JHEP01(2025)057
Published for SISSA by
Springer
Received:
November 28, 2024
Accepted:
December 9, 2024
Published:
January 9, 2025
Entanglement asymmetry and symmetry defects in
boundary conformal field theory
Yuya Kusuki
,
a,b,c
Sara Murciano
,
d,e
Hirosi Ooguri
d,f
and Sridip Pal
d
a
Institute for Advanced Study, Kyushu University,
Fukuoka 819-0395, Japan
b
Department of Physics, Kyushu University,
Fukuoka 819-0395, Japan
c
RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS),
Wako, Saitama 351-0198, Japan
d
Walter Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, U.S.A.
e
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology,
Pasadena, CA 91125, U.S.A.
f
Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo,
Kashiwa 277-8583, Japan
E-mail:
kusuki.yuya@phys.kyushu-u.ac.jp
, smurcian@scaltech.edu
,
ooguri@caltech.edu
, sridip@caltech.edu
Abstract:
A state in a quantum system with a given global symmetry,
G
, can be sensitive
to the presence of boundaries, which may either preserve or break this symmetry. In
this work, we investigate how conformal invariant boundary conditions influence the
G
-
symmetry breaking through the lens of the entanglement asymmetry, a quantifier of the
“distance” between a symmetry-broken state and its symmetrized counterpart. By leveraging
2D boundary conformal field theory (BCFT), we investigate the symmetry breaking for
both finite and compact Lie groups. Beyond the leading order term, we also compute the
subleading corrections in the subsystem size, highlighting their dependence on the symmetry
group
G
and the BCFT operator content. We further explore the entanglement asymmetry
following a global quantum quench, where a symmetry-broken state evolves under a symmetry-
restoring Hamiltonian. In this dynamical setting, we compute the entanglement asymmetry
by extending the method of images to a BCFT with non-local objects such as invertible
symmetry defects.
Keywords:
Field Theories in Lower Dimensions, Global Symmetries, Scale and Conformal
Symmetries
ArXiv ePrint:
2411.09792
Open Access
,
©
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP01(2025)057
JHEP01(2025)057
Contents
1 Introduction
1
1.1 Review of known results about entanglement asymmetry
1
1.2 Take-home message
2
2 Definitions and connections to BCFT
4
2.1 General discussion
5
2.2 Finite groups
7
2.3 Compact Lie groups
9
2.4 Example: symmetry breaking in the Potts model
11
2.5 Example: U
(1)
symmetry breaking
13
3 Symmetry defects in a dynamical setting
15
3.1 Finite interval
15
3.2 Semi-infinite interval
23
4 Discussion
23
1 Introduction
How conservation laws influence the entanglement structure is a subject that has attracted a
lot of attention in recent years. Indeed, in equilibrium settings, studying the entanglement in
the presence of a conserved charge can reveal non-trivial information about the underlying
symmetry of the Hilbert space. Out-of-equilibrium, conservation laws in many-body quantum
systems can impact the time evolution of entanglement, thermalization, operator growth.
Therefore, a natural question one can ask is what happens when the symmetry is broken
and, especially, how it is possible to quantify the breaking effects. In [
1
], this topic has been
investigated by introducing the concept of
entanglement asymmetry
.
In this paper, we initiate the study of entanglement asymmetry through the lens of
symmetry defects in boundary conformal field theory (BCFT). The formalism of BCFT has
been proven to be useful in studying various quantum information motivated quantities such
as entanglement entropy [
2
,
3
], symmetry resolved entanglement [
4
–
7
], which has further
been generalized to the setup of non-invertible symmetry in [
8
–
10
].
1.1 Review of known results about entanglement asymmetry
We refer to the following section for a formal definition of the entanglement asymmetry and
here we review the interesting physics that this quantity can probe. The notion of
asymmetry
was first given in the context of quantum information and resource theory [
11
–
14
]. Only
several years later, ref. [
1
] introduced the concept of entanglement asymmetry in many-body
quantum systems. Beyond providing a new framework to study symmetry breaking through
the lens of entanglement, it can also detect interesting phenomena. The first application
– 1 –
JHEP01(2025)057
concerned the evolution after a quantum quench of a spin chain from a state that breaks
a global
U
(1)
symmetry to a Hamiltonian that preserves it. It turns out that the more
the symmetry is initially broken, the smaller the time to restore it is and this phenomenon
has been dubbed the
quantum Mpemba effect
. Since then, the entanglement asymmetry for
arbitrary groups has been studied in several contexts, such as generic integrable systems [
15
–
24
], mixed states [
25
,
26
], random or dual unitary circuits [
27
–
31
], higher dimensions [
32
,
33
],
holography [
34
], in the presence of confinement [
35
] or many-body localization [
36
]. Moreover,
the definition of the asymmetry using the replica trick has made it suitable for an experiment
in an ion trap simulator, where the quantum Mpemba effect has also been observed for the
first time [
37
]. However, dynamics is not the only setup in which it is interesting to study
symmetry breaking. In this spirit, ref. [
38
] found a universal behavior of the entanglement
asymmetry in matrix product states for both finite and continuous groups, and this result
has been extended to compact Lie groups in conformal field theories in [
39
]. Other field
theoretical results concern the behavior of the entanglement asymmetry in certain coherent
states of the massless compact boson [
40
], and the
SU
(2)
to
U
(1)
symmetry breaking in the
critical XXZ spin chain [
41
]. The main upshot of these computations in macroscopic models
is that the asymmetry exhibits a leading-order behavior which is fixed by the dimension
of the group under study and, at least for continuous symmetries, conformal invariance
imprints subleading corrections in the subsystem size
ℓ
of the form
log
ℓ/ℓ
. Finally, we note
that using algebraic QFT approach the refs. [
42
–
45
] studied entropic parameter, which can
detect the breaking of symmetry.
1.2 Take-home message
After a broad review of the main results related to the entanglement asymmetry, we explain
here its formal definition and the main conclusions of the present paper. We consider a
conformal field theory (CFT) with global symmetry
G
. The interplay of CFT data such as
asymptotic density of states and symmetry defects has been explored in [
46
–
50
]. Here our
goal is to understand the entanglement structure of a state
|
Ψ
⟩
in CFT, in relation to the
global symmetry
G
, which we assume to be non-anomalous.
A way to quantify the entanglement structure of a state is to bi-partite the CFT Hilbert
space and consider the reduced density matrix
ρ
A
supported on some interval
A
ρ
A
:= Tr
A
c
|
Ψ
⟩⟨
Ψ
|
,
(1.1)
where
A
c
is complementary to
A
. The global symmetry
G
is implemented by unitary
operators
U
(
g
)
acting on the Hilbert space of states. Assuming
U
(
g
)
can be factorized as
U
A
(
g
)
×
U
A
c
(
g
)
, there is a well defined action of
U
A
(
g
)
on the reduced density matrix
ρ
A
.
If there exists
g
∈
G
such that
U
A
(
g
)
ρ
A
U
A
(
g
)
†
̸
=
ρ
A
,
(1.2)
we can infer that the symmetry is broken by
ρ
A
. The entanglement asymmetry is a quantitative
measure of how much the symmetry generated by
G
is broken. Given
ρ
A
, we construct a
new density matrix
ρ
A
;
sym
:=
1
|
G
|
X
g
U
A
(
g
)
ρ
A
U
A
(
g
)
†
,
(1.3)
– 2 –
JHEP01(2025)057
which satisfies
U
A
(
g
)
ρ
A;
sym
U
A
(
g
)
†
=
ρ
A;
sym
.
(1.4)
The
n
-th Renyi entanglement asymmetry is defined by
∆
S
(
n
)
A
:=
1
1
−
n
log
"
Tr
ρ
n
A
;
sym
Tr
ρ
n
A
#
,
(1.5)
and the replica limit
n
→
1
yields
∆
S
A
:= lim
n
→
1
∆
S
(
n
)
A
= Tr(
ρ
A
log
ρ
A
)
−
Tr(
ρ
A
log
ρ
A
;
sym
)
.
(1.6)
By using the linearity in the definition
(1.3)
,
∆
S
A
can rewritten as the relative entropy
between
ρ
A
;
sym
and
ρ
A
, whose positivity implies that
∆
S
A
≥
0
. This property is valid also for
∆
S
(
n
)
A
and the equality
∆
S
(
n
)
A
= 0
is satisfied if and only if
ρ
A
=
ρ
A
;
sym
.
The unitary operator
U
(
g
)
is a topological codimension
1
operator. However, one needs
to be careful about the topological property of
U
A
(
g
)
since the subsystem
A
has boundaries.
For a system with a boundary, we also need to specify whether
U
A
(
g
)
ends topologically on
the boundary. Thus we can envision a scenario where
U
A
(
g
)
is topological in the bulk, but it
does not end topologically on the boundary. As a result, we can deform such codimension
1
operator in the bulk freely, but keeping the anchoring points of the operator on the boundary
fixed. The non-topological nature on the boundary leads to eq.
(1.2)
. An example of this
setup can be the ground state of a Hamiltonian with boundary terms breaking the symmetry,
G
: in their presence, the ground state is no longer an eigenstate of
G
, even though the
symmetry is only locally broken at the boundary.
In this paper, we show that the entanglement asymmetry is a way to quantify the failure
of a bulk topological line to end topologically. Subsequently, we compute the entanglement
asymmetry coming from the breaking of the symmetry by the boundary condition using
the framework of BCFT. Using this tool to compute
∆
S
(
n
)
A
allows us to find a result for
a
generic
CFT. In particular, if
G
is a finite group and
A
is an interval of size
ℓ
attached
to the symmetry breaking boundary, we find that
∆
S
A
= log
|
G
|−
ε
ℓ
2∆
∗
W
(∆
∗
) +
o
(
ℓ
−
2∆
∗
)
,
(1.7)
where
|
G
|
is the order of the group,
∆
∗
is a universal number depending on the broken
symmetry and on the CFT,
ε
is an ultra-violet cutoff and finally
W
(∆
∗
)
is a quantity that
depends on the broken symmetry and on the CFT, which can be read from eq.
(2.19)
. The
term
(
ε/ℓ
)
2∆
∗
W
(∆
∗
)
depends on the cutoff as a power law. Hence, beyond the leading order,
the only other universal information is the fact that subleading correction to
∆
S
A
decays as
a power law and the corresponding exponent of the power law. Up to this last non-universal
contribution, our result is independent of the microscopic details of the theory. Moreover,
this result shows that the first correction to the leading order term
log
|
G
|
is negative, which
is consistent with the inequality
∆
S
(1)
A
≤
log
|
G
|
proven in [
19
].
– 3 –
JHEP01(2025)057
We can repeat a similar analysis if
G
is a compact Lie group of dimension
dim
(
G
)
and we find that
∆
S
(
n
)
A
=
dim(
G
)
2
log(log(
ℓ/ε
)) +
O
(1)
.
(1.8)
We can also reliably compute the
O
(1)
terms, which are universal and explicitly given in
eq.
(2.26)
and they depend on the volume of the group
G
as well as on the symmetry breaking
pattern. The key point that we want to highlight is that in contrast to the
log
(
ℓ
)
scaling
found in ref. [
39
], the leading order behavior here scales as
log
(
log
(
ℓ
))
. This difference is
due to the fact that the symmetry is broken only at the boundary, while it is still preserved
in the bulk of the system.
We also study the dynamics of the entanglement asymmetry after a global quantum
quench, i.e. we start from a state that breaks a discrete symmetry
G
, and we let it evolve
with a CFT Hamiltonian
H
, that, on the other hand, respects the symmetry. We show
that, under reasonable approximations that we will describe in the dedicated section, the
entanglement asymmetry of a subsystem of size
ℓ
behaves as
∆
S
A
(
t
)
≃
log
|
G
|
,
0
< t <
ℓ
2
,
0
.
ℓ
2
< t
(1.9)
At the initial time, the entanglement asymmetry is non-zero, reflecting the symmetry breaking.
On the other hand, the state is indistinguishable from a thermal state at late time, which
implies the restoration of the symmetry. For this purpose, we extended the method of images
to a BCFT with non-local objects. We expect that this idea will have a wide range of
applications, not limited to entanglement asymmetry.
The structure of the paper is as follows: in section
2 we introduce the definition of the
entanglement asymmetry, and we focus on a conformal invariant system with a symmetry
breaking boundary. In this setup, we leverage BCFT to compute the asymmetry both for
finite and compact Lie groups. We corroborate our findings with some examples for the Potts
model (
G
=
Z
3
) and for a compact boson (
G
=
U
(1)
). In section
3 we study the asymmetry
after a global quantum quench by connecting this quantity to invertible symmetry defects.
We finally draw our conclusions in section
4 .
2 Definitions and connections to BCFT
The goal of the next section is to define the setup and the strategy we have adopted in
this paper to derive the result
(1.7)
. We consider a CFT with global symmetry
G
, on a
semi-infinite line
[
−∞
,
0]
. We have a physical boundary at
x
= 0
and choose a boundary
condition at
x
= 0
such that the symmetry is broken. The subsystem under consideration is
on
[
−
ℓ,
0]
. We aim to quantify the breaking of the symmetry at the level of the subsystem by
computing
∆
S
A
in eq.
(1.6)
. Note that the boundary condition imposed at
x
=
−
ℓ
, i.e. the
boundary coming from the entanglement cut [
2
,
3
], is chosen to preserve the symmetry. Here
we assume the symmetry is not anomalous, hence such a choice is feasible.
1
1
It might be possible to define the entanglement asymmetry, even when the symmetry is anomalous, using
the framework of algebraic quantum field theory (AQFT), as advocated in [
45
] in the context of symmetry
resolved entanglement entropy (see also ref. [
51] for a previous study of the symmetry resolution in AQFT).
– 4 –
JHEP01(2025)057
2.1 General discussion
We start by rewriting eq.
(1.3)
as
Tr(
ρ
n
A
;
sym
)
Tr(
ρ
n
A
)
=
1
|
G
|
n
Z
n
X
g
i
∈
G
Z
n
(
{
g
i
}
)
,
(2.1)
where
Z
n
(
{
g
i
}
) := Tr[
U
(
g
1
)
†
ρ
A
U
(
g
1
)
U
(
g
2
)
†
ρ
A
U
(
g
2
)
···
U
(
g
n
)
†
ρ
A
U
(
g
n
)]
, Z
n
=
Z
n
(
{
g
i
=
e
}
)
,
(2.2)
and
e
denotes the identity. As mentioned in the introduction, we need to be careful about
the endpoint of the topological defects
U
(
g
i
)
.
It turns out that the computation can be conveniently explained in the following con-
formal frame:
z
7→
w
=
z
+
ℓ
ℓ
−
z
1
n
.
(2.3)
Under this mapping, the
n
-sheeted Riemann surface gets mapped to a disk with a hole, with
the inner circle i.e the circle bounding the hole, representing the regularizing circle around
the entanglement cut [
2
] while the outer circle i.e. the boundary of the disk representing the
physical boundary. To fix the ideas, we focus on
n
= 3
and illustrate the computation of
Z
n
(
{
g
i
}
)
in figure
1 and 2 , using the above mentioned conformal frame, to better understand
what happens. In the left panel of figure
1 , the brown lines are the insertions of
U
†
(
g
1
)
and
U
(
g
1
)
, the red lines are the insertions of
U
†
(
g
2
)
and
U
(
g
2
)
while the green lines are the
insertions of
U
†
(
g
3
)
and
U
(
g
3
)
. The curly lines represent the branch cuts. Since the operators
U
(
g
i
)
are topological in the bulk, we can deform them such that they fuse to the identity, but
without moving the end-points, which do not end topologically, as depicted in figure
1 . As a
result, we obtain the insertions of boundary condition changing operators (see figure
2 ): if
g
i
=
g
j
, we do not need to change the boundary condition, while if
g
i
̸
=
g
j
, since the boundary
state is non-invariant under the action of
G
because of the presence of a symmetry breaking
physical boundary, a nontrivial boundary operator must be inserted. Given this analysis, the
ratio
Z
n
(
{
g
i
}
)
/Z
n
can be expressed as a correlation function of boundary-changing operators
depending on the configuration of the group elements
{
g
i
}
.
The result
(2.1)
can be easily generalized if
G
is a compact Lie group, i.e.
Tr(
ρ
n
A
;
sym
)
Tr(
ρ
n
A
)
=
1
[Vol(
G
)]
n
Z
G
n
dg
1
...dg
n
Z
n
(
{
g
i
}
)
Z
n
,
(2.4)
where
R
G
dg
denotes the (unnormalized) Haar measure and the integrand can be written as
a correlation function of boundary-changing operators. By using the cyclicity of the trace
and doing a change of variables, the integral above can be rewritten as
Tr(
ρ
n
A
;
sym
)
Tr(
ρ
n
A
)
=
1
[Vol(
G
)]
n
−
1
Z
G
n
d
̃
g
1
...d
̃
g
n
Tr(
ρ
A
U
( ̃
g
1
)
···
ρ
A
U
( ̃
g
n
))
Z
n
δ
n
Y
i
=1
̃
g
i
−
e
!
.
(2.5)
In the next section, we explicitly compute the entanglement asymmetry for an arbitrary
finite group and CFT by rewriting eq.
(2.1)
as a correlation function.
– 5 –
JHEP01(2025)057
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(
g
2
)
†
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U
(
g
1
)
<latexit sha1_base64="x+S86cvCvhhPO1cLeyfP3qTgcCs=">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</latexit>
U
(
g
2
)
Figure 1.
In the left panel, the bold black lines of the left denote the physical boundary, the brown
lines are the insertions of
U
†
(
g
1
)
and
U
(
g
1
)
, the red lines are the insertions of
U
†
(
g
2
)
and
U
(
g
2
)
while
the green lines are the insertions of
U
†
(
g
3
)
and
U
(
g
3
)
. The curly lines represent the branch cuts. Since
the operators
U
(
g
i
)
and
U
(
g
i
)
†
are topological in the bulk, we can deform them without moving the
endpoints. Here we moved
U
(
g
i
)
while fixing insertions of
U
(
g
i
)
†
, as depicted in the picture on the
right panel. At this point, in the bulk, we can fuse
U
(
g
i
)
†
with
U
(
g
i
)
to get identity. What remains is
the action of the defect line on the boundary, denoted as colored lines going parallel to the boundary
of the sheets.
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