PHYSICAL REVIEW B
110
, 165144 (2024)
Fusion of one-dimensional gapped phases and their domain walls
David T. Stephen
1
,
2
and Xie Chen
2
,
3
1
Department of Physics and Center for Theory of Quantum Matter,
University of Colorado Boulder
, Boulder, Colorado 80309, USA
2
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology
,
Pasadena, California 91125, USA
3
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
, Pasadena, California 91125, USA
(Received 15 August 2024; revised 1 October 2024; accepted 3 October 2024; published 18 October 2024)
Finite depth quantum circuits provide an equivalence relation between gapped phases. Moreover, there can
be nontrivial domain walls either within the same gapped phase or between different gapped phases, whose
equivalence relations are given by finite depth quantum circuits in one lower dimension. In this paper, we use
such unitary equivalence relations to study the fusion of one-dimensional gapped phases. In particular, we use
finite depth circuits to fuse two gapped phases, local unitaries to fuse two domain walls, and a combination of
both to fuse gapped phases with domain walls. This provides a concrete illustration of some simple aspects of
the higher-category structure of gapped defects in a higher-dimensional trivial gapped bulk state.
DOI:
10.1103/PhysRevB.110.165144
I. INTRODUCTION
Defects in higher-dimensional quantum phases have re-
cently received a lot of attention [
1
–
12
]. Some of them satisfy
a noninvertible fusion rule [
1
–
4
,
6
,
7
,
11
] and generalize the
usual notion of symmetry transformations which are unitary
/
antiunitary and invertible. Gapped defects in nontrivial topo-
logical phases have been proposed to have a category or higher
category structure [
5
,
9
,
10
,
12
]. In particular, a 2-category
structure has been proposed to describe one-dimensional (1D)
defects where the objects of the 2-category are 1D gapped
defects and the morphisms between the objects are the zero-
dimensional (0D) domain walls either within the same defect
or between different defects.
Gapped defects and excitations in a topologically trivial
phase should also have a higher category structure. 1D gapped
defects in a higher-dimensional trivial bulk state are simply
1D gapped phases and their morphisms are domain walls ei-
ther within the same phase or between different phases. In this
paper, we study some simple aspects of the 2-category struc-
ture of 1D gapped phases and their 0D domain walls. On the
one hand, this provides a concrete and simple physics context
to illustrate some aspects of the higher-category structure dis-
cussed in the math literature. On the other hand, understanding
defects in trivial phases is a prerequisite for the proper under-
standing of defects in topologically nontrivial phases as the
1D phases exist as decoupled defects in topological phases
and more importantly, they show up as coefficients in the
fusion of nondecoupled defects (see for example Refs. [
1
,
13
].
Discussions of the 2-category structure of 1D gapped phases
can be found in Refs. [
6
,
14
].
Our analysis uses unitary quantum circuits and is based on
the following rule:
The equivalence relation between 1D gapped defects is
given by 1D finite depth circuits while the equivalence relation
between 0D domain walls is given by local unitary operations.
Figure
1
is an illustration of (a) 0D local unitary operations
and (b) 1D finite-depth circuits. These equivalence relations
are natural extensions of the definition for superselection
sectors of quasiparticle excitations [
15
] and the finite depth
circuit equivalence between 1D gapped phases [
16
]. We use
these equivalence relations to study the equivalence classes
of 1D gapped phases and their domain walls. Moreover, we
derive fusion rules between 1D gapped phases, between their
domain walls, and between 1D gapped phases with domain
walls.
When the higher-dimensional bulk state has a global sym-
metry, the 1D gapped phases as gapped defects in the bulk
share the same symmetry. When fusing 1D gapped phases
with a global symmetry, the circuits we use need to preserve
the global symmetry. This involves two different cases, as
discussed in detail in the following sections: in one case, each
gate in the circuit commutes with the symmetry and the circuit
is locally symmetric; in the second case, the circuit decouples
some degrees of freedom from the bulk but retains the overall
form of the symmetry on the remaining degrees of freedom.
The second case is necessary when the fusion is noninvertible.
The decoupled degrees of freedom form the coefficient of the
fusion result.
Some of the interesting results we get include:
(i) Fusion of a symmetry breaking phase with a symmetric
phase results in a symmetry breaking phase.
(ii) Nontrivial domain walls on symmetry breaking phases
are flux domain walls between order parameters, while non-
trivial domain walls on symmetric phases are symmetry
charges.
(iii) Fusion of two symmetry breaking phases results in
a symmetry breaking phase with a coefficient that is a 1D
system with degeneracy not protected by the symmetry.
(iv) When one of the symmetry breaking phases contains
a domain wall, the fusion coefficient contains a domain wall.
2469-9950/2024/110(16)/165144(14)
165144-1
©2024 American Physical Society
DAVID T. STEPHEN AND XIE CHEN
PHYSICAL REVIEW B
110
, 165144 (2024)
The paper is structured as follows. In Sec.
II
, we discuss
possible 1D gapped phases together with their domain walls,
especially in the presence of a symmetry. In Sec.
III
,weuse
finite depth quantum circuits (FDQCs) to fuse 1D gapped
phases without domain walls. In Sec.
IV
, we use 0D local
unitaries to fuse domain walls. In Sec.
V
, we discuss how to
use a combination of 0D local unitaries and 1D finite depth
circuits to fuse 1D gapped phases with domain walls. The
situation in higher dimensions is much more complicated with
the appearance of nontirival topological order, but we discuss
some simple cases in Sec.
VI
.
Throughout the discussion, we will make frequent use of
operators of the form
e
−
i
π
4
O
. We define short-hand notation
R
(
O
)
≡
e
−
i
π
4
O
, which has the property that for Pauli operators
P
and
Q
,
R
(
Q
)
PR
(
Q
)
†
=
P
;[
P
,
Q
]
=
0
iPQ
;
{
P
,
Q
}=
0
.
(1)
II. 1D GAPPED PHASES AND DOMAIN WALLS
1D gapped phases have been completely classified with or
without global symmetry [
17
–
19
]. Without global symmetry,
there are no nontrivial phases, whereas with global symmetry,
there are different possibilities. First the global symmetry
G
can be spontaneously broken if it is finite, giving rise to a
symmetry broken (SB) phase with nontrivial ground state
degeneracy. When the symmetry is not broken, there is the
possibility of having a symmetry protected topological (SPT)
phase classified by the cocycle group
H
2
(
G
,
U
(1)). The bulk
of the SPT phases are gapped and nondegenerate while the
edge carries nontrivial degeneracy. Finally, there is the pos-
sibility of partial symmetry breaking from
G
to a subgroup
H
combined with symmetry protected topological order of
H
classified by
H
2
(
H
,
U
(1)). In this work, we will focus our
attention on cases where the symmetry is either preserved or
completely broken. We make some comments on the case of
partial symmetry breaking in Sec.
II B
.
It is known that finite depth quantum circuits connect
ground states within the same gapped 1D phase [
16
]. For a
system with global symmetry
G
, the circuit is symmetric in
the sense that each local gate is invariant under the symmetry.
Ground states of different gapped phases on the other hand,
cannot be mapped into each other through finite depth circuit
[
16
]. Instead, a sequential quantum circuit is needed [
20
,
21
].
What kinds of 0D domain walls exist within each 1D
gapped phase? A domain wall is a local excitation on top of
the ground state. On the two sides of the domain wall, the
reduced density matrix looks exactly like that in the ground
state while on the domain wall the reduced density matrix
can be different. A domain wall is nontrivial if the reduced
density matrix around the domain wall is different from that
of the ground state and if it cannot be created with (symmetric)
local unitaries at the location of the domain wall. Otherwise,
the domain wall is trivial. Two domain walls are equivalent
to each other if they can be mapped to each other through a
(symmetric) local unitary.
Applying this rule, we can find domain walls for different
gapped phases.
FIG. 1. (a) 0D local unitary operations and (b) 1D finite depth
quantum circuit.
(i) In a symmetric phase (trivial or nontrivial SPT), a non-
trivial domain wall is an isolated charge.
(ii) In the symmetry-breaking phase, a nontrivial domain
wall is a domain wall between different values of the order
parameter, which we refer to as a flux.
Figure
2
gives the graphical representation of (a) a
symmetric phase with a charged domain wall, and (b) a
symmetry-breaking phase with a flux domain wall. In general,
a charged domain wall can be generated from the ground state
at some site
k
by applying a local charged operator around
k
,
while a flux domain wall can be generated by applying the
broken symmetry to all sites
i
>
k
. Neither of these corre-
spond to local symmetric operators, and hence they generate
nontrivial domain walls.
Prototypical examples of these two cases can be given by
the transverse field Ising model. In the symmetric phase with
the Hamiltonian
H
=−
i
X
i
,
(2)
and ground state wave function
|
...
+++++
...
,
(3)
a charged domain wall at site
k
corresponds to changing the
sign of the Hamiltonian term at site
k
to
+
X
k
H
=−
i
=
k
X
i
+
X
k
.
(4)
The ground state wave function takes the form
|
...
++−++
...
.
(5)
FIG. 2. Gapped 1D phases with domain wall: (a) A symmetric
phase with a charged domain wall labeled by
e
. (b) A symmetry
breaking phase with a flux domain wall labeled by
m
. (c) A degener-
ate domain wall between different SPT phases. (d) A nondegenerate
domain wall between symmetry breaking and SPT phases.
165144-2
FUSION OF ONE-DIMENSIONAL GAPPED PHASES AND ...
PHYSICAL REVIEW B
110
, 165144 (2024)
This charge can be generated by acting with a charged opera-
tor
Z
at site
k
.
In the symmetry breaking phase with the Hamiltonian
H
=−
i
Z
i
Z
i
+
1
(6)
and symmetrized ground state,
|
...
0000
...
+|
...
1111
...
,
(7)
a flux domain wall between sites
k
and
k
+
1 corresponds to
changing the sign of the corresponding Hamiltonian term to
+
Z
k
Z
k
+
1
,
H
=−
i
=
k
Z
i
Z
i
+
1
+
Z
k
Z
k
+
1
.
(8)
The (symmetrized) ground state wave function takes the form
|
...
0011
...
+|
...
1100
...
.
(9)
The flux domain wall can be generated between sites
k
and
k
+
1 by acting with the broken symmetry on all sites
i
>
k
.
A prototypical nontrivial SPT phases is given by the cluster
state with
Z
2
×
Z
2
symmetry [
22
,
23
]. The system contains
two sets of qubits, one on integer lattice sites and one on half
integer ones, each transforming under a
Z
2
symmetry
i
X
i
and
i
X
i
+
1
/
2
. The Hamiltonian of the cluster state is
H
=
i
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
(10)
and the ground state wave function is given by
i
CZ
i
−
1
2
,
i
CZ
i
,
i
+
1
2
|
...
++++++
...
.
(11)
Because the symmetry is now
Z
2
×
Z
2
, there are two different
symmetry charges. A domain wall charged under the first
Z
2
can be generated at site
k
by acting with
Z
, corresponding
to changing the sign of the Hamiltonian term at site
k
to
+
Z
k
−
1
/
2
X
k
Z
k
+
1
/
2
:
H
=
i
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
+
2
Z
k
−
1
2
X
k
Z
k
+
1
2
.
(12)
The ground state changes to
i
CZ
i
−
1
2
,
i
CZ
i
,
i
+
1
2
|
...
++−+++
...
.
(13)
We can similarly generate a domain wall charged under the
second
Z
2
by acting with
Z
on a site
k
+
1
/
2. We can also
try to generate flux domain walls in the cluster state by, e.g.,
acting with one of the
Z
2
symmetries on all half integer sites to
the right of site
k
. But, this will have the same effect as acting
with
Z
on site
k
, i.e., inserting a charged domain wall, so this
does not generate a new kind of domain wall.
There are no other nontrivial class of domain walls: a
symmetry flux in a symmetric state gives rise to either a
trivial domain wall in a trivial symmetric state or a charged
domain wall in a symmetry protected topological state, as
seen above. On the other hand, a charged domain wall in the
symmetry-breaking phase disappears into the bulk and is not
locally detectable and is hence a trivial domain wall. We give
general arguments for these claims using matrix product states
in Appendix
A
.
If the system contains fermions, the Majorana chain gives
a nontrivial phase [
24
]. It is similar to the bosonic SPT phases
in that a fermion parity symmetry flux is equivalent to a
symmetry charge. Therefore, there is one type of nontrivial
domain wall on the Majorana chain in the form of a fermion.
There are also domain walls between different phases. Be-
tween different SPT phases, there is a degenerate projective
edge mode denoted by the circle in Fig.
2(c)
. We denote such
domain walls as
p
. For example, between the cluster state at
i
0 and the trivial phase at
i
>
0 with the Hamiltonian
H
=
i
<
0
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
+
i
>
0
−
X
i
−
1
2
−
X
i
,
(14)
the twofold degenerate edge mode is given by the anticom-
muting operators
Z
−
1
/
2
X
0
and
Z
0
which both commute with
H
and are each charged under one of the
Z
2
symmetries. The
wave function across the domain wall cannot be invariant un-
der the full
Z
2
×
Z
2
group. One possible form that is invariant
under
i
X
i
but not under
i
X
i
+
1
/
2
is given by
CZ
−
1
2
,
0
i
<
0
CZ
i
−
1
2
,
i
CZ
i
,
i
+
1
2
|
...
+++
...
.
(15)
In fermion systems, the domain wall between the Majorana
chain and a trivial chain contains a Majorana zero mode.
When an SPT phase is connected to a symmetry-breaking
phase, the edge mode of the SPT phase can be coupled to
the order parameters in the symmetry-breaking phase, thereby
hiding the degeneracy of the projective edge mode behind
the bulk degeneracy in the symmetry-breaking phase. This is
shown in Fig.
2(d)
with a solid point and is denoted by
q
.Ifthe
SPT phase is on the left-hand side and the symmetry-breaking
phase on the right-hand side, we denote the domain wall as
̄
q
. For example, between the cluster state at
i
0 and the
symmetry breaking phase at
i
>
0 with the Hamiltonian
H
=
i
<
0
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
+
i
>
0
−
Z
i
Z
i
+
1
−
Z
i
−
1
2
Z
i
+
1
2
,
(16)
a symmetric term
Z
−
1
/
2
X
0
Z
1
/
2
can be added at the domain
wall to couple the edge mode (acted upon by
Z
−
1
/
2
X
0
)to
the order parameter
Z
1
/
2
and merge the degeneracy. This
reduces the eightfold degeneracy (the twofold degeneracy of
the SPT edge combined with the fourfold degeneracy of the
symmetry-breaking phase) to a net fourfold degeneracy. The
four degenerate states across the domain wall are then given
by
CZ
−
1
2
,
0
i
<
0
CZ
i
−
1
2
,
i
CZ
i
,
i
+
1
2
×|
...
++
00
...
,
(17)
|
...
++
01
...
,
|
...
+−
10
...
,
|
...
+−
11
...
.
Since the degenerate bulk states of the symmetry breaking
phase combine into different integer charged states under the
165144-3
DAVID T. STEPHEN AND XIE CHEN
PHYSICAL REVIEW B
110
, 165144 (2024)
FIG. 3. Fusion result of 1D gapped phases. (a) SB and SPT fuse into SB. (b) SPT and SPT fuse into SPT. (c) SB and SB fuse into SB with
nontrivial coefficient (the dashed line).
global symmetry, it cannot screen the projective mode on the
SPT edge. The degeneracy associated with the projectiveness
of the SPT edge will resurface when we study the fusion of
such domain walls in the Sec.
IV
.
There are of course other ways of connecting the sym-
metry breaking side and SPT side, for example with
Hamiltonian terms
−
Z
1
/
2
X
0
Z
1
/
2
,
−
Z
0
Z
1
,
+
Z
0
Z
1
. They can
all be mapped to each other through symmetric local uni-
tary transformations, hence confirming that there is only one
types of domain wall between an SB and an SPT state.
To map between
+
Z
1
/
2
X
0
Z
1
/
2
and
−
Z
1
/
2
X
0
Z
1
/
2
, we can
use
Z
0
Z
1
; to map between
−
Z
0
Z
1
and
+
Z
0
Z
1
, we can use
Z
1
/
2
X
0
Z
1
/
2
; to map between
Z
1
/
2
X
0
Z
1
/
2
and
Z
0
Z
1
, we can use
R
(
Z
0
Z
1
)
R
(
Z
1
/
2
X
0
Z
1
/
2
).
III. FUSION OF GAPPED PHASES
We can fuse gapped phases by stacking them on top of each
other and applying a finite depth circuit. When the system
has a global symmetry, each local gate in the circuit needs
to be symmetric. In the following subsections, we are going
to explicitly construct the circuit that realizes the fusion for
different pairs of phases. In particular, we are going to choose
a standard form for each phase and use finite depth circuits to
map the stack of two phases into the standard form of a third.
The result of the fusion is shown in Fig.
3
.
(i) The fusion of a symmetry-breaking with a symmetric
phase results in a symmetry breaking phase. This holds no
matter what SPT order the symmetric phase has.
(ii) The fusion of two symmetric phases results in a sym-
metric phase.
(iii) The fusion of two symmetry breaking phases results
in a symmetry breaking phase but with a nontrivial coefficient.
These fusion results hold even when the symmetry is only
partially broken. We explain each case in the following sub-
sections.
A. SB
×
SPT
To demonstrate the fusion circuit in this case, we are going
to use the symmetry-breaking phase on the upper chain and
the symmetric phase of the 1D Ising model on the lower chain
with the Hamiltonian
H
u
=−
i
Z
u
i
Z
u
i
+
1
and
H
l
=−
i
X
l
i
(18)
where the superscripts
l
and
u
denote operators acting on the
upper and lower chain. The two chains can be fused with the
circuit shown in Fig.
4
. The first step involves
R
(
ZZ
) gates on
all vertical pairs connected by green dashed lines. The second
step involves
R
(
X
) gates on the qubits in the lower chain. The
Hamiltonian terms in the symmetry breaking chain remain
invariant while the
−
X
l
i
Hamiltonian terms in the symmetric
chain are mapped to
−
Z
u
i
Z
l
i
terms between each vertical pair
of qubits. The wavefunction transforms as
(
|
00
...
0
+|
11
...
1
)
⊗|++
...
+
→|
00
...
0
⊗|
00
...
0
+|
11
...
1
⊗|
11
...
1
.
(19)
That is, the tensor product of a SB GHZ state in one chain
and a symmetric product state in the other is mapped to a
GHZ state on two chains such that the order parameters from
the two chains match. The fusion circuit consists of gates that
commute with the global symmetry.
Of course, if we think of these gapped phases as defects
within a higher dimensional trivial symmetric bulk state, the
symmetric phase of the Ising chain is a trivial defect and
nothing needs to be done to fuse it with the symmetry breaking
defect. Instead, we can view the inverse of the above circuit
as a way to map the GHZ state on two chains to the GHZ
state on one chain. This step can be combined with the cir-
cuits discussed in the following subsections to put the 2-chain
symmetry breaking state into the standard form on one chain
only.
When the symmetric phase has a nontrivial SPT order, it
corresponds to a nontrivial defect in the higher dimensional
trivial symmetric bulk. Its fusion with a symmetry breaking
FIG. 4. Fusion of the symmetry breaking ground state and the
symmetric ground state of the Ising model into the symmetry break-
ing state with a symmetric finite depth circuit. The first step involves
R
(
ZZ
) gates on all vertical pairs connected by green dashed lines.
The second step involves
R
(
X
) gates on the qubits in the second line.
165144-4
FUSION OF ONE-DIMENSIONAL GAPPED PHASES AND ...
PHYSICAL REVIEW B
110
, 165144 (2024)
FIG. 5. Fusion of the symmetry breaking ground state and the
nontrivial SPT state with
Z
2
×
Z
2
symmetry into the symmetry
breaking state with a symmetric finite depth circuit. The first step
involves
R
(
ZZ
) gates on all vertical pairs connected by green dashed
lines. The second step involves
R
(
ZXZ
) gates generated by the
ZXZ
Hamiltonian terms in the lower chains (green triangles).
phase still results in a symmetry breaking state, as demon-
strated below with the symmetry breaking phase
H
u
=
i
−
Z
u
i
Z
u
i
+
1
−
Z
u
i
−
1
2
Z
u
i
+
1
2
,
(20)
and the cluster state
H
l
=
i
−
Z
l
i
−
1
2
X
l
i
Z
l
i
+
1
2
−
Z
l
i
X
l
i
+
1
2
Z
l
i
+
1
(21)
with
Z
2
×
Z
2
symmetry. The circuit, as shown in Fig.
5
, keeps
the Hamiltonian terms in the symmetry breaking chain invari-
ant while mapping the Hamiltonian terms in the cluster state
chain to vertical
−
Z
u
Z
l
pairs, hence resulting in a combined
symmetry breaking state of
Z
2
×
Z
2
on two chains. Applying
(two copies of) the inverse of the circuit in Fig.
4
, we can
further map it back to the
Z
2
×
Z
2
symmetry breaking state
on one chain with symmetric product state in the other. The
whole process again consists of gates that commute with the
global symmetry.
To summarize, we find
SB
×
SPT
→
SB
.
(22)
B. SPT
×
SPT
It is well known that SPT phases under stacking form
an Abelian group given by
H
2
(
G
,
U
(1)) [
17
,
18
]. We briefly
review the argument.
Take two SPT phases in the standard form of two projective
representations on site. The left spin with basis states
|
g
,
g
∈
G
, transforms under the left half of the symmetry as
V
L
(
g
)
|
g
=
ω
(
g
,
g
)
|
g
g
,
(23)
while the right spin transforms under the right half of the
symmetry as
V
R
(
g
)
|
g
=
ω
∗
(
g
,
g
)
|
g
g
.
(24)
The on-site symmetry
U
(
g
)
=
V
L
(
g
)
⊗
V
R
(
g
) forms a linear
representation of
G
,
U
(
g
1
)
U
(
g
2
)
=
U
(
g
1
g
2
)
,
(25)
FIG. 6. Fusion of two SPT states into one SPT state. The two
SPT states have projective edge states given by
ω
1
and
ω
2
, respec-
tively. The fusion of the two SPTs gives a third SPT with edge state
given by
ω
1
ω
2
.
while
V
L
and
V
R
each forms a projective representation
V
L
(
g
1
)
V
L
(
g
2
)
=
ω
(
g
1
,
g
2
)
V
L
(
g
1
g
2
)
,
V
R
(
g
1
)
V
R
(
g
2
)
=
ω
∗
(
g
1
,
g
2
)
V
R
(
g
1
g
2
)
.
(26)
The pair of spins connected between nearest-neighbor sites
are in the entangled state of
g
|
gg
.
(27)
To map the two chains to one SPT chain, we can simply map
the two entangled pairs in the upper and lower chain into a
diagonal entangled state
⎛
⎝
g
u
|
g
u
g
u
⎞
⎠
⊗
⎛
⎝
g
l
|
g
l
g
l
⎞
⎠
→
̃
g
|
̃
g
̃
g
,
(28)
where ̃
g
=
g
u
g
l
. Because the entangled states before and after
the map are both singlets under the global symmetry, this step
can be achieved with local symmetric unitary transformations.
Therefore, we find a finite depth symmetric circuit to map the
two SPT chains into one with projective edge state associated
with
̃
w
(
g
1
,
g
2
)
=
ω
u
(
g
1
,
g
2
)
ω
l
(
g
1
,
g
2
)
.
(29)
This can be seen from
̃
V
(
g
)
|
̃
g
=
V
u
(
g
)
V
l
(
g
)
|
g
u
g
l
=
ω
u
(
g
,
g
)
ω
l
(
g
,
g
)
|
̃
g
g
(30)
and
̃
V
(
g
1
)
̃
V
(
g
2
)
=
ω
u
(
g
1
,
g
2
)
ω
l
(
g
1
,
g
2
)
̃
V
(
g
1
g
2
)
.
(31)
Therefore, we can write
SPT
1
×
SPT
2
→
SPT
1
+
2
,
(32)
where the sum in the subscript 1
+
2 corresponds to
the Abelian composition of projective representations in
H
2
(
G
,
U
(1)). This fusion is shown in Fig.
6
C. SB
×
SB
The fusion of two symmetry breaking phases results in a
symmetry-breaking phase with a nontrivial fusion coefficient.
Let us discuss this case carefully.
Consider two chains both in the symmetry breaking phase
of the Ising chain
H
u
=−
i
Z
u
i
Z
u
i
+
1
,
H
l
=−
i
Z
l
i
Z
l
i
+
1
.
(33)
165144-5
DAVID T. STEPHEN AND XIE CHEN
PHYSICAL REVIEW B
110
, 165144 (2024)
FIG. 7. Fusion of two symmetry breaking phases. The circuit
consists of pairwise contolled-NOT (CX) gates with the spin in the
first chain as control and the corresponding spin the second chain as
target.
We take the wave function in both chains to be the sym-
metrized GHZ state while keeping in mind that there is
another degenerate state with nontrivial total charge.
The circuit shown in Fig.
7
fuses the two chains together.
The circuit is composed of controlled-Not gates between pairs
of spins in the two chains, defined as
CX
ct
=|
0
0
|
c
⊗
I
t
+
|
1
1
|
c
⊗
X
t
acting on control (c) and target (t) qubits. The
spins in the top chain are used as control while the spins in
the bottom chain are used as target. The controlled-Not gates
are not symmetric. Instead, we are going to think of it as
implementing a local change of basis and let the symmetry
operator transform with it:
i
CX
ul
i
i
X
u
i
X
l
i
i
CX
ul
i
=
i
X
u
i
.
(34)
The wave function remains invariant but now the interpreta-
tion is different.
(
|
00
...
0
+|
11
...
1
)
⊗
(
|
00
...
0
+|
11
...
1
)
→
(
|
00
...
0
+|
11
...
1
)
⊗
(
|
00
...
0
+|
11
...
1
)
.
(35)
The transformed symmetry operator acts on the first chain
only and the first GHZ state is the fusion result, which is
still in the SB phase. The second chain does not transform
under the symmetry any more. It is also in the GHZ state, but
with a degeneracy not protected by the symmetry. This is the
coefficient in front of the fusion result
SB
×
SB
→
Z
2
×
SB,
(36)
and is represented by the dashed red line in Fig.
3(c)
.This
coefficient is going to play an interesting role in the fusion of
symmetry breaking phases with domain walls, as we discuss
in Sec.
V
.
IV. FUSION OF DOMAIN WALLS
Domain walls on the same 1D chain can be fused with 0D
symmetric local unitary gates when they are at a finite distance
from each other. The result is summarized in Fig.
8
.
Some of these fusion results are straightforward to see. For
example, case (a) and (b)
SPT
−
e
−
SPT
−
e
−
SPT
→
SPT,
SB
−
m
−
SB
−
m
−
SB
→
SB
.
(37)
We are only going to discuss cases (c) through (f) in detail
below.
FIG. 8. Fusion of domain walls. (a) Fusion of charge domain
walls. (b) Fusion of flux domain walls. (c) Fusion of a charge with a
projective domain wall into the same projective domain wall between
SPT phases. (d) Fusion of a flux domain wall with a
q
domain wall
from SB to SPT phase into the same
q
domain wall. (e) A
q
and
p
domain wall fuse into a
q
domain wall with a degeneracy given by
the
p
domain wall. (f) A ̄
q
and
q
domain wall fuse into a
p
domain
wall with a degeneracy given by the SB phase. (g) A
q
and ̄
q
domain
wall fuse into all possible flux domain walls.
A. SPT1-e-SPT1-p-SPT2
We consider case (c) where a symmetry charge is fused into
the projective edge state between different SPT phases.
Suppose the right half of the system is in the trivial sym-
metric phase of
Z
2
×
Z
2
symmetry while the left half of the
system is in the nontrivial SPT phase,
H
=
i
>
0
−
X
i
−
X
i
−
1
2
+
i
<
0
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
.
(38)
The domain wall is acted upon by a pair of anti-commuting
operators
Z
−
1
2
X
0
,
Z
0
,
(39)
neither of which is symmetric.
With a symmetry charge on the trivial symmetric side, the
Hamiltonian term at, for example
i
=
1 changes sign
H
=
2
X
1
+
i
>
0
−
X
i
−
X
i
−
1
2
+
i
<
0
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
.
(40)
To map Eq. (
40
) back to Eq. (
38
) without the changing de-
generate space described by Eq. (
39
), we can use the local
symmetric unitary operator
Z
0
Z
1
.
(41)
In this way, we fuse a
p
domain wall between two SPT phases
with an
e
domain wall on one of the SPT phase into a
p
domain
165144-6
FUSION OF ONE-DIMENSIONAL GAPPED PHASES AND ...
PHYSICAL REVIEW B
110
, 165144 (2024)
wall:
SPT
1
−
e
−
SPT
1
−
p
−
SPT
2
→
SPT
1
−
p
−
SPT
2
.
(42)
B. SB-m-SB-q-SPT
Let us discuss case (d) where an
m
domain wall on a
symmetry-breaking state fuses into the
q
domain wall between
a symmetry-breaking and SPT state. The fusion result has to
be a
q
domain wall. We will see how that happens through
local unitary transformations on the domain walls.
Consider the
Z
2
×
Z
2
symmetry-breaking state on the right
half and the cluster state on the left half
H
=
i
>
0
−
Z
i
Z
i
+
1
−
Z
i
−
1
2
Z
i
+
1
2
+
i
<
0
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
(43)
and a coupling term
Z
−
1
2
X
0
Z
1
2
.
(44)
A domain wall on the SB side corresponds to flipping the sign
of one of the
ZZ
terms, say between
1
2
and 3
/
2:
H
=
2
Z
1
2
Z
2
/
3
+
i
>
0
−
Z
i
Z
i
+
1
−
Z
i
−
1
2
Z
i
+
1
2
+
i
<
0
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
X
i
+
1
2
Z
i
+
1
.
(45)
To map Eq. (
45
) back to Eq. (
43
) without changing Eq. (
44
),
we can use the local symmetric unitary operator
Z
0
X
1
2
Z
1
.
(46)
In this way, we fuse a
q
domain wall between the SB and SPT
phase with an
m
domain wall on the SB phase into a
q
domain
wall:
SB
−
m
−
SB
−
q
−
SPT
→
SB
−
q
−
SPT
.
(47)
C. SB-q-SPT1-p-SPT2
We consider case (e) where the
q
domain wall from an SB
phase to an SPT phase is fused with the
p
domain wall from
the first SPT phase to another SPT phase. We expect the two
to fuse into a single
q
domain wall but with the multiplicity of
the
p
domain wall contributing a prefactor to the fusion result.
Suppose that the system has a symmetry breaking state on
the left, and cluster state on the right, and a trivial state in the
middle
H
=
i
<
−
1
−
Z
i
−
1
Z
i
−
Z
i
−
1
2
Z
i
+
1
2
+
i
=−
1
,
0
,
1
−
X
i
−
X
i
+
1
2
+
i
>
2
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
−
1
X
i
−
1
2
Z
i
+
1
2
.
(48)
The projective edge state between the two SPT states is acted
upon by
X
2
Z
5
/
2
and
Z
2
. Applying the local symmetric unitary
transformation
i
=−
1
,
0
,
1
R
(
X
i
)
R
(
Z
i
−
1
Z
i
)
R
(
X
i
+
1
2
)
R
(
Z
i
−
1
2
Z
i
+
1
2
) (49)
maps the Hamiltonian to
H
=
i
<
2
−
Z
i
−
1
Z
i
−
Z
i
−
1
2
Z
i
+
1
2
+
i
>
2
−
Z
i
−
1
2
X
i
Z
i
+
1
2
−
Z
i
−
1
X
i
−
1
2
Z
i
+
1
2
.
(50)
The twofold degeneracy can be separated by the eigenvalue of
Z
1
Z
2
=±
1. The two cases differ by the action of
Z
3
/
2
X
2
Z
5
/
2
.
In either case, the
q
domain wall from symmetry breaking
to trivial symmetric phases fuses with the
p
domain wall
between the trivial and nontrivial SPT phases and becomes
a
q
domain wall from the symmetry breaking phase to the
nontrivial SPT phases. Therefore,
SB
−
q
−
SPT
1
−
p
−
SPT
2
→
n
p
×
SB
−
q
−
SPT
2
,
(51)
where
n
p
is the degeneracy of the projective domain wall
between the two SPT phases.
D. SPT1- ̄
q
-SB-
q
-SPT2
Consider case (f) where a ̄
q
domain wall from one SPT
phase to the SB phase is fused with a
q
domain wall from
the SB phase to another SPT phase. ̄
q
and
q
fuse into the
p
domain wall between SPT1 and SPT2, but with a twofold
degeneracy that comes from squeezing the intermediate SB
region.
Suppose that the Hamiltonian takes the form
H
=
i
<
−
1
−
X
i
−
X
i
+
1
2
−
Z
−
1
Z
0
−
Z
0
Z
1
−
Z
−
1
2
Z
1
2
+
i
>
1
−
Z
i
−
1
X
i
−
1
2
Z
i
−
Z
i
−
1
2
X
i
Z
i
+
1
2
.
(52)
Applying the local symmetric transformation,
R
(
X
−
1
)
R
(
Z
−
1
Z
0
)
R
(
X
0
)
R
(
Z
0
Z
1
)
R
(
X
−
1
2
)
R
(
Z
−
1
2
Z
1
2
) (53)
maps the Hamiltonian to
H
=
i
<
0
−
X
i
−
X
i
+
1
2
−
X
0
+
i
>
1
−
Z
i
−
1
X
i
−
1
2
Z
i
−
Z
i
−
1
2
X
i
Z
i
+
1
2
.
(54)
The low energy space is a direct sum of two parts, one with
X
1
/
2
=
1, one with
X
1
/
2
=−
1. In the first case, the ̄
q
and
q
domain walls fuse into a
p
domain wall between the two SPT
phases. In the second case, there is an extra charge at site 1
/
2
which can be merged into the
p
domain wall with an operator
Z
1
/
2
X
1
Z
3
/
2
. That is,
SPT
1
−
̄
q
−
SB
−
q
−
SPT
2
→
n
SB
×
SPT
1
−
p
−
SPT
2
,
(55)
where
n
SB
is the degeneracy of the symmetry breaking phase.
165144-7
DAVID T. STEPHEN AND XIE CHEN
PHYSICAL REVIEW B
110
, 165144 (2024)
E. SB-
q
-SPT- ̄
q
-SB
Consider case (g) where a
q
domain wall from the SB to
SPT phase is fused with a ̄
q
domain wall from the SPT phase
back to the SB phase. After the fusion, the SB phases on
the two sides are connected, but there are two possibilities at
the domain wall. There is either no nontrivial domain wall in
between or there is a flux domain wall in between. The fusion
result is the direct sum of these two.
To derive this result, we take the
Z
2
symmetry breaking
phase with the on the two sides with the symmetric phase in
between. The Hamiltonian for the whole chain is
H
=
i
<
−
1
−
Z
i
−
1
Z
i
+
i
=−
1
,
0
,
1
−
X
i
+
i
>
1
Z
i
Z
i
+
1
(56)
with symmetrized wave function
(
|
...
00
+|
...
11
)
⊗ | + ++ ⊗
(
|
00
...
+|
11
...
)
.
(57)
Applying local unitary gates
R
(
Z
−
2
Z
−
1
)
R
(
X
−
1
)
R
(
Z
−
1
Z
0
)
R
(
X
0
)
R
(
Z
0
Z
1
)
R
(
X
1
) (58)
maps the Hamiltonian to
H
=
i
1
−
Z
i
−
1
Z
i
+
i
>
1
Z
i
Z
i
+
1
(59)
and the wave function to
(
|
...
00
+|
...
11
)
⊗
(
|
00
...
+|
11
...
)
.
(60)
The fusion result is hence a direct sum of two possibilities, one
corresponding to
Z
1
Z
2
=
1, the other corresponding to
Z
1
Z
2
=
−
1–an
m
domain wall. In terms of wave functions, the two
parts in direct sum are
|
...
0000
...
+|
...
1111
...
(61)
and
|
...
0011
...
+|
...
1100
...
.
(62)
Therefore,
SB
−
q
−
SPT
−
̄
q
−
SB
→
SB
+
SB
−
m
−
SB
.
(63)
V. FUSION OF PHASES WITH DOMAIN WALLS
When the 1D phases to be fused have domain walls on
them, they can again be fused with symmetric finite-depth
circuits, but we need to be careful in choosing the finite-depth
circuit. This is because in many situations domain walls can
be created
/
annihilated with finite depth circuits. For example,
the flux domain wall in a symmetry breaking state can be
created by applying symmetry to a segment on the chain,
which is a depth one circuit. The charge domain wall on sym-
metric phases can also be created with a finite depth circuit
simply by applying the charge hopping operator to a segment.
Therefore, in order to properly discuss the fusion of 1D phases
with domain wall, we need to use a circuit that preserves the
existence of a domain wall. To that end, we can use the circuit
discussed in Sec.
II
to fuse the bulk of the gapped phase and
add extra symmetric local unitary to fuse the domain walls.
That is, on the two sides of the domain walls, we use the same
circuit used for fusing gapped phases without domain walls.
At the domain wall, we have the freedom to change the circuit
by a symmetric local unitary. By doing so, we make sure we
do not have the freedom to create
/
remove domain walls with
the fusion circuit.
Figure
9
summarizes the fusion result of gapped phases
with domain walls. The results in Figs.
9(a)
and
9(b)
are
straightforward. We will discuss Figs.
9(c)
–
9(e)
in the follow
subsections. We remark that an interesting subtlety can occur
in case (b) when the symmetry is only partially broken, in
which the SPT is absorbed at the cost of creating a charge
domain wall. This is discussed in Appendix
B
.
A. SB
×
SB
We discuss case (c) of a symmetry breaking phase fused
with another symmetry breaking phase carefully when one or
both of them carry flux domain walls. As shown in Fig.
9(c)
,
the fusion result might depend on the ordering of the defects
to be fused.
First, we consider the case where the first chain has a flux
defect while the second does not. The wave function before
fusion is
(
|
...
0011
...
+|
...
1100
...
)
⊗
(
|
...
0000
...
+|
...
1111
...
)
.
(64)
Applying the same fusion circuit as in Sec.
III C
when two
symmetry breaking phases without domain walls are fused,
we get
(
|
...
0011
...
+|
...
1100
...
)
⊗
(
|
...
0011
...
+|
...
1100
...
)
.
(65)
As the transformed symmetry operator only acts on the first
chain, we interpret the first chain as the fusion result while the
second chain is the coefficient. We see that the fusion result is
a symmetry breaking chain with flux domain wall, while the
coefficient is also in a GHZ state with domain wall.
When the two chains are exchanged, the fusion has a dif-
ferent result:
(
|
...
0000
...
+|
...
1111
...
)
⊗
(
|
...
0011
...
+|
...
1100
...
)
(66)
is mapped to
(
|
...
0000
...
+|
...
1111
...
)
⊗
(
|
...
0011
...
+|
...
1100
...
)
.
(67)
The fusion result is hence the symmetry breaking chain with-
out domain wall, while the coefficient stays the same as the
previous case which is a GHZ state with domain wall.
Finally, when both chains contain flux domain walls,
(
|
...
0011
...
+|
...
1100
...
)
⊗
(
|
...
0011
...
+|
...
1100
...
)
(68)
it maps to
(
|
...
0011
...
+|
...
1100
...
)
⊗
(
|
...
0000
...
+|
...
1111
...
)
.
(69)
The fusion result is a symmetry breaking chain with domain
wall, while the coefficient does not have a domain wall.
To summarize, we find
SB
−
m
−
SB
×
SB
→
Z
2
−
m
−
Z
2
×
SB
−
m
−
SB,
SB
×
SB
−
m
−
SB
→
Z
2
−
m
−
Z
2
×
SB,
SB
−
m
−
SB
×
SB
−
m
−
SB
→
Z
2
×
SB
−
m
−
SB
.
(70)
165144-8
FUSION OF ONE-DIMENSIONAL GAPPED PHASES AND ...
PHYSICAL REVIEW B
110
, 165144 (2024)
FIG. 9. Fusion of 1D gapped phases with domain walls. (a) SPT phases and their charge domain walls fuse according to their additive
group structure. (b) Fusion of SB phase with a flux domain wall and SPT phase with a charge domain wall results in SB phase with a flux
domain wall. (c) Fusion of two SB phases with or without domain walls; the dashed line is the fusion coefficient. (d) Fusion of SB phase with
two segments of SPT phases with a projective domain wall in between;
n
p
is the degeneracy of the projective domain wall. (e) Fusion of a
chain half in SB phase and half in SPT phase with a SB chain results in a SB chain with all possible flux domain walls summed over. The
dotted line between the domain wall and the fusion coefficient indicate their coupling. (f) Fusion of two chains with
q
and ̄
q
domain walls.
B. SB
×
SPT-p-SPT
Consider case (d) involving the fusion of a symmetry
breaking chain with an SPT chain but with two different SPT
orders on the two sides and a
p
domain wall in between. As
the SB order eats up the SPT orders, we expect the
p
domain
wall to disappear after the fusion.
To see how explicitly that happens, consider the setup in
Fig.
10
with a top chain with
Z
2
×
Z
2
symmetry breaking
order. The Hamiltonian in the top chain is given by
H
u
=
i
−
Z
u
i
Z
u
i
+
1
−
Z
u
i
−
1
2
Z
u
i
+
1
2
.
(71)
FIG. 10. Fusion of SB and SPT phases in the presence of a
p
domain wall.
The bottom chain has trivial SPT order on the left-hand side
and nontrivial SPT order on the right-hand side. The Hamilto-
nian in the bottom chain is given by
H
l
=
i
<
0
−
X
l
i
−
X
l
i
+
1
/
2
+
i
>
0
−
Z
l
i
−
1
2
X
l
i
Z
l
i
+
1
2
−
Z
l
i
X
l
i
−
1
2
Z
l
i
+
1
.
(72)
On the two sides of the domain wall, we can use the circuit
in Figs.
4
and
5
, respectively, to fuse the trivial and nontrivial
SPTs into the symmetry breaking state. In particular, step 1
involves the gate
R
(
ZZ
) on each vertical pair connected by
dashed green lines, and step 2 involves
R
(
X
) on the blue dots
and
R
(
ZXZ
) centered on the orange dots in the lower chain.
After these two steps, the horizontal
ZZ
terms remain while
the
X
and
ZXZ
terms are replaced by vertical
ZZ
terms on
pairs of spins connected by the green dashed line. Therefore,
all qubits are merged into the symmetry-breaking state except
for the yellow dot.
The degeneracy associated with the edge state on the do-
main wall can be removed by the
ZZ
term in the black box,
which commutes with all other terms in the Hamiltonian.
The low energy space of the system is a direct sum of two
165144-9
DAVID T. STEPHEN AND XIE CHEN
PHYSICAL REVIEW B
110
, 165144 (2024)
FIG. 11. Fusion of a SB-SPT chain with a
q
domain wall and a
SB chain.
parts, one with eigenvalue
+
1 under the
ZZ
term in the
black box, the other with a
−
1 eigenvalue. With either eigen-
value, the yellow dot merges into the symmetry breaking state
and the two differ only by the local symmetric operation of
X
on the yellow dot. Therefore, there is a factor of 2 in the
fusion result in Fig.
10
, but otherwise we see the projective
edge state between two SPT states disappears when fused with
a symmetry-breaking state. That is,
SB
×
SPT
1
−
p
−
SPT
2
→
n
p
×
SB
.
(73)
C. SB-
q
-SPT
×
SB
We consider case (e) where a SB-SPT chain is fused with a
SB chain. We expect the whole system to fuse into a SB state
but with the possibility of having or not having a
m
domain
wall in between.
Suppose that both chains have a
Z
2
symmetry. The upper
chain contains a symmetry breaking phase on the left half and
a symmetric phase on the right half,
H
u
=−
i
<
0
Z
u
i
Z
u
i
+
1
−
i
>
0
X
u
i
.
(74)
The lower chain is in a symmetry breaking phase,
H
l
=
−
Z
l
i
Z
l
i
+
1
.
(75)
Fusion can be realized with a circuit shown in Fig.
11
.
The orange and green boxes indicate the Hamiltonian terms
before the fusion. The circuit on the left-hand side follows the
one in Fig.
7
and is composed of CX gates from the top chain
to the bottom chain.
After the circuit, all the Hamiltonian terms remain invariant
on the two sides of the domain wall. Global
Z
2
symmetry does
not act on the left half of the lower chain any more and it
becomes a fusion coefficient. The left half of the upper chain
and the right half of the lower chain merge into a SB chain
which is the fusion result. At the domain wall, there is now a
three body
Z
1
Z
2
Z
3
term, indicated by the dashed black box in
Fig.
11
. This term couples the order parameter of the fusion
coefficient
Z
1
with the domain wall on the fusion result
Z
2
Z
3
.
Therefore, we can write
SB
−
q
−
SPT
×
SB
→
m
SB
−
m
−
SB
,
(76)
although this way of writing does not make explicit the cou-
pling of
m
to the fusion coefficient.
FIG. 12. Fusion of phases 1 and 2 separated by a domain wall
a
and phases 3 and 4 separated by a domain wall
b
. We can first
separate the domain walls a bit to the left and right, which does not
change anything. Then we fuse the two systems on either side of the
dotted line independently (each containing only one domain wall).
This results in a 1D system with two domain walls
a
and
b
separated
by an intermediate phase (2
×
3), which can then be fused into one
domain wall
c
. For simplicity, we do not draw fusion coefficients,
degeneracy, or sums over domain walls, all of which are possible in
general.
D. Other fusions
Other more complex fusions can be understood in terms
of simpler fusions using the following trick. In Fig.
12
,we
show how a fusion of two systems, each containing a domain
wall, can be understood as two separate phase fusions, each
involving only a single domain wall, followed by a fusion of
domain walls.
As an application of this trick, consider the fusion SB
−
q
−
SPT
1
×
SPT
2
−
̄
q
−
SB shown in Fig.
9(f)
. First, we note
the simple fusion result
SB
−
q
−
SPT
1
×
SPT
2
→
SB
−
q
−
SPT
1
+
2
,
(77)
which can be explicitly obtained using similar circuits as
in previous examples. Using this, and applying the trick in
Fig.
12
, we find
SB
−
q
−
SPT
1
×
SPT
2
−
̄
q
−
SB
→
SB
−
q
−
SPT
1
+
2
−
̄
q
−
SB
→
m
SB
−
m
−
SB
,
(78)
where the last step used Eq. (
63
)tofusethe
q
and ̄
q
domain
walls.
VI. SOME RESULTS IN HIGHER DIMENSIONS
Gapped phases in two or higher dimensions form cate-
gories of even higher order [
10
,
25
,
26
]. For example, in 2D
gapped phases there are 1D domain walls and there can further
be 0D domain walls on top of the 1D domain walls. A com-
plete discussion of the fusion of higher dimensional gapped
phases and their domain walls is much more complicated and
beyond the scope of this paper, but we do want to discuss a
few simple cases, and show how things work in an analogous
way as their 1D counterparts.
A. 1D domain walls in 2D SPTs
Consider the 2D SPT state with
Z
2
symmetry. One type of
1D defect is a symmetry breaking defect where the symmetry
charge condenses. One can then ask if there are any other
165144-10
FUSION OF ONE-DIMENSIONAL GAPPED PHASES AND ...
PHYSICAL REVIEW B
110
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FIG. 13. Symmetry defect in 2D
Z
2
SPT state. Each lattice site
(blue discs) hosts four qubits. Every four qubits connected into a
square are in the
|
0000
+|
1111
state. The
Z
2
symmetry action on
each site is given in (b). Applying the symmetry in the lower half
plane induces transformation along the boundary line (dotted red) as
shown in (a). The same transformation can be induced by applying
the gate set in (c) to each site which are
Z
2
symmetric.
types of interesting 1D defects. In this section we discuss
whether a 1D symmetry defect line is a nontrivial defect inside
the SPT state. That is, suppose the
Z
2
symmetry is applied
to a region inside the 2D state. As the state is symmetric, it
remains invariant both inside and outside the region, but can
change along the boundary. In this section, we are going to
show using the fixed-point form of the SPT state discussed
in Refs. [
27
,
28
] that the changes along the boundary can be
induced with a finite depth symmetric circuit. Therefore, a
symmetry defect line is not a nontrivial 1D defect in the SPT
state.
Consider the 2D state as shown in Fig.
13
, where the each
lattice site (blue discs) hosts four qubits. The
Z
2
symmetry on
each lattice site is given in (b) which involves
X
on all the
qubits as well as phase factor
αα
̄
α
̄
α
over connected pairs. In
the ground state, every four qubits connected into a square
are in the local entangled state
|
0000
+|
1111
.Thewave
function remains invariant if the
Z
2
symmetry is applied to
all lattice sites. If the symmetry is applied only to a subregion
(the lower half plane for example), the wave function changes
along the boundary of the subregion. The change in the wave
function corresponds to applying
X
to all the black dots
in (a) and
α
to all the red bonds in (a). To realize this
change with a symmetric finite depth circuit, we can apply
the transformation shown in (c) to the lattice sites along the
dotted boundary line. It can be explicitly checked that this
realizes the same unitary transformation as (a) and the unitary
in each lattice site commutes with the
Z
2
symmetry and is
hence symmetric.
B. Fusion of 1-form symmetry breaking phases
1-form symmetries start to play an interesting role in 2D
gapped phases. In particular, the breaking of 1-form symmetry
results in topological order. For example, the 2D Toric code
with
Z
2
topological order breaks a
Z
2
1-form symmetry of the
Wilson lines. The point defect of this symmetry are
Z
2
gauge
charge excitations. We show in this section how the fusion of
two 1-form symmetry breaking Toric code states with or with-
out symmetry defect follows similar rules as that discussed in
FIG. 14. Toric code model on the square lattice. The model has
a 1-form symmetry given by
X
on all closed loops including the
nontrivial ones (think horizontal and vertical lines).
Sec.
VA
for 1D 0-form symmetry breaking phases with or
without flux domain wall.
Consider the 2D Toric Code state defined on square lattice
as shown in Fig.
14
.The
Z
2
gauge field degrees of freedom are
on the edges. The Hamiltonian terms include the four-body
plaquette terms of
e
∈
p
X
e
and four-body vertex terms of
v
∈
e
Z
e
. The 1-form symmetry is given by
X
on all closed
loops, including nontrivial loops in the
x
and
y
directions. The
ground state spontaneously breaks this 1-form symmetry. If
the
|
0
state on each edge is regarded as no string and the
|
1
state on each edge is regarded as having a string, the
symmetrized ground state wave function is an equal weight
superposition of all closed-loop configurations, including the
nontrivial ones.
|
ψ
=
C
:closed loop configurations
|
C
.
(79)
With two copies of the Toric Code state, there is a
Z
2
×
Z
2
1-form symmetry. We consider the situation where only the
diagonal
Z
2
1-form symmetry is preserved, which may come
from the 1-form symmetry of a higher dimensional bulk.
Applying pairwise controlled-Not gates between correspond-
ing qubits in the two toric code states keeps the two states
invariant but changes the diagonal 1-form symmetry to act
on only the first copy of Toric Code. Therefore, after the
transformation, two copies of the Toric Code fuse into one
copy, with the coefficient also being a Toric Code.
T.C.
×
T.C.
→
t.c.
×
T.C.,
(80)
where the lower case t.c. represents the coefficient state not
acted upon by the 1-form symmetry.
Now consider the situation where either one or both of the
Toric Code to be fused has a 1-form symmetry defect. When
the action of the 1-form symmetry operator is to add closed
loops in the wave function, a 1-form symmetry defect corre-
sponds to end of string, which we label as the
e
excitation.
When the first Toric Code has a defect while the second
does not, applying the controlled-Not circuit adds the end of
string to the second wave function without changing the first
165144-11