of 30
CALT-TH 2024-045
IPMU 24-0038
KYUSHU-HET-301
RIKEN-iTHEMS-Report-24
Entanglement asymmetry and symmetry defects in
boundary conformal field theory
Yuya Kusuki
1
,
2
,
3
, Sara Murciano
4
,
5
, Hirosi Ooguri
4
,
6
and Sridip Pal
4
1
Institute for Advanced Study, Kyushu University, Fukuoka 819-0395, Japan
2
Department of Physics, Kyushu University, Fukuoka 819-0395, Japan
3
RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), Wako, Saitama 351-
0198, Japan
4
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA
91125, USA
5
Department of Physics and Institute for Quantum Information and Matter, California Institute
of Technology, Pasadena, CA 91125, USA
6
Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo,
Kashiwa 277-8583, Japan
Abstract:
A state in a quantum system with a given global symmetry,
G
, can be sen-
sitive 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 leverag-
ing 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 asymme-
try 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.
arXiv:2411.09792v1 [hep-th] 14 Nov 2024
Contents
1 Introduction
1
1.1 Review of known results about entanglement asymmetry
2
1.2 Take-home message
2
2 Definitions and connections to BCFT
4
2.1 General discussion
5
2.2 Finite groups
6
2.3 Compact Lie groups
10
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
16
3.1.1 Entanglement entropy
17
3.1.2 Entanglement asymmetry
22
3.2 Semi-infinite interval
23
4 Discussion
24
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 under-
lying 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 [4–6], which has
further been generalized to the setup of non-invertible symmetry in [7–9].
– 1 –
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 [10–
13]. 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 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 inte-
grable systems [14–22], mixed states [23, 24], random or dual unitary circuits [25–29], higher
dimensions [30, 31], holography [32], in the presence of confinement [33] or many-body lo-
calization [34]. 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 [35]. However, dynamics is not the only setup in
which it is interesting to study symmetry breaking. In this spirit, Ref. [36] found a universal
behavior of the entanglement asymmetry in matrix product states for both finite and con-
tinuous groups, and this result has been extended to compact Lie groups in conformal field
theories in [37]. Other field theoretical results concern the behavior of the entanglement
asymmetry in certain coherent states of the massless compact boson [38], and the
SU
(2)
to
U
(1)
symmetry breaking in the critical XXZ spin chain [39]. The main upshot of these
computations in macroscopic models is that the asymmetry exhibits a leading-order behav-
ior 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
.
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 [40–44]. 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
.
– 2 –
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 quantita-
tive 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)
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
– 3 –
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 [18].
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. [37], 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 break-
ing. 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
– 4 –
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
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 Fig. 1 and 2, using the above mentioned conformal frame, to better understand
what happens. In the left panel of Fig. 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 Fig. 1. As a result, we obtain the insertions of boundary condition changing operators
(see Fig. 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)
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. [46] for a previous study of the symmetry resolution in AQFT).
– 5 –
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g
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Figure 1
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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.
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.
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
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