# Charge correlator expansion for free fermion negativity

Yang-Yang Tang<EMAIL_ADDRESS>Institute for Advanced Study,
Tsinghua University, Beijing 100084, China

###### Abstract

Logarithmic negativity is a widely used entanglement measure in quantum
information theories, which can also be efficiently computed in quantum many-
body systems by replica trick or by relating to correlation matrices. In this
paper, we demonstrate that in free-fermion systems with conserved charge,
Rényi and logarithmic negativity can be expanded by connected charge
correlators, analogous to the case for entanglement entropy in the context of
full counting statistics (FCS). We confirm the rapid convergence of this
expansion in random all-connected Hamiltonian through numerical verification,
especially for systems with only local hopping. We find that the replica trick
that get logarithmic negativity from the limit of Rényi negativity is valid in
this method only for translational invariant systems. Using this expansion, we
analyze the scaling behavior of negativity in extensive free-fermion systems.
In particular, in 1+1 dimensional free-fermion systems, we observe that the
scaling behavior of negativity from our expansion is consistent with known
results from the method with Toeplitz matrix. These findings provide insights
into the entanglement properties of free-fermion systems, and demonstrate the
efficacy of the expansion approach in studying entanglement measures.

††preprint: APS/123-QED

##  I Introduction

Entanglement, which refers to the non-separability of multipartite quantum
states [1], is widely regarded as the most non-classical phenomenon in quantum
mechanics [2]. Due to the significant utility of entangled states in quantum
information theories and experiments, various entanglement detections and
measures have been proposed to detect and quantify the entanglement of quantum
states [3, 4]. For pure states, the well-known von Neumann entropy uniquely
captures their entanglement [4]. However, for mixed states, the condition is
complicated, as entanglement entropy contains both classical and quantum
correlations. It is widely believed that there is no unique entanglement
measure for mixed states, and therefore various entanglement measures are
defined for different purposes [4].

Among the various entanglement measures, the (logarithmic) negativity, also
known as partial transpose (PT) negativity [5], is the most commonly employed
one in the study of quantum many-body systems. This measure is particularly
useful as it can be computed efficiently compared to other measures that rely
on variational expressions [4]. The negativity measure is inspired by the
positive partial transpose (PPT) criterion [6, 7], which claims that for
separable (non-entangled) states $\rho$, the partial transposition of the
density matrix $\rho^{T_{1}}$ have no negative eigenvalues. Negativity
quantifies how negative the eigenvalues of $\rho^{T_{1}}$ are, and has been
shown to bound the utility of the state $\rho$ in quantum teleportation [5].
Moreover, the logarithmic negativity is additive for product states and thus
is extensively used in quantum many-body systems [8, 9]. It has also been
proven to bound the entanglement that can be asymptotically distilled from
$\rho$ [5].

Fermionic systems possess an alternative definition for negativity, known as
partial time-reversal (TR) negativity, which exhibits superior properties
compared to the fermionic PT negativity based on the corresponding bosonic
density matrix [10, 11]. The key advantage of partial TR negativity is that
the partial TR of a fermionic Gaussian state remains Gaussian. This feature is
particularly useful for free-fermion systems, as the partial TR negativity for
the ground state or thermal equilibrium state at finite temperature can then
be efficiently computed in terms of two-point correlator matrices [10, 12].
Therefore, partial TR negativity offers a more efficient and accurate method
for studying negativity in fermionic systems.

Negativity can also be computed by replica trick, where the logarithmic
negativity $\mathcal{E}$ is the replica limit $n_{e}\rightarrow 1$ of Rényi
negativity $\mathcal{E}_{n_{e}}$ [8, 9, 10, 12], analogous to the case for von
Neumann entropy and Rényi entropy. For certain geometry and dimension, both
bosonic [8, 9] and fermionic [10, 12] negativity $\mathcal{E}_{n_{e}}$ can be
computed by CFT methods, and therefore the scaling behavior of logarithmic
negativity $\mathcal{E}$ can also be analyzed.

Alternatively, in free-fermion system with conserved charge, we can expand
these computable entanglement measures into connected charge correlators by
cumulant expansion. It can be proved that $n$-Rényi entropy can be expanded by
the charge (partial number) fluctuation as

$\displaystyle S_{A}^{(n)}=-\frac{1}{1-n}\sum_{\begin{subarray}{c}M=2\\\
\mathrm{even}\ M\end{subarray}}^{\infty}$ $\displaystyle
2\zeta\left(-M,\frac{n+1}{2}\right)$ (1)
$\displaystyle\times\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}$

(see Sec. II.1 for derivation), where $\zeta(s,a)$ is the generalization of
the Riemann zeta function known as the Hurwitz zeta function, and
$\braket{Q_{A}^{M}}_{c}$ is the $M$-th order connected charge correlator in
region $A$. Taking the replica limit $n\rightarrow 1$ we get the cumulant
expansion for von Neumann entropy [13]

$\displaystyle S_{A}=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2\zeta(M)\braket{Q_{A}^{M}}_{c},$ (2)

where $\zeta(s)=\zeta(s,1)$ is the Riemann zeta function. This result was
equivalently expressed in the context of full counting statistics (FCS) [14,
15, 16] as

$\displaystyle S_{A}=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}(2\pi)^{M}\frac{|B_{M}|}{M!}\braket{Q_{A}^{M}}_{c},$
(3)

where $B_{M}$ are Bernoulli numbers.

Recently, it has been demonstrated [13, 17] that in $D$-dimensional free-
fermion systems with Fermi sea, which requires conserved $U(1)$ charge,
translational symmetry and zero temperature, the $(D+1)$-th order density
correlator in momentum space is proportional to the Euler characteristic
$\chi_{F}$ of the Fermi sea for small $\bm{q}$

$\int\frac{\mathrm{d}^{D}\bm{q}_{D+1}}{(2\pi)^{D}}\braket{\rho_{\bm{q}_{1}}\cdots\rho_{\bm{q}_{D+1}}}_{c}=\frac{\chi_{F}}{(2\pi)^{D}}|\det\mathbb{Q}|,$
(4)

where $\rho_{\bm{q}}=\int{dx}\rho(\bm{x})e^{-i\bm{q}\cdot\bm{x}}$ is the
charge density operator in momentum space, and $\mathbb{Q}$ is the matrix
composed of the momenta $\bm{q}_{1},\cdots,\bm{q}_{D}$. $\chi_{F}$ is a
topological invariant only change at Lifshitz transition. Combining (2) and
(4), it is proved [13] that the multipartite mutual information
$\mathcal{I}_{D+1}$ (topological entanglement entropy [18, 19]) is
proportional to $\chi_{F}$ of Fermi sea. This proposes a way to probe the
topology of Fermi sea both theoretically [13, 17] and experimentally [20, 21,
22], and further expands our understanding of the entanglement patterns
related to topology and phase transitions in quantum many-body systems [23,
24, 25, 26].

Inspired by this, we derive the charge correlator expansion for partial TR
negativity [10]

$\mathcal{E}=\log\mathrm{Tr}\sqrt{\rho^{R_{A}}\rho^{R_{A}\dagger}}.$ (5)

in this paper. We briefly summarize the results here and leave the definition
and derivation in Sec. II.2. For free-fermion systems with conserved $U(1)$
charge, the cumulant expansion for Rényi partial TR negativity [10] with even
$n_{e}$ defined as

$\mathcal{E}_{n_{e}}=\log\mathrm{Tr}(\rho^{R_{A}}\rho^{R_{A}\dagger})^{n_{e}/2}$
(6)

is

$\displaystyle\mathcal{E}_{n_{e}}=$
$\displaystyle\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}\frac{1}{M!}(\pi i)^{M}$ (7)
$\displaystyle\times\left[\sum_{p=-\frac{n_{e}-1}{2}}^{-1/2}\left<{\left(-\frac{2p}{n_{e}}Q_{A}+\left(\frac{2p}{n_{e}}+1\right)Q_{B}\right)^{M}}\right>_{c}\right.$
$\displaystyle\left.+\sum_{p=\frac{1}{2}}^{\frac{n_{e}-1}{2}}\left<{\left(-\frac{2p}{n_{e}}Q_{A}+\left(\frac{2p}{n_{e}}-1\right)Q_{B}\right)^{M}}\right>_{c}\right].$

Written the expansion by orders of $M$ as

$\displaystyle\mathcal{E}_{n_{e}}=\sum_{\begin{subarray}{c}M=2\\\
\mathrm{even}\ M\end{subarray}}^{\infty}\mathcal{E}_{n_{e}}^{(M)},$ (8)

by direct calculation we get the first few terms

$\displaystyle\mathcal{E}_{n_{e}}^{(2)}=-\pi^{2}$
$\displaystyle\left[\frac{n_{e}^{2}-1}{6n_{e}}(\braket{Q_{A}^{2}}_{c}+\braket{Q_{B}^{2}}_{c})\right.$
(9)
$\displaystyle\left.+\frac{n_{e}^{2}+2}{6n_{e}}\braket{Q_{A}Q_{B}}_{c}\right],$
$\displaystyle\mathcal{E}_{n_{e}}^{(4)}=\pi^{4}$
$\displaystyle\left[\frac{3n_{e}^{4}-10n_{e}^{2}+7}{360n_{e}^{3}}(\braket{Q_{A}^{4}}_{c}+\braket{Q_{B}^{4}}_{c})\right.$
(10)
$\displaystyle+\frac{3n_{e}^{4}+10n_{e}^{2}-28}{360n_{e}^{3}}(\braket{Q_{A}^{3}Q_{B}}_{c}+\braket{Q_{A}Q_{B}^{3}}_{c})$
$\displaystyle\left.+\frac{n_{e}^{4}+14}{120n_{e}^{3}}\braket{Q_{A}^{2}Q_{B}^{2}}_{c}\right]$

and similarly for higher orders. We find that the convergence of the expansion
depends significantly on the choice of the range of the replica momentum $p$
in cumulant expansion (see Sec. II.1 and II.2 for details). There is a unique
choice for fermionic entropy or negativity to ensure the best convergence of
the expansion, while for bosonic systems there seems to be no natural way to
achieve this. Therefore we suppose that the connected correlator expansion for
computable entanglement measures is limited to fermionic systems.

In Sec. III, we numerically check the validity of this expansion and the
quality of convergence on thermal equilibrium states of a 100-fermion system
with random free Hamiltonian and conserved charge. The fermionic negativity
and charge correlators are both calculated by the relation with the two-point
correlator matrices for Gaussian states. We check both Hamiltonian with all-
connected hopping and local hopping to study the effect of locality on the
convergence. The convergence for Rényi negativity is generally rapid even for
all-connected free-fermion Hamiltonian, and is especially accurate when there
is only local hopping terms, which is the normal case for extensive systems.
Therefore this expansion can be used to analyze the scaling behavior of Rényi
negativity in high-dimensional free-fermion systems, especially with local
Hamiltonian.

In the researches about negativity [8, 9], it is usually assumed that by
taking the replica limit $n_{e}\rightarrow 1$ while keeping $n_{e}$ even we
would get the logarithmic negativity. Taking this limit in (9) and (10) we get

$\displaystyle\mathcal{E}_{n_{e}\rightarrow
1}^{(2)}=-\frac{\pi^{2}}{2}\braket{Q_{A}Q_{B}}_{c}$ (11)

and

$\displaystyle\mathcal{E}_{n_{e}\rightarrow 1}^{(4)}=\pi^{4}$
$\displaystyle\left[-\frac{1}{24}(\braket{Q_{A}^{3}Q_{B}}_{c}+\braket{Q_{A}Q_{B}^{3}}_{c})\right.$
(12) $\displaystyle\left.+\frac{1}{8}\braket{Q_{A}^{2}Q_{B}^{2}}_{c}\right].$

However, taking the limit $n_{e}\rightarrow 1$ is mathematically ambiguous, as
arbitrary periodic functions can be included while doing analytic continuation
from integers to real numbers. Therefore it is questionable whether
$\mathcal{E}_{n_{e}\rightarrow 1}=\mathcal{E}$ in a specified method using
replica trick. In Sec. III we also check the validity of the expansion for
logarithmic negativity. We find that the quality of the expansion is good only
for systems with translational invariant local Hamiltonian. Fortunately
translational invariance is common for quantum many-body systems we study.
Therefore the expansion for $\mathcal{E}_{n_{e}\rightarrow 1}$ converges to
$\mathcal{E}$ and captures its scaling behavior in translational invariant
free-fermion systems with local Hamiltonian, as also confirmed by numerical
results in [10]. We leave this as an open question for further discussion and
exploration.

In Sec. IV we give some applications of (8) in free-fermion systems of various
dimensions. By applying (4) we see in (1+1)-d fermion gas (which is also a
(1+1)-d CFT system) it corresponds with the result in [10] from the method
with Toeplitz matrix.

##  II Correlator expansion for fermion negativity

###  II.1 Review: Charge cumulant expansion for fermion entropy

Charge cumulant expansion for entropy was first derived in the context of FCS
[14]. Here we briefly review the alternative derivation by replica method
[13]. The von Neumann entropy and $n$-th Rényi entropy of region $A$ in a
bipartite system $A\cup B$ is defined as

$S_{A}=-\mathrm{Tr}(\rho_{A}\log\rho_{A})$ (13)

and

$S_{A}^{(n)}=\frac{1}{1-n}\log\mathrm{Tr}(\rho_{A}^{n}),$ (14)

where $\rho_{A}=\mathrm{Tr}_{B}(\rho_{AB})$ is the reduced density matrix on
subsystem $A$. In replica methods we calculate $\mathrm{Tr}(\rho_{A}^{n})$,
and to get von Neumann entropy we take the limit

$S_{A}=\lim_{n\rightarrow 1}S_{A}^{(n)}=\lim_{n\rightarrow
1}\frac{1}{1-n}\log\mathrm{Tr}(\rho_{A}^{n})$ (15)

or

$S_{A}=-\left.\frac{\partial\mathrm{Tr}(\rho_{A}^{n})}{\partial
n}\right|_{n=1}=-\left.\frac{\partial\log\mathrm{Tr}(\rho_{A}^{n})}{\partial
n}\right|_{n=1}.$ (16)

$\mathrm{Tr}(\rho_{A}^{n})$ can be written in the form of the expectation
value of a twist operator by inserting the overcomplete bases of coherent
states. For a single-fermion system,

$\mathrm{Tr}(\rho)=\sum_{n}\braket{n}{\rho}{n}=\int d\bar{\eta}d\eta
e^{-\bar{\eta}\eta}\braket{-\bar{\eta}}{\rho}{\eta}$ (17)

and

$\displaystyle\mathrm{Tr}(\rho^{n})=$
$\displaystyle\sum_{n_{i}}\braket{n_{1}}{\rho}{n_{2}}\cdots\braket{n_{n}}{\rho}{n_{1}}$
(18) $\displaystyle=$ $\displaystyle\int d\bar{\eta}_{1}d\eta_{1}\cdots
d\bar{\eta}_{n}d\eta_{n}e^{-\sum_{i}\bar{\eta}_{i}\eta_{i}}$
$\displaystyle\times\braket{-\bar{\eta}_{1}}{\rho}{\eta_{2}}\braket{\bar{\eta}_{2}}{\rho}{\eta_{3}}\cdots\braket{\bar{\eta}_{n}}{\rho}{\eta_{1}}$
$\displaystyle=$ $\displaystyle\int d\bar{\eta}_{1}d\eta_{1}\cdots
d\bar{\eta}_{n}d\eta_{n}e^{-\sum_{i}\bar{\eta}_{i}\eta_{i}}$
$\displaystyle\times\braket{-\bar{\eta}_{1}}{\rho}{-\eta_{2}}\braket{-\bar{\eta}_{2}}{\rho}{-\eta_{3}}\cdots\braket{-\bar{\eta}_{n}}{\rho}{\eta_{1}},$

where $\ket{\eta}$ and $\bra{\bar{\eta}}$ are independent coherent states of
fermion satisfying $\psi\ket{\eta}=\eta\ket{\eta}$ and
$\bra{\bar{\eta}}\psi=\bra{\bar{\eta}}\bar{\eta}$, with $\eta,\bar{\eta}$
being independent grassmann numbers. The minus sign comes from the
anticommutation relation of fermion operator $\psi$. Therefore for a bipartite
system, we can view $\mathrm{Tr}(\rho_{A}^{n})$ as the expectation value of a
twist operator $T_{n}$ on $n$ replicas of the original Hilbert space as

$\mathrm{Tr}(\rho_{A}^{n})=\braket{T_{n}},$ (19)

where the operation of $T_{n}$ on the $n$-fold Hilbert space in region $A$ is

$T_{n}(A)\ket{\eta_{1},\cdots,\eta_{n-1},\eta_{n}}=\ket{-\eta_{2},\cdots,-\eta_{n},\eta_{1}}$
(20)

or equivalently for the fermion operators

$\begin{aligned} T_{n}(A)^{\dagger}\psi_{a}T_{n}(A)=\begin{cases}-\psi_{a+1}\
&a<n\\\ \psi_{1}\ &a=n\end{cases}\end{aligned}.$ (21)

We can also write $T_{n}$ in the matrix form as

$\begin{aligned} T_{n}(A)=\begin{pmatrix}0&-1\\\ \ &0&-1\\\ \ &\
&\ddots&\ddots\\\ \ &\ &\ &0&-1\\\ 1&\ &\ &\ &0\end{pmatrix}\end{aligned}.$
(22)

In region $B=\bar{A}$ the operation is the identity matrix

$\begin{aligned} T_{n}(B)=\begin{pmatrix}1\\\ \ &1\\\ \ &\ &\ddots\\\ \ &\ &\
&1\\\ \ &\ &\ &\ &1\end{pmatrix}\end{aligned}.$ (23)

To calculate $\braket{T_{n}}$, we diagonalize $T_{n}(A)$ by introducing
Fourier transformation in replica space

$\tilde{\psi}_{p}(\bm{r})=\frac{1}{\sqrt{n}}\sum_{a=1}^{n}e^{-\frac{2\pi
ip}{n}a}\psi_{a}(\bm{r}),$ (24)

where

$p=-\frac{n-1}{2},-\frac{n-3}{2},\cdots,\frac{n-1}{2}\ (\mathrm{mod}\ n).$
(25)

Here we have the freedom to adjust the choice of the range of $p$ by adding
integer multiplies of $n$ to any $p$ (e.g. change $-(n-1)/2$ to $(n+1)/2$). In
the ”momentum” replica space $p$, the operation of $T_{n}$ is

$T_{n}^{\dagger}\tilde{\psi}_{p}(\bm{r})T_{n}=\tilde{\psi}_{p}(\bm{r})\begin{cases}-e^{\frac{2\pi
ip}{n}}&\bm{r}\in A\\\ 1&\bm{r}\in B.\end{cases}$ (26)

and through the anticommutation relation of fermion operators we can get

$T_{n}=\prod_{p=-\frac{n-1}{2}}^{\frac{n-1}{2}}e^{(\frac{2\pi ip}{n}+\pi
i)Q_{A,p}},$ (27)

where

$Q_{A,p}=\int_{\bm{r}\in
A}d^{D}\bm{r}\tilde{\psi}_{p}^{\dagger}(\bm{r})\tilde{\psi}_{p}(\bm{r})$ (28)

is the total charge operator in $A$ region of the replica $p$.

In some texts there is another form of $T_{n}(A)$

$\begin{aligned} T_{n}(A)=\begin{pmatrix}0&1\\\ \ &0&1\\\ \ &\
&\ddots&\ddots\\\ \ &\ &\ &0&1\\\ (-1)^{n+1}&\ &\ &\
&0\end{pmatrix}\end{aligned},$ (29)

which will result in

$T_{n}=\prod_{p=-\frac{n-1}{2}}^{\frac{n-1}{2}}e^{\frac{2\pi ip}{n}Q_{A,p}}.$
(30)

For free-fermion systems with conserved $U(1)$ charge (i.e. with Hamiltonian
in the form $H=\sum_{i,j}t_{ij}c_{i}^{\dagger}c_{j}$), (27) and (30) are
equivalent, as the Hamiltonian for all momentum replica space $p$ are
independent and equals to the original Hamiltonian. Thus all $Q_{A,p}=Q_{A}$,
and to get (30) from (27) we just switch the range of $p$. But for interactive
systems they may differ and possibly have some effects.

As mentioned above, for free-fermion systems with conserved charge, the
momentum replica spaces with different $p$ are independent, therefore the
expectation value of $T_{n}$ decouples into each momentum replica space

$\braket{T_{n}}=\prod_{p}\braket{e^{\frac{2\pi ip}{n}Q_{A,p}}}.$ (31)

Then we use the cumulant expansion

$\log\braket{e^{ikx}}=\exp\left[\sum_{n=1}^{\infty}\frac{(ik)^{n}}{n!}\braket{x^{n}}_{c}\right]$
(32)

to expand the expectation value and get

$\displaystyle\log\braket{T_{n}}$
$\displaystyle=\sum_{p}\sum_{M=1}^{\infty}\frac{1}{M!}\left(\frac{2\pi
ip}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}$ (33)
$\displaystyle=\sum_{M=1}^{\infty}\left(\sum_{p}p^{M}\right)\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}.$

For the range of $p$ in summation $\sum_{p}p^{M}$, we choose

$p=-\frac{n-1}{2},-\frac{n-3}{2},\cdots,\frac{n-1}{2}.$ (34)

Note that this is the only natural choice to make $\sum_{p}p^{M}$ smallest for
even $M$ and 0 for odd $M$. Although the choice of range have no effect for
(31) as the eigenvalues of charge operator $Q_{A,p}$ are all integers, it
matters for (33). A different choice from (34) will lead to nonzero imaginary
odd-$M$ terms in (33), and the convergence for even-$M$ terms will also become
bad. After truncation to a finite order of $M$, it will fail to give the
correct coefficient, especially when we use the expansion to analyze the
scaling behavior of entropy in extensive systems. This comes from the
multivaluedness of log function, analogous to the expansion for 1 as

$\displaystyle 1=e^{2\pi iN}=\sum_{M=1}^{\infty}\frac{1}{M!}(2\pi iN)^{M}$
(35)

where $N$ is an integer, which is also bad for $N\neq 0$.

(33) and (34) gives the cumulant expansion for Rényi entropy

$\displaystyle S_{A}^{(n)}=-\frac{1}{1-n}\sum_{\begin{subarray}{c}M=2\\\
\mathrm{even}\ M\end{subarray}}^{\infty}$ $\displaystyle
2\zeta\left(-M,\frac{n+1}{2}\right)$ (36)
$\displaystyle\times\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}$

(see Appendix A for derivation). By taking limit $n\rightarrow 1$ we get the
cumulant expansion for von Neumann entropy

$\displaystyle S_{A}=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2\zeta(M)\braket{Q_{A}^{M}}_{c}$ (37)

(see Appendix A or [13] for derivation), where $\zeta(M)$ is the Riemann zeta
function.

###  II.2 Charge correlator expansion for fermion negativity

For a bipartite bosonic system $A\cup B$, (logarithmic) PT negativity is
defined as

$\mathcal{E}=\log||\rho^{T_{A}}||_{1}=\log\mathrm{Tr}\sqrt{(\rho^{T_{A}})^{2}}.$
(38)

where $\rho^{T_{A}}$ is the partial transposition of the density matrix $\rho$
on the subsystem $A$, defined in the occupation number basis as

$\displaystyle(\ket{n_{A}}$
$\displaystyle\otimes\ket{n_{B}}\bra{m_{A}}\otimes\bra{m_{B}})^{T_{A}}$ (39)
$\displaystyle=\ket{m_{A}}\otimes\ket{n_{B}}\bra{n_{A}}\otimes\bra{m_{B}}.$

For fermionic systems, we use the alternative definition [10] called partial
TR negativity

$\mathcal{E}=\log||\rho^{R_{A}}||_{1}=\log\mathrm{Tr}\sqrt{\rho^{R_{A}}\rho^{R_{A}\dagger}},$
(40)

where $\rho^{R_{A}}$ is the partial time-reversal of $\rho_{AB}$ on subsystem
$A$, defined in the coherent basis as

$U(\ket{\eta}\bra{\bar{\eta}})^{R}U^{\dagger}=\ket{i\bar{\eta}}\bra{i\eta},$
(41)

where $U$ is the unitary operator in time-reversal operator and have no effect
in the trace. In the occupation number basis it is

$\displaystyle U(\ket{n_{A}}$
$\displaystyle\otimes\ket{n_{B}}\bra{m_{A}}\otimes\bra{m_{B}})^{R_{A}}U^{\dagger}$
(42)
$\displaystyle=(-1)^{\phi(\\{n\\},\\{m\\})}\ket{m_{A}}\otimes\ket{n_{B}}\bra{n_{A}}\otimes\bra{m_{B}},$

where the phase factor is

$\displaystyle\phi(\\{n\\},\\{m\\})=$
$\displaystyle\frac{\tau_{1}(\tau_{1}+2)}{2}+\frac{\bar{\tau}_{1}(\bar{\tau}_{1}+2)}{2}+\tau_{2}\bar{\tau}_{2}$
(43)
$\displaystyle+\tau_{1}\tau_{2}+\bar{\tau}_{1}\bar{\tau}_{2}+(\bar{\tau}_{1}+\bar{\tau}_{2})(\tau_{1}+\tau_{2}),$

and $\tau_{1(2)}=\sum_{i\in A(B)}n_{i}$, $\bar{\tau}_{1(2)}=\sum_{i\in
A(B)}m_{i}$.

In the following we consider a fermionic field system devided into three parts
$A$, $B$ and $C$, and still use (40) to define the negativity of $A$ and $B$
by $\rho_{AB}=\mathrm{Tr}_{C}\rho_{ABC}$ in order to measure the entanglement
between region $A$ and $B$. We can also use replica method [8, 9, 10] to
calculate negativity. The $n$-th Rényi negativity is defined as [10]

$\mathcal{E}_{n}=\begin{cases}\log\mathrm{Tr}\left(\rho^{R_{A}}\rho^{R_{A}\dagger}\cdots\rho^{R_{A}}\rho^{R_{A}\dagger}\right)\
&n\ \mathrm{even}\\\
\log\mathrm{Tr}\left(\rho^{R_{A}}\rho^{R_{A}\dagger}\cdots\rho^{R_{A}}\right)\
&n\ \mathrm{odd}\end{cases}$ (44)

and it is usually assumed that by taking limit $n_{e}\rightarrow 1$ while
keeping $n_{e}$ even we would get the negativity $\mathcal{E}$. In [10] it is
proved that Rényi negativity can also be written as the expectation value of a
twist operator

$\mathcal{E}_{n}=\braket{T_{n}},$ (45)

where the operation of the twist operator on the three regions is

$\begin{aligned} T_{n}(A)=\begin{pmatrix}0&\ &\ &\ &(-1)^{n-1}\\\ 1&0\\\ \
&1&0\\\ \ &\ &\ddots&\ddots\\\ &\ &\ &1&0\end{pmatrix}\end{aligned},$ (46)
$\begin{aligned} T_{n}(B)=\begin{pmatrix}0&-1\\\ \ &0&-1\\\ \ &\
&\ddots&\ddots\\\ \ &\ &\ &0&-1\\\ 1&\ &\ &\ &0\end{pmatrix}\end{aligned},$
(47)

and

$\begin{aligned} T_{n}(C)=\begin{pmatrix}1\\\ \ &1\\\ \ &\ &\ddots\\\ \ &\ &\
&1\\\ \ &\ &\ &\ &1\end{pmatrix}\end{aligned}.$ (48)

The matrices (46) and (47) can be simultaneously diagonalized only when $n$ is
even. Similarly with (27), for even $n_{e}$ it is direct to get

$T_{n_{e}}=\prod_{p=-\frac{n_{e}-1}{2}}^{\frac{n_{e}-1}{2}}e^{-\frac{2\pi
ip}{n_{e}}Q_{A,p}+(\frac{2\pi ip}{n_{e}}+\pi i)Q_{B,p}},$ (49)

which is equivalent to (57) of [10] for free-fermion systems with conserved
$U(1)$ charge, since they differ with only a shift of the range of $p$.
Similarly with the case for Rényi entropy, for free-fermion systems with
conserved charge the expectation value of $T_{n}$ decouples into each momentum
replica space

$\braket{T_{n_{e}}}=\prod_{p=-\frac{n_{e}-1}{2}}^{\frac{n_{e}-1}{2}}\braket{e^{-\frac{2\pi
ip}{n_{e}}Q_{A,p}+(\frac{2\pi ip}{n_{e}}+\pi i)Q_{B,p}}}.$ (50)

Now in order to get the cumulant expansion, we shift the phase before
$Q_{B,p}$ by $2\pi i$ for the positive half of $p$, so

$\displaystyle\braket{T_{n_{e}}}=$
$\displaystyle\prod_{p=-\frac{n_{e}-1}{2}}^{-\frac{1}{2}}\braket{e^{-\frac{2\pi
ip}{n_{e}}Q_{A,p}+(\frac{2\pi ip}{n_{e}}+\pi i)Q_{B,p}}}$ (51)
$\displaystyle\times\prod_{p=\frac{1}{2}}^{\frac{n_{e}-1}{2}}\braket{e^{-\frac{2\pi
ip}{n_{e}}Q_{A,p}+(\frac{2\pi ip}{n_{e}}-\pi i)Q_{B,p}}}.$

Then we apply the cumulant expansion and get

$\displaystyle\mathcal{E}_{n_{e}}=$
$\displaystyle\sum_{M=1}^{\infty}\frac{1}{M!}(\pi i)^{M}$ (52)
$\displaystyle\times\left[\sum_{p=-\frac{n_{e}-1}{2}}^{-1/2}\left<{\left(-\frac{2p}{n_{e}}Q_{A}+\left(\frac{2p}{n_{e}}+1\right)Q_{B}\right)^{M}}\right>_{c}\right.$
$\displaystyle\left.+\sum_{p=\frac{1}{2}}^{\frac{n_{e}-1}{2}}\left<{\left(-\frac{2p}{n_{e}}Q_{A}+\left(\frac{2p}{n_{e}}-1\right)Q_{B}\right)^{M}}\right>_{c}\right].$

Here the range of $p$ is chosen for a similar reason as (33). This is also the
only way to make the coefficient smallest for even $M$ and 0 for odd $M$,
which is the key point for the convergence of the expansion. Written the
expansion by orders of $M$ as

$\displaystyle\mathcal{E}_{n_{e}}=\sum_{\begin{subarray}{c}M=2\\\
\mathrm{even}\ M\end{subarray}}^{\infty}\mathcal{E}_{n_{e}}^{(M)},$ (53)

Summing over $p$ we get the first two terms

$\displaystyle\mathcal{E}_{n_{e}}^{(2)}=-\pi^{2}$
$\displaystyle\left[\frac{n_{e}^{2}-1}{6n_{e}}(\braket{Q_{A}^{2}}_{c}+\braket{Q_{B}^{2}}_{c})\right.$
(54)
$\displaystyle\left.+\frac{n_{e}^{2}+2}{6n_{e}}\braket{Q_{A}Q_{B}}_{c}\right],$

and

$\displaystyle\mathcal{E}_{n_{e}}^{(4)}=\pi^{4}$
$\displaystyle\left[\frac{3n_{e}^{4}-10n_{e}^{2}+7}{360n_{e}^{3}}(\braket{Q_{A}^{4}}_{c}+\braket{Q_{B}^{4}}_{c})\right.$
(55)
$\displaystyle+\frac{3n_{e}^{4}+10n_{e}^{2}-28}{360n_{e}^{3}}(\braket{Q_{A}^{3}Q_{B}}_{c}+\braket{Q_{A}Q_{B}^{3}}_{c})$
$\displaystyle\left.+\frac{n_{e}^{4}+14}{120n_{e}^{3}}\braket{Q_{A}^{2}Q_{B}^{2}}_{c}\right].$

It is also direct for higher orders. We have numerically checked that this
expansion converges rapidly for thermal equilibrium states of randomly
generated all-connected free Hamiltonian, see Sec. III. For Hamiltonian with
local random hopping, the expansion is extremely accurate at low temperature.

Taking the limit $n_{e}\rightarrow 1$ we get

$\displaystyle\mathcal{E}_{n_{e}\rightarrow
1}^{(2)}=-\frac{\pi^{2}}{2}\braket{Q_{A}Q_{B}}_{c}$ (56)

and

$\displaystyle\mathcal{E}_{n_{e}\rightarrow 1}^{(4)}=\pi^{4}$
$\displaystyle\left[-\frac{1}{24}(\braket{Q_{A}^{3}Q_{B}}_{c}+\braket{Q_{A}Q_{B}^{3}}_{c})\right.$
(57) $\displaystyle\left.+\frac{1}{8}\braket{Q_{A}^{2}Q_{B}^{2}}_{c}\right].$

The analytic continuation is ambiguous mathematically, and it is questionable
whether $\mathcal{E}_{n_{e}\rightarrow 1}=\mathcal{E}$. In Sec. III we find
that the correspondence of the expansion for $\mathcal{E}_{n_{e}\rightarrow
1}$ with $\mathcal{E}$ requires translational invariance in the Hamiltonian.

##  III Numerical verification

Figure 1: Numerical verification for the validity of charge correlator
expansion (52) for $n_{e}=2,4$ and $n_{e}\rightarrow 1$ on thermal equilibrium
states of 100-fermion system with conserved charge, i.e. $\rho=e^{-\beta
H}/Z$, $\beta=1/T$ and $H=\sum_{i,j}t_{ij}c_{i}^{\dagger}c_{j}$. The amplitude
of each $t_{ij}$ is randomly generated from a normal distribution, and the
phase is generated from an even distribution in $[0,2\pi)$. The first two
terms of the expansion (54) and (55) is calculated by (59) to (66), and the
exact values of negativity are calculated by (70) and (71). The regions $A$
and $B$ are chosen to be the 21 to 30-th and 34 to 54-th fermions
respectively. The quality of convergence depends on the pattern of the random
Hamiltonian generated. (a) With random all-connected hopping $t_{ij}$,
generated from the same distribution between all sites. (b) With random local
hopping $t_{ij}$ for 1+1-d system. The amplitude of $t_{ij}$ exponentially
decays with the distance $|i-j|$. (c) With random translational invariant
local hopping $t_{ij}=t_{i-j,0}$ for 1+1-d system, within the distance of 3
atoms $|i-j|\leq 3$.

To check the validity of (52), we numerically calculate the expansion (54),
(55), (56) and (57) in thermal equilibrium states of a 100-fermion system with
randomly generated free Hamiltonian and conserved charge. It is well known
that Rényi and von Neumann entropy of Gaussian states can be computed
efficiently by relating to the two-point correlator matrices [27, 28, 29].
Both Rényi and logarithmic negativity of Gaussian states can also be related
to correlator matrices, and so do the connected charge correlators. The
correlator matrix of a bipartite system $A\cup B$ can be partitioned to block
matrices as

$C=\begin{pmatrix}C^{11}&C^{12}\\\ C^{21}&C^{22}\end{pmatrix},$ (58)

where the matrix elements are $C_{ij}=\braket{c_{i}^{\dagger}c_{j}}$. $C^{11}$
and $C^{22}$ are the reduced correlator matrices on the subsystems $A$ and
$B$, and $C^{12}$ and $C^{21}$ contain correlation between two regions. Using
Wick’s theorem, the connected charge correlators in the expansion can be
reduced to the trace of the product of the block matrices as

$\braket{Q_{A}^{2}}_{c}=\mathrm{Tr}C^{11}-\mathrm{Tr}(C^{11})^{2},$ (59)
$\braket{Q_{A}Q_{B}}_{c}=-\mathrm{Tr}(C^{12}C^{21}),$ (60)
$\braket{Q_{B}^{2}}_{c}=\mathrm{Tr}C^{22}-\mathrm{Tr}(C^{22})^{2},$ (61)
$\braket{Q_{A}^{4}}_{c}=\mathrm{Tr}C^{11}-7\mathrm{Tr}(C^{11})^{2}+12\mathrm{Tr}(C^{11})^{3}-6\mathrm{Tr}(C^{11})^{4},$
(62) $\displaystyle\braket{Q_{A}^{3}Q_{B}}_{c}=$
$\displaystyle-\mathrm{Tr}(C^{12}C^{21})+6\mathrm{Tr}(C^{11}C^{12}C^{21})$
(63) $\displaystyle-6\mathrm{Tr}(C^{11}C^{11}C^{12}C^{21}),$
$\displaystyle\braket{Q_{A}^{2}Q_{B}^{2}}_{c}=$
$\displaystyle-\mathrm{Tr}(C^{12}C^{21})+2\mathrm{Tr}(C^{11}C^{12}C^{21})$
(64)
$\displaystyle+2\mathrm{Tr}(C^{22}C^{21}C^{12})-4\mathrm{Tr}(C^{11}C^{12}C^{21}C^{12})$
$\displaystyle-2\mathrm{Tr}(C^{12}C^{21})^{2},$
$\displaystyle\braket{Q_{A}Q_{B}^{3}}_{c}=$
$\displaystyle-\mathrm{Tr}(C^{12}C^{21})+6\mathrm{Tr}(C^{22}C^{21}C^{12})$
(65) $\displaystyle-6\mathrm{Tr}(C^{22}C^{22}C^{21}C^{12}),$
$\braket{Q_{B}^{4}}_{c}=\mathrm{Tr}C^{22}-7\mathrm{Tr}(C^{22})^{2}+12\mathrm{Tr}(C^{22})^{3}-6\mathrm{Tr}(C^{22})^{4}.$
(66)

To calculate the negativity of Gaussian states, we follow the method developed
in [10] and [12]. Define the covariance matrices

$\Gamma=\mathbb{I}-2C=\begin{pmatrix}\Gamma^{11}&\Gamma^{12}\\\
\Gamma^{21}&\Gamma^{22}\end{pmatrix}$ (67)

(where $\mathbb{I}$ is the identity matrix), the corresponding transformed
matrices

$\Gamma_{\pm}=\begin{pmatrix}-\Gamma^{11}&\pm i\Gamma^{12}\\\ \pm
i\Gamma^{21}&\Gamma^{22}\end{pmatrix},$ (68)

and a new single particle correlation function

$C_{\Xi}=\frac{1}{2}[\mathbb{I}-(\mathbb{I}+\Gamma_{+}\Gamma_{-})^{-1}(\Gamma_{+}+\Gamma_{-})].$
(69)

Denoting $\zeta_{i}$ and $\xi_{i}$ as the eigenvalues of $C$ and $C_{\Xi}$
respectively, the logarithmic and Rényi negativity for free system with
conserved charge is

$\mathcal{E}=\sum_{j}\log[\xi_{j}^{1/2}+(1-\xi_{j})^{1/2}]+\frac{1}{2}\sum_{j}\log[\zeta_{j}^{2}+(1-\zeta_{j})^{2}]$
(70)

and

$\displaystyle\mathcal{E}_{n_{e}}=$
$\displaystyle\sum_{j}\log[\xi_{j}^{n_{e}/2}+(1-\xi_{j})^{n_{e}/2}]$ (71)
$\displaystyle+\frac{n_{e}}{2}\sum_{j}\log[\zeta_{j}^{2}+(1-\zeta_{j})^{2}].$

Using (59) to (66), we calculate the first two orders of the charge correlator
expansion (54) and (55) and compare it with the exact $\mathcal{E}$ and
$\mathcal{E}_{n_{e}}$ calculated by (70) and (71), see Fig. 1. From Fig. 1
(a)(b) we see that the convergence of Rényi negativity is generally fast even
for all-connected Hamiltonian. When there are only local hopping terms that
decays exponentially with the distance, which is the normal case for extensive
systems, the convergence is extremely accurate at low temperature. For high
temperature the convergence becomes slow, particularly for higher $n_{e}$.
Note that these phenomena are all similar to the case of the expansion for
Rényi and von Neumann entropy (36) and (37), where the convergence is better
for systems with low temperature and local hopping terms.

However, the correspondence of the expansion at replica limit
$n_{e}\rightarrow 1$ with logarithmic negativity requires more constraint.
Fig.1 (b) shows the bad convergence even for local random Hamiltonian, from
Fig.1 (c) we see that the convergence is valid for translational invariant
systems. With translational symmetry, the convergence is also better in
systems with local hopping terms.

Therefore, from numerical verification we conclude that the charge correlator
expansion can be used to analyze the scaling behavior of Rényi negativity for
local free Hamiltonian with conserved charge, while for logarithmic negativity
it further requires translational invariance. In [10] it is numerically
verified that the scaling behavior of negativity in Kitaev chain and SSH model
(both with translational invariant Hamiltonian) is consistent with the CFT
result from replica trick.

##  IV Applications

###  IV.1 (1+1)-d free fermion

(1+1)-d free-fermion systems with translational symmetry at zero temperature,
with the Euler characteristic of the Fermi sea being $\chi_{F}$ (equals the
number of segments of the Fermi sea), is also a CFT with central charge
$c=\chi_{F}$ [13]. For (1+1)-d free-fermion systems, the $M=2$ term is related
to $\chi_{F}$ and (4) becomes the universal second-order density correlator

$\int\frac{dq^{\prime}}{2\pi}\braket{\rho_{q}\rho_{q^{\prime}}}_{c}=\chi_{F}|q|,$
(72)

for small $q$. where $\rho_{q}$ is the charge density in momentum space

$\rho_{q}=\int{dx}\rho(x)e^{-iqx}=\int\frac{dk}{2\pi}c_{k}^{\dagger}c_{k+q}.$
(73)

After Fourier transformation we get density correlator
$\braket{\rho(x_{1})\rho(x_{2})}_{c}$ in real space. Here we have to apply the
invariance of correlator under translation in systems with translational
symmetry by

$\displaystyle\braket{\rho(x_{1})\rho(x_{2})}_{c}$
$\displaystyle=\braket{\rho(x_{1}-x_{2})\rho(0)}_{c}$ (74)
$\displaystyle=\int\frac{dq}{2\pi}\frac{dq^{\prime}}{2\pi}\braket{\rho_{q}\rho_{q^{\prime}}}_{c}e^{iq(x_{1}-x_{2})}$
$\displaystyle=-\frac{\chi_{F}}{2\pi^{2}}\frac{1}{(x_{1}-x_{2})^{2}}.$

(74) applies for large $x_{1}-x_{2}\gg\varepsilon$, where the short-distance
cutoff $\varepsilon$ is about the inverse of the curvature of Fermi surface
[13] (in 1d the length of the shortest segment or interval of Fermi sea).

To get charge correlator of regions we integrate (74) in real space. Now
suppose we take the region $A:\ -l_{1}\leq x\leq 0$, $B:\ d\leq x\leq d+l_{2}$
and $C$ being the rest parts. For $\braket{Q_{A}^{2}}_{c}$ we encounter the
case for short distance with small $x_{1}-x_{2}$. To solve this problem we
consider a simplified model, where for $|x_{1}-x_{2}|>\varepsilon$

$\braket{\rho(x_{1})\rho(x_{2})}_{c}=-\frac{\chi_{F}}{2\pi^{2}}\frac{1}{(x_{1}-x_{2})^{2}},$
(75)

and for $|x_{1}-x_{2}|\leq\varepsilon$ it is a constant

$\braket{\rho(x_{1})\rho(x_{2})}_{c}=-\frac{\chi_{F}}{2\pi^{2}}\frac{1}{\varepsilon^{2}}.$
(76)

We consider the case that the regions $A,B$ are much larger than the short-
distance cutoff, i.e. $l_{1},l_{2}\gg\varepsilon$. Therefore

$\displaystyle\braket{Q_{A}^{2}}_{c}$
$\displaystyle=\int_{-l_{1}}^{0}dx_{1}\int_{-l_{1}}^{0}dx_{2}\braket{\rho(x_{1})\rho(x_{2})}_{c}$
(77)
$\displaystyle=\frac{\chi_{F}}{2\pi^{2}}\left[2\log\frac{l_{1}}{\varepsilon}+O\left(\frac{l_{1}}{\varepsilon}\right)\right].$

Similarly for $\braket{Q_{B}^{2}}_{c}$ we have

$\displaystyle\braket{Q_{B}^{2}}_{c}$
$\displaystyle=\frac{\chi_{F}}{2\pi^{2}}\left[2\log\frac{l_{2}}{\varepsilon}+O\left(\frac{l_{2}}{\varepsilon}\right)\right].$
(78)

For $\braket{Q_{A}Q_{B}}_{c}$, at $d\geq\varepsilon$ the integration get

$\displaystyle\braket{Q_{A}Q_{B}}_{c}$
$\displaystyle=\int_{-l_{1}}^{0}dx_{1}\int_{d}^{d+l_{2}}dx_{2}\braket{\rho(x_{1})\rho(x_{2})}_{c}$
(79)
$\displaystyle=-\frac{\chi_{F}}{2\pi^{2}}\log\frac{(l_{1}+d)(l_{2}+d)}{d(l_{1}+l_{2}+d)}$

and for $d<\varepsilon\ll l_{1},l_{2}$,

$\displaystyle\braket{Q_{A}Q_{B}}_{c}$
$\displaystyle=\int_{-l_{1}}^{0}dx_{1}\int_{d}^{d+l_{2}}dx_{2}\braket{\rho(x_{1})\rho(x_{2})}_{c}$
(80)
$\displaystyle=-\frac{\chi_{F}}{2\pi^{2}}\left[\log\frac{(l_{1}+d)(l_{2}+d)}{\varepsilon(l_{1}+l_{2}+d)}+O\left(\frac{d}{\varepsilon}\right)\right]$
$\displaystyle=-\frac{\chi_{F}}{2\pi^{2}}\log\frac{l_{1}l_{2}}{\varepsilon(l_{1}+l_{2})}.$

In the last step we ignore the small $O(d/\varepsilon)$ terms. With the above
results for charge correlators, for $d\gg\varepsilon$ (54) becomes

$\displaystyle\mathcal{E}_{n_{e}}^{(2)}=$
$\displaystyle\chi_{F}\left[-\frac{n_{e}^{2}-1}{6n_{e}}\left(\log\frac{l_{1}l_{2}}{\varepsilon^{2}}+O\left(\frac{l}{\varepsilon}\right)\right)\right.$
(81)
$\displaystyle+\left.\frac{n_{e}^{2}+2}{12n_{e}}\log\frac{(l_{1}+d)(l_{2}+d)}{d(l_{1}+l_{2}+d)}\right].$

And for $d\rightarrow 0$ it becomes

$\displaystyle\mathcal{E}_{n_{e}}^{(2)}=$
$\displaystyle\chi_{F}\left[-\frac{n_{e}^{2}-1}{6n_{e}}\left(\log\frac{l_{1}l_{2}}{\varepsilon^{2}}+O\left(\frac{l}{\varepsilon}\right)\right)\right.$
(82)
$\displaystyle+\left.\frac{n_{e}^{2}+2}{12n_{e}}\log\frac{l_{1}l_{2}}{\varepsilon(l_{1}+l_{2})}\right].$

First we consider the case $d\rightarrow 0$, i.e. regions $A$ and $B$ are
adjacent, (82) can also be written as

$\displaystyle\mathcal{E}_{n_{e}}^{(2)}=\chi_{F}$
$\displaystyle\left[-\frac{n_{e}^{2}-4}{12n_{e}}\log\frac{l_{1}l_{2}}{\varepsilon^{2}}-\frac{n_{e}^{2}+2}{12n_{e}}\log\frac{l_{1}+l_{2}}{\varepsilon^{2}}\right.$
(83)
$\displaystyle\left.-\frac{n_{e}^{2}-1}{6n_{e}}O\left(\frac{l}{\varepsilon}\right)\right],$

which is identical to (85) in [10] at even $n_{e}$ except for the short
distance cutoff (note that the term for short distance cutoff
$O\left(l/{\varepsilon}\right)$ is generally large and cannot be neglected).
Taking the limit $n_{e}\rightarrow 1$ we get

$\displaystyle\mathcal{E}_{n_{e}\rightarrow
1}^{(2)}=\frac{\chi_{F}}{4}\log\frac{l_{1}l_{2}}{\varepsilon(l_{1}+l_{2})}.$
(84)

Then the first term related to divergent short-distance cutoff is eliminated,
which is consistent with what we expect for negativity. Numerical checks in
Sec. III have confirmed that for system with translational symmetry, the
expansion of $\mathcal{E}_{n_{e}\rightarrow 1}$ converges to $\mathcal{E}$, so
(84) captures the scaling property of $\mathcal{E}$.

We can also consider the cases that region $A$ and $B$ are far apart, i.e.
$d\gg l_{1},l_{2}$. Then (81) can be expanded to the second order of $l_{i}/d$
as

$\displaystyle\mathcal{E}_{n_{e}}^{(2)}=\chi_{F}$
$\displaystyle\left[-\frac{n_{e}^{2}-1}{6n_{e}}\left(\log\frac{l_{1}l_{2}}{\varepsilon^{2}}+O\left(\frac{l}{\varepsilon}\right)\right)\right.$
(85)
$\displaystyle+\left.\frac{n_{e}^{2}+2}{12n_{e}}\frac{l_{1}l_{2}}{d^{2}}\right].$

Taking the limit $n_{e}\rightarrow 1$ the first term vanishes and we also get
rid of the short-distance cutoff

$\displaystyle\mathcal{E}_{n_{e}\rightarrow
1}^{(2)}=\frac{\chi_{F}}{4}\frac{l_{1}l_{2}}{d^{2}}.$ (86)

Therefore it is proportional to the length of the two regions and decays with
the distance by a power law.

###  IV.2 free fermion in higher dimension

Similar to the 1+1-dimensional case, we consider free-fermion system with
conserved charge in $(D+1)$-dimension. Here we consider region $A$ and $B$ to
be two balls with radius $R_{1},R_{2}$ and the distance between their centers
$d\gg R_{1},R_{2}$. In higher dimensions the $M$-th order connected charge
correlator satisfies

$\int\frac{d\bm{q}_{M}}{2\pi}\braket{\rho_{\bm{q}_{1}}\cdots\rho_{\bm{q}_{M}}}_{c}\sim
k_{F}^{D+1-M}q^{M-1}$ (87)

for small $\bm{q}$ [13], where $q$ is the typical length of $\bm{q}_{i}$,
$i=1,\cdots M-1$. In (54) the second order correlator is

$\int\frac{d\bm{q}^{\prime}}{2\pi}\braket{\rho_{\bm{q}}\rho_{\bm{q}^{\prime}}}_{c}\sim
k_{F}^{D-1}|\bm{q}|.$ (88)

by Fourier transformation it gives the real space correlator

$\braket{\rho(\bm{r}_{1})\rho(\bm{r}_{2})}_{c}\sim\frac{k_{F}^{D-1}}{|\bm{r}_{1}-\bm{r}_{2}|^{D+1}},$
(89)

for $|\bm{r}_{1}-\bm{r}_{2}|\gg\varepsilon$. Taking the similar model for
short distance cutoff as the 1+1-d case, the integral in real space gives

$\displaystyle\braket{Q_{A}^{2}}_{c}$
$\displaystyle\sim(k_{F}R_{1})^{D-1}\left(1+O\left(\frac{R_{1}}{\varepsilon}\right)\right),$
(90) $\displaystyle\braket{Q_{B}^{2}}_{c}$
$\displaystyle\sim(k_{F}R_{2})^{D-1}\left(1+O\left(\frac{R_{2}}{\varepsilon}\right)\right),$
(91)

and

$\displaystyle\braket{Q_{A}Q_{B}}_{c}$ $\displaystyle\sim
k_{F}^{D-1}\frac{R_{1}^{D}R_{2}^{D}}{d^{D+1}}.$ (92)

Therefore (54) becomes

$\displaystyle\mathcal{E}_{n_{e}}^{(2)}\sim$ $\displaystyle
k_{F}^{D-1}\left[c_{1}\frac{n_{e}^{2}-1}{6n_{e}}\left(R_{1}^{D-1}+R_{2}^{D-1}+O\left(\frac{R}{\varepsilon}\right)\right)\right.$
(93)
$\displaystyle\left.+c_{2}\frac{n_{e}^{2}+2}{6n_{e}}\frac{R_{1}^{D}R_{2}^{D}}{d^{D+1}}\right],$

where $c_{1},c_{2}$ are some finite coefficient. We see that when regions $A$
and $B$ are far apart, the terms from $\braket{Q_{A}^{2}}_{c}$ and
$\braket{Q_{B}^{2}}_{c}$ contain a constant term with short-distance cutoff
that do not depend on the distance between $A$ and $B$, and is much larger
than the terms from $\braket{Q_{A}Q_{B}}_{c}$. This incidates that Rényi
negativity is not a good entanglement measure. However, at replica limit
$n_{e}\rightarrow 1$, the first term with short distance cutoff is also
eliminated, and we get

$\displaystyle\mathcal{E}_{n_{e}\rightarrow 1}^{(2)}\sim$ $\displaystyle
k_{F}^{D-1}\frac{R_{1}^{D}R_{2}^{D}}{d^{D+1}}.$ (94)

This is consistent with our expectation for logarithmic negativity as an
entanglement measure. The negativity is proportional to the area of two
regions and decays also by a power law with the distance between them.

##  V Conclusion and discussion

In this paper, we derive the connected charge correlator expansion for Rényi
and logarithmic negativity in free-fermion systems with conserved $U(1)$
charge. We numerically check the validity of the expansion, and find that the
convergence at replica limit requires translational symmetry. Using this
expansion, we analyze the scaling behavior of negativity in extensive free-
fermion systems. From the expansion, it is direct to note that there are terms
that diverge for short distance cutoff in the expansion for Rényi negativity,
but they vanish at the limit $n_{e}\rightarrow 1$. This is consistent with our
expectation for logarithmic negativity rather than Rényi negativity as an
entanglement measure.

Our research in charge correlator expansion is generally limited to free-
fermion systems. One can also consider canonical bosonic systems where the
charge operator is similarly defined, but will encounter difficulty in the
range of $p$ (34). For bosonic systems, the range of $p$ is given by
$p=0,1,\cdots,n-1\ (\mathrm{mod}\ n)$. There seems to be no natural choice of
the range to make the summation $\sum_{p}p^{M}$ smallest for even $M$ and 0
for odd $M$, as required for the convergence in fermionic systems. As a
consequence, we cannot obtain the correct coefficient in the expansion after
the terms with higher $M$ are truncated. Spin systems may also be considered,
but the charge operator is hard to define, and free-spin systems (without
spin-spin interaction) may not be intriguing for research.

The charge expansion may also be applied to other computable entanglement
measures besides PT negativity, e.g. the computable cross norm or realignment
(CCNR) negativity [30, 31, 32]. Recently, it has been shown that Rényi and
logarithmic CCNR negativity of two intervals that are not adjacent has a
universal expression [33] in (1+1)-d CFT. The twist operators for CCNR
negativity cannot be simultaneously diagonalized, but they can be
simultaneously block diagonalized in the form of $2\times 2$ matrices.
Therefore CCNR negativity can be in free systems can be related to the
expectation value of $2\times 2$ operators instead of $U(1)$ charge operators.
We suppose that the charge expansion for these computable entanglement
measures can provide a deeper understanding of the entanglement properties of
quantum systems.

Formula (4) shows that the universal $(D+1)$-th order charge correlator in
$D$-dimensional free-fermion systems is proportional to the Euler
characteristic of the Fermi sea. The leading term in the expansion for Rényi
and logarithmic negativity (54) and (56) contains second-order charge
correlators, and therefore in (1+1)-d system, it extracts the Euler
characteristic, as shown in Sec. IV.1. However, for higher dimensions, the
divergent terms always exist in the leading term. It is an interesting
question whether there are ways to subtract the divergent terms and extract
topological terms in the expansion for negativity, like the case for
topological entanglement entropy [13].

###### Acknowledgements.

We thank Yingfei Gu for suggesting the topic, advising on various stages of
this project, and comments on the draft. We thank Pok Man Tam for detailed
discussion on the derivation and constructive suggestions. We thank Liang Mao,
Zhenhuan Liu, Chao Yin, Yi-Fei Wang, Yuzhen Zhang and Zhenhua Zhu for valuable
discussions.

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##  Appendix A Cumulant expansion for Rényi and von Neumann entropy

From (33)

$\displaystyle\log\braket{T_{n}}$
$\displaystyle=\sum_{M=1}^{\infty}\left(\sum_{p}p^{M}\right)\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c},$ (95)

the summation of $p$ is

$C_{n,M}=\sum_{p=-\frac{n-1}{2}}^{\frac{n-1}{2}}p^{M}$ (96)

It can be written in the form of generalized harmonic number [34]

$H_{n}^{(s)}(z)=\sum_{j=1}^{n}\frac{1}{(j+z)^{s}}\ \ \ \
(n\in\mathbb{Z}^{+},s\in\mathbb{C},z\in\mathbb{C}\backslash\mathbb{Z}^{-}).$
(97)

First consider the case of even $n$, the result is

$\displaystyle\sum_{p=-\frac{n-1}{2}}^{\frac{n-1}{2}}p^{M}$
$\displaystyle=\left[1+(-1)^{M}\right]\times\left[\left(\frac{1}{2}\right)^{M}+\left(\frac{3}{2}\right)^{M}+\cdots+\left(\frac{n-1}{2}\right)^{M}\right]$
(98) $\displaystyle=\begin{cases}2H_{n/2}^{(-M)}\left(-\frac{1}{2}\right)\
&\mathrm{even}\ M\\\ 0\ &\mathrm{odd}\ M.\end{cases}$

For the range of arguments we consider here, the generalized harmonic number
is related to the generalized Riemann zeta function (this is generally not
correct).

$\zeta(n,a)=\sum_{k=0}^{\infty}\frac{1}{(k+a)^{n}}$ (99)

by

$H_{n}^{(s)}(z)=\zeta(s,z+1)-\zeta(s,z+n+1)$ (100)

and $\zeta(n)=\zeta(n,1)$. Thus

$\displaystyle\log\braket{T_{n}}$
$\displaystyle=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2H_{n/2}^{(-M)}\left(-\frac{1}{2}\right)\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}$ (101)
$\displaystyle=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2\left[\zeta\left(-M,\frac{1}{2}\right)-\zeta\left(-M,\frac{n+1}{2}\right)\right]\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}.$

For even $M\geq 0$, $\zeta\left(-M,1/2\right)=0$, we get

$\displaystyle\log\braket{T_{n}}$
$\displaystyle=-\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2\zeta\left(-M,\frac{n+1}{2}\right)\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}.$ (102)

For odd $n$, it is easy to check that we get

$\displaystyle\log\braket{T_{n}}$
$\displaystyle=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2H_{(n-1)/2}^{(-M)}\left(0\right)\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}$ (103)
$\displaystyle=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2\left[\zeta\left(-M,1\right)-\zeta\left(-M,\frac{n+1}{2}\right)\right]\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}.$

For even $M\geq 0$, $\zeta\left(-M,1\right)=0$, thus we get the same result

$\displaystyle\log\braket{T_{n}}$
$\displaystyle=-\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2\zeta\left(-M,\frac{n+1}{2}\right)\frac{1}{M!}\left(\frac{2\pi
i}{n}\right)^{M}\braket{Q_{A}^{M}}_{c}.$ (104)

This gives the cumulant expansion for Rényi entropy (36).

Since the right hand side of (104) is real for all positive integer $n$, we
now extend it from $n\in\mathbb{Z}^{+}$ to $n\in\mathbb{R}^{+}$ and calculate
the von Neumann entropy by taking the $n\rightarrow 1$ limit. The analytic
continuation can differ with periodic functions and affect the result, but we
assume it is correct here. By using

$\frac{d}{dn}\zeta(k,n)=-k\zeta(k+1,n),$ (105)

we get

$\displaystyle S_{A}$ $\displaystyle=\lim_{n\rightarrow
1}\frac{\log\braket{T_{n}}}{1-n}=\sum_{\begin{subarray}{c}M=2\\\
\mathrm{even}\
M\end{subarray}}^{\infty}2\left.\frac{d}{dn}\zeta\left(-M,\frac{n+1}{2}\right)\right|_{n=1}\frac{1}{M!}({2\pi
i})^{M}\braket{Q_{A}^{M}}_{c}$ (106)
$\displaystyle=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}M\left.\zeta\left(1-M,\frac{n+1}{2}\right)\right|_{n=1}\frac{1}{M!}({2\pi
i})^{M}\braket{Q_{A}^{M}}_{c}.$

And by using the relation

$\zeta(1-M)=2(2\pi)^{-M}\cos(\pi M/2)(M-1)!\zeta(M),$ (107)

we get

$\displaystyle S_{A}$ $\displaystyle=\sum_{\begin{subarray}{c}M=2\\\
\mathrm{even}\
M\end{subarray}}^{\infty}M\zeta\left(1-M,1\right)\frac{1}{M!}({2\pi
i})^{M}\braket{Q_{A}^{M}}_{c}$ (108)
$\displaystyle=\sum_{\begin{subarray}{c}M=2\\\ \mathrm{even}\
M\end{subarray}}^{\infty}2\zeta(M)\braket{Q_{A}^{M}}_{c}.$

This is the cumulant expansion for von Neumann entropy (37).


– Under Review 2024 Full Paper – MIDL 2024 submission Tianyi RenContributed
equally1<EMAIL_ADDRESS>
Abhishek Sharma1<EMAIL_ADDRESS>
Juampablo Heras Rivera1<EMAIL_ADDRESS>
Harshitha Rebala 2<EMAIL_ADDRESS>
Ethan Honey1<EMAIL_ADDRESS>
Agamdeep Chopra1<EMAIL_ADDRESS>
Jacob Ruzevick3<EMAIL_ADDRESS>
Mehmet Kurt1<EMAIL_ADDRESS>
1 Department of Mechanical Engineering, University of Washington
2 Paul G. Allen School of Computer Science, University of Washington
3 Department of Neurological Surgery, University of Washington

# Re-DiffiNet: Modeling discrepancies loss in tumor segmentation using
diffusion models

###### Abstract

Identification of tumor margins is essential for surgical decision-making for
glioblastoma patients and provides reliable assistance for neurosurgeons.
Despite improvements in deep learning architectures for tumor segmentation
over the years, creating a fully autonomous system suitable for clinical
floors remains a formidable challenge because the model predictions have not
yet reached the desired level of accuracy and generalizability for clinical
applications. Generative modeling techniques have seen significant
improvements in recent times. Specifically, Generative Adversarial Networks
(GANs) and Denoising-diffusion-based models (DDPMs) have been used to generate
higher-quality images with fewer artifacts and finer attributes. In this work,
we introduce a framework called Re-Diffinet for modeling the discrepancy
between the outputs of a segmentation model like U-Net and the ground truth,
using DDPMs. By explicitly modeling the discrepancy, the results show an
average improvement of 0.55% in the Dice score and 16.28% in HD95 from cross-
validation over 5-folds, compared to the state-of-the-art U-Net segmentation
model. The code is available: https://github.com/KurtLabUW/Re-DiffiNet.git.

###### keywords:

Tumor segmentation, DDPMs, MRI, Deep learning

††editors: Under Review for MIDL 2024

##  1 Introduction

### Background: Brain Tumor Segmentation

Glioblastoma is the most frequent primary malignant brain tumor in adults,
representing approximately 57% of all gliomas and 48% of all primary malignant
central nervous system (CNS) tumors [Ostrom et al.(2018)Ostrom, Gittleman,
Truitt, Boscia, Kruchko, and Barnholtz-Sloan, Tan et al.(2020)Tan, Ashley,
López, Malinzak, Friedman, and Khasraw]. This heterogeneous group of tumors is
characterized by their resemblance to glia that perform a variety of important
functions including support to neurons [Isensee et al.(2021)Isensee, Jaeger,
Kohl, Petersen, and Maier-Hein, Ahuja et al.(2020)Ahuja, Panigrahi, and
Gandhi].

The treatment for glioma patients generally consists of surgery, radiotherapy,
and chemotherapy and the outcomes of patients with gliomas vary widely
according to the glioma type and prognostic factors. Due to the superior soft
tissue contrast, multimodal MRI images which allow the complexity and the
heterogeneity of the tumor lesion to be better visualized than a CT scan have
become the golden standard for surgical decision-making for glioma patients
[Hanif et al.(2017)Hanif, Muzaffar, Perveen, Malhi, and Simjee, Keunen et
al.(2014)Keunen, Taxt, Grüner, Lund-Johansen, Tonn, Pavlin, Bjerkvig, Niclou,
and Thorsen, van Dijken et al.(2019)van Dijken, van Laar, Smits, Dankbaar,
Enting, and van der Hoorn]. However, visual identification of tumor margins in
CT or MRI still remains a challenge for neurosurgeons and researchers [Wang et
al.(2019)Wang, Liang, Li, Liu, Huang, and Li]. Clinically, brain tumor masks
are often obtained through Magnetic Resonance Imaging (MRI) scans, which
require experienced radiologists to manually segment tumor sub-regions [Baid
et al.(2021b)Baid, Ghodasara, Mohan, Bilello, Calabrese, Colak, Farahani,
Kalpathy-Cramer, Kitamura, Pati, et al.]. This is a long process that is
unscalable to the needs of all patients. Thus, the recent growth of machine
learning technologies holds promise to provide a reliable and automated
solution to segmentation to save time and help medical professionals with this
process [Luu and Park(2021)].

Deep learning techniques have been widely used in brain tumor segmentation.
U-Net is the state of art for tumor segmentation. U-Net and its variants have
been used in brain tumor segmentation. such as U-Net++ [Zhou et al.(2018)Zhou,
Siddiquee, Tajbakhsh, and Liang], 3D U-Net [Çiçek et al.(2016)Çiçek,
Abdulkadir, Lienkamp, Brox, and Ronneberger], V-Net [Milletari et
al.(2016)Milletari, Navab, and Ahmadi], and Attention-U-Net [Oktay et
al.(2018)Oktay, Schlemper, Folgoc, Lee, Heinrich, Misawa, Mori, McDonagh,
Hammerla, Kainz, Glocker, and Rueckert]. Transformer architectures has also
been applied in brain tumor segmentation. TransU-Net and Swin-U-Net show
potential to predict accurate tumor margins. However, the state-of-the-art
models in brain tumor segmentation are still based on the encoder-decoder
architectures such as U-Net [Isensee et al.(2021)Isensee, Jaeger, Kohl,
Petersen, and Maier-Hein] and its variations. For instance, Luu et. al [Luu
and Park(2021)] modified the nnU-Net model by adding an axial attention in the
decoder. Futrega et. al [Futrega et al.(2021)Futrega, Milesi, Marcinkiewicz,
and Ribalta] optimized the U-Net model by adding foreground voxels to the
input data, increasing the encoder depth and convolutional filters. Siddiquee
et. al [Siddiquee and Myronenko(2021)] applied adaptive ensembling to minimize
redundancy under perturbations.

While U-Net-based architecture have led to significant improvements in region-
based metrics for tumor segmentation e.g. Dice scores, it is also important to
improve boundary-distance metrics like HD scores [Karimi and Salcudean(2019),
Yeghiazaryan and Voiculescu(2018)]. Being able to locate boundaries of tumors
is crucial for surgical planning. Thus, modeling techniques that are able to
capture finer details and high frequency information at the boundaries, are
desirable. One of the critical factors that makes predicting tumor boundaries
difficult, is the inherent variability in tumor attributes at the boundaries.
Thus, the modeling techniques also need to be able to capture the variability
in tumor shapes.

Generative modeling techniques have seen great improvements in recent times.
Specifically, Generative Adversarial Networks and Denoising-Diffusion based
models have been used to generate desired images of greater quality. While
GANs are able to generate images of high fidelity, they are also prone to mode
collapse. Thus, they often fail to capture the variability of the data they
seek to model. On the other hand, Denoising-Diffusion based models have been
shown to be good at both mode coverage i.e. capturing the variability in the
data [Kingma et al.(2021)Kingma, Salimans, Poole, and Ho], as well as at
generating high quality images [Dhariwal and Nichol(2021)]. There have been
very few instances of Diffusion models being used for brain tumor
segmentation, that have shown promising results. [Xing et al.(2023)Xing, Wan,
Fu, Yang, and Zhu, Wolleb et al.(2022)Wolleb, Sandkühler, Bieder, Valmaggia,
and Cattin, Wu et al.(2022)Wu, Fang, Zhang, Yang, and Xu].

In this work, we introduce a framework called Re-Diffinet, for modeling
discrepancy between the outputs of a segmentation model like U-Net and the
ground truth, using the advantages of Denoising Diffusion Probabilistic Models
and U-Net model. By explicitly modeling the discrepancy, we intend to build
upon previous segmentation models, force diffusion models to focus explicitly
on the regions that other models miss, and exploit diffusion models’ ability
to capture finer details and variability in the data.

##  2 Methods

###  2.1 Dataset

####  2.1.1 BraTS2023

The training dataset provided for the BraTS23 challenge [Baid et
al.(2021a)Baid, Ghodasara, Mohan, Bilello, Calabrese, Colak, Farahani,
Kalpathy-Cramer, Kitamura, Pati, et al.] consists of 1251 brain MRI scans
along with segmentation annotations of tumorous regions. The 3D volumes were
skull-stripped and resampled to 1$mm^{3}$ isotropic resolution, with
dimensions of (240, 240, 155) voxels. For each example, four modalities were
given: native (T1), post-contrast T1-weighted (T1Gd), T2-weighted (T2), and T2
Fluid Attenuated Inversion Recovery (T2-FLAIR). Segmentation labels were
annotated manually by one to four experts. Annotations consist of three
disjoint classes: enhancing tumor (ET), peritumoral edematous tissue (ED), and
necrotic tumor core (NCR). To get the ground truth labels for these datasets,
all imaging volumes have then been segmented using the STAPLE [Warfield et
al.(2004)Warfield, Zou, and Wells] fusion of previous top-ranked BraTS
algorithms, such as nnU-Net [Isensee et al.(2021)Isensee, Jaeger, Kohl,
Petersen, and Maier-Hein]. These segmented labels were then refined manually
by volunteer neuroradiology experts following a consistently communicated
annotation protocol. The manually refined annotations were finally approved by
experienced board-certified attending neuro-radiologists.

####  2.1.2 Data Reprocessing

For all the MRI contrasts in the BraTS2023 dataset, we rescale the voxel
intensity after Z-Score normalization as the preprocessing protocol.

fig:fig2

Figure 1: Re-DiffiNet uses MRI and predictions from baseline U-Net as inputs
to generate predictions about incorrect voxels in U-Net predictions and
corrects those voxels to generate redefined tumor masks.

####  2.1.3 Evaluation Metrics

The model will be evaluated on two metrics: Dice similarity coefficient (DICE)
measures the similarity between the model prediction and the ground truth;
Hausdorff distance (95%) measures the boundary distance between the model
prediction and the ground truth.

###  2.2 Model Architectures

####  2.2.1 Baseline U-Net

We adapted the optimized U-Net [Futrega et al.(2021)Futrega, Milesi,
Marcinkiewicz, and Ribalta] as our baseline model architecture [Ren et
al.(2024)Ren, Honey, Rebala, Sharma, Chopra, and Kurt] for comparison
purposes. U-Net has a symmetric U-shape that characterizes architecture and
can be divided into two parts, i.e., encoder and decoder. The encoder
comprises 5 levels of same-resolution convolutional layers with strided
convolution downsampling. The decoder follows the same structure with
transpose convolution upsampling and convolution operating on concatenated
skip features from the encoder branch at the same level. The training dataset
is comprised of the pairs $\\{(I,x_{0})\\}_{i=1}^{N}$, where
$I\in\mathbb{R}^{4\times D\times W\times H}$ represents the four 3D-MRI
contrast as multi-channel input, $x_{0}\in\mathbb{R}^{3\times D\times W\times
H}$ represents the associated one-hot encoded segmentation mask, with 3 tumor
labels: 1) Whole Tumor, 2) Enhancing Tumor, and 3) Necrotic Tumor Core. The
baseline U-Net predicts the tumor labels $\hat{x}_{0}$ given the input $I$:

$\hat{x}_{0}=U(I)$ (1)

####  2.2.2 U-Net augmented Diffusion (UA-Diffusion)

This model builds upon the Diff-U-Net model proposed by [Xing et
al.(2023)Xing, Wan, Fu, Yang, and Zhu]. Diff-U-Net conditions DDPMs with MRIs
at each denoising step to generate the corresponding tumor masks. In
comparison, we condition our diffusion model with predictions ($U(I)$) from
baseline U-Net along with MRIs ($I$) as shown in figure LABEL:fig:fig2. We
tested 3 variants of this approach (3 different inputs) as explained in
section 3. Among the 3 variants, we chose the best performing UA-Diffusion
approach (Table LABEL:tab:results1) and used it for the remaining experiments.
The expression for the best performing variant of UA-Diffusion is shown in
equation 2:

$\hat{x}_{0}=DU(\text{cat}(U(I),I,x_{t}),t,\hat{I}_{f})$ (2)

where t is the time embedding, $x_{t}$ is the corresponding noise masks,
$\hat{I}_{f}=\xi(\text{cat}(U(I),I))$ are the multi-scale features extracted
using a trainable copy ($\xi$) of the encoder of the denoising-U-Net ($DU$).
These multi-scale features are added to the outputs of the corresponding
layers in the denoising-U-Net, $DU$ (see Figure LABEL:fig:fig2).

####  2.2.3 Re-DiffiNet

Our proposed Re-DiffiNet model architecture uses the same architecture as the
U-Net augmented Diffusion (UA-Diffusion). However, instead of trying to
generate ground truth segmentation masks ${x}_{0}$, we generate the absolute
discrepancy between the ground truth segmentation masks and baseline U-Net’s
predictions i.e. $\Delta{x}_{0}=abs(U(I)-x_{0})$ (Figure LABEL:fig:fig2).
These discrepancy masks ($\Delta{x}_{0}$) will have a value of $1$ for each
voxel that is predicted incorrectly by the baseline U-Net ($U$), and 0 for
voxels where the predictions are correct. Once, we have generated the
estimated discrepancies $\Delta{\hat{x}}_{0}$, the tumor mask predictions can
be obtained as shown in equation 3 :

$\Delta{\hat{x}}_{0}=DU(\text{cat}(U(I),I,x_{t}),t,\hat{I}_{f})\\\
\Rightarrow{\hat{x}}_{0}=abs(U(I)-\Delta{\hat{x}}_{0})\\\ $ (3)

Subtracting the estimated discrepancy $\Delta{\hat{x}}_{0}$ from $U(I)$ and
taking the absolute, we flip every incorrect voxel (as per our estimate) in
$U(I)$ i.e. $1\rightarrow 0$ and $0\rightarrow 1$. While, the correct voxels
(as per our estimation) in the baseline U-Net, remain the same.

tab:results1  Model Dice HD95(mm) WT ET TC Avg WT ET TC Avg Baseline U-Net
92.63% 86.87% 93.28% 90.93% 1.06 1.62 1.57 1.42 Diff-U-Net 87.98% 83.92%
86.25% 86.05% 2.46 3.56 3.32 3.11 UA-Diffusion 90.72% 83.76% 86.57% 87.02%
1.12 1.90 2.56 1.86 (Input: U(I)) UA-Diffusion 92.86% 85.08% 91.43% 89.79%
1.39 1.93 1.59 1.63 (Input: concat(I, U(I)) UA-Diffusion 91.32% 84.46% 91.18%
88.99% 1.46 1.74 1.89 1.70 (Input: mask$(I,U(I))$) see eq.4 Re-DiffiNet 93.23%
86.79% 93.98% 91.33% 0.87 1.27 1.34 1.16

Table 1: Comparison of the proposed model architecture in section 2.2.

###  2.3 Training details

Our models were implemented in Pytorch and MONAI, and trained on 2 NVIDIA A40
GPUs. The model was trained on overlapping regions, whole tumor (WT), tumor
core (TC), and enhancing tumor(ET). TC entails the ET, as well as the necrotic
(NCR) parts of the tumor, and WT describes the complete extent of the disease.
The diffusion models was trained using a compound loss function including DICE
loss, Binary cross entropy (BCE) loss, and Mean square error(MSE) loss. The
model was trained using the AdamW optimizer with a learning rate of 0.0001 and
a weight decay equal to 0.0001. The network’s performance was evaluated using
5-fold cross-validation. The data were randomly shuffled and equally split
into 5 groups for cross-validation.

##  3 Experiment and Results

We trained 3 models 1) Baseline U-Net (section 2.2.1), 2) U-Net augmented
diffusion or UA-Diffusion (section 2.2.2), and 3) _Re-DiffiNet_ (section
2.2.3). We first trained the baseline U-Net model. Then, the predictions of
the baseline U-Net model were used as inputs in U-Net augmented diffusion (UA-
Diffusion), and Re-DiffiNet. In a preliminary study, we trained 3 variants of
the U-Net augmented diffusion (UA-Diffusion) on a random train-test split and
compared them with the baseline-U-Net,: 1) Conditioning the diffusion model
with only the U-Net output $U(I)$, 2) Conditioning the diffusion model with a
concatenation of MRI contrasts and baseline U-Net predictions $U(I)$, 3)
conditioning the diffusion model with MRIs $I$ masked by U-Net predictions
$U(I)$. A mask which has a value 1 for each tumor voxel and 0.2 for non-tumor
voxel is applied to each of the 4 MRI contrasts, which are concatenated and
used as inputs. The resulting masked input is represented as shown in equation
4 :

$\aligned M^{\prime}(x,y,z)&=\cases{1}\text{if}U(I)[x,y,z]>0,\\\
0.2\text{if}U(I)[x,y,z]=0,\text{wherex,y,xarevoxelindices}\\\
mask(I,U(I))=concat(I_{i}\circ
M^{\prime}|i=1,2,3,4),\text{whereidenotesanMRIcontrast}$ (4)

tab:results2  Fold # Model Dice HD95(mm) WT ET TC Avg WT ET TC Avg fold1
Baseline U-Net 92.63% 86.87% 93.28% 90.93% 1.06 1.62 1.57 1.42 UA-Diffusion
92.72% 86.76% 93.57% 91.02% 1.12 1.40 1.56 1.36 Re-DiffiNet 93.23% 86.79%
93.98% 91.33% 0.87 1.27 1.34 1.16 fold2 Baseline U-Net 92.60% 88.30% 93.79%
91.56% 1.18 1.77 1.24 1.40 UA-Diffusion 92.62% 87.86% 94.09% 91.52% 1.19 1.73
1.18 1.37 Re-DiffiNet 93.04% 87.34% 94.48% 91.62% 0.97 1.67 0.85 1.16 fold3
Baseline U-Net 92.40% 87.04% 92.47% 90.64% 1.41 1.78 1.46 1.55 UA-Diffusion
92.93% 87.22% 93.21% 91.12% 1.05 1.62 1.14 1.27 Re-DiffiNet 92.86% 87.23%
93.11% 91.07% 1.07 1.60 1.21 1.29 fold4 Baseline U-Net 91.21% 86.90% 92.66%
89.06% 1.62 1.74 1.30 1.55 UA-Diffusion 91.32% 86.25% 92.99% 88.79% 1.61 1.73
1.26 1.53 Re-DiffiNet 91.73% 87.18% 92.91% 89.46% 1.58 1.64 1.21 1.48 fold5
Baseline U-Net 91.30% 86.61% 93.25% 90.39% 1.34 1.72 1.16 1.41 UA-Diffusion
91.43% 87.01% 93.56% 90.67% 1.30 1.68 1.18 1.39 Re-DiffiNet 92.72% 87.81%
94.30% 91.61% 1.15 1.37 1.06 1.04

Table 2: 5 fold cross-validation results for 3 models: Baseline U-Net, U-Net
augmented Diffusion (UA-Diffusion) with concatenation of MRI and Baseline
U-Net predictions as input, and Re-DiffiNet

We found that using diffusion directly to predict tumor masks doesn’t lead to
any significant performance gains over the baseline U-Net, as shown in Table
LABEL:tab:results1. On the other hand, using diffusion model to predict
discrepancies and using them to correct U-Net’s outputs leads to significant
performance gains specially in terms of HD95 score. Among the 3 UA-Diffusion
approaches concatenating the U-Net prediction and MRI yielded the best
performance. Thus, we use the UA-Diffusion with concatenation of MRI and U-Net
prediction, for 5-fold cross-validation in Table LABEL:tab:results2.

The results of 5-fold cross-validation are shown in Table LABEL:tab:results2,
which reports the Dice Score (DICE) and 95 percentile Hausdorff distance
(HD95) and the average scores of all methods on the three overlapping regions
whole tumor (WT), tumor core (TC) and Enhancing tumor (ET) for the BraTS2023
dataset (Figure LABEL:fig:fig3). We found that while using the diffusion model
to directly output the tumor segmentation mask does lead to improvements over
the U-Net model, the improvements are modest: 0.12% improvement in Dice, and
5.61% improvement in HD95 score. On the other hand, with Re-DiffiNet we found
a 16.28% improvement in HD95 score, indicating the benefits of modeling
discrepancy using diffusion models, while simultaneously the Dice score was
comparable with the baseline U-Net (0.55% improvement). Figure LABEL:fig:fig3
shows an example of the segmented masks of baseline U-Net and Re-DiffiNet.

fig:fig3

Figure 2: A comparison between the segmentations generated by baseline U-Net,
and Re-DiffiNet. In this example, Re-DiffiNet can predict the false positive
lesion on Tumor core masks that was predicted by baseline U-Net. Meanwhile,
Re-DiffiNet predicts a smoother boundary

##  4 Discussion and Conclusion

In this research, we proposed a tumor segmentation framework Re-DiffiNet,
which uses diffusion models to refine and improve predictions of a tumor
segmentation model (like optimized U-Net). Most tumor segmentation studies
optimize for region-based metrics like Dice scores, and have been able to show
high Dice score in the range of 90% or greater. However, boundary-distance
metrics like HD scores are also critical, and being able to improve upon these
score while not sacrificing performance on Dice score is highly desirable. In
this work, we investigated the potential to refine predictions generated by
state-of-the-art U-Net models using diffusion models.

We found that while using diffusion models to directly generate tumor masks
did lead to improvements in performance over the baseline U-Net, it was the
use of discrepancy modeling i.e. predicting the differences between ground
truth masks and baseline U-Net’s outputs, that led to most significant
improvements. This was indicated by 16.28% improvement in HD-95 score,
highlighting significant improvements on the boundaries of tumors. While,
discrepancies can be modeled by any other modeling technique (even a U-Net),
effectively acting as a boosting method, we chose diffusion primarily because
of its ability to generate high-fidelity visual attributes, as well as capture
variability in the data distribution, both of which are exhibited by brain
tumors. Another benefit of using diffusion models instead of a U-Net to
improve the baseline U-Net’s predictions, would be the potential to learn more
robust and diverse representations from the data, due to the inherently
different mechanism using which diffusion models are trained.

Our work shows the potential of further improving tumor segmentation by
combining diffusion models and discrepancy modeling. In this work, we
investigated Re-DiffiNet for the segmentation of gliomas. In the future, we
intend to test our approach to improve the segmentation of other kinds of
tumors like meningioma, and pediatric brain tumors.

Juampablo Heras Rivera is supported by the U.S. Department of Energy, Office
of Science, Office of Advanced Scientific Computing Research, Department of
Energy Computational Science Graduate Fellowship under Award Number DE-
SC0024386.

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