory to construct the entanglement entropy.
One can calculate the correlators in some specified approximation scheme, but
one can also substitute in this expansion experimentally measured correlators,
or the same measured in numerical experiments like Monte Carlo methods. This
thus provides an alternative to constructing REE from the knowledge of the
exact quantum state. Note that reconstructing a generic many-body quantum
state requires the knowledge of an exponential number of variables ${\cal
O}(e^{V})$, while $S_{m\mathrm{P}}^{(n)}$ only requires knowledge of upto $m$
particle connected correlators which have $\order{V_{A}^{m}}$ variables.
Further, REE and EE are non-linear non-local functions of the RDM; there are
very few ways of computing them, whereas a large number of analytic and
numerical approximation schemes are known for calculating correlation
functions. This connection between correlators and entanglement in Fermionic
systems will allow these approximation methods to be used in calculating REE
of interacting fermions.

Figure 4:  (a) Interaction vertices in Keldysh field theory corresponding to a
pair wise interaction
$\bar{\psi}_{r}\bar{\psi}_{r^{\prime}}U(r,r^{\prime})\psi_{r^{\prime}}\psi_{r}$.
There are eight vertices in the $\psi_{\mathrm{s,a}}$ basis, each with an odd
number of symmetric and antisymmetric fields, and a vertex factor of $\iota
U(r,r^{\prime})/2$ (represented as a green wavy line). We use fields with
dotted lines as a placeholder for both symmetric(solid) and
antisymmetric(dashed) fields for the sake of brevity.(b) The presence of
interactions induce self energy corrections $\Sigma$ which in turn dress the
propagators. We use thick solid lines to signify the interacting Keldysh
propagators. (c) & (d) Interaction dressing converts the free propagators in
the ring diagrams to their interacting counterparts. (e) The one-particle
contribution to $S^{(n)}$, $S^{(n)}_{1\mathrm{P}}$ is entirely determined by
the sum of ring diagrams with the interacting Keldysh propagators.

###  III.1  One-particle correlators and $S^{(n)}_{1\mathrm{P}}$

We will first focus on calculating $S^{(n)}_{1\mathrm{P}}$. To do this, let us
first consider the REE of a system of free fermions, where
$S^{(n)}_{m\mathrm{P}}=0$ for $m>1$, i.e. $S^{(n)}=S^{(n)}_{1\mathrm{P}}$. For
the diagrammatic expansion, we denote the symmetric fermionic fields by a
solid line, and the anti-symmetric fields by dashed lines, as shown in Fig.
3(a). We have omitted spin indices, but they can be easily added to this
description. Each field carries a replica index along with usual space-time
indices. We will expand the free energy around the theory with $n$ independent
identical replicas and treat $\mathcal{S}_{{\text{ent}}}$ as an additional
coupling. In the independent replica theory, the propagators connect fields
with same replica indices. The propagators for each replica are exactly the
same as that of a single replica ( aka standard) Keldysh field theory. There
are three independent propagators, the retarded propagator
$G^{R}=-\iota\expectationvalue*{\psi_{\text{s}}\bar{\psi}_{\text{a}}}$, the
advanced propagator
$G^{A}=-\iota\expectationvalue*{\psi_{\text{a}}\bar{\psi}_{\text{s}}}$, and
the Keldysh propagator
$G^{K}=-\iota\expectationvalue*{\psi_{\text{s}}\bar{\psi}_{\text{s}}}$, as
shown in Fig 3(b). Note
$\expectationvalue*{\psi_{\text{a}}\bar{\psi}_{\text{a}}}=0$ in the single
replica Keldysh field theory. These definitions hold for both the interacting
and non-interacting theories. The quadratic coupling in
$\mathcal{S}_{{\text{ent}}}$ is represented by an open circle, with the ${\cal
J}$ vertex given by the matrix $\mathbb{J}\otimes\hat{P}_{A}$ as shown in Fig
3(c). Note that this vertex couples symmetric fields in different replicas on
the same site in subsystem $A$ at the time when REE is computed.

The first point to note is that since the Rényi entropy is a free energy (in
presence of inter-replica currents), it can be written as a sum of all fully
connected diagrams in the replica field theory with no external legs. The
second point to note is that the Keldysh free energy of the independent
replicas is $0$, so the diagrams which contribute to REE must have one or more
$\mathcal{S}_{{\text{ent}}}$ vertices. Finally, since
$\mathcal{S}_{{\text{ent}}}$ is quadratic, one can easily re-sum all the
diagrams here. Since the diagonal elements of $\mathbb{J}$ are $0$, there are
no diagrams with a single ${\cal J}$ vertex. The first non-trivial diagram,
which has two ${\cal J}$ vertices is shown in Fig. 3(d), together with its
explicit evaluation. In fact, the set of ring diagrams shown in Fig. 3(e)
exhausts all the free energy diagrams for the free theory. Note that only the
Keldysh propagator, which carries information about distribution functions
appear in this series. We note that the diagram with $p$ circles have a
symmetry factor of $1/p$ ( $1/p!$ from the exponential and $(p-1)!$ from
permutation of the $\cal J$ vertices). Defining
$\iota\hat{G}^{K}_{0;A}=\hat{P}_{A}\iota\hat{G}^{K}(t_{o},t_{o})\hat{P}_{A}$
as the projection of the equal time Keldysh correlator onto the subsystem $A$,
it is easy to see that

$\displaystyle(1-n)S^{(n)}_{0}$ $\displaystyle=(1-n)V_{A}\ln
2-\sum_{p=1}^{\infty}\frac{1}{p}\Tr_{\cal
R}[{\mathbb{J}}^{p}]\Tr_{A}[(\iota\hat{G}_{0;A}^{K})^{p}]$
$\displaystyle=\ln(\frac{1}{2^{(n-1)V_{A}}}\det[\mathbb{I}\otimes\hat{1}-\mathbb{J}\otimes\iota\hat{G}^{K}_{0;A}]),$
(18)

where $\mathbb{I}$ is the $n\times n$ identity matrix and $\hat{1}$ is the
identitty operator on the subsystem $A$. This determinant is known exactly
(see Appendix B of Ref. [38]), and it reduces to a closed form expression for
$S^{(n)}_{0}$,

$S^{(n)}_{0}(t_{o})=\frac{1}{1-n}\Tr_{A}\ln\left[(\hat{C}_{0}(t_{o}))^{n}+(\hat{1}-\hat{C}_{0}(t_{o}))^{n}\right],$
(19)

where
$C_{0}(i,j;t_{o})\equiv\expectationvalue*{\hat{c}^{\dagger}_{i}(t_{o})\hat{c}^{\vphantom{\dagger}}_{j}(t_{o})}_{0}$
is the non-interacting correlation matrix restricted to $A$, and is related to
the equal-time Keldysh correlator by
$\hat{C}_{0}(t_{o})=(\hat{1}-\iota\hat{G}^{K}_{0,A})/2$. Here
$\hat{c}^{\dagger}_{i}$ creates a Fermion on site $i$. Eq. (19) matches with
formulae derived earlier for non-interacting fermionic systems [59, 60, 38].
This formula for REE of free fermions in terms of the one-particle correlation
function is well known in the literature, and this acts as a check on our new
method to calculate entanglement.

Let us now turn our focus to interacting Fermionic systems, where, for the
sake of convenience, we assume a pairwise interaction,

$\mathcal{S}_{{\text{int}}}^{\pm}=\pm\\!\int\\!\\!\differential{t}\sum_{r,r^{\prime}}\bar{\psi}_{\pm}(r,t)\bar{\psi}_{\pm}(r^{\prime},t)\,U(r,r^{\prime})\psi_{\pm}(r^{\prime},t)\psi_{\pm}(r,t).$

Here $r,r^{\prime}$ run over degrees of freedom in the whole system. We note
that the final formulae we derive is agnostic to the precise form of the
interaction; however assuming a representative form of
$\mathcal{S}_{{\text{ent}}}$ helps us draw Feynman diagrams and clearly
illustrate some of the properties of the formalism. In Sec. III.4, we outline
a derivation of the same formulae without making reference to any particular
form of interaction. For such a pairwise form the typical interaction
vertices, shown in Fig 4(a), consist of three symmetric and one antisymmetric
fields, or vice versa. Note that interaction vertex couples fields with the
same replica index. It is still true that the Keldysh free energy of $n$
independent interacting replicas is $0$, so we only need to worry about
connected diagrams with one or more open $\cal J$ vertices. It is instructive
to think about the modifications of the ring diagrams due to interaction
vertices. There is a set of diagrams where the interaction vertices couple
fields on the same propagator. Some representative diagrams are shown on Fig.
4(c). These are essentially self-energy corrections to the propagators, shown
in Fig. 4(b). The net effect of summing up all such diagrams is to convert the
non-interacting one-particle correlator $\hat{C}_{0}$ to the _interacting_ one
particle correlator $\hat{C}$ in the expressions given in Eq. (19), as seen in
Fig. 4(c) and Fig. 4(d). These dressed diagrams, depicted in Fig. 4(e),
exhaust $S^{(n)}_{1\mathrm{P}}$ and other possible diagrams for $S^{(n)}$
depend on higher particle correlators. So, for interacting Fermions, we have

$\displaystyle S^{(n)}_{1\mathrm{P}}$
$\displaystyle=\frac{1}{1-n}\Tr_{A}\ln\left[\hat{C}^{n}+(\hat{1}-\hat{C})^{n}\right].$
(20)

This is the analytic form of $S^{(n)}_{1\mathrm{P}}$ in terms of the _exact
interacting_ one-particle correlator (two-point function).

Figure 5: Feynman diagrams for the two-particle contribution to REE: (a)
Propagators in two separate rings are connected by interaction vertices.
Dressing of this pair of propagators converts them to an interacting connected
two-particle correlator $G^{(2)}_{c}$ (shown in (c)). The rings are now
connected and lead to a valid connected diagram for $S^{(n)}_{2\mathrm{P}}$.
(b) A pair of propagator within the same ring is connected by interaction
vertices and gets dressed to $G^{(2)}_{c}$. (a) and (b) can be viewed as
diagrams where external legs of $G^{(2)}_{c}$ are connected by chains of
entangling vertices ${\cal J}$ interlinked with single-particle propagators.
(c) Details of the two particle connected correlator $G^{(2)}_{c}$. Note that
it is diagonal in replica space. (d) Chains with different numbers of
entangling vertices can be resummed into a single connector ${\cal\hat{V}}$.
(e) The series for constructing ${\cal\hat{V}}$. The solid lines are
interacting Keldysh propagators in a single replica. Detailed expressions for
${\cal\hat{V}}$ are provided in Eq. (22) and Eq. (23)

###  III.2  Two-particle correlators and $S^{(n)}_{\mathrm{2\mathrm{P}}}$

In a non-interacting system the one particle correlators define the density
matrix and hence the REE [60]. However this is no longer true for interacting
systems, where multi-particle connected correlators also contribute to the
REE. To see this, let us focus on the first diagram shown in Fig. 5(a). Here
we consider two otherwise disconnected rings, where an interaction line
connects a propagator in one ring to a propagator in the other ring. This is
now a connected diagram which contributes to $S^{(n)}$, but does not belong to
the series of ring diagrams in $S^{(n)}_{1\mathrm{P}}$. The set of diagrams in
Fig. 5 (a) shows how one can add more complicated motifs between the pair of
propagators to convert them to the fully interacting connected two particle
propagator (shaded region)

$\displaystyle\iota^{2}G^{(2)}_{c}(i,j|k,l;t_{o})$
$\displaystyle=\expectationvalue{\psi_{\text{s}}(i,t_{o})\psi_{\text{s}}(j,t_{o})\bar{\psi}_{\text{s}}(l,t_{o})\bar{\psi}_{\text{s}}(k,t_{o})}_{c}$
$\displaystyle=2^{2}\expectationvalue{\hat{c}^{\dagger}_{k}\hat{c}^{\dagger}_{l}\hat{c}^{\vphantom{\dagger}}_{j}\hat{c}^{\vphantom{\dagger}}_{i}}_{c}.$

An equivalent way of getting this diagram is to first consider the connected
correlator $G^{(2)}_{c}$ and join its external legs through ${\cal J}$
vertices with interacting one particle propagators in between them. Note that
each chain connecting the ends of $G^{(2)}_{c}$ should have at least one
${\cal J}$ vertex. Otherwise the two particle connected correlator would
effectively reduce to a single particle propagator and reproduce diagrams in
$S^{(n)}_{1P}$. It is simple to generalize to the case of $p$ rings where pair
of propagators from the different rings are connected by interaction lines,
leading to diagrams with multiple $G^{2}_{c}$. A second class of diagrams
contributing to $S^{(n)}_{2P}$ can be obtained by considering pairs of
propagators in a single ring and connecting them by interaction lines,
effectively converting this pair to a $G^{(2)}_{c}$. A series of connections
leading to such a diagram is shown in Fig. 5(b). In both cases (a) and (b),
the resultant diagram can be reproduced by first considering $G^{(2)}_{c}$ and
connecting its legs by chains of ${\cal J}$ vertices coupled by one particle
propagators.

Figure 6: (a) The diagrams for $S^{(n)}_{2\mathrm{P}}$, which involve $1$ and
$2$ $G^{(2)}_{c}$ s. (i),(ii),(iii) The diagrams shown exhaust all
topologically distinct possibilities up to two instances of $G^{(2)}_{c}$.
$\alpha,\beta$ are replica indices, and lattice indices are implied to be
traced over the subsystem $A$. (b) The details of a $k$-particle connected
correlator including replica and site indices. Note that all incoming and
outgoing lines represent symmetric fields and carry the same replica index.
(c)-(e) Some representative diagrams contributing to $S^{(n)}_{3\mathrm{P}}$.
(c) involves a single $G^{(3)}_{c}$, (d) involves a $G^{(3)}_{c}$ and a
$G^{(2)}_{c}$, while (e) involves two $G^{(3)}_{c}$s.

At this point, it is useful to note that each chain connecting the external
legs of $G^{(2)}_{c}$ can have $1$, $2$, $3$, any number of ${\cal J}$
vertices. This is depicted in Fig. 5(d). One can then use a resummed one-
particle connector ${\cal\hat{V}}$ to join the external legs of $G^{(2)}_{c}$
(see Sec. III.4 for details). The diagrammatic series corresponding to this
resummation is shown in Fig. 5(e), which evaluates to the following for
$\cal\hat{V}$,

$\displaystyle\mathcal{\hat{V}}_{\alpha\beta}$
$\displaystyle=\sum_{p=1}^{\infty}\left[{\,\mathbb{J}^{p}\,}\right]_{\alpha\beta}\,\left[\iota\hat{G}_{A}^{K}(t_{o},t_{o})\right]^{p-1},$
$\displaystyle=\left[{\,\mathbb{J}\otimes\hat{1}\left(\mathbb{I}\otimes\hat{1}-\mathbb{J}\otimes\iota\hat{G}^{K}_{A}\right)^{-1}}\right]_{\alpha\beta},$
(21)

where $\iota\hat{G}^{K}_{A}=\hat{P}_{A}\hat{G}^{K}(t_{o},t_{o})\hat{P}_{A}$ is
the interacting one particle Keldysh correlator at equal times, restricted to
the subsystem $A$. It is possible to analytically obtain an expression for the
matrix elements of $\cal\hat{V}$, namely,

$[\mathcal{\hat{V}}_{\alpha\beta}]=\begin{bmatrix}\hat{v}_{0}&-\hat{v}_{n-1}&-\hat{v}_{n-2}&\cdots&-\hat{v}_{1}\\\
\hat{v}_{1}&\hat{v}_{0}&\ddots&\ddots&\vdots\\\
\hat{v}_{2}&\ddots&\ddots&\ddots&-\hat{v}_{n-2}\\\
\vdots&\ddots&\ddots&\hat{v}_{0}&-\hat{v}_{n-1}\\\
\hat{v}_{n-1}&\cdots&\hat{v}_{2}&\hat{v}_{1}&\hat{v}_{0}\end{bmatrix},$ (22)

where

$\displaystyle\hat{v}_{0}$
$\displaystyle=\frac{1}{2}\frac{\hat{C}^{n-1}-(\hat{1}-\hat{C})^{n-1}}{\hat{C}^{n}+(\hat{1}-\hat{C})^{n}},$
(23) $\displaystyle\hat{v}_{k}$
$\displaystyle=\frac{1}{2}\frac{\hat{C}^{n-k-1}(\hat{1}-\hat{C})^{k-1}}{\hat{C}^{n}+(\hat{1}-\hat{C})^{n}}{\text{
for }}1\leq k\leq n-1.$

This resummed connector is denoted diagrammatically by two concentric open
circles with the inner one filled in with a cross-hatch pattern. Note that
while the $\mathcal{J}(=\mathbb{J}\otimes\hat{P}_{A})$ vertex had only off-
diagonal components in replica space, $\mathcal{\hat{V}}$ can couple fields
with same replica indices as well. In this new language, we consider the
interacting two-particle connected correlator and join its external legs by
the $\mathcal{\hat{V}}$ connectors to rewrite the diagrams of Fig. 5(a) and
(b). These are shown in Fig. 6(a-i).

Now, one can construct a larger class of diagrams which only involve the fully
interacting $G^{(2)}_{c}$ and $\mathcal{\hat{V}}$s. To construct these
diagrams consider $p$ different $G^{(2)}_{c}$ s, and connect their external
legs by $\mathcal{\hat{V}}$s, so that no external leg remains unconnected. One
can show (see Sec. III.4) that the symmetry factor of the diagram is simply
determined by considering the possible permutations of the $G^{(2)}_{c}$
blocks. All the diagrams involving one or two $G^{(2)}_{c}$ blocks are shown
in Fig. 6(a). The sum of all such diagrams which depend only on the 1-particle
and 2-particle correlators, with the number of the latter varying from $1$
through $\infty$, forms $S^{(n)}_{2\mathrm{P}}$. This can be neatly summarized
in the Feynman rules for diagrams in $S^{(n)}_{2\mathrm{P}}$:

  * •

Consider $p$ two particle connected correlators and join their external legs
in all topologically distinct ways.

  * •

For each connected correlator, put a factor of
$\iota^{2}G^{(2)}_{c}(i,j|k,l;t_{o})$, for each line joining external legs,
put a factor of the resummed connector
${\cal\hat{V}}_{\alpha\beta}(i,j;t_{o})$.

  * •

Sum over possible replica indices (noting that external legs of $G^{(2)}_{c}$
belong to the same replica).

  * •

Sum over site indices in the subsystem $A$.

  * •

Multiply by $(-1)^{N_{L}}$, where $N_{L}$ is the number of Fermion loops.

  * •

Multiply each diagram by its symmetry factor (see Sec. III.4 for details)

Using these Feynman rules one can see that the first diagram in Fig 6(a-i)
evaluates to

$-n\sum_{ijkl\in
A}G^{(2)}_{c}(i,j|k,l;t_{o})\,v_{0}(k,i;t_{o})\,v_{0}(l,j;t_{o}).$ (24)

Similarly the first diagram (of ladder topology) shown in Fig. 6(a-ii) is
given by

$\hphantom{-}\frac{1}{2}\sum_{\alpha\beta}\sum_{\begin{subarray}{c}ijkl\\\
xyzw\end{subarray}\in A}G^{(2)}_{c}(i,j|k,l)\,G^{(2)}_{c}(x,y|z,w)\,{\cal
V}_{\alpha\beta}(k,x)\,{\cal V}_{\alpha\beta}(l,y)\,{\cal
V}_{\beta\alpha}(z,i)\,{\cal V}_{\beta\alpha}(w,j),$ (25)

while the first diagram (of bubble topology) in Fig. 6(a-iii) evaluates to

$-\frac{1}{2}\sum_{\alpha\beta}\sum_{\begin{subarray}{c}ijkl\\\
xyzw\end{subarray}\in A}G^{(2)}_{c}(i,j|k,l)\,G^{(2)}_{c}(x,y|z,w)\,{\cal
V}_{\alpha\alpha}(k,i)\,{\cal V}_{\beta\beta}(z,x)\,{\cal
V}_{\alpha\beta}(l,y)\,{\cal V}_{\beta\alpha}(w,j).$ (26)

We have suppressed the time index in the above two equations for the sake of
clarity. Summing over all such diagrams yields $S^{(n)}_{2\mathrm{P}}$. We
thus have a prescription for construction of $S^{(n)}_{2P}$ from the one-
particle and two-particle correlators.

###  III.3  $m$-particle correlators and $S^{(n)}_{m\mathrm{P}}$

Once we have established the Feynman rules for constructing the two-particle
contribution to REE, $S^{(n)}_{2\mathrm{P}}$, it is easy to extend it to the
case of $S^{(n)}_{m\mathrm{P}}$. Note that by definition, diagrams for
$S^{(n)}_{m\mathrm{P}}$ must have at least one instance of the $m$-particle
connected correlator

$\displaystyle\iota^{m}\,G^{(m)}_{c}(i_{1},i_{2},\cdots
i_{m}|j_{1},j_{2},\cdots j_{m};t_{o})$ (27) $\displaystyle\hskip
5.0pt=\expectationvalue{\psi_{\text{s}}(i_{1},t_{o})\cdots\psi_{\text{s}}(i_{m},t_{o})\bar{\psi}_{\text{s}}(j_{m},t_{o})\cdots\bar{\psi}_{\text{s}}(j_{1},t_{o})}_{c}$
$\displaystyle\hskip
5.0pt=(-2)^{m}\expectationvalue{\hat{c}^{\dagger}_{j_{m}}\cdots\hat{c}^{\dagger}_{j_{2}}\hat{c}^{\dagger}_{j_{1}}\hat{c}^{\vphantom{\dagger}}_{i_{1}}\hat{c}^{\vphantom{\dagger}}_{i_{2}}\cdots\hat{c}^{\vphantom{\dagger}}_{i_{m}}}_{c}.$

The diagrams cannot contain higher order ($k>m$) correlators. Now,
$G^{(m)}_{c}$ has $m$ incoming lines and $m$ outgoing lines. Each of these
outgoing lines can be (a) connected to the incoming lines of same
$G^{(m)}_{c}$. Fig. 6(c) shows such a diagram for $G^{(3)}_{c}$. (b) connected
to incoming lines of some other $G^{(k)}_{c}$ with $k<m$. Fig. 6(d) shows such
a diagram where some of the outgoing lines of $G^{(3)}_{c}$ are connected to
the incoming lines of a $G^{(2)}_{c}$. (c) connected to incoming lines of
another $G^{(m)}_{c}$. Fig. 6(e) shows such a diagram where two $G^{(3)}_{c}$
s are connected to each other. We draw topologically distinct connected
diagrams where all the incoming and outgoing lines are joined with
$\cal\hat{V}$ vertices, making it a free energy diagram. The symmetry factor
is determined by considering permutations of the connected correlators in the
usual way. The sum of all such diagrams give $S^{(n)}_{m\mathrm{P}}$. The
Feynman rules for these diagrams are simple extensions of those for
$S^{(n)}_{2\mathrm{P}}$:

  * •

Consider at least one $m$-particle connected correlator and any number of
$k$-particle connected correlators with $k<m$. Join their external legs in all
topologically distinct ways, so that no external lines remain hanging.

  * •

Draw only fully connected diagrams.

  * •

For each $k$-particle connected correlator, put a factor of
$\iota^{k}G^{(k)}_{c}(i_{1},..i_{k}|j_{1},..j_{k};t_{o})$, for each line
joining external legs, put a factor of the resummed connector
${\cal\hat{V}}_{\alpha\beta}(i,j;t_{o})$.

  * •

Sum over possible replica indices (noting that external legs of $G^{(k)}_{c}$
belong to the same replica).

  * •

Sum over site indices in the subsystem $A$.

  * •

multiply by $(-1)^{N_{L}}$, where $N_{L}$ is the number of Fermion loops.

  * •

Multiply each diagram by its symmetry factor (see Sec. III.4 for details).

This provides a general prescription to construct the $m$-particle
contribution to the Rényi entropy $S^{(n)}_{m\mathrm{P}}$. Some representative
diagrams in the construction of $S^{(n)}_{3\mathrm{P}}$, which involve
$G^{(2)}_{c}$ and $G^{(3)}_{c}$ are shown in Fig. 6(b-d). As an instance of
employing the Feynman rules, the diagram in Fig. 6(b) is given by (time index
suppressed)

$n\sum_{\begin{subarray}{c}i,j,k\\\ x,y,z\end{subarray}\in
A}\iota^{3}G^{(3)}_{c}(i,j,k|x,y,z)\,v_{0}(x,i)\,v_{0}(y,j)\,v_{0}(z,k).$ (28)

One can now use the Feynman rules given above to construct diagrams and
convert them into integral contributions to $S^{(n)}_{m\mathrm{P}}$. We note
here for completeness that diagrams constructed following the aforementioned
rules will contribute to $(1-n)S^{(n)}_{m\mathrm{P}}$. We have thus provided a
general prescription of constructing an estimate of Rényi entanglement entropy
if we only have knowledge of upto $m$-particle connected correlation
functions.

###  III.4 General Derivation

Figure 7:  Feynman diagrams from cumulant expansion: (a)-(c) Diagrams
corresponding to the first three cumulants of
$\iota\mathcal{S}_{{\text{ent}}}$. Replica indices are explicitly marked on
each propagator whereas lattice indices are implied to be traced over. (d)
Diagrams corresponding to the fourth cumulant of
$\iota\mathcal{S}_{{\text{ent}}}$. Note that this involves diagrams which form
part of $S^{(n)}_{1\mathrm{P}}$ (first diagram) as well as diagrams which form
part of $S^{(n)}_{2\mathrm{P}}$ (last 4 diagrams). Similarly the sixth
cumulant will have terms belonging to $S^{(n)}_{3\mathrm{P}}$ as well as
$S^{(n)}_{1\mathrm{P}}$ and $S^{(n)}_{2\mathrm{P}}$. Note that each of
$S^{(n)}_{m\mathrm{P}}$ has contribution from an infinite set of cumulants
starting at order $2m$. The cumulants simply correspond to a different
grouping of all the diagrams discussed earlier.

In the discussion so far, we started by inspecting what effect a “replica
current” term $\mathcal{S}_{{\text{ent}}}$ would have on the free energy
diagrams of $n$ independent free theories $\mathcal{S}_{0}$. We then proceeded
to add interactions $\mathcal{S}_{{\text{int}}}$ in each replica to argue for,
and illustrate, various non-perturbative effects. In this section we instead
choose to formally expand Eq. (13) in cumulants of
$\iota\mathcal{S}_{\text{ent}}$ and evaluate them in terms of connected
correlators in the (un-replicated) interacting theory
$\mathcal{S}_{0}+\mathcal{S}_{{\text{int}}}$. This will provide a first
principles derivation of the previously stated results which makes no appeal
to the form or nature of $\mathcal{S}_{{\text{int}}}$.

We crucially use three facts mentioned earlier, but reiterated here for
emphasis: (i) REE $S^{(n)}$ is equivalent to a Keldysh free energy of $n$
independent replicas in presence of the entangling action
$\mathcal{S}_{{\text{ent}}}$ (ii) Keldysh partition functions are inherently
normalised to unity in the absence of sources which implies that all diagrams
must include at least one instance of $\mathcal{S}_{{\text{ent}}}$ (iii) All
correlators which occur in the diagrammatic expansion are “standard”
Schwinger-Keldysh correlators of a single replica; i.e. these are the
correlation functions one finds in standard textbooks [69]. Owing to the
structure of $\mathcal{S}_{{\text{ent}}}$, only equal time correlators of the
symmetric fields, restricted to the subsystem $A$, will make an appearance.
With this in mind, $\Tr[\hat{\rho}_{A}^{n}]$ can be viewed as an expectation
value of $\,\mathrm{e}^{\iota\mathcal{S}_{\text{ent}}}$ in $n$ independent
copies of the interacting theory,

$\,\mathrm{e}^{-(n-1)(S^{(n)}-V_{A}\ln
2)}=\expectationvalue{\,\mathrm{e}^{\iota{\cal S}_{\text{ent}}}}_{n},$

where the subscript on $\expectationvalue{\bullet}_{n}$ denotes the number of
independent copies of the action $\mathcal{S}_{K}$. The REE $S^{(n)}$ can be
obtained as the cumulant expansion of the above,

$\displaystyle(1-n)(S^{(n)}-V_{A}\ln 2)$
$\displaystyle\hphantom{-}=\expectationvalue{\iota\mathcal{S}_{\text{ent}}}+\frac{1}{2!}\expectationvalue{(\iota\mathcal{S}_{\text{ent}})^{2}}_{c}+\frac{1}{3!}\expectationvalue{(\iota\mathcal{S}_{\text{ent}})^{3}}_{c}+\cdots,$
(29)

where $\expectationvalue{\bullet}_{c}$ represents the connected part of the
expectation value calculated with respect to the $n$ independent replicas.For
example, in case of the second cumulant
$\expectationvalue{(\iota\mathcal{S}_{\text{ent}})^{2}}_{c}=\expectationvalue{(\iota\mathcal{S}_{\text{ent}})^{2}}_{n}-\expectationvalue{\iota\mathcal{S}_{\text{ent}}}^{2}_{n}$
. We will employ the diagrammatic rules set up previously to evaluate these
cumulants. A $p$th order cumulant will involve all possible topologically
distinct connected diagrams made out of $p$ number of $\mathcal{J}$ vertices.
The first few orders are depicted in Fig. 7. Since the $\cal J$ vertex doesn’t
connect fields in the same replica, the first cumulant
$\expectationvalue{\iota\mathcal{S}_{{\text{ent}}}}_{n}$ is exactly zero. This
simplifies the diagrams at higher orders. In particular, till the third
cumulant, the only possible fully connected diagrams are ring diagrams as
shown in Fig. 7(a)-(c). Explicitly evaluating the diagram in Fig. 7(b) and (c)
we get,

$\displaystyle\expectationvalue{(\iota\mathcal{S}_{{\text{ent}}})^{2}}_{c}$
$\displaystyle=-\Tr_{\cal
R}[\mathbb{J}^{2}]\Tr_{A}[(\iota\hat{G}^{K}_{A})^{2}],$
$\displaystyle\expectationvalue{(\iota\mathcal{S}_{{\text{ent}}})^{3}}_{c}$
$\displaystyle=-2!\Tr_{\cal
R}[\mathbb{J}^{3}]\Tr_{A}[(\iota\hat{G}^{K}_{A})^{3}].$

where
$\iota\hat{G}^{K}_{A}=\hat{P}_{A}\iota\hat{G}^{K}(t_{o},t_{o})\hat{P}_{A}$ is
the equal time two-point Keldysh correlator evaluated in the interacting
theory and restricted to subsystem $A$. In a similar fashion, every higher
order cumulant will give rise to a ring diagram of $\mathcal{J}$ vertices,
with the number of cyclic permutations at order $p$ being $(p-1)!$. Taken
together with the combinatorial factor of $1/p!$ in the definition of the
cumulant expansion, the weight of a ring diagram at order $p$ is $1/p$.
Grouping all order ring diagrams together we have recovered the _$1$ -particle
contribution to $S^{(n)}$_, $S^{(n)}_{1\mathrm{P}}$ as depicted in Fig. 4(e),
and consequently the analytic form of the same in terms of the interacting
correlation function $C$ in Eq. (20).

In higher order cumulants, connected diagrams of $\cal J$ vertices can also be
constructed using multiparticle connected correlators. For example Fig. 7(d)
shows the diagrams with $G^{(2)}_{c}$ that contribute to the fourth order
cumulant. In fact, the $p$th order cumulant will involve all possible fully
connected diagrams with $p$ instances of the entangling vertex $\mathcal{J}$
joined by all possible $k$-particle equal time connected Keldysh correlators
$G^{(k)}_{c}$ with $2k\leq p$.

We now turn to classifying the diagrams based on their correlator content.
Diagrams with atleast one instance of the 2 body connected correlator
$G^{(2)}_{c}$, but no higher body correlators are clubbed together as the
“$2$-particle contribution”, $S^{(n)}_{2\mathrm{P}}$. Similarly we define the
“$m$-particle contribution” to $S^{(n)}$ as the collection of all diagrams
which contain atleast one instance of $G^{(m)}_{c}$ but none of the higher
body correlators. This formal regrouping lets us write a
“$m$-particle”-decomposition for EE as stated in Eq. (17). Given the structure
of $S^{(n)}_{m\mathrm{P}}$, if the $m$th correlator is factorisable, the
entire collection of diagrams get decimated.

Each of these $m$-particle contributions contain a multitude of diagrams with
different correlator content (number and type of $G^{(m)}_{c}$ boxes) and
topologies. They also include infinitely many diagrams of the same topology
and correlator content but with a variable number of $\cal J$ vertices on the
lines between the $G^{(m)}_{c}$ boxes. Such diagrams can be clubbed together
to get a resummed connector $\cal V$ on _each line_ since the relative weights
of the diagrams turn out to be exactly one. Assume a diagram of a given
structure with a total of $p$ number of $\cal J$ vertices, distributed across
$q$ different lines with the number in each line being
$n_{1},n_{2},\cdots,n_{q}$. The ways of distributing $p$ identical vertices
into $q$ lines with the number of vertices in each line fixed, gets exactly
cancelled by the $1/p!$ from the cumulant expansion and $n_{j}!$ permutations
from the $j$th line,

${\text{Weight}}=\frac{1}{p!}\times\frac{p!}{n_{1}!\,n_{2}!\,\cdots
n_{q}!}\times(n_{1}!\,n_{2}!\,\cdots n_{q}!)=1.$

This fact reproduces the series depicted in Fig. 5(d) and the functional forms
in Eq. (23). This also makes it apparent that any and all symmetry factors for
the diagram is determined from exchange of the correlator blocks. In summary,
all diagrams are constructed following the Feynman rules laid down in the
previous sections.

We have thus provided a constructive prescription for evaluating
$S^{(n)}_{m\mathrm{P}}$ in a manner independent of the underlying theory of
the problem. We note that if one is interested in the exact answer for
entanglement, one has to compute upto $S^{(n)}_{V_{A}\mathrm{P}}$ and the
complexity of the problem is same as exact diagonalization. However field
theoretic methods are rarely good for exact answers, they are usually geared
to provide useful approximate answers to various quantities. In this case, the
decomposition of $S^{(n)}$ into $S^{(n)}_{m\mathrm{P}}$ is useful when the
series can be truncated after a few terms to yield good estimates of
entanglement. This will happen if the higher order connected correlators are
parametrically small, i.e. the system is connected to a Gaussian theory by
small couplings. This can happen in a weakly interacting Fermi liquid, where
the higher order correlators occur at higher orders in the interaction
strength. It can also happen if the Gaussian theory represents a symmetry
broken mean field state, e.g. in a large $N$ theory of a superconductor or
magnets, where higher order connected correlators have larger powers of $1/N$.
In general, if one only has information about few body correlators, one can
use this decomposition to obtain estimates of REE. The question of specific
approximation schemes for particular situations is not discussed here. It will
be taken up in future works.

Figure 8: Examples of diagrams which do not contribute to EE, i.e. do not
survive under analytic continuation $n\to 1$. (a)-(b) Diagrams with a single
multi-particle correlator with external legs connected amongst themselves
through $\hat{v}_{0}$. (a) scales like $\sim(n-1)^{2}$ whereas (b) scales as
$\sim(n-1)^{3}$. (c) Diagram with mixed correlators scaling as $\sim(n-1)^{2}$
despite having only one instance of $\hat{v}_{0}$ . (d)-(e) Series of diagrams
with recurrent motifs which get decimated in the $n\to 1$ limit. In (d) each
successive diagram with $p$ motifis scales as atleast $\order{(n-1)^{p}}$,
whereas in (e) each successive diagram scales as atleast $\order{(n-1)^{2}}$.
Note that the first diagram in both series are topologically equivalent. It is
included twice to emphasize the series structure. (f) A given diagram can be
modified to generate new diagrams by replacing a $k$-particle connected
correlator by a $(k+1)$-particle correlator with a contracted pair of legs.
The extended diagram preserves the replica structure of connectors as in the
original but scales with one higher power of $n-1$ in the $n\to 1$ limit. Such
extensions are “irrelevant” for the analytic continuation, in the sense that
if the original diagram survived the $n\to 1$ limit, the extended diagram
would not. The diagram in (c) is such an irrelevant extension of a diagram
which has a nonzero analytic continuation, and hence has a zero contribution
itself.

##  IV Analytic continuation and von-Neumann Entropy

In the previous section we have shown a way to construct the $n$th order Rényi
entropy in terms of the multi-particle correlators of an interacting Fermionic
system. In this section we will discuss how this can be used to obtain a
construction of the von Neumann entropy (EE) $S=\lim_{n\to 1}S^{(n)}$. The
diagrammatic construction discussed so far is given for $(1-n)S^{(n)}$. Hence,
only diagrams which scale as $\sim(n-1)$ will contribute to $S$ in this
expansion; terms $\sim\order{(n-1)^{2}}$ will not survive the analytic
continuation.

Let us first consider the one-particle contribution $S_{1\mathrm{P}}$. Taking
the analytic continuation of Eq. (20), one can easily show that

$S_{1\mathrm{P}}=-\Tr[\hat{C}\ln\hat{C}+(\hat{1}-\hat{C})\ln(\hat{1}-\hat{C})].$
(30)

It is hard to formulate such a general expression for analytic continuation of
$S^{(n)}_{m\mathrm{P}}$. This is primarily because of the fact that elements
of the matrix $\mathcal{\hat{V}}_{\alpha\beta}$ have explicit $n$ dependence
as well as dependence on $\alpha-\beta$. Thus the $n$ dependence of different
diagrams, which involve summing over replica indices, have to be calculated
individually for each diagram and a priori cannot be captured by a general
formula. However, a large class of diagrams vanish when the analytic
continuation is taken and the diagrammatic expansion for $S$ has many diagrams
less than that for $S^{(n)}$. To see this, note that in the limit $n\to 1$,

$\hat{v}_{0}\sim\frac{n-1}{2}\ln\left[\frac{\hat{C}}{1-\hat{C}}\right]+\order{(n-1)^{2}}.$
(31)

This has the immediate consequence that any term involving more than one
instance of $\hat{v}_{0}$ must scale at least as $\sim(n-1)^{2}$ and hence
have vanishing contribution in the $n\to 1$ limit. These $\hat{v}_{0}$
connectors may connect the external legs on the same correlator, or the legs
from different correlators with the same replica index. As an example,
consider diagrams like those in Fig. 8(a) & (b), where the external legs of a
single multi-particle correlator are joined by the resummed connector
$\mathcal{\hat{V}}$. Such diagrams necessarily contain more than one factor of
$\hat{v}_{0}$ and are decimated under analytic continuation. Similarly one can
show that entire series of diagrams vanish under analytic continuation due to
this criterion. We depict two examples of such series in Fig. 8(d) & (e) which
have the topology of Harteee corrections and RPA diagrams of many body theory
respectively.

The leading scaling of $\hat{v}_{0}$ in the $n\to 1$ limit can also be used to
constrain possible diagrams which have non-zero contribution to EE. Given a
legitimate diagram with a $k$-particle correlator, we can generate another
equally legitimate diagram from it by replacing the chosen correlator with a
$(k+1)$-particle one with a pair of external legs self-contracted through
$\hat{v}_{0}$, as shown in Fig. 8(f). This procedure keeps the replica
structure of the connectors in the original diagram intact. However the
presence of the extra $\hat{v}_{0}$ in the extended diagram increases the
leading $(n-1)$ scaling by one power as compared to the original. If the
original diagram had a non-zero contribution to EE, the diagram born from such
an “irrelevant” extension will not contribute to EE. As an example, the
diagram in Fig. 8(c) is an irrelevant extension of the diagram in Fig. 6(a-ii)
and doesn’t contribute in the $n\to 1$ limit for EE.

We now turn to focus on diagrams which do survive in the $n\to 1$ limit. As an
example, consider the analytic continuation of the diagrams for
$S^{(n)}_{2\mathrm{P}}$ shown in Fig. 6(a). The diagrams of Fig. 6(a-i) and
(a-iii) vanish in the $n\to 1$ limit, while the first non-trivial diagram for
$S_{2\mathrm{P}}$ is given by the analytic continuation of the diagram in Fig
6(a-ii). For the evaluation of this diagram, it is more convenient to work in
the eigenbasis of the interacting correlation matrix $C(i,j)$ restricted to
the subsystem $A$, defined as $\hat{C}\ket{c_{m}}=c_{m}\ket{c_{m}}$ where
$c_{m}$ and $\ket{c_{m}}$ are respectively the eigenvalues and eigenvectors of
$\hat{C}$. For $n\to 1$, this diagram gives (see Appendix B for details)

$-\frac{1}{32}\sum_{c_{1},c_{2},c_{3},c_{4}}\frac{\ln(x)-\ln(y)}{x-y}\left|\matrixelement{c_{1},c_{2}}{\hat{G}^{(2)}_{c}}{c_{3},c_{4}}\right|^{2},$
(32)

where $x=c_{1}c_{2}(1-c_{3})(1-c_{4})$ and $y=(1-c_{1})(1-c_{2})c_{3}c_{4}$,
and $\ket{c_{1},c_{2}}=\ket{c_{1}}\otimes\ket{c_{2}}$. The arguments presented
in Appendix B can be readily adapted to analytically continue other diagrams
with the same replica connection structure, such as the diagram in Fig. 6(e).

So far we have assumed that the expression in Eq. (32) is finite. Indeed the
$(\ln x-\ln y)/(x-y)$ factor in the summand of Eq. (32) is finite for $x=y$
but diverges when either of $x,y\to 0$. It is a priori unclear if the matrix
elements of $G^{(2)}_{c}$ are sufficiently small in this regime to result in a
finite answer for this diagram. In case they do not converge, the program of
diagram-by-diagram analytic continuation falls under suspicion and some
appropriate subset of diagrams in $S^{(n)}_{m\mathrm{P}}$ might have to be
resummed first and then analytically continued. Such considerations and its
implications for entanglement are left as topics of future work.

Figure 9:  Feynman diagrams for Rényi entanglement entropy using propagators
in the replicated theory. (a): Single particle propagators $\tilde{G}$ in the
non-interacting theory with the replica coupling term
$\mathcal{S}_{{\text{ent}}}$ included. We use double lines for these
propagators and single lines for propagators in a single replica ($G$).
$\alpha$ $\beta$ are the replica indices. Note that $\tilde{G}^{\rm\,aa}$ is
finite in this theory, in contrast to a single replica Keldysh theory. (b):
The Dyson series for the $\tilde{G}$ in terms of $G$ and the entanglement
vertex ${\cal J}$. The structure of the Dyson equation is similar for all
Keldysh components. Hence a single diagram with dotted lines is used to show
this. The specific Keldysh components can easily be obtained from this by
replacing the dotted external lines with a solid (symmetric) or a dashed
(antisymmetric) line. Note that for $\tilde{G}^{\rm\,aa}$, the first term
$\expectationvalue*{\psi_{\text{a}}\bar{\psi}_{\text{a}}}_{0}=0$. (c):
Diagrams showing the first order correction due to interaction vertices. The
diagrams are drawn with double dotted lines to show their structural
similarity with standard free energy diagrams in equilibrium field theory
(Direct and Exchange terms). However, the propagators here carry Keldysh
indices and only particular combinations are allowed by interaction vertices.
This is explicitly shown in (c) for the first order diagrams. Note that lines
emanating from an interaction vertex must have the same replica index. (d):
Feynman diagrams corresponding to second order corrections in the interaction
strength. Only the structure of diagrams is shown with dotted lines for the
sake of brevity.

##  V Beyond Standard Keldysh Field Theory

In section III.1, we have shown that the $n$th Rényi entropy of a system of
interacting Fermions is the Keldysh free energy of n replicas governed by an
action, $\mathcal{S}_{0}+\mathcal{S}_{\text{int}}+\mathcal{S}_{{\text{ent}}}$,
where $\mathcal{S}_{0}$ is the non-interacting (quadratic) action for
decoupled replicas, $\mathcal{S}_{\text{int}}$ is the interacting part of the
action for decoupled replicas and $\mathcal{S}_{{\text{ent}}}$ is the
quadratic action which couples the different replicas and generates
entanglement. In section III.4, we expanded the free energy around
$\mathcal{S}_{0}+\mathcal{S}_{\text{int}}$ and connected entanglement entropy
with interacting connected correlators in the usual single replica theory.
However, one can also think of solving
$\mathcal{S}_{0}+\mathcal{S}_{{\text{ent}}}$ exactly since this is still a
quadratic action, and use the resulting propagators to expand in
$\mathcal{S}_{\text{int}}$. Note that this is a regrouping of the terms worked
out earlier. In this section we will work out this expansion and resultant
diagrammatics.

The earlier expansion around $\mathcal{S}_{0}+\mathcal{S}_{\text{int}}$ had
the advantage that it used propagators and correlators which can be related to
observable correlations in the system. The one-particle propagators, for
example, can be grouped into retarded, advanced or Keldysh propagators with
well known properties and relations to measured quantities. The current
expansion will work with propagators which are matrices in replica space and
cannot be immediately related to observables. They will not follow the clear
demarcation into retarded, advanced or Keldysh propagators, at least they will
not inherit their well known properties. However, there are two advantages to
the current expansion: (i) the organization of diagrams is simpler, since the
replica space propagators already incorporate resummations due to
$\mathcal{S}_{{\text{ent}}}$ and (ii) this expansion will tie up vertex
functions rather than correlation functions. This can be of relevance if one
is interested in calculating “effective entanglement actions” under various
circumstances with the “effective entanglement action” playing a similar role
as that of modular Hamiltonians [76] used to study entanglement entropy. Since
this maintains the language of an effective action, this is also the natural
language to introduce techniques like auxiliary fields, saddle points etc.
This is also the natural language to think about renormalization group in this
context.

Let us first focus on $\mathcal{S}_{0}+\mathcal{S}_{{\text{ent}}}$. In this
case both the fields and the propagators carry space-time, Keldysh and replica
indices (one can add spin or other quantum numbers as well). Using
$\mathcal{S}_{{\text{ent}}}$ as a self energy correction, one can solve the
Dyson series exactly (see Fig. 9(b)), and define the propagators

$\displaystyle\tilde{G}^{\rm\,sa}_{\alpha\beta}(r,t;r^{\prime},t^{\prime})$
$\displaystyle=\delta_{\alpha\beta}G^{R}_{0}(r,t;r^{\prime},t^{\prime})+\iota\sum_{i,j\in
A}G^{K}_{0}(r,t;i,t_{o})\,{\cal
V}^{0}_{\alpha\beta}(i,j)\,G^{R}_{0}(j,t_{o};r^{\prime},t^{\prime}),$ (33)
$\displaystyle\tilde{G}^{\rm\,as}_{\alpha\beta}(r,t;r^{\prime},t^{\prime})$
$\displaystyle=\delta_{\alpha\beta}G^{A}_{0}(r,t;r^{\prime},t^{\prime})+\iota\,\sum_{i,j\in
A}G^{A}_{0}(r,t;i,t_{o})\,{\cal
V}^{0}_{\alpha\beta}(i,j)\,G^{K}_{0}(j,t_{o};r^{\prime},t^{\prime}),$
$\displaystyle\tilde{G}^{\rm\,ss}_{\alpha\beta}(r,t;r^{\prime},t^{\prime})$
$\displaystyle=\delta_{\alpha\beta}G^{K}_{0}(r,t;r^{\prime},t^{\prime})+\iota\,\sum_{i,j\in
A}G^{K}_{0}(r,t;i,t_{o})\,{\cal
V}^{0}_{\alpha\beta}(i,j)\,G^{K}_{0}(j,t_{o};r^{\prime},t^{\prime}),$
$\displaystyle\tilde{G}^{\rm\,aa}_{\alpha\beta}(r,t;r^{\prime},t^{\prime})$
$\displaystyle=\iota\,\sum_{i,j\in A}G^{A}_{0}(r,t;i,t_{o})\,{\cal
V}^{0}_{\alpha\beta}(i,j)\,G^{R}_{0}(j,t_{o};r^{\prime},t^{\prime}),$

where $G^{R}_{0}$, $G^{A}_{0}$ and $G^{K}_{0}$ are the retarded, advanced, and
Keldysh propagators of the single replica Keldysh field theory without
interactions. $\mathcal{\hat{V}}^{0}$ has the same structure as in Eq. (22)
but with the non-interacting correlation matrix determining the blocks in Eq.
(23). We denote this difference by using concentric empty circles to represent
$\cal\hat{V}_{0}$ in contrast to the cross-hatched concentric circles used for
$\cal\hat{V}$. These propagators and the Dyson series for them is shown in
Fig. 9(a) and (b). Although these propagators are not directly related to
observables, they can be constructed out of the standard one-particle
propagators. We note that in this case $\tilde{G}^{\rm\,sa}$ is not a retarded
propagator, nor is $\tilde{G}^{\rm\,as}$ an advanced propagator, although the
relation $\left(\tilde{G}^{\rm\,as}\right)^{\dagger}=\tilde{G}^{\rm\,sa}$
still holds. Further, unlike the standard Keldysh theory,
$\tilde{G}^{\rm\,aa}$ is non-zero due to the presence of
$\mathcal{S}_{{\text{ent}}}$ in the action. It is easy to show that
$G^{\rm\,ss}$ and $G^{\rm\,aa}$ are both anti-Hermitian in this replicated
theory. These propagators are represented by a double line in the diagrams.

One can now work out the diagrammatic expansion of the free energy in terms of
the original interaction vertices in $\mathcal{S}_{{\text{int}}}$ and the new
propagators in the usual way:(a) draw all topologically distinct connected
diagrams. (b) For each interaction vertex, put $\iota U(r,r^{\prime})/2$,
where $U(r,r^{\prime})$ is the matrix element of the interaction, for each
propagator put a factor of $\iota\tilde{G}$. (c) multiply by symmetry factor
and $(-1)^{n_{L}}$, where $n_{L}$ is the number of Fermion loops (d) Sum over
all internal indices (over all space and time, _not_ only in the subsystem).
Three important things need to be kept in mind: (i) The fields coming out of
any interaction vertex belongs to the same replica, (ii) The propagators
$\tilde{G}(r,t;r^{\prime},t^{\prime})$ are supported over the entire system
and generically lack translational invariance due to the presence of the
entanglement cut, and (iii) Certain diagrams which vanish in standard Keldysh
field theory give finite contribution, as $\tilde{G}^{\rm\,aa}$ are finite in
this theory. The first order correction to $S^{n}$ is shown in Fig. 9(c). The
first of these diagrams (the direct contribution) evaluates to

$\delta
S^{(n)}=(-\iota)\,\sum_{\alpha}\,\int\differential{t}\sum_{r,r^{\prime}}\left[\tilde{G}^{\rm\,ss}_{\alpha\alpha}(r,t;r,t)+\tilde{G}^{\rm\,aa}_{\alpha\alpha}(r,t;r,t)\right]U(r,r^{\prime})\left[\tilde{G}^{\rm\,as}_{\alpha\alpha}(r^{\prime},t;r^{\prime},t)+\tilde{G}^{\rm\,sa}_{\alpha\alpha}(r^{\prime},t;r^{\prime},t)\right],$
(34)

while the second diagram (the exchange contribution) is given by

$\delta
S^{(n)}=(\iota)\,\sum_{\alpha}\,\int\differential{t}\sum_{r,r^{\prime}}\left[\tilde{G}^{\rm\,ss}_{\alpha\alpha}(r,t;r^{\prime},t)+\tilde{G}^{\rm\,aa}_{\alpha\alpha}(r,t;r^{\prime},t)\right]U(r,r^{\prime})\left[\tilde{G}^{\rm\,as}_{\alpha\alpha}(r^{\prime},t;r,t)+\tilde{G}^{\rm\,sa}_{\alpha\alpha}(r^{\prime},t;r,t)\right].$
(35)

One can similarly evaluate other diagrams, some of which are shown in Fig.
9(d). One can thus reconstruct the diagrammatic series in terms of the replica
propagators and the original interaction vertices. We note that while the
propagators cannot be related to anything physical, the number of diagrams
reduce considerably in this way of grouping the terms. However, in absence of
physical correlators, one needs to construct useful approximate truncations of
the diagrams. While a perturbation theory immediately provides a truncation,
one should be more careful about constructing non-perturbative approximations
in this non-standard Keldysh field theory.

##  VI Conclusions

In this paper, we have formulated a new way of calculating entanglement
entropy of a generic interacting Fermionic system from the knowledge of
correlation functions in the subsystem. Using a Wigner function based method,
coupled with Schwinger Keldysh field theory, we show that the $n$th Rényi
entropy $S^{(n)}$ is the Keldysh free energy of a theory of $n$ replicas which
are coupled by inter-replica currents, which exist in the subsystem. These
currents are local in space-time, i.e. they are turned on between same degrees
of freedom at the time of measurement of the entanglement theory. These
currents have a structure which is not allowed in usual Keldysh field theory
with a single replica, and hence we do not have an equivalent formulation in
usual single-contour field theories. These currents implement the boundary
condition matching required in standard replica formulation of entanglement
theory.

Starting from this description of EE as a free energy in presence of inter-
replica currents, we show that the EE can be written as a sum of terms which
require knowledge of progressively higher order connected correlators in the
system. These correlators are usual field theoretic observables, calculated in
a standard ffield theory with no replicas and correspond to measureables in
the system. We provide an analytic formula for the single-particle
contribution to entanglement; we also provide a diagrammatic construction for
the contribution of higher particle correlators to EE. We thus relate the
correlation functions in a system to its . These constructions are agnostic to
how the correlators are calculated, and hence forms a universal basis for
further approximate calculations. One can also use experimentally measured
correlation functions in these formulae to calculate entanglement. They
provide estimates for entanglement when only a few order correlation functions
are known. This reduces the complexity of calculating entanglement entropies
vis a vis direct methods which requires the knowledge of the full quantum many
body state.

We have considered how one can implement the analytic continuation required to
obtain the von-Neumann entropy $S$ from the Rényi entropy. We obtain analytic
formula for the single-particle contribution to $S$ in terms of interacting
one-particle distributions. We show that a large class of diagrams for multi-
particle contributions vanish under the analytic continuation. We calculate an
analytic formula for the first non-trivial diagram for two-particle
contribution to $S$.

Technically we achieve this in two different ways,(a) by constructing an
expansion around an interacting theory with independent replicas. This
provides a relation between observables and entanglement and is useful for
gaining insights. (b) by constructing an expansion around a non-interacting
theory of coupled replicas. While this method is less insightful, it provides
a simpler construction of diagrams, since large class of individual diagrams
in the first method are re-summed into single objects in this case. While the
first method provides relation between correlations and entanglement, the
second method provides a relation between vertex functions and entanglement,
and may be more suitable for treatments like renormalization group analysis.

We note that what we have done here is akin to setting up a general
diagrammatic expansion and writing down the Feynman diagrams and Feynman rules
for the calculation of entanglement entropy. Calculations for particular
systems would require further approximations. One can ask the following
question: In general correlations will be calculated using approximation
methods. One would have to further truncate/approximate using a subset of the
diagrams we have drawn here. For simple approximations like perturbation
theory or large $N$ approximations, it is clear how such a truncation will
happen. However for non-perturbative approximations, it may turn out that
certain approximations for correlators are compatible with certain subsets of
these diagrams. Is there a general rule for such compatibility? We do not take
up this question in this work, but leave it as a general question to be
answered in future works.

###### Acknowledgements.

The authors acknowledge useful discussions with Subir Sachdev, Mohit Randeria,
Gautam Mandal, Sandip Trivedi, Kedar Damle and Onkar Parrikkar. SM and RS
acknowledge support of the Department of Atomic Energy, Government of India,
for support under Project Identification No. RTI 4002.

##  Appendix A Equal time Keldysh correlators in terms of operators

In this section we provide the explicit form of the many-particle equal time
Keldysh correlators in terms of electron operators. These will be useful when
connecting our formalism to numerical simulations or experiments measuring
such correlators.

Like in usual quantum field theory, the Schwinger Keldysh partition function
in presence of sources, $\mathcal{Z}[J_{\mathrm{s,a}},\bar{J}_{\mathrm{s,a}}]$
is the generating function of $k$-particle correlators, and $\ln\mathcal{Z}$
is the generating function for connected correlators. In particular, to get
the $k$-particle Keldysh correlator involving all symmetric fields, we take
derivatives of $\ln\mathcal{Z}$ w.r.to the antisymmetric sources,

$\displaystyle\expectationvalue{\psi_{\mathrm{s}}(1)\cdots\psi_{\mathrm{s}}(k)\bar{\psi}_{\mathrm{s}}(k^{\prime})\cdots\bar{\psi}_{\mathrm{s}}(1^{\prime})}_{c}$
(36)
$\displaystyle=\left.\frac{\delta^{2k}\ln\mathcal{Z}[J_{\mathrm{a,s}},\bar{J}_{\mathrm{a,s}}]}{\delta\bar{J}_{\mathrm{a}}(1)\cdots\delta\bar{J}_{\mathrm{a}}(k)\,\delta
J_{\mathrm{a}}(k^{\prime})\cdots\delta
J_{\mathrm{a}}(1^{\prime})}\right|_{J_{\mathrm{a,s}},\bar{J}_{\mathrm{a,s}}=0},$

where the field arguments are shorthand for coordinates,
$(1)\equiv(i_{1},t_{1})$, $(1^{\prime})\equiv(i_{1}^{\prime},t_{1}^{\prime})$
etc. We now use the fact that the Wigner Characteristic function
$\chi({\bm{\bar{\zeta}}},{\bm{\zeta}};t_{o})$ is a Keldysh partition function
in the presence of instantaneous sources[37, 38]. In case of the equal time
Keldysh correlator, Eq. (11) allows us to replace $\cal Z$ in Eq. (36) with
$\chi({\bm{\bar{\zeta}}},{\bm{\zeta}};t_{o})$, and employing Eq. (12), the
functional derivatives w.r.to the sources $J_{\text{a}}$ simplify to partial
derivatives w.r.to the Grassman variables ${\bm{\zeta}},{\bm{\bar{\zeta}}}$,

$\displaystyle\iota^{k}G^{(k)}_{c}(i_{1}\cdot\cdot\,i_{k}|i^{\prime}_{1}\cdot\cdot\,i^{\prime}_{k},t_{o})=2^{k}\\!\left.\frac{\partial^{2k}\ln\chi({\bm{\bar{\zeta}}},{\bm{\zeta}};t_{o})}{\partial\bar{\zeta}_{1}\cdot\cdot\,\partial\bar{\zeta}_{k}\,\partial\zeta_{k^{\prime}}\cdot\cdot\,\partial\zeta_{1^{\prime}}}\right|_{\begin{subarray}{c}{\bm{\zeta}}=0\\\
{\bm{\bar{\zeta}}}=0\end{subarray}}.$ (37)

Here $\zeta_{k}$ is shorthand notation for the Grassman variable at site
$i_{k}$, $\zeta_{i_{k}}$. For the rest of this section we suppress the time
label $t_{o}$ for brevity. To illustrate how this expression with partial
derivatives simplifies it is convenient to first consider the full correlation
function $G^{(k)}$ generated from $\chi_{A}$. The resultant expression’s
connected piece will then reproduce the connected correlation function
$G^{(k)}_{c}$.

From the definition of $\chi$ in Eq. (6) as an expectation value of the
Fermionic displacement operator $\hat{D}({\bm{\bar{\zeta}}},{\bm{\zeta}})$,
the correlation function $G^{(k)}$ can be written as

$\displaystyle\iota^{k}G^{(k)}(i_{1}\cdot\cdot\,i_{k}|i^{\prime}_{1}\cdot\cdot\,i^{\prime}_{k})=2^{k}\\!\left.\expectationvalue{\frac{\partial^{2k}\hat{D}({\bm{\bar{\zeta}}},{\bm{\zeta}})}{\partial\bar{\zeta}_{1}\cdot\cdot\,\partial\bar{\zeta}_{k}\,\partial\zeta_{k^{\prime}}\cdot\cdot\,\partial\zeta_{1^{\prime}}}}\right|_{\begin{subarray}{c}{\bm{\zeta}}=0\\\
{\bm{\bar{\zeta}}}=0\end{subarray}}.$ (38)

We can use the anti-commutation relations amongst the Grassman variables and
Fermion operators to get a simplified form for $\hat{D}$ [38],

$\displaystyle\hat{D}({\bm{\bar{\zeta}}},{\bm{\zeta}})$
$\displaystyle\equiv\,\mathrm{e}^{\sum_{i}\hat{c}^{\dagger}_{i}\zeta_{i}-\bar{\zeta}_{i}\hat{c}^{\vphantom{\dagger}}_{i}}$
$\displaystyle=\prod_{i}\left[1+\hat{c}^{\dagger}_{i}\zeta_{i}-\bar{\zeta}_{i}\hat{c}^{\vphantom{\dagger}}_{i}+\zeta_{i}\bar{\zeta}_{i}(\hat{c}^{\vphantom{\dagger}}_{i}\hat{c}^{\dagger}_{i}-\tfrac{1}{2})\right].$
(39)

It is then immediate to read off the partial derivatives,

$\begin{gathered}\left.\frac{\partial\hat{D}}{\partial\zeta_{m}}\right|_{{\bm{\zeta}},{\bm{\bar{\zeta}}}=0}=-\hat{c}^{\dagger}_{i_{m}},\quad\left.\frac{\partial\hat{D}}{\partial\bar{\zeta}_{m}}\right|_{{\bm{\zeta}},{\bm{\bar{\zeta}}}=0}=-\hat{c}^{\vphantom{\dagger}}_{i_{m}},\\\\[2.0pt]
{\text{and}}\quad\left.\frac{\partial^{2}\hat{D}}{\partial\bar{\zeta}_{m}\partial\zeta_{m}}\right|_{{\bm{\zeta}},{\bm{\bar{\zeta}}}=0}=\hat{c}^{\vphantom{\dagger}}_{i_{m}}\hat{c}^{\dagger}_{i_{m}}-\tfrac{1}{2}.\end{gathered}$
(40)

In the case where none of the $(m)$ and $(m^{\prime})$ coordinates coincide,
it is easy to see that the $k$-particle correlator is,

$\iota^{k}G^{(k)}(i_{1}\cdots i_{k}|i^{\prime}_{1}\cdots
i^{\prime}_{k})=2^{k}\expectationvalue{\hat{c}^{\vphantom{\dagger}}_{i\vphantom{{}^{\prime}}_{1}}\cdots\hat{c}^{\vphantom{\dagger}}_{i\vphantom{{}^{\prime}}_{k}}\hat{c}^{\dagger}_{i^{\prime}_{k}}\cdots\hat{c}^{\dagger}_{i^{\prime}_{1}}},$
(41)

which in turn implies that the connected correlator is given by

$\iota^{k}G^{(k)}_{c}(i_{1}\cdots i_{k}|i^{\prime}_{1}\cdots
i^{\prime}_{k})=(-2)^{k}\expectationvalue{\hat{c}^{\dagger}_{i^{\prime}_{k}}\cdots\hat{c}^{\dagger}_{i^{\prime}_{1}}\hat{c}^{\vphantom{\dagger}}_{i\vphantom{{}^{\prime}}_{1}}\cdots\hat{c}^{\vphantom{\dagger}}_{i\vphantom{{}^{\prime}}_{k}}}_{c}.$
(42)

The extra $(-1)^{k}$ results from normal ordering the operators. In case of
coincident coordinates, the expression for the $k$-particle correlator picks
up extra terms with lower order correlators ($k-1$, $k-2$, etc.). However
these extra pieces get cancelled in the subtractions to get the connected
$k$-particle correlator, making Eq. (42) valid for arbitrary coordinates.

##  Appendix B Analytic continuation of two $G^{(2)}_{c}$ diagram

The objective of this section is to work out the analytic continuation of a
particular diagram given in Fig. 6(a-ii), henceforth referred to as
$\mathfrak{D}$.

For the following it is convenient ot define $\\{\ket{i}\\}$ as a complete set
of states in $A$, with $\ket{i}$ localised on degree of freedom $i$. We can
then make the identifications

$\matrixelement{i}{\mathcal{\hat{V}}_{\alpha\beta}}{j}\equiv\mathcal{V}_{\alpha\beta}(i,j),\quad\matrixelement{ij}{\hat{G}^{(2)}_{c}}{kl}\equiv
G^{(2)}_{c}(i,j|k,l;t_{o}).$

where $\ket{ij}=\ket{i}\otimes\ket{j}$. It is then immediate to rewrite Eq.
(25) in more compact notation as

$\mathfrak{D}=\frac{1}{2}\sum_{\alpha,\beta=1}^{n}\Tr_{A}[\mathcal{\hat{V}}_{\alpha\beta}\otimes\mathcal{\hat{V}}_{\alpha\beta}\hat{G}^{(2)}_{c}\,\mathcal{\hat{V}}_{\beta\alpha}\otimes\mathcal{\hat{V}}_{\beta\alpha}\hat{G}^{(2)}_{c}],$
(43)

where $\Tr_{A}$ is now understood to be over two copies of $A$. From Eq. (22)
it is clear that the matrix
$[\mathcal{\hat{V}}_{\alpha\beta}\otimes\mathcal{\hat{V}}_{\alpha\beta}]$ is
block circulant in replica indices, and hence the sum over the same in Eq.
(43) can be simplified to read

$\displaystyle\mathfrak{D}$
$\displaystyle=\frac{n}{2}\Tr_{A}[\hat{v}_{0}^{\otimes
2}\,\hat{G}^{(2)}_{c}\,\hat{v}_{0}^{\otimes 2}\hat{G}^{(2)}_{c}]$ (44)
$\displaystyle\hphantom{=}+\frac{n}{2}\sum_{k=1}^{n-1}\Tr_{A}[\hat{v}_{k}^{\otimes
2}\,\hat{G}^{(2)}_{c}\,\hat{v}_{n-k}^{\otimes 2}\hat{G}^{(2)}_{c}],$

where $\hat{v}_{k}$ are as defined in Eq. (23) and $\hat{v}_{k}^{\otimes
2}=\hat{v}_{k}\otimes\hat{v}_{k}$. We immediately note that the first term in
the sum will not contribute in the $n\to 1$ limit since $\hat{v}_{0}\sim(n-1)$
from Eq. (31). Evaluating the rest of the sum is not a priori straightforward
as $\hat{v}_{k}^{\otimes 2}$ and $\hat{G}^{(2)}_{c}$ do not commute in
general. It is convenient to switch to the basis in which $\hat{v}_{k}$ is
diagonal, namely the eigenbasis of the $\hat{C}$ operator (correlation matrix
restricted to the subsystem $A$), $\\{\ket{c}\\}$ defined as
$\hat{C}\ket{c}=c\ket{c}$. In this basis, $\hat{v}_{k}^{\otimes 2}$ takes the
form,

$\hat{v}_{k}^{\otimes
2}=\sum_{c_{1},c_{2}}\frac{(c_{1}c_{2})^{n-k-1}[(1-c_{1})(1-c_{2})]^{k-1}}{4[c_{1}^{n}+(1-c_{1})^{n}][c_{2}^{n}+(1-c_{2})^{n}]}\outerproduct{c_{1}c_{2}}{c_{1}c_{2}}.$
(45)

Here $c_{1}$ and $c_{2}$ are eigenvalues of $\hat{C}$, each running over the
entire spectrum of the same. Substituting this form into Eq. (44) and ignoring
the leading piece with $\hat{v}_{0}$, we get

$\mathfrak{D}=\frac{n}{32}\sum_{\begin{subarray}{c}c_{1},c_{2}\\\
c_{3},c_{4}\end{subarray}}\frac{\matrixelement{c_{1}c_{2}}{\hat{G}^{(2)}_{c}}{c_{3}c_{4}}\matrixelement{c_{3}c_{4}}{\hat{G}^{(2)}_{c}}{c_{1}c_{2}}}{\prod_{j=1}^{4}[c_{j}^{n}+(1-c_{j})^{n}]}\\\
\sum_{k=1}^{n-1}x^{n-k-1}y^{k-1},$ (46)

where we have defined

$x=c_{1}\,c_{2}\,(1-c_{3})(1-c_{4}),\quad y=(1-c_{1})(1-c_{2})\,c_{3}\,c_{4}.$
(47)

The last sum over replica blocks can now be done trivially,

$\sum_{k=1}^{n-1}x^{n-k-1}y^{k-1}=\frac{x^{n-1}-y^{n-1}}{x-y}.$

To analytically continue the contribution of this diagram to $S$, we look at
$\lim_{n\to 1}\mathfrak{D}/(1-n)$, which gives back the result quoted in Eq.
(32).

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