# Two-fermion negativity and confinement in the Schwinger model

Adrien Florio  0000-0002-7276-4515<EMAIL_ADDRESS>Department of Physics,
Brookhaven National Laboratory, Upton, NY 11973

###### Abstract

We consider the fermionic (logarithmic) negativity between two fermionic modes
in the Schwinger model. Recent results pointed out that fermionic systems can
exhibit stronger entanglement than bosonic systems, exhibiting a negativity
that decays only algebraically. The Schwinger model is described by fermionic
excitations at short distances, while its asymptotic spectrum is the one of a
bosonic theory. We show that the two-mode negativity detects this confining,
fermion-to-boson transition, shifting from an algebraic decay to an
exponential decay at distances of the order of the de Broglie wavelength of
the first excited state. We derive analytical expressions in the massless
Schwinger model and confront them with tensor network simulations. We also
perform tensor network simulations in the massive model, which is not solvable
analytically, and close to the Ising quantum critical point of the Schwinger
model, where we show that the negativity behaves as its bosonic counterpart.

Introduction. Entanglement is a defining property of the quantum world. It is
key to understand its structural impact on quantum many-body systems and
quantum field theories. This was realized in the early days of modern quantum
field theory Reeh and Schlieder (1961). It has led to a plethora of works,
starting from the conjecture of the area law for the entanglement entropy
Srednicki (1993), computation of the entanglement entropy in CFT Calabrese and
Cardy (2004), and in holography Ryu and Takayanagi (2006). More than their
intrinsic interests, these studies have led to reconsidering and gaining
insights into old field theoretic questions and considering new problems.
Examples range from understanding high-energy scattering in terms of
entanglement Kharzeev and Levin (2017, 2021); Gong  _et al._ (2022); de Jong
_et al._ (2022); Florio  _et al._ (2023); Belyansky  _et al._ (2023); Barata
_et al._ (2023); Lee  _et al._ (2023), entanglement in pair production Florio
and Kharzeev (2021); Dunne  _et al._ (2023); Grieninger  _et al._ (2023a, b),
the impact of entanglement structures on the systematic construction of
effective field theories Beane  _et al._ (2019); Klco and Savage (2021a, b);
Klco  _et al._ (2023), links to thermalization Kharzeev and Tuchin (2005);
Berges  _et al._ (2018); Zhou  _et al._ (2022); Mueller  _et al._ (2022);
Ebner  _et al._ (2023); Yao (2023); Grieninger  _et al._ (2023c), and to
partial and dynamical symmetry breaking and restoration Ares  _et al._ (2023a,
b, c), see also Nishioka (2018); Klco  _et al._ (2022); Bauer  _et al._ (2023)
for reviews on the interface of quantum information science and field theory.

To fully apprehend entanglement, the study of mixed states is crucial. From a
many-body perspective, it allows, for instance, to probe entanglement between
subsystems. In this case, entanglement measures are not unique. The
(logarithmic) negativity is a useful one as it provides an upper bound on the
amount of distillable entanglement, i.e. the amount of entanglement that can
be recovered using local observables Peres (1996); Horodecki (1997); Simon
(2000); Vidal and Werner (2002); Plenio (2005). Its behavior has been
extensively studied in CFTs Calabrese  _et al._ (2012, 2013); Murciano  _et
al._ (2021, 2022) and was also explored in free scalar field theories Klco and
Savage (2021a, b); Klco  _et al._ (2023). Recently, Ref. Parez and Witczak-
Krempa (2023) pointed out the interesting fact that fermionic degrees of
freedom can carry more entanglement than bosonic degrees of freedom. In
particular, they exhibited systems where the fermionic negativity decays only
as a power law. This is in contrast to bosonic systems, as even close to
critical points, the logarithmic negativity decays faster than any power
Marcovitch  _et al._ (2009); Calabrese  _et al._ (2012). Having fermionic
excitations potentially more strongly entangled than bosonic excitations is
interesting in relation to confining theories, where the spectrum of the
theory is not made out of trivial excitations of the microscopic particles. It
is suggestive that when a fermionic theory gives rise only to bosonic
excitations, the spatial distribution of negativity should probe the scale at
which the theory confines. As a result, an intriguing problem to investigate
is the relationship between entanglement measures and confinement.

This work explores these ideas within the Schwinger model, quantum
electrodynamics in $1+1$ dimensions:

$H=\int\mathrm{d}x\left[\overline{\psi}\left(\gamma^{1}(-i\partial_{1}+A_{1})+m\right)\psi\right]+\tfrac{g^{2}}{2}E^{2}\
,$ (1)

with $A_{1}$ the gauge potential, $E$ is the canonically conjugate operator
for the electric field and $\psi^{T}=(\psi_{1}\ \psi_{2})$ is a two component
spinor. Gauss’s constraint supplements it,

$\partial_{1}E=\overline{\psi}\gamma^{0}\psi\ \equiv\psi^{\dagger}\psi\ ,$ (2)

with $\bar{\psi}\equiv\psi^{\dagger}\gamma^{0}$. It is the manifestation of
the remaining “gauge invariance” of the Hamiltonian formalism. We express the
$\gamma$ matrices in terms of Pauli matrices as

$\gamma^{0}=\sigma_{z}\ ,\quad\gamma^{1}=-i\sigma_{y}\
,\quad\gamma^{5}=\gamma^{0}\gamma^{1}=-\sigma_{x}\ .$ (3)

It is an ideal playground to test these ideas as it is known to be dual to a
bosonic scalar theory and that asymptotic excitations of the model, “mesons”,
are bosons (they are solitons made out of the scalar field) Coleman  _et al._
(1975); Coleman (1976). In other words, it behaves as a fermionic theory only
over the shortest scales and transitions to a bosonic behavior at larger
scales. Moreover, when the mass parameter $m$ vanishes, the bosonic theory is
simply a massive free scalar field with mass $m_{b}=g/\sqrt{\pi}$ Schwinger
(1962). In this case, the theory is analytically solvable.

We will study a standard fermionic chain that approaches the Schwinger model
in the continuum. It can be constructed by introducing “staggered fermions”
Kogut and Susskind (1975); Susskind (1977)

$\chi_{2j}=\sqrt{2a}\psi_{1}(x_{j}),\ \chi_{2j+1}=\sqrt{2a}\psi_{2}(x_{j})\ .$
(4)

As depicted in Fig. 1, lattice sites are labeled by an integer $n=1,\dots,2N$.
A physical site is made out of two lattice sites. The upper component of the
continuum spinors lives on odd sites, while the down component lives on even
sites. Physical distances are measured in units of $1/2a$ with $a$ the lattice
spacing. We further impose open-boundary conditions on the fermions
$\chi_{0}=\chi_{2N+1}=0$. In this setting, the remaining gauge freedom allows
to set the gauge potential to zero and express the electric field only in
terms of fermionic operators by solving Gauss law, see for instance Ikeda  _et
al._ (2021) for explicit expressions. The resulting Hamiltonian reads

$\displaystyle H_{S}^{L}$
$\displaystyle=-\frac{i}{2a}\sum_{n=1}^{N-1}\big{[}\chi^{\dagger}_{n}\chi_{n+1}-\chi^{\dagger}_{n+1}\chi_{n}\big{]}+m\sum_{n=1}^{N}(-1)^{n}\chi^{\dagger}_{n}\chi_{n}$
$\displaystyle+\frac{ag^{2}}{2}\sum_{n=1}^{N-1}\left(\sum_{i=1}^{n}Q_{i}\right)^{2}\
,$ (5)

with $Q_{i}=n_{i}\left(1-(-1)^{i}\right)$ the lattice charge density operator
and $n_{i}=\chi^{\dagger}_{i}\chi_{i}$ the fermion number operator.

We will investigate the two-fermion (logarithmic) negativity. After defining
it and studying its behavior in the continuum, we show that it indeed
transitions from an algebraic decay at small distances to an exponential decay
at large distances. Moreover, the transition happens precisely at the de
Broglie wavelength of the first bosonic excited state. We conclude by
investigating the behavior of the logarithmic fermionic negativity at the
quantum Ising critical point of the Schwinger model. We show that even when
using a fermionic description, the bosonic expectation of the theory is met,
and the negativity decays exponentially fast.

Two-fermion negativity and the continuum limit. We start from the simple
realization that a tractable expression for the density matrix for two
staggered modes can be written down explicitly in terms of the staggered
fields’ two- and four-point functions Javanmard  _et al._ (2018); Berthiere
and Witczak-Krempa (2022); Parez and Witczak-Krempa (2023). We consider the
reduced density matrix between modes whose physical sites are separated by a
physical distance $r$ and are equally distant from the center of the lattice,
as depicted in Fig. 1. Because of the staggering, the sites are not related by
a reflection symmetry, and we need a generalization of Parez and Witczak-
Krempa (2023). Consider the state
$|\psi\rangle=\sum_{i_{1},\dots,i_{2N}=0,1}c_{i_{1},\dots,i_{2N}}|i_{1},\dots,i_{2N}\rangle$.
The reduced density matrix obtained after tracing all sites except $j$ and $k$
reads

$\displaystyle\rho_{j\leftrightarrow k}$
$\displaystyle=\sum_{i_{j}^{\prime},i_{k}^{\prime},i_{j},i_{k}=0,1}\Lambda_{i_{j}^{\prime}i_{k}^{\prime},i_{j}i_{k}}|i_{j}^{\prime}i_{k}^{\prime}\rangle\langle
i_{j}i_{k}|\ ,$ (6)
$\displaystyle\Lambda_{i_{j}^{\prime}i_{k}^{\prime},i_{j}i_{k}}$
$\displaystyle=\sum_{\begin{subarray}{c}\\{i_{l}\\}=0,1\\\ l\neq
j,k\end{subarray}}c_{i_{1},\dots i_{j},\dots
i_{k},\dots,i_{2N}}^{*}c_{i_{1},\dots i_{j}^{\prime},\dots
i_{k}^{\prime},\dots,i_{2N}}\ .$

The matrix elements $\Lambda_{i_{j}^{\prime}i_{k}^{\prime},i_{j}i_{k}}$ can be
expressed in terms of the fermion fields’ two- and four-point functions. We
restrict ourselves to states with zero total charge $Q=0$. As a result, the
density matrix takes the following form

$\rho_{j\leftrightarrow k}=\begin{pmatrix}\Lambda_{00,00}&0&0&0\\\
0&\Lambda_{01,01}&\Lambda_{01,10}&0\\\
0&\Lambda_{01,10}^{*}&\Lambda_{10,10}&0\\\
0&0&0&\Lambda_{11,11}\end{pmatrix}.$ (7)

Direct computations give

$\displaystyle\langle n_{j}\rangle_{\psi}=\Lambda_{10,10}+\Lambda_{11,11},$
$\displaystyle\ \ \langle
n_{k}\rangle_{\psi}=\Lambda_{01,01}+\Lambda_{11,11},$ $\displaystyle\langle
n_{j}n_{k}\rangle_{\psi}=\Lambda_{11,11},$ $\displaystyle\ \
\langle\chi_{j}^{\dagger}\chi_{k}\rangle_{\psi}=\Lambda_{01,10},$ (8)

using the shorthand notation
$\langle\dots\rangle_{\psi}\equiv\langle\psi|\dots|\psi\rangle$. Thus, using
(8) and $\mathrm{Tr}\left(\rho_{j\leftrightarrow k}\right)=1$ in the
occupation number basis $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ ,
we have,

$\displaystyle\Lambda_{00,00}=1-\langle n_{j}\rangle_{\psi}-\langle
n_{k}\rangle_{\psi}+\langle n_{j}n_{k}\rangle_{\psi}\ ,$
$\displaystyle\Lambda_{10,10}=\langle n_{j}\rangle_{\psi}-\langle
n_{j}n_{k}\rangle_{\psi}\ ,$ $\displaystyle\Lambda_{11,11}=\langle
n_{j}n_{k}\rangle_{\psi}\ ,$ $\displaystyle\Lambda_{01,01}=\langle
n_{k}\rangle_{\psi}-\langle n_{j}n_{k}\rangle_{\psi}\ ,$
$\displaystyle\Lambda_{01,10}=\langle\chi_{j}^{\dagger}\chi_{k}\rangle_{\psi}\
.$

We need the fermionic partial transpose of (10) to compute the negativity.
Following Shapourian  _et al._ (2017), it acts on our two-fermion reduced
Hilbert space as

$\displaystyle(|i_{j}^{\prime}i_{k}^{\prime}\rangle\langle
i_{j}i_{k}|)^{T^{f}_{j}}$
$\displaystyle=(-1)^{\alpha}|i_{j}i_{k}^{\prime}\rangle\langle
i_{j}^{\prime}i_{k}|\ ,$ (9) $\displaystyle\alpha$
$\displaystyle=\frac{i_{j}^{\prime}(i_{j}^{\prime}+2)}{2}+\frac{i_{j}(i_{j}+2)}{2}$
$\displaystyle+i_{k}i_{k}^{\prime}+i_{j}^{\prime}i_{k}^{\prime}+i_{j}i_{k}+(i_{j}+i_{k})(i_{j}^{\prime}+i_{k}^{\prime})\
.$

It leads to

$\rho_{j\leftrightarrow
k}^{T^{j}_{j}}=\begin{pmatrix}\Lambda_{00,00}&0&0&i\Lambda_{01,10}^{*}\\\
0&\Lambda_{01,01}&0&0\\\ 0&0&\Lambda_{10,10}&0\\\
i\Lambda_{01,10}&0&0&\Lambda_{11,11}\end{pmatrix},$ (10)

with eigenvalues $\lambda_{1},\lambda_{2},\lambda_{+},\lambda_{-}$ :

$\displaystyle\lambda_{1}$ $\displaystyle=\Lambda_{01,01}\ \ ,\ \ \ \ \ \ \
\lambda_{2}=\Lambda_{10,10}\ ,$ $\displaystyle\lambda_{\pm}$
$\displaystyle=\frac{1}{2}\big{(}1-\Lambda_{01,01}-\Lambda_{10,10}\ $ (11)
$\displaystyle\pm\sqrt{-4|\Lambda_{01,10}|^{2}+(-\Lambda_{00,00}+\Lambda_{11,11})^{2})}\big{)}\
.$

The eigenvalues $\lambda_{1,2}$ are real and positive. We can take
$\lambda_{\pm}=\lambda_{R}\pm i\lambda_{I}$ with $\lambda_{I}\neq 0$ as when
they are real, they are also positive, as shown in App. A. Then,

$\displaystyle\mathcal{N}^{2f}_{\alpha\beta}$
$\displaystyle=|\lambda_{\pm}|-\lambda_{R}$ (12)
$\displaystyle\mathcal{\epsilon}_{\mathcal{N}_{2f}}$
$\displaystyle=\log(1+2(|\lambda_{\pm}|-\lambda_{R}))\ .$ (13)

These simple expressions allow us to investigate what happens in the continuum
limit. We focus on the ground state of the Schwinger model. First, we recover
translational invariance. Then, $\langle n_{x}\rangle=1/2$ as the vacuum state
of the field theory corresponds to half-filling in terms of the staggered
fermions. For the same reason, the correlator that has a continuum limit is
the connected two-point function of $\langle\delta n_{j}\delta n_{k}\rangle$
(equivalent to “normal ordering” in the continuum theory). Taking $x_{j}$ and
$x_{k}$ the physical points associated with $j$ and $k$, see Fig. 1, we write
its continuum limit $n_{2}(r)$, with $r=x_{j}-x_{k}$ (it can be expressed in
terms of the vector charge and the chiral condensate, but it won’t be used
here). Lastly, $\langle\chi^{\dagger}_{j}\chi_{k}\rangle$ converges to
appropriate components of the continuum, full equal time correlator of the
theory $D_{\alpha\beta}(r)$. In more detail, we have, depending on their
parity,

$\begin{pmatrix}\langle\chi^{\dagger}_{2x_{j}}\chi_{2x_{k}}\rangle&\langle\chi^{\dagger}_{2x_{j}}\chi_{2x_{k}+1}\rangle\\\
\langle\chi^{\dagger}_{2x_{j}+1}\chi_{2x_{k}}\rangle&\langle\chi^{\dagger}_{2x_{j}+1}\chi_{2x_{k}+1}\rangle\end{pmatrix}{\to}\begin{pmatrix}D_{11}(r)&D_{12}(r)\\\
D_{21}(r)&D_{22}(r)\end{pmatrix}\ .$ (14)

This is how the fact that the Dirac propagator in the continuum is a two-by-
two matrix manifests itself in the staggered formulation.

Reinstating the correct factors of $a$ according to (4), we have

$\displaystyle\Lambda_{00,00},\Lambda_{11,11}\to 2a^{2}n_{2}+\frac{1}{4}\ ,\ $
(15) $\displaystyle\Lambda_{10,10},\Lambda_{01,01}\to\frac{1}{4}-2a^{2}n_{2}\
\ ,\ \Lambda_{01,10}\to 2a^{2}D_{\alpha\beta}\ \ $ (16)
$\displaystyle\lambda_{\pm}\to\frac{1}{4}-2a^{2}n_{2}\pm
2ia^{2}D_{\alpha\beta}\ .$ (17)

Note that, notwithstanding the continuum analysis, it matches the results of
Parez and Witczak-Krempa (2023) around half-filling.

Now, we are in a position to study the continuum behavior of the two-fermion
negativity. Plugging this in (12), we find

$\displaystyle\mathcal{N}^{2f}_{\alpha\beta}/a^{2}\to
8|D_{\alpha\beta}|^{2}+O(a^{2})\ .$ (18)

The fact that the two-fermion negativity vanishes as $a^{2}$ can easily be
explained. The quantity with a continuum limit without any rescaling by
factors of $a$ is the one between two subsystems of a given physical length.
As $a\to 0$, the number of modes contained in each subsystem grows as $1/a$.
By insisting on computing the negativity between single modes, we need to pay
a factor of $1/a^{2}$ in (18). The indices $\alpha,\beta$ reflect the fact you
can consider the negativity between the same component modes or opposite
component modes. Note that this relation to the propagator is reminiscent of
the results of Malla  _et al._ (2023), where the Quantum Fisher Information,
another measure of multipartite entanglement, was successfully probed using
single particle propagators only.

What is insightful is that the continuum limit of
$\mathcal{N}^{2f}_{\alpha\beta}/a^{2}$ is simply proportional to the square
propagator between the two modes. It gives a new interpretation to the space-
like part of the propagator in this kind of field theory: it measures the two-
mode entanglement in the system. It also strengthens the speculation made in
the introduction. The two-fermion negativity detects confinement, at least in
one dimension. More generally, it is suggestive that entanglement measures can
be used to probe confinement, exhibiting different behavior when probing
asymptotically free degrees of freedom with respect to scattering states.

Figure 1: Staggered set-up in the Schwinger model. Physical sites are made out
of two lattice sites. The difference from a spinful “electron” is that gauge
fields, before being integrated out, also link fermions within a physical
site. We consider here the negativity between even/odd modes; the same
analysis can be repeated for the odd/odd or even/even correlations.

Massless Schwinger Model. We exemplify these ideas by studying first the
massless Schwinger model. The continuum theory is dual to the theory of a
massive free boson $\phi$ of mass $m_{b}=g/\sqrt{\pi}$ Schwinger (1962). The
fermionic propagator is known explicitly Casher  _et al._ (1973); Steele  _et
al._ (1995)

$\displaystyle D_{11}(r)$ $\displaystyle=D_{22}(r)\equiv
d_{1}(r)=\frac{m_{b}}{4\pi}e^{G(r)+\gamma_{E}},$ (19) $\displaystyle
D_{12}(r)$ $\displaystyle=D_{21}(r)\equiv
d_{2}(r)=\frac{m_{b}}{4\pi}e^{G(r)+K_{0}(m_{b}r)+\gamma_{E}},$ (20)
$\displaystyle G(r)$
$\displaystyle=-m_{b}^{2}\int_{0}^{\infty}\frac{\mathrm{d}k}{\sqrt{k^{2}+m_{b}^{2}}}\frac{\sin^{2}\left(\frac{kr}{2}\right)}{k^{2}}$
(21)
$\displaystyle=-\frac{m_{b}r}{8}G_{2,4}^{2,2}\left(\frac{m_{b}^{2}r^{2}}{4}\left|\begin{array}[]{c}\frac{1}{2},1\\\
\frac{1}{2},\frac{1}{2},-\frac{1}{2},0\\\ \end{array}\right.\right)\ ,$ (24)

with $r=x-y$, $K_{0}$ the $0^{th}$ Bessel function of the second kind and $G$
is the Meijer $G$-function. They both admit simple asymptotes

$\displaystyle d_{1}(r)\sim\begin{cases}\frac{m_{b}}{4\pi}e^{\gamma_{E}}\ \ \
\ \ \ \ \ \ \ \ \ ,\ \ r\to 0\\\
\frac{m_{b}}{4\pi}e^{\frac{1}{2}+\gamma_{E}-\frac{m_{b}\pi r}{4}}\ ,~{}\
r\to\infty\end{cases},$ (25) $\displaystyle
d_{2}(r)\sim\begin{cases}\frac{1}{2\pi r}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ,\ \
r\to 0\\\ \frac{m_{b}}{4\pi}e^{\frac{1}{2}+\gamma_{E}-\frac{m_{b}\pi r}{4}}\
,~{}\ r\to\infty\end{cases}\ ,$ (26)

with $\gamma_{E}$ the Euler-Mascheroni constant. As was pointed out early on
Casher  _et al._ (1973), the fermionic propagator captures the transition from
a free constituent fermionic theory at small distances to a massive bosonic
free bosonic theory at large distances, where the propagator is screened.

We now show that the two-fermion negativity defined from the lattice model
indeed approaches the square propagator. To compute the negativity from the
lattice model, we performed tensor network simulations using the ITensor
library Fishman  _et al._ (2022a, b). We compute the ground state using DMRG
and evaluate the negativity with (12). To minimize finite-size effects, and
for all simulations presented in this work, we use the improved lattice mass
parameter of Dempsey  _et al._ (2022). We show the even/odd negativity
$\mathcal{N}^{2f}_{12}$ normalized by $a^{2}$ for a fixed coupling in Fig. 2.
We also fix the physical volume such that finite-volume effects are
negligible. The colored curves correspond to different lattice spacings and
represent taking the continuum limit (with $N=100,200$, and $400$ staggered
sites). The grey curve is the analytical prediction (18). As claimed in the
previous section, the negativity systematically approaches the norm square of
the propagator. The transition from the algebraic fermionic behavior to the
exponentially decreasing bosonic one is sharp and happens at the boson radius
$1/m_{b}$. Note the deviations from the analytical prediction at a large
radius are due to approaching the boundaries of our lattice (we use open-
boundary conditions). It is a concrete example where the two sites’ fermionic
negativity transitions from an algebraic decay to an exponential one. This
transition probes confinement. We will see that it still happens in massive
case which is not analytically solvable.

Figure 2: The deviations at small $r$ are lattice artifacts, as demonstrated
by the apparent convergence to the analytical result as $a\to 0$. The
deviations from the analytical prediction at large $r$ are due to boundary
effects. The different lattice spacings correspond to lattices of
$N=100,200,400$ staggered sites.

The same analysis can, in principle, be repeated for the even/even or odd/odd
case. In practice and because of the use of staggered fermions, it turns out
to be more complicated. The cancellation between $\Lambda_{00,00}$ and
$\Lambda_{11,11}$ is crucial to get a non-zero imaginary part and happens only
when $\langle n_{j}\rangle+\langle n_{k}\rangle$ is sufficiently close to $1$.
We show the typical behavior of $\langle n_{x}\rangle$ in Fig. 3. Both odd and
even site densities approach $1/2$ rather slowly in an alternating pattern. On
the other hand, the sum of even and odd densities approaches one much faster
as $a\to 0$. As a result, the even/even and odd/odd negativities are zero for
the range of lattice spacings we studied and become non-zero only closer to
the continuum. Consequently, we will restrict ourselves to the study of the
even/odd negativity for the rest of this work.

Figure 3: Number density as a function of distance. The continuum vacuum is at
half-filling. Odd sites approach half-filling from above and even sites from
below.

Massive Schwinger model. We now move on to the massive Schwinger model, which
is not known to be analytically solvable. It is dual to an interacting bosonic
theory Coleman (1976). We repeat our DMRG simulation for the same $g$ value as
in the massless case for different mass parameters. We show the resulting
negativities in Fig 4. The transition from an algebraic behavior to an
exponential decay is still clear. It happens at shorter distances for larger
masses. We show the same data in Fig. 5, rescaling the $x$ and $y$ axis in
terms of the mass gap. For the three smaller ratios of $m/g$, we used the
values of Adam (2003), which we checked against our own determination of the
mass gap obtained by targeting the first excited state. For the larger mass,
the leading prediction $m_{1st}^{2}=m_{b}^{2}+m^{2}$ is accurate at the
percent level Adam (2003) and was used as an estimate for the gap. We observe
a relatively good collapse, showing that the leading dependence of the
negativity is the mass of the first excited state. In particular, the
transition from an algebraic behavior to an exponential decay happens in all
cases at $r\sim 1/m_{1st}$, and thus is sensitive to the typical confinement
length at which the theory transitions from being fermionic to bosonic.

Figure 4: Two-fermion negativity as a function of distance, for different
$m/g$ ratios. All curves transition from an algebraic to an exponential decay.
Figure 5: Two-fermion negativity as a function of distance, for different
$m/g$ ratios, with the axis scaled by appropriate powers of the mass gap. The
curves’ approximate collapse indicates that the negativity’s main dependence
is on the mass gap. The transition from the algebraic fermionic decay to the
bosonic exponential one happens at $r\sim 1/m_{1st}$, indicated by a dotted
black line.

Massive Schwinger model at criticality. The Schwinger is critical for negative
masses at the critical mass to coupling ratio $m_{c}/g\approx 0.3335$ Byrnes
_et al._ (2002a, b) and is in the $2D$ Ising universality class. Ref. Parez
and Witczak-Krempa (2023) showed that fermionic CFTs can exhibit logarithmic
negativity that decays algebraically, in contrast to bosonic CFTs where the
logarithmic negativity is known to decay faster than any polynomials
Marcovitch  _et al._ (2009); Calabrese  _et al._ (2012). An interesting
feature of the massive Schwinger model is that it is a fermionic
representation of a bosonic CFT, and it is thus of value to investigate the
behavior of the fermionic logarithmic negativity in this model. Results are
shown in Fig. 6. The system is tuned at its second-order critical point. The
two-fermion logarithmic negativity is plotted as a function of distance. The
different curves correspond to different volumes. The dashed line is a fit to
an exponential decay. In this case, the fermionic negativity is also
exponentially suppressed at large distances; it behaves in the same way as the
bosonic one. This matches expectations from universality, as the underlying
CFT has no knowledge of fermionic operators.

We found the small distance behavior to be out of reach of the specific
numerical methods used in this work; we show the small $r$ behavior of our
data in the inset. We also plot the universal Ising expectation for very small
$r$. While far from the regime of applicability of this prediction, it appears
that our data do not contradict this small distance behavior. This regime can
be better studied by using dedicated tensor network simulations more
appropriate to critical systems Vidal (2009); Bridgeman and Chubb (2017).

Figure 6: Two-fermion logarithmic negativity at the critical mass. At large
distances, the fermionic negativity still decays exponentially, as expected
for the $2D$ Ising CFT. The dashed line is the result of a fit to an
exponential $\alpha e^{-\beta gr}$ with $\alpha=0.00786\pm 0.00023$ and
$\beta=0.20051\pm 0.00075$. Inset: while out-of-reach of the data presented
here, the behavior of $\mathcal{\epsilon}_{\mathcal{N}}$ also seems consistent
with the small $r$ prediction
$\mathcal{\epsilon}_{\mathcal{N}}\sim-c/4\log(gr)$ with $c=1/2$ the central
charge of the $2D$ Ising CFT.

Discussion. We studied in this work the two-fermion (logarithmic) negativity
of fermionic spin chains and its behavior in the continuum limit. In
particular, we focused on the Schwinger model. While expressible purely as a
fermionic chain, it has the interesting property of being fermionic only at
short distances, while its asymptotic spectrum is described by a bosonic
scalar theory. We showed that the negativity is sensitive to this “confinement
radius”, at which it transitions from an algebraic decay to an exponential
one. We also showed that the fermionic negativity behaves as its bosonic
counterpart at the Ising critical point of the massive Schwinger model, as
universality would suggest.

These results are interesting from two different perspectives. First, it is an
explicit example where the fermionic negativity exhibits a transition from a
more entangling, in the sense of Parez and Witczak-Krempa (2023), fermionic
behavior to a bosonic behavior at large distances, as would be expected by
bosonization. It shows that the fermionic negativity captures the system’s
fundamental statistics and does not measure an artifact related to the
representation of the system’s degrees of freedom. Second, it further
motivates us to think about how confinement relates to entanglement and how
entanglement measures can be used to probe it. Moreover, the explicit relation
between the two-fermion negativity and the full propagator of the theory is
intriguing and invites reflection on how entanglement measures between
physical subregions can be systematically related to field theoretical
$n$-point functions.

This work opens other natural outlooks. A simple extension would be to
consider how the two-mode negativity is affected when computed between two
full Dirac fermions (four modes negativity in a staggered formulation or two-
site negativity for Wilson fermions). Considering its behavior in the presence
of localized states is also an interesting avenue. More challenging,
investigating the two-mode density matrix in the presence of dynamical gauge
fields will be an essential step in generalizing these results to higher
dimensions and more realistic theories.

Acknowledgment. The author thanks João Barata for interesting discussions,
Erik Gustafson for speeding up his code, and Robert Szafron for insightful
discussions on negativity. The author is also grateful to Dmitri Kharzeev,
Robert Pisarski, Robert Szafron, and Raju Venugopalan for comments on a draft
of this manuscript. The author is supported by the U.S. Department of Energy
under contract DE-SC0012704 and by the U.S. Department of Energy, Office of
Science, National Quantum Information Science Research Centers, Co-design
Center for Quantum Advantage (C2QA), under contract number DE-SC0012704.

##  Appendix A Details on the derivation of the negativity

To compute the (logarithmic) negativity, we consider the sum of the absolute
values of these eigenvalues. We start by noting that
$\Lambda_{00,00},\Lambda_{01,01},\Lambda_{10,10},\Lambda_{11,11}\geq 0$ as
they are all sums of positive numbers by definition (6) in the main text. This
implies $\lambda_{1},\lambda_{2}>0$. Now assume $\lambda_{\pm}$ are real. We
have the following inequalities

$\displaystyle\lambda_{+}$
$\displaystyle=\frac{1}{2}\left(1-\Lambda_{01,01}-\Lambda_{10,10}+\sqrt{-4|\Lambda_{01,10}|^{2}+(-\Lambda_{00,00}+\Lambda_{11,11})^{2})}\right)$
(27)
$\displaystyle\geq\frac{1}{2}\left(1-\Lambda_{01,01}-\Lambda_{10,10}\right)=\frac{1}{2}\left(\Lambda_{00,00}+\Lambda_{11,11}\right)\geq
0$ (28) $\displaystyle\lambda_{-}$
$\displaystyle=\frac{1}{2}\left(1-\Lambda_{01,01}-\Lambda_{10,10}-\sqrt{-4|\Lambda_{01,10}|^{2}+(-\Lambda_{00,00}+\Lambda_{11,11})^{2})}\right)$
(29)
$\displaystyle\geq\frac{1}{2}\left(1-\Lambda_{01,01}-\Lambda_{10,10}-|-\Lambda_{00,00}+\Lambda_{11,11}|\right)$
(30) $\displaystyle=\begin{cases}\Lambda_{00,00}\geq 0\ ,\text{ if
}\Lambda_{00,00}\leq\Lambda_{11,11}\\\ \Lambda_{11,11}\geq 0\ ,\text{ if
}\Lambda_{00,00}\geq\Lambda_{11,11}\end{cases}\ .$ (31)

In this case, the (logarithmic) negativity vanishes.

The other case to consider is when $\lambda_{\pm}=\lambda_{R}\pm i\lambda_{I}$
are complex conjugate to each other with $\lambda_{I}\neq 0$. We then have

$\displaystyle|\lambda_{1}|+|\lambda_{2}|+2|\lambda_{\pm}|$
$\displaystyle=1-\lambda_{+}-\lambda_{-}+2|\lambda_{\pm}|$ (32)
$\displaystyle=1-2(|\lambda_{\pm}|-\lambda_{R})$ (33)

and Eqs. (12)-(13) trivially follow from the definition of the (logarithmic)
negativity.

## References

  * Reeh and Schlieder (1961) H. Reeh and S. Schlieder, “Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern,” Nuovo Cim. 22, 1051–1068 (1961).
  * Srednicki (1993) Mark Srednicki, “Entropy and area,” Phys. Rev. Lett. 71, 666–669 (1993), arXiv:hep-th/9303048 .
  * Calabrese and Cardy (2004) Pasquale Calabrese and John L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat. Mech. 0406, P06002 (2004), arXiv:hep-th/0405152 .
  * Ryu and Takayanagi (2006) Shinsei Ryu and Tadashi Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006), arXiv:hep-th/0603001 .
  * Kharzeev and Levin (2017) Dmitri E. Kharzeev and Eugene M. Levin, “Deep inelastic scattering as a probe of entanglement,” Phys. Rev. D 95, 114008 (2017), arXiv:1702.03489 [hep-ph] .
  * Kharzeev and Levin (2021) Dmitri E. Kharzeev and Eugene Levin, “Deep inelastic scattering as a probe of entanglement: Confronting experimental data,” Phys. Rev. D 104, L031503 (2021), arXiv:2102.09773 [hep-ph] .
  * Gong  _et al._ (2022) Wenjie Gong, Ganesh Parida, Zhoudunming Tu, and Raju Venugopalan, “Measurement of Bell-type inequalities and quantum entanglement from $\Lambda$-hyperon spin correlations at high energy colliders,” Phys. Rev. D 106, L031501 (2022), arXiv:2107.13007 [hep-ph] .
  * de Jong  _et al._ (2022) Wibe A. de Jong, Kyle Lee, James Mulligan, Mateusz Płoskoń, Felix Ringer, and Xiaojun Yao, “Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model,” Phys. Rev. D 106, 054508 (2022), arXiv:2106.08394 [quant-ph] .
  * Florio  _et al._ (2023) Adrien Florio, David Frenklakh, Kazuki Ikeda, Dmitri Kharzeev, Vladimir Korepin, Shuzhe Shi, and Kwangmin Yu, “Real-Time Nonperturbative Dynamics of Jet Production in Schwinger Model: Quantum Entanglement and Vacuum Modification,” Phys. Rev. Lett. 131, 021902 (2023), arXiv:2301.11991 [hep-ph] .
  * Belyansky  _et al._ (2023) Ron Belyansky, Seth Whitsitt, Niklas Mueller, Ali Fahimniya, Elizabeth R. Bennewitz, Zohreh Davoudi, and Alexey V. Gorshkov, “High-Energy Collision of Quarks and Hadrons in the Schwinger Model: From Tensor Networks to Circuit QED,” (2023), arXiv:2307.02522 [quant-ph] .
  * Barata  _et al._ (2023) João Barata, Wenjie Gong, and Raju Venugopalan, “Realtime dynamics of hyperon spin correlations from string fragmentation in a deformed four-flavor Schwinger model,” (2023), arXiv:2308.13596 [hep-ph] .
  * Lee  _et al._ (2023) Kyle Lee, James Mulligan, Felix Ringer, and Xiaojun Yao, “Liouvillian dynamics of the open Schwinger model: String breaking and kinetic dissipation in a thermal medium,” Phys. Rev. D 108, 094518 (2023), arXiv:2308.03878 [quant-ph] .
  * Florio and Kharzeev (2021) Adrien Florio and Dmitri E. Kharzeev, “Gibbs entropy from entanglement in electric quenches,” Phys. Rev. D 104, 056021 (2021), arXiv:2106.00838 [hep-th] .
  * Dunne  _et al._ (2023) Gerald V. Dunne, Adrien Florio, and Dmitri E. Kharzeev, “Entropy suppression through quantum interference in electric pulses,” Phys. Rev. D 108, L031901 (2023), arXiv:2211.13347 [hep-ph] .
  * Grieninger  _et al._ (2023a) Sebastian Grieninger, Dmitri E. Kharzeev, and Ismail Zahed, “Entanglement in a holographic Schwinger pair with confinement,” Phys. Rev. D 108, 086030 (2023a), arXiv:2305.07121 [hep-th] .
  * Grieninger  _et al._ (2023b) Sebastian Grieninger, Dmitri E. Kharzeev, and Ismail Zahed, “Entanglement entropy in a time-dependent holographic Schwinger pair creation,” (2023b), arXiv:2310.12042 [hep-th] .
  * Beane  _et al._ (2019) Silas R. Beane, David B. Kaplan, Natalie Klco, and Martin J. Savage, “Entanglement Suppression and Emergent Symmetries of Strong Interactions,” Phys. Rev. Lett. 122, 102001 (2019), arXiv:1812.03138 [nucl-th] .
  * Klco and Savage (2021a) Natalie Klco and Martin J. Savage, “Geometric quantum information structure in quantum fields and their lattice simulation,” Phys. Rev. D 103, 065007 (2021a), arXiv:2008.03647 [quant-ph] .
  * Klco and Savage (2021b) Natalie Klco and Martin J. Savage, “Entanglement Spheres and a UV-IR Connection in Effective Field Theories,” Phys. Rev. Lett. 127, 211602 (2021b), arXiv:2103.14999 [hep-th] .
  * Klco  _et al._ (2023) Natalie Klco, D. H. Beck, and Martin J. Savage, “Entanglement structures in quantum field theories: Negativity cores and bound entanglement in the vacuum,” Phys. Rev. A 107, 012415 (2023), arXiv:2110.10736 [quant-ph] .
  * Kharzeev and Tuchin (2005) Dmitri Kharzeev and Kirill Tuchin, “From color glass condensate to quark gluon plasma through the event horizon,” Nucl. Phys. A 753, 316–334 (2005), arXiv:hep-ph/0501234 .
  * Berges  _et al._ (2018) Jürgen Berges, Stefan Floerchinger, and Raju Venugopalan, “Dynamics of entanglement in expanding quantum fields,” JHEP 04, 145 (2018), arXiv:1712.09362 [hep-th] .
  * Zhou  _et al._ (2022) Zhao-Yu Zhou, Guo-Xian Su, Jad C. Halimeh, Robert Ott, Hui Sun, Philipp Hauke, Bing Yang, Zhen-Sheng Yuan, Jürgen Berges, and Jian-Wei Pan, “Thermalization dynamics of a gauge theory on a quantum simulator,” Science 377, abl6277 (2022), arXiv:2107.13563 [cond-mat.quant-gas] .
  * Mueller  _et al._ (2022) Niklas Mueller, Torsten V. Zache, and Robert Ott, “Thermalization of Gauge Theories from their Entanglement Spectrum,” Phys. Rev. Lett. 129, 011601 (2022), arXiv:2107.11416 [quant-ph] .
  * Ebner  _et al._ (2023) Lukas Ebner, Berndt Müller, Andreas Schäfer, Clemens Seidl, and Xiaojun Yao, “Eigenstate Thermalization in 2+1 dimensional SU(2) Lattice Gauge Theory,” (2023), arXiv:2308.16202 [hep-lat] .
  * Yao (2023) Xiaojun Yao, “SU(2) gauge theory in 2+1 dimensions on a plaquette chain obeys the eigenstate thermalization hypothesis,” Phys. Rev. D 108, L031504 (2023), arXiv:2303.14264 [hep-lat] .
  * Grieninger  _et al._ (2023c) Sebastian Grieninger, Kazuki Ikeda, Dmitri E. Kharzeev, and Ismail Zahed, “Entanglement in massive Schwinger model at finite temperature and density,” (2023c), arXiv:2312.03172 [hep-th] .
  * Ares  _et al._ (2023a) Filiberto Ares, Sara Murciano, and Pasquale Calabrese, “Entanglement asymmetry as a probe of symmetry breaking,” Nature Commun. 14, 2036 (2023a), arXiv:2207.14693 [cond-mat.stat-mech] .
  * Ares  _et al._ (2023b) Filiberto Ares, Sara Murciano, Eric Vernier, and Pasquale Calabrese, “Lack of symmetry restoration after a quantum quench: An entanglement asymmetry study,” SciPost Phys. 15, 089 (2023b), arXiv:2302.03330 [cond-mat.stat-mech] .
  * Ares  _et al._ (2023c) Filiberto Ares, Sara Murciano, Lorenzo Piroli, and Pasquale Calabrese, “An entanglement asymmetry study of black hole radiation,” (2023c), arXiv:2311.12683 [hep-th] .
  * Nishioka (2018) Tatsuma Nishioka, “Entanglement entropy: holography and renormalization group,” Rev. Mod. Phys. 90, 035007 (2018), arXiv:1801.10352 [hep-th] .
  * Klco  _et al._ (2022) Natalie Klco, Alessandro Roggero, and Martin J. Savage, “Standard model physics and the digital quantum revolution: thoughts about the interface,” Rept. Prog. Phys. 85, 064301 (2022), arXiv:2107.04769 [quant-ph] .
  * Bauer  _et al._ (2023) Christian W. Bauer, Zohreh Davoudi, Natalie Klco, and Martin J. Savage, “Quantum simulation of fundamental particles and forces,” Nature Rev. Phys. 5, 420–432 (2023).
  * Peres (1996) Asher Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996), arXiv:quant-ph/9604005 .
  * Horodecki (1997) Pawel Horodecki, “Separability criterion and inseparable mixed states with positive partial transposition,” Phys. Lett. A 232, 333 (1997), arXiv:quant-ph/9703004 .
  * Simon (2000) R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. 84, 2726–2729 (2000), arXiv:quant-ph/9909044 .
  * Vidal and Werner (2002) G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002), arXiv:quant-ph/0102117 .
  * Plenio (2005) M. B. Plenio, “Logarithmic Negativity: A Full Entanglement Monotone That is not Convex,” Phys. Rev. Lett. 95, 090503 (2005), arXiv:quant-ph/0505071 .
  * Calabrese  _et al._ (2012) Pasquale Calabrese, John Cardy, and Erik Tonni, “Entanglement negativity in quantum field theory,” Phys. Rev. Lett. 109, 130502 (2012), arXiv:1206.3092 [cond-mat.stat-mech] .
  * Calabrese  _et al._ (2013) Pasquale Calabrese, John Cardy, and Erik Tonni, “Entanglement negativity in extended systems: A field theoretical approach,” J. Stat. Mech. 1302, P02008 (2013), arXiv:1210.5359 [cond-mat.stat-mech] .
  * Murciano  _et al._ (2021) Sara Murciano, Riccarda Bonsignori, and Pasquale Calabrese, “Symmetry decomposition of negativity of massless free fermions,” SciPost Phys. 10, 111 (2021), arXiv:2102.10054 [cond-mat.stat-mech] .
  * Murciano  _et al._ (2022) Sara Murciano, Vittorio Vitale, Marcello Dalmonte, and Pasquale Calabrese, “Negativity Hamiltonian: An Operator Characterization of Mixed-State Entanglement,” Phys. Rev. Lett. 128, 140502 (2022), arXiv:2201.03989 [cond-mat.stat-mech] .
  * Parez and Witczak-Krempa (2023) Gilles Parez and William Witczak-Krempa, “Are fermionic conformal field theories more entangled?” (2023), arXiv:2310.15273 [cond-mat.str-el] .
  * Marcovitch  _et al._ (2009) S. Marcovitch, A. Retzker, M. B. Plenio, and B. Reznik, “Critical and noncritical long-range entanglement in Klein-Gordon fields,” Phys. Rev. A 80, 012325 (2009), arXiv:0811.1288 [quant-ph] .
  * Coleman  _et al._ (1975) Sidney R. Coleman, R. Jackiw, and Leonard Susskind, “Charge Shielding and Quark Confinement in the Massive Schwinger Model,” Annals Phys. 93, 267 (1975).
  * Coleman (1976) Sidney R. Coleman, “More About the Massive Schwinger Model,” Annals Phys. 101, 239 (1976).
  * Schwinger (1962) Julian S. Schwinger, “Gauge Invariance and Mass. 2.” Phys. Rev. 128, 2425–2429 (1962).
  * Kogut and Susskind (1975) John B. Kogut and Leonard Susskind, “Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,” Phys. Rev. D 11, 395–408 (1975).
  * Susskind (1977) Leonard Susskind, “Lattice Fermions,” Phys. Rev. D 16, 3031–3039 (1977).
  * Ikeda  _et al._ (2021) Kazuki Ikeda, Dmitri E. Kharzeev, and Yuta Kikuchi, “Real-time dynamics of Chern-Simons fluctuations near a critical point,” Phys. Rev. D 103, L071502 (2021), arXiv:2012.02926 [hep-ph] .
  * Javanmard  _et al._ (2018) Younes Javanmard, Daniele Trapin, Soumya Bera, Jens H. Bardarson, and Markus Heyl, “Sharp entanglement thresholds in the logarithmic negativity of disjoint blocks in the transverse-field Ising chain,” New J. Phys. 20, 083032 (2018), arXiv:1805.06935 [cond-mat.str-el] .
  * Berthiere and Witczak-Krempa (2022) Clément Berthiere and William Witczak-Krempa, “Entanglement of Skeletal Regions,” Phys. Rev. Lett. 128, 240502 (2022), arXiv:2112.13931 [cond-mat.str-el] .
  * Shapourian  _et al._ (2017) Hassan Shapourian, Ken Shiozaki, and Shinsei Ryu, “Partial time-reversal transformation and entanglement negativity in fermionic systems,” Phys. Rev. B 95, 165101 (2017), arXiv:1611.07536 [cond-mat.str-el] .
  * Malla  _et al._ (2023) Rajesh K. Malla, Andreas Weichselbaum, Tzu-Chieh Wei, and Robert M. Konik, “Detecting Multipartite Entanglement Patterns using Single Particle Green’s Functions,” (2023), arXiv:2310.05870 [quant-ph] .
  * Casher  _et al._ (1973) A. Casher, John B. Kogut, and Leonard Susskind, “Vacuum polarization and the quark parton puzzle,” Phys. Rev. Lett. 31, 792–795 (1973).
  * Steele  _et al._ (1995) James V. Steele, J. J. M. Verbaarschot, and I. Zahed, “The Invariant fermion correlator in the Schwinger model on the torus,” Phys. Rev. D 51, 5915–5923 (1995), arXiv:hep-th/9407125 .
  * Fishman  _et al._ (2022a) Matthew Fishman, Steven R. White, and E. Miles Stoudenmire, “The ITensor Software Library for Tensor Network Calculations,” SciPost Phys. Codebases , 4 (2022a).
  * Fishman  _et al._ (2022b) Matthew Fishman, Steven R. White, and E. Miles Stoudenmire, “Codebase release 0.3 for ITensor,” SciPost Phys. Codebases , 4–r0.3 (2022b).
  * Dempsey  _et al._ (2022) Ross Dempsey, Igor R. Klebanov, Silviu S. Pufu, and Bernardo Zan, “Discrete chiral symmetry and mass shift in the lattice Hamiltonian approach to the Schwinger model,” Phys. Rev. Res. 4, 043133 (2022), arXiv:2206.05308 [hep-th] .
  * Adam (2003) C. Adam, “Improved vector and scalar masses in the massive Schwinger model,” Phys. Lett. B 555, 132–137 (2003), arXiv:hep-th/0212171 .
  * Byrnes  _et al._ (2002a) T. Byrnes, P. Sriganesh, R. J. Bursill, and C. J. Hamer, “Density matrix renormalization group approach to the massive Schwinger model,” Nucl. Phys. B Proc. Suppl. 109, 202–206 (2002a), arXiv:hep-lat/0201007 .
  * Byrnes  _et al._ (2002b) T. Byrnes, P. Sriganesh, R. J. Bursill, and C. J. Hamer, “Density matrix renormalization group approach to the massive Schwinger model,” Phys. Rev. D 66, 013002 (2002b), arXiv:hep-lat/0202014 .
  * Vidal (2009) Guifre Vidal, “Entanglement Renormalization: an introduction,” (2009), arXiv:0912.1651 [cond-mat.str-el] .
  * Bridgeman and Chubb (2017) Jacob C. Bridgeman and Christopher T. Chubb, “Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks,” J. Phys. A 50, 223001 (2017), arXiv:1603.03039 [quant-ph] .