of 30
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
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
attached
to the symmetry breaking boundary, we find that
S
A
= log
|
G
|−

ε

2∆
W
(∆
) +
o
(
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
(
)
scaling
found in ref. [
39
], the leading order behavior here scales as
log
(
log
(
))
. 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
behaves as
S
A
(
t
)
log
|
G
|
,
0
< t <
2
,
0
.
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
=
, 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
+
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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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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b,
1
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b,
2
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b,
3
Figure 2.
Following the procedure, explained in figure
1 , we are left with colored defect lines going
parallel to the boundary of the sheets. The action of the symmetry defects on the boundary can
potentially produce different boundary states if the initial boundary state is not invariant under
the group
G
. In this example,
g
i
acting on the left boundary produces three different boundary
states, colored by brown, red, and green. Correspondingly, we have inserted three boundary-changing
operators
φ
b,i
. Note, if we have
g
i
=
g
j
, then we should take
φ
b
as the identity operator i.e. we do not
have any insertion of nontrivial boundary-changing operator. The right boundary state is taken to be
invariant under group
G
. Thus we can simply remove the defect lines without inserting any nontrivial
boundary-changing operators.
– 6 –
JHEP01(2025)057
2.2 Finite groups
Let us consider the case where
G
is a finite group of order
|
G
|
. On each sheet of the Riemann
surface, the geometry of an interval
A
attached to the boundary can be described by the
complex coordinate
z
=
x
+
, with
τ
= 0
and
x
0
. According to the general
discussion, on the
z
-plane, we have boundary-changing operators inserted at the boundary,
i.e.
z
= 0
in this case. The number of such boundary-changing operators depends on the
Rényi index
n
and on the number of distinct insertions of defect lines
g
i
̸
=
g
j
. First of all,
note that for a given
n
, the total number of summands appearing in
(2.1)
is
|
G
|
n
. Every term
can be denoted by a configuration
(
g
1
,g
2
,
···
g
n
)
and each of the entries can be chosen from
|
G
|
possibilities leading to
|
G
|
n
terms in the sum. For instance, in the case of Ising CFT, we
have
Z
2
symmetry-group, with 2 elements
e
and
g
, i.e.
|
G
|
= 2
. For
n
= 2
we have
2
2
= 4
terms appearing in the sum:
(
e,e
)
,
(
g,g
)
,
(
e,g
)
,
(
g,e
)
. Only the last two configurations give
rise to
2
boundary-changing operators. For
n
= 3
, we have
2
3
= 8
configurations:
(
e,e,e
)
,
(
g,g,g
)
,
(
e,e,g
)
,
(
e,g,e
)
,
(
g,e,e
)
,
(
g,g,e
)
,
(
e,g,g
)
,
(
g,e,g
)
. The last 6 configurations give
rise to 2 boundary-changing operators. In both cases, there are two configurations with
no insertion of boundary operators.
In general, the number of configurations involving boundary-changing operators is
(
|
G
|
)
n
− |
G
|
. The leading contribution to eq.
(2.1)
in the large
limit comes from the
configurations with no insertion of boundary-changing operators, and there are
|
G
|
of them.
Now we aim to figure out the subleading corrections due to boundary-changing operators.
This leads to the following question: how many boundary-changing operators are inserted in
each of the
(
|
G
|
)
n
−|
G
|
configurations? This number depends on
|
G
|
, e.g. if
|
G
|
= 2
, we have
n
(
n
1)
configurations with two insertions of boundary-changing operators. Even though it
is not easy to identify these numbers in general, we can make progress in the
→∞
limit.
For a generic group
G
, the contributions coming from insertions of more than two
boundary-changing operators are subleading in the large
limit. Thus we should focus on
the configuration with two boundary-changing operators. Given the symmetry breaking
boundary state
|
b
, for each pair of elements
g
i
and
g
j
with
g
j
|
b
⟩ ̸
=
g
i
|
b
, we will have
n
(
n
1)
configurations with two insertions of boundary-changing operators with scaling
dimension that depends on
g
i
g
1
j
. Among all such pairs of elements of the group
G
, the
leading contributions come from the pairs that correspond to boundary-changing operators
having the lowest scaling dimension. We can compute how each of these
n
(
n
1)
terms
contributes to the entanglement asymmetry.
Given a pair of elements
g
i
and
g
j
with
g
i
|
b
⟩̸
=
g
j
|
b
, for each
k
∈{
1
,
···
n
1
}
, we have
k-type I
: (
g
i
,g
j
,g
j
,
···
g
j
|
{z
}
k
times
,g
i
,
···
g
i
)
appearing
(
n
k
)
times,
k-type II
: (
g
j
,g
i
,g
i
,
···
g
i
|
{z
}
k
times
,g
j
,
···
g
j
)
appearing
(
n
k
)
times.
(2.6)
As a consistency check, we have
n
1
X
k
=1
2(
n
k
) =
n
(
n
1)
,
(2.7)
– 7 –
JHEP01(2025)057
which is the total number of configurations with two insertions of boundary-changing operator,
given a pair of distinct group elements.
Now the contribution coming from each of the k-type configurations is given by two point
functions of boundary-changing operators. In order to compute this, we consider a conformal
map from the
n
-sheeted Riemann surface to a complex
w
-plane:
w
=

z
+
z

1
/n
.
(2.8)
The boundary-changing operators inserted at
z
= 0
on various sheets get mapped to
w
r
=
e
2
πir/n
,
r
= 0
,...n
1
.
Both in the k-type I and II configurations, the boundary-changing operators are inserted
at
w
r
and
w
r
+
k
, and
r
runs over
{
1
,
2
,
···
n
k
}
. Explicitly, the two point function of
boundary-changing operators
φ
b
is given by
φ
b
(
z
(
w
r
))
φ
b
(
z
(
w
r
+
k
)
=

ε
nℓ

2∆
b
[sin(
πk/n
)]
2∆
b
.
(2.9)
The presence of the cuoff
ε
requires some explanation. To do this, it is better to view
the path-integral defining
Z
n
(
{
g
i
}
)
on a discrete lattice with finite lattice spacing
a
. The
boundary-changing operators defined on the lattice
φ
lat
b
have the following large distance
behavior (assuming the lattice model flows to a CFT in the continuum limit)
φ
lat
b,j
φ
lat
b,j
+
s
⟩ ∼
a/s
0

Aa
s

2∆
b
,
(2.10)
for some numerical coefficient
A
, which should be computed from the lattice. We define
ε
:=
Aa
and this serves as a UV-regulator. Here we are indexing the lattice sites with
j
.
Usually, we define the operators
̃
φ
b
in the continuum limit in a way so that the correlator
is non-zero. This amounts to the following definition
̃
φ
b
:=
ε
b
φ
lattice
b
.
(2.11)
Here we do not use the rescaled
̃
φ
b
operator, but rather the operator defined on the lattice
and we omit the superscript
lattice
for brevity. In other words, the correction coming from
the correlators of boundary-changing operators can be viewed as an effect away from the
continuum limit. In the strict continuum limit, i.e.
ε/ℓ
0
, the correction vanishes.
If we take into account the contribution of all the pairs of boundary-changing operators,
we get a total contribution of
2(
n
k
)
ε
nℓ

2∆
b
[sin(
π
(
k
)
/n
)]
2∆
b
and, overall, we have
n
1
X
k
=1
2(
n
k
)

ε
nℓ

2∆
b
[sin(
π
(
k
)
/n
)]
2∆
b
=

ε

2∆
b
n
1
2∆
b
n
1
X
k
=1
[sin(
π
(
k
)
/n
)]
2∆
b
.
(2.12)
Note that
b
is a function of
g
i
g
1
j
. To get the leading answer, we should consider the
pairs with the smallest
b
, which we denote as
. Moreover, we assume that there are
N
such pairs with the same scaling dimension
. Hence we have
S
(
n
)
A
=
1
1
n
log
"
1
|
G
|
n
|
G
|
+
N

ε
nℓ

2∆
n
n
1
X
k
=1
[sin(
π
(
k
)
/n
)]
2∆
+
o
(
2∆
)
!#
.
(2.13)
– 8 –
JHEP01(2025)057
For the Ising CFT,
G
=
Z
2
,
N
= 1
and
= 1
/
2
. Since we are considering only the
large
-limit, we can further expand the result above to get
S
(
n
)
A
= log
|
G
|
+
N
1
n

ε
nℓ

2∆
n
|
G
|
n
1
X
k
=1
[sin(
π
(
k
)
/n
)]
2∆
+
o
(
2∆
)
.
(2.14)
Analytic continuation.
We aim to perform an analytic continuation in
n
of eq.
(2.14)
to get the replica limit
(1.6)
. If we denote by
s
(
n
) =
1
2
n
1
X
k
=1
[sin(
π
(
k
)
/n
)]
2∆
,
(2.15)
its analytic continuation has been done in [
52
] for
<
1
/
4
. It reads
s
(
n
) =
n
1
2
g
(0) +
X
k
=1
[
ng
(
nk
)
g
(
k
)]
,
(2.16)
where
g
(
k
) =
4
sin(
π
)Γ(
k
+ ∆
)Γ(
k
+ ∆
)Γ(1
2∆
) sin(
π
(∆
−|
k
|
))
π
2
.
(2.17)
By plugging this result in eq.
(1.6)
and using [
52
]
s
(1) =
π
Γ(∆
+ 1)
4Γ(∆
+ 3
/
2)
,
(2.18)
in the limit
n
1
we get
S
(1)
A
= log
|
G
|−

ε

2∆
N
π
Γ(1 + ∆
)
2
|
G
|
Γ(∆
+ 3
/
2)
.
(2.19)
Even though this result has been derived for
<
1
/
4
, the final form of
s
(1)
is analytic all
the way up to
=
[
52
], so we expect our result holds also for larger values of
.
2.3 Compact Lie groups
If a 2d CFT has a compact Lie
G
symmetry,
G
is always upgraded to the corresponding
Kac-Moody algebra
ˆ
G
of some level
k
Z
+
. This is because the Noether theorem for
the
G
symmetry implies the existence of the Lie algebra-valued currents
J
a
and
̄
J
a
(
a
=
1
,
···
,
dim
G
)
with conformal weights
(1
,
0)
and
(0
,
1)
, respectively.
The topological line defect
U
(
g
)
for
g
G
is endable at a twist field
φ
g
(
z,
̄
z
)
, and its
conformal dimension
g
depends on
k
and the conjugacy class of
g
as follows. By bosonization,
the currents
J
a
and
̄
J
a
can be expressed in terms of free bosons
X
i
(
i
= 1
,
···
,r
, where
r
is the rank of
G
) and parafermions. In this realization, the current
J
i
in the maximum
torus direction of
G
is expressed as
J
i
=
i
k∂X
i
, where we assume that
X
i
is normalized
as
X
i
(
z
)
X
j
(
w
) =
δ
ij
log
(
z
w
)
with the energy-momentum tensor
T
(
z
) =
1
2
P
i
(
∂X
i
)
2
.
Any
g
G
can be moved to the maximum torus by conjugation
hgh
1
with some
h
G
.
Since the dimension
g
of the twist field
φ
g
is invariant under the conjugation, it is sufficient
to calculate it along the maximum torus. If
g
=
e
i
P
i
x
i
J
i
0
, the bosonization formula gives
– 9 –
JHEP01(2025)057
g
=
1
2
k
P
i
(
x
i
)
2
. In general,
g
=
1
2
k
(
d
(
g
))
2
, where
d
(
g
)
is the shortest distance from the
identity to
g
measured by the Killing metric on
G
.
We can now carry out the integral over
g
1
,
···
,g
n
in
(2.5)
. To do it, we will follow a
procedure similar to the one adopted in [
39
], with the main difference that, within our setup,
the symmetry is broken only at the boundary. Since the theory is critical, we can assume that
Z
n
(
{
g
i
}
)
Z
n
=

ε

β
n
(
{
g
i
}
)
,
(2.20)
where the coefficient
β
n
is universal, i.e. cutoff-independent. As we have highlighted before,
it depends on the specific CFT and symmetry we are considering. In ref. [
39
] it has been
computed for a massless Majorana fermion field theory and a
U
(1)
group. The
n
-dimensional
integral can be done by a saddle point approximation around the points belonging to the
set
{
(
h
1
,h
2
,
···
,h
n
)
,h
i
H
}
, where
H
is a finite symmetry subgroup of
G
i.e.
U
A
(
h
)
ρ
A
U
A
(
h
)
=
ρ
A
,
h
H .
(2.21)
Since all the saddle points equally contribute to the integral, we can just evaluate it around the
identity and multiply the final result by the total number of the saddle points, which is
|
H
|
n
1
.
Using the local coordinate chart
x
=
{
x
1
,...,x
dim(
G
)
} ∈
R
dim(
G
)
g
(
x
) =
e
i
P
a
x
a
J
a
G
,
we can expand the coefficient
β
n
around the vector
x
= 0
as [
39
]
β
n
(
g
(
x
)) =
1
2
x
T
H
β
x
+
O
(
x
3
)
,
(2.22)
where
H
β
is the Hessian matrix and
x
=
{
x
1
,...,
x
n
}
. Therefore, the integral
(2.5)
can
be rewritten as
Tr(
ρ
n
A
;
sym
)
Tr(
ρ
n
A
)
1
Vol(
G
)
n
1
μ
(0)
n
1
|
H
|
n
1
Z
R
n
dim(
G
)
d
x
δ
n
X
i
=1
x
i
!
e
log
2
x
T
H
β
x
,
(2.23)
where
μ
(0)
comes from the expansion of the Haar measure
μ
(
x
)
around
x
= 0
. By performing
the change of variables
x
j
ω
j
=
1
n
P
n
1
p
=0
x
p
e
2
πijp/n
, the Hessian matrix can be diagonalized
in blocks
D
p
,
p
= 0
,...,n
1
by exploiting the fact that it is a block-circulant matrix. Thus,
eq.
(2.23)
becomes
Tr(
ρ
n
A
;
sym
)
Tr(
ρ
n
A
)
1
Vol(
G
)
n
1
μ
(0)
n
1
|
H
|
n
1
n
n
1
Y
p
=1
Z
R
dim(
G
)
dωe
log(
ℓ/ε
)
2
ω
T
D
p
ω
,
(2.24)
and the Gaussian integral can be easily performed, assuming
D
p
is a positive definite matrix.
It yields
Tr(
ρ
n
A
;
sym
)
Tr(
ρ
n
A
)
1
Vol(
G
)
n
1

log(
ℓ/ε
)
2
π

dim(
G
)(1
n
)
2
μ
(0)
n
1
|
H
|
n
1
n
n
1
Y
p
=1
s
1
det
D
p
.
(2.25)
We can write down the final result for the entanglement asymmetry, which simply reads
S
(
n
)
A
=
dim(
G
)
2
log(log(
ℓ/ε
))
dim(
G
)
2
log (2
π
) + log Vol(
G
)
log (
μ
(0)
|
H
|
)
+
1
2
log

n
1
n
1

+
1
2(
n
1)
n
1
X
p
=1
Tr(log
D
p
) +
...
(2.26)
– 10 –
JHEP01(2025)057
The approach above is general and can be applied to an arbitrary form of the universal
coefficients
β
n
. The main difference with respect to the setup in which the symmetry is
broken both in the bulk and in the boundary [
39
] is that the leading order term grows as
log
(
log
(
))
, and not as
log
(
)
. The reason is that in eq.
(2.20)
, we do not have a term that
decays exponentially with the volume of the defect, since the symmetry is preserved in the
bulk. Moreover, in contrast to the symmetry breaking of a finite group, at leading order, the
asymmetry is not constant but it shows a sublogarithmic growth with system size.
2.4 Example: symmetry breaking in the Potts model
As a non-trivial example of the breaking of the symmetry generated by a finite group, we
consider the
Z
3
breaking in the 3-state Potts model, a conformal field theory with central
charge
4
/
5
. A possible microscopic description of the model is given by [
53
]
H
Potts
=
X
j
=0
[
U
j
U
j
+1
+
U
j
U
j
+1
+
V
j
+
V
j
]
,
(2.27)
where
U
j
=
1 0 0
0
ω
0
0 0
ω
2
, V
j
=
0 0 1
1 0 0
0 1 0
,
(2.28)
with
ω
=
e
2
πi/
3
. To study the
Z
3
-breaking, we consider a semi-infinite line and impose
fixed boundary condition at the end-point. In [
54
,
55
], these fixed boundary conditions were
identified with the Cardy states
|
e
0
,
|
e
ψ
,
|
e
ψ
:
|
e
0
has been identified with
|
A
, which means
that all sites on the boundary are in state
A
, the other two states can be identified with
B
and
C
, respectively. Here
A,B
, and
C
correspond to the three possible states of the Potts
model, and they can be implemented by adding a boundary field [
53
,
56
]. For instance,
B
0
=
h
1 0 0
0 0 0
0 0 0
(2.29)
favors
A
or
|
e
0
, one of the three Potts states. The
Z
3
group consists of three elements,
e,g
1
and
g
2
=
g
2
1
=
g
1
1
, whose action on the boundary states are specified by
e
|
A
=
|
A
, g
1
|
A
=
|
B
, g
2
|
A
=
|
C
.
(2.30)
The corresponding partition functions
Z
AB
,Z
AC
fix the scaling dimension of the boundary-
changing operators. The operator
φ
AB
(
φ
AC
) producing a transition from
A
to
B
(
C
)
transforms under the representation of weight
2
/
3
of the Virasoro algebra. Therefore, the
entanglement asymmetry reads
S
(1)
A
(
|
A
) = log(3)

ε

4
/
3
π
Γ(5
/
3)
12Γ(13
/
6)
+
O
(
γ
)
, γ >
4
/
3
.
(2.31)
We can also choose a different boundary condition, e.g.
|
B
+
C
, in which one of the three
spin states is forbidden at the boundary and the Potts spins on the boundary fluctuate
– 11 –