SciPost Phys. 17, 013 (2024)

Decorated defect construction of gapless-SPT states
Linhao Li1 , Masaki Oshikawa1,2,3 and Yunqin Zheng1,2
1 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
2 Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
3 Trans-scale Quantum Science Institute, University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan

Abstract
Symmetry protected topological (SPT) phases are one of the simplest, yet nontrivial,
gapped systems that go beyond the Landau paradigm. In this work, we study an extension of the notion of SPT for gapless systems, namely, gapless symmetry protected
topological states. We construct several simple gapless-SPT models using the decorated
defect construction, which allow analytical understanding of non-trivial topological features including the symmetry charge under twisted boundary conditions, and boundary(quasi)-degeneracy under open boundary conditions. We also comment on the stabilityof the gapless-SPT models under symmetric perturbations, and apply small-scale exact
diagonalization when direct analytic understanding is not available.
Copyright L. Li et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 29-04-2022
Accepted 06-06-2024
Check forupdatesPublished 15-07-2024
doi:10.21468/SciPostPhys.17.1.013

Contents
1 Introduction and summary
1.1 Gapped quantum matter
1.2 Properties of gapless SPT states
1.3 Decorated defect construction
1.3.1 Constructing gapped SPT
1.3.2 Constructing gapless-SPT
1.4 Probing gSPT and igSPT
1.5 Organization of the paper

2
2
3
5
5
6
7
8

2 gSPT: (1 + 1)d spin chains With Z2 × Z2 symmetry
2.1 Spin chain construction
2.1.1 Domain wall decoration
2.1.2 ZA2 × Z2G gSPT
2.1.3 More on U DW
2.2 Trivializability upon stacking gapped SPTs
2.3 Symmetry features of ZA2 × Z2G gSPT
2.3.1 Periodic boundary condition
2.3.2 Twisted boundary condition
2.3.3 Open boundary condition
2.4 Instability of gSPT

8
8
9
10
10
11
12
12
13
14
15

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SciPost Phys. 17, 013 (2024)

3 igSPT: (1 + 1)d spin chain with Z4 symmetry
3.1 Spin chain construction
3.1.1 Domain wall decoration and induced anomaly
3.1.2 The model
3.2 Symmetry features of ZΓ4 igSPT
3.2.1 Periodic boundary condition
3.2.2 Twisted boundary condition
3.2.3 Open boundary condition
3.3 Stability of igSPT

16
16
16
17
18
18
20
22
22

4 gSPT and IgSPT in the spin-1 system
4.1 The model and phase diagramy
4.2 Zz2 ⋉ Z4 × ZT igSPT
y
4.3 Zz2 ⋉ Z4 × ZT gSPT

22
23
23
25

A Stability of boundary degeneracy of Z2A × ZG
gapped SPT
225

B Spectrum of Levin-Gu model under different boundary conditions
B.1 Exact solutions under PBC by Jordan-Wigner transformation
B.2 Mapping to XX chain and charge of ground state
B.3 Spectrum under open boundary condition

26
26
28
30

C Equivalence between ground sector of Z4 igSPT and Levin-Gu model

30

D Edge Degeneracy of gSPT and igSPT
D.1 Edge Degeneracy of Z2 × Z2 gSPT
D.2 Edge degeneracy of Z4 igSPT

31
32
32

E ZT4 × Z2 igSPT
E.1 Lattice Hamiltonian
E.2 Charge of twisted boundary condition
E.3 Open boundary condition

32
33
34
34

F

35

Small-scale numerical study for igSPT under perturbation

References

1

Introduction and summary

1.1

Gapped quantum matter

37

The study of topological phases of quantum matter has led to tremendous progress in understanding quantum many body systems beyond the Landau paradigm. The gapped phases are
so far relatively well understood. Based on their symmetry and entanglement properties, thegapped phases can be classified into the following categories [1]:1. Trivially gapped phase: There is a single ground state on an arbitrary spatial manifold,
and a finite energy gap from the first excited state in the thermodynamic limit. The
ground state preserves the global symmetry, and can be deformed to the trivial productstate through finite depth locally-symmetric unitary transformation without closing the2

SciPost Phys. 17, 013 (2024)

energy gap. Its entanglement entropy obeys area law while the subleading contributions
vanish in the thermodynamic limit. The ground state is short range entangled [1].
2. Symmetry protected topological (SPT) phase: Similarly to the trivially gapped phase,
there is still a single ground state on an arbitrary closed spatial manifold and a finite
energy gap from the first excited state in the thermodynamic limit. The ground state
preserves the symmetry and is short-range entangled. The global symmetry should beanomaly free. However, unlike in the trivially gapped phase, when placing the system ona spatial manifold with nontrivial boundaries, due to the nontrivial physics appearing at
the boundaries, there are either multiple ground states, or the energy spectrum becomesgapless in the thermodynamic limit. There is no finite depth locally-symmetric unitarytransformation that maps the ground state to a trivial product state.1 A systematic construction of gapped SPT phases is the decorated defect construction [7–9].
3. Topological ordered (TO) phases and symmetry enriched topological (SET) phases:
The low energy is described by a symmetric topological quantum field theory (TQFT).
The number of ground states depends on the topology of the spatial manifold. In particular when the spatial manifold is S d there is only one ground state. The ground states
also have a finite energy gap from the first excited states in the thermodynamic limit. The
entanglement entropy of the ground state has a constant contribution besides the area
law part, which survives in the thermodynamic limit. This is termed topological entanglement entropy [10–13]. There are also nontrivial physics (e.g. gapless edge modes,
spontaneous symmetry breaking or gapped TQFT) on the boundary when the spatial
manifold is open. Finally, as the line operators (worldlines of anyons) are topological,
they do not obey area law, and the theory is deconfined.
4. Symmetry breaking phases: There are multiple ground states even when the spatial
manifold is S d , due to spontaneous breaking of the global symmetry. These phases arewithin the Landau paradigm. There are also phases where the Landau symmetry breaking order and SPT/TO/SET orders coexist.
From the description above, it is clear that the SPT phase is the simplest, yet nontrivial, generalization of trivially gapped phase that goes beyond the Landau paradigm. We use gapped
SPT phases to emphasize that the conventional SPT phases are for gapped systems.

1.2

Properties of gapless SPT states

In contrast to the gapped topological phases of quantum matter which are relatively wellunderstood, a systematic understanding of gapless quantum systems is still under development. See [14–26] for recent developments. The simplest type of gapless systems with nontrivial topological features are the so-called gapless symmetry protected topological states,
studied in [14–16, 27, 28]. Let’s summarize their common properties:
1. The gapless system has the global symmetry Γ . Γ should be anomaly free and is not
spontaneously broken by the ground state under periodic boundary conditions.
2. When placing the system on an arbitrary spatial manifold with periodic boundary conditions, the system should have a non-degenerate ground state with a finite size bulk gap
which decays polynomially with respect to the system size.
1

There are also exotic phases that do not require onsite unitary symmetries, but still satisfy the above properties,
i.e. no degeneracy on closed manifolds and nontrivial boundary physics. They include Kitaev’s E8 state in 2 + 1d[2,3] and w2 w3 theory in 4 + 1d [4–6]. We also consider them as SPT phases where the symmetry is the spacetime
diffeomorphism.

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SciPost Phys. 17, 013 (2024)

Table 1: Classification of gapless SPTs by whether they are purely gapless (horizontal
direction) and intrinsically gapless (vertical direction).
Contains gapped sector

No gapped sector

Non-intrinsic

gapless SPT [14, 15, 30]

purely gapless SPT [15]Intrinsic

intrinsically gapless SPT [16, 30, 31]

intrinsically purely gapless SPT

3. When placing the system on a spatial manifold with nontrivial boundaries, there aredegenerate ground states with a finite size splitting decaying qualitatively faster (e.g.
exponentially, or polynomially with a larger decaying constant) with respect to the system size.
4. When placing the system on a closed spatial manifold where the boundary conditions
are twisted by the global symmetry Γ , a.k.a. twisted boundary conditions, the groundstate carries nontrivial Γ symmetry charge.
5. The criticality is confined. In particular, if the criticality has a 1-form symmetry, it should
not be spontaneously broken.
The above properties are similar to those of the gapped SPT states, but there are major differences. For instance, the gap of gapless-SPT vanishes in the thermodynamical limit, while
it remains open for the gapped-SPT. Moreover, the number of nearly degenerate states under
OBC may differ from that of the gapped SPT. 2
The examples of gapless-SPT states studied so far can be schematically organized by two
features, as shown in Table 1.
• The vertical direction is distinguished by whether the gapless-SPT is intrinsic or nonintrinsic. If the topological features mentioned in the previous paragraph is can be realized by a gapped-SPT, then the gapless-SPT is non-intrinsic. Otherwise, it is intrinsic [16].
• The horizontal direction is distinguished by whether the gapless-SPT has a gapped sector. When there is a gapped sector, the degeneracy under OBC has at most exponential
splitting decay. Otherwise, the splitting can be polynomial decaying, and was named as
purely gapless-SPT.
The first example of gapless-SPT, which is non-intrinsic and contains a gapped sector, was firststudied in [14]. The intrinsic gapless-SPT with a gapped sector was first studied in [16]. The
gapless-SPT states without gapped sector (i.e. purely gapless SPT) was much less studied.
The first example was found in [15] involving the time reversal symmetry and an on-site Z2
symmetry, and the terminology “purely” was proposed in [32], and examples with only onsite symmetries are demanding. A more systematic treatment of purely gapless-SPT states,
both non-intrinsic and intrinsic, with on-site symmetries will be discussed in an upcoming
2

We would like to comment that the fifth property is not implied by the first four. One example is the second
order phase transition between a (2 + 1)d topological order and a trivially gapped phase. This system does nothave any 0-form global symmetry and thus trivially satisfies the first four properties. Yet, as discussed in [29], thismodel has an emergent 1-form symmetry which is numerically demonstrated to be spontaneously broken, hence
is deconfined. The fifth property is introduced to exclude this possibility.

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SciPost Phys. 17, 013 (2024)

For simplicity, we will use the following short hand notations to label the four classes of gaplessSPTs respectively:• gSPT = gapless-SPT,• igSPT = intrinsically gapless-SPT,• pgSPT = purely gapless-SPT,• ipgSPT = intrinsically purely gapless-SPT .
This work will focus on systems with a gapped sector, i.e. gSPT and igSPT.3

1.3

Decorated defect construction

A useful method to construct the gSPT and igSPT with a gapped sector is the decorated defectconstruction (DDC). The DDC was first used to construct gapped SPT states [7–9]. Applying the same construction to gapless system inspired the discovery of the first examples ofgapless-SPT [14]. Later, by incorporating the symmetry extension method [34], the DDC also
inspired the discovery of first examples of intrinsic gapless-SPT [16]. Our goal of this paper
is to review this construction, and apply it to constructing bosonic spin models with on-site
symmetries. Such models are simple, from which certain analytic results concerning theirsymmetry properties can be achieved. These models will also play an important role in our
upcoming works [33, 35].
1.3.1

Constructing gapped SPT

The decorated defect construction was first devised to systematically construct gapped SPT
phases, starting from the known lower dimensional gapped SPTs [7–9]. Suppose one would
like to construct a gapped SPT system with global symmetry Γ . Assume Γ fits into the symmetry
extension
1 → A → Γ → G → 1,
(1)where A is the normal subgroup of Γ , and G := Γ /A. For simplicity, we assume that theextension is central, i.e. G does not act on A.4 One starts with a phase where G is spontaneously
broken, and on each codimension p G-defect one decorates a (d + 1 − p) dimensional gapped
SPT protected by symmetry A (i.e. A gapped SPT). As we would like to eventually proliferate
the G-defect network to restore the entire Γ symmetry, the decorations should be consistentsuch that G-defect of each codimension should be free of A-anomaly, and in particular, there
are no gapless modes localized on G-defects. Otherwise, if there are nontrivial gapless degrees
of freedom localized on the G-defects, proliferation would not yield a gapped phase with one
ground state. After defect proliferation, the resulting theory is a gapped SPT protected by theΓΓ symmetry. The topological action of Γ gapped SPT is given by the Γ cocycle Fd+1
which is a5
representative element in the cohomology group [36]Γ[Fd+1
] ∈ H d+1 (Γ , U(1)) .
3(2)

Throughout this paper, “gSPT” specifically refers to non-intrinsic and not purely gapless-SPT. When we don’t
want to specify whether it is intrinsic or not, and would like to emphasize its gaplessness (to contrast with the
gapped systems), we will use “gapless-SPT”.
4
The decorated defect construction of gapped SPTs was first discussed [7] in the special situation where theextension (1) is trivial, i.e. Γ = A × G. The construction was later generalized to non-trivial extension (1) in [9].
5If Γ is a continuous symmetry, the cohomology group should be H d+1 (BΓ , U(1)) where BΓ is the classifying
space of Γ .

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SciPost Phys. 17, 013 (2024)

We remark that a given Γ can fit into multiple symmetry extensions with different pairs(A, G). For a given extension (A, G), as long as we exhaust all possible ways of decorating
A gapped SPT on G-defects, proliferating the G-defects exhausts all possible Γ gapped SPTs.
Hence different choices of (A, G) yield the same set of Γ gapped SPTs, and one can choose themost convenient pair (A, G).
1.3.2

Constructing gapless-SPT

Let us proceed to construct the Γ symmetric gapless-SPT states by modifying the decorated
defect construction reviewed in section 1.3.1. We still assume that the global symmetry Γ fits
into the symmetry extension (1), and start with a gapped phase where G is spontaneously
broken. On each codimension p G-defect, one decorates a (d + 1 − p) dimensional A gapped
SPT. We finally fluctuate the G-defect network to the critical point, and define the critical point
to be the gapless-SPT.
Comparing with the decorated defect construction of the gapped-SPT, the construction of
the gapless-SPT has several important new features. As one no longer demands that fully
proliferating the G-defect network leads to a gapped SPT phase, the consistency condition forthe decoration can be relaxed. Depending on whether the consistency condition is preservedor relaxed, the resulting gapless-SPT are non-intrinsic and intrinsic respectively.
1. gSPT: The A gapped SPTs decorated on the G-defects satisfy the same consistency condition as those for constructing the gapped SPT. Concretely, the G-defect of each codimension is free of A anomaly. This means that further increasing the G-defect fluctuating
strength leads to a Γ gapped SPT, and gSPT is the phase transition between G spontaneously broken phase and Γ gapped SPT. In particular, when the extension (1) is trivial,
i.e. Γ = A × G, the construction was discussed in [14, 15]. See the left panel of figure 1
for the schematic phase diagram of gSPT.
2. igSPT: The A gapped SPT decorated on the G-defects satisfies only a weaker, modified
consistency condition. Concretely, the symmetry breaking phase we started with has aparticular anomaly of a particular quotient group bΓ of Γ , where G ⊂ b
Γ . The choice ofbΓ and its anomaly should be considered as part of input data of the construction. The
defect decoration is constrained such that the anomaly of b
Γ in the G symmetry breaking
phase is precisely cancelled against the anomaly induced by the defect decoration.6 After
decoration, the total symmetry group Γ is anomaly free, and fluctuating the G-defectnetwork to the critical point yields a Γ anomaly free igSPT [16] . See the right panel of
figure 1 for the schematic phase diagram of igSPT
It is natural to assume that the process of defect decoration and the process of G-defectfluctuation commute with each other. Then we may simplify the decorated defect construction
by directly starting with a gapless critical system and decorating its G-defects. The gapless critical system is obtained by fluctuating the G-defects of the G symmetry breaking phase before
decorating the A gapped SPTs, and from section 1.2 we require such critical system before decoration should have a non-degenerate ground state under periodic boundary condition, andis confined.7 For the gSPT, we need to start with a critical point without any anomaly. Whilefor the igSPT, we need to start with a critical point with a particular bΓ anomaly.
As commented in section 1.3.1, for a given Γ , there can be multiple choices of the symmetry
extension (1). We noticed that the gapped SPT can be constructed using arbitrary (A, G).
The phenomenon of induced anomaly also appear in the discussion of anomalous-SPT [9, 37] and symmetry
extended boundary of gapped SPT [34, 38].
7
We will see in later sections that the defect decoration can be implemented by a unitary operation, which does
not change the energy spectrum. This implies that the ground state degeneracy should be one both before andafter defect decoration.
6

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SciPost Phys. 17, 013 (2024)

Γ gSPT
G SSB
Phase

Γ igSPT

Γ Gapped
SPT

G SSB
Phase
λ

Non-SPT
λFigure 1: Phase diagram of non-intrinsically and intrinsically gapless-SPT. The horizontal axis is the strength of G-defect fluctuation. For the non-intrinsic case (leftpanel), the G-defects can be fully proliferated and one obtains Γ gapped SPT. For
the intrinsic case, one can only fluctuate the G-defects to the critical point. Further
increase the fluctuation will not drive the system to Γ symmetric gapped SPT phase.
However, this is no longer true for the igSPT. Note that one needs to specify an anomaly of bΓ(which includes G) as an input data of the decorated defect construction of igSPT. By definition,
the resulting igSPT depends on the choice of symmetry extension (1), b
Γ and the anomaly of bΓ.
In this work, we will use the DDC to construct “canonical” bosonic spin models of gSPT
and igSPT and discuss their topological properties. We also briefly comment on the stability
under perturbations, while leaving an analytic study to an upcoming work [33].

1.4

Probing gSPT and igSPT

Given a gapless system with a non-degenerate ground state in the bulk with finite size, how
can we tell whether it is a nontrivial gapless-SPT? If it is nontrivial, how can we tell whether it
is intrinsic or non-intrinsic? There are several features commonly discussed in the literature:
1. degenerate ground states under OBC,
2. non-trivial symmetry charge of the ground state under the twisted boundary condition.
It is well-known that these features are useful in probing non-trivial gapped SPT phases [39–42]. The first feature is limited in two aspects: (1) It is useful for (1 + 1)d systems [39, 40],
but for higher dimensions the boundary is extensive and the degeneracy on the boundary depends on the boundary dynamics. (2) For a generic Hamiltonian respecting the symmetry, theground states on a finite open chain are only quasi-degenerate with exponentially small splittings, instead of being exactly degenerate [14]. This makes the identification of degenerate
ground states subtle, especially in the gapless systems. While we can still separate the quasidegenerate ground states with exponentially small finite-size excitation energies from gaplessexcitations with power-law finite-size excitation energies, the distinction can be challenging
in practical numerical calculations.
We highlight that the second feature is merely based on the global symmetry, hence (1)can be applied to arbitrary spacetime dimension, and (2) is expected to be stable and exact
for a generic Hamiltonian in the given gSPT and igSPT phase. This stability is also helpful fornumerical calculations, as we will see later. See [43] for an application of twisted boundary
condition to Lieb-Schultz-Mattis ingappability. Moreover, as discussed in [15], the charge
under the twisted boundary condition is equivalent to the charge on the edge of the string orderparameter for CFTs, and the latter is more commonly discussed in the literature. We preferto discuss the twisted boundary condition rather than the string order parameter because thetwisted boundary condition is less well-explored in the literature, and having a systematicand elementary discussion here should be more beneficial. Moreover, the twisted boundary
condition can be generalized more naturally to higher dimensions.

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1.5

Organization of the paper

We emphasize that the concepts and methods to be discussed in this paper, including thedecorated defect construction, and the application of twisted boundary conditions to probe
the gapless SPT, have been discussed in previous works already, in particular [14–16]. The
goal of this paper is to apply the decorated defect construction to build concrete (1 + 1)dlattice spin models of gSPT and igSPT and study their properties in great detail. Our models
are simple enough so that one can extract the ground state symmetry charges under various
boundary conditions analytically, although the models are not exactly solvable.8 Although the
DDC was both applied to constructing gSPT in [14] and igSPT in [16], we believe that it iseducational to present the construction of both gSPT and igSPT in a single place, highlighting
the usefulness of DDC. It turns out that the examples constructed in this work pave the way
to our later explorations of unified treatment of gSPT, igSPT, pgSPT and ipgSPT [33].
This paper is organized as follows. In section 2, we discuss in detail an analyticallytractable example of gSPT, where Γ = Z2 × Z2 , A = Z2 , G = Z2 and the spacetime dimension is d = 1 + 1. In section 3, we discuss in detail an analytically tractable example of igSPT,
where Γ = Z4 , A = Z2 , G = b
Γ = Z2 and d = 1 + 1. We discuss a more realistic spin-1 modelin section 4, which hosts both gSPT and igSPT simultaneously. There are several appendices.
Appendix A shows the stability of boundary degeneracy of Z2 × Z2 gapped SPT. Appendices
B, C and D are devoted to further detailed discussions in section 3. Appendix E discusses anexample of igSPT which involves time reversal symmetry. Appendix F shows the numericalresult on the stability of igSPT under a certain symmetric perturbation.

gSPT: (1 + 1)d spin chains With Z2 × Z2 symmetry2

In this section, we study a concrete lattice model of gSPT: (1 + 1)d spin chain with global
symmetry Γ = Z2 × Z2 . We let A = Z2 , G = Z2 , and the symmetry extension in (1) is trivial.
For clarity, we use the superscript A and G to label the two Z2 ’s.

2.1

Spin chain construction

We construct the 1 + 1d spin chain with Γ = ZA2 × Z2G global symmetry. Since there are two Z2
symmetries, it is natural to assign two spin- 21 ’s per unit cell: the spin- 21 ’s living on the sites arecharged under Z2G while those living in between the sites are charged under ZA2 . The symmetry
operators are defined to be
UA =

L
Y

τx 1 ,

i=1

UG =

i+ 2

L
Y

σix ,

(3)

i=1

where σia and τa 1 , a = x, y, z, are Pauli matrices acting on the two spin- 12 ’s, and L is thei+ 2number of unit cells. Both symmetry operators are on-site9 and therefore Γ is anomaly free.
As explained in the introduction, we would like to start with a Z2G spontaneously broken phase,with the Hamiltonian
H0 = −

LX
i=1

τ x 1 + σzi σzi+1 .
i+ 2

8

(4)

Our models do not involve fermions, but it can be shown that under Jordan-Wigner transformation, our modelare equivalent to one of the models discussed in [44].
9
A symmetry operator is on-site
if it can be written as a product of local operators on mutually adjacent butQ
un-overlapping patches, U = i Ui , where i labels the patches.

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si

gi j

sj
[g]

Figure 2: Triangulation of 2d spacetime. The black and red solid links are where
the background field g i j = 0, 1 respectively. The red dashed line in the dual latticeis the spacetime trajectory of the Z2 domain wall [g], i.e. Z2 symmetry defect line.
Flatness of g ensures that [g] forms loops.
It has two ground states|±〉 =X{τz 1 }i+ 2|{τz 1 }, {σzi = ±1}〉 .

(5)

i+ 2

Each of them spontaneously breaks Z2G but preserves ZA2 .
2.1.1

Domain wall decoration

To construct a ZA2 ×Z2G gSPT, we decorate each Z2G domain wall by a 0+1d ZA2 SPT in a consistentway.10 Each Z2G domain wall is associated with a Z2G group element g. g = 0, 1 means the domain wall is trivial/nontrivial, i.e. the adjacent σz spin configurations are the same/opposite,
respectively. We present the domain wall configuration using both the spacetime picture andthe Hamiltonian picture.
The spacetime picture: It is useful to first discuss the domain wall in the spacetime picture.
The spacetime is triangulated into 2-simplices. See figure 2 for an illustration. Each site i isassigned a Z2 group element si = 0, 1, which corresponds to σzi = (−1)si in the Hamiltonian
picture. Each link is assigned a Z2 1-cochain g i j = s j − si . The g i j is understood as a flat
background field for the Z2G symmetry, and it measures the local domain wall excitation on thelink. The locus where g i j = 1 form a closed loop [g] in the dual spacetime lattice, representing
the worldline of the domain wall, a.k.a. the Z2G symmetry defect line. Decorating the Z2G
domain wall by a 1d ZA2 SPT [7, 14] means that we insert a ZA2 Wilson line, a.k.a. 1d ZA2 SPT,
supported on [g]
 Z

 Z

a = exp iπ

exp iπ
[g]

a∪g

,

(6)

M2

in the path integral. The flatness of the ZA2 background field a ensures that the decoration isconsistent: the domain wall junctions do not have ZA2 anomaly. This fits into the constructionof gSPT mentioned in section 1.3.2. The equality in (6) used the Poincare duality to transformthe integral on [g] into the integral over the entire 2d spacetime M2 . The topological term on
the right hand side of (6) is precisely the effective action of ZA2 × Z2G gapped SPT.
The Hamiltonian picture: In the Hamiltonian picture, domain wall decoration is implemented as follows [14]. We first identify the configuration representing the Z2G domain
10
In (1 + 1)d, we only have codimension 1 defects, i.e. the domain walls. For this reason, the decorated defectconstruction is more conventionally called the decorated domain wall construction.

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SciPost Phys. 17, 013 (2024)

wall, i.e. σzi σzi+1 = −1. Then on the link (i, i + 1), we stack a ZA2 SPT (6), which assigns the wavefunction a minus sign if τz 1 = −1 (i.e. ai,i+1 = 1 in the spacetime picture)i+ 2

on the wall. Combining the two steps, one assigns a minus sign to the two configurations(σzi , τz 1 , σzi+1 ) = (1, −1, −1), (−1, −1, 1) and leaves the wavefunction unchanged for otheri+ 2

configurations. This operation can be realized by acting the unitary operator
L
Y






πi
πi
z
z
z
z
U DW =
exp,(1 − σi ) 1 − τ 1 exp(1 − σi+1 ) 1 − τ 1
i+ 2
i+ 2
4
4
i=1


(7)

on the states (5) [14]. In terms of the Hamiltonian, domain wall decoration just amounts toconjugating the original Hamiltonian (4) by U DW , yielding†
H1 := U DW H0 U DW
=−

L 

X
z x
z
z z
σi τ 1 σi+1 + σi σi+1 .
i+ 2

i=1

(8)

The ground states of H1 are still (5), but the first excited states associated with the domain
wall excitations are decorated.
2.1.2 Z2A × ZG
gSPT
2The next step is to fluctuate the decorated domain walls. It is helpful to discuss the fluctuation
without decoration
P L first. The fluctuation is well-known to be achieved by adding a transverse
field ∆H = −λ i=1 σix , so that the Z2G spontaneously broken ferromagnetic phase of the Isingmodel (when λ < 1) is driven to the Z2G preserving paramagnetic phase (when λ > 1) where
the domain walls are fully proliferated. The transition happens at λ = 1, which is of second
order, and is described by a critical Ising CFT.
After domain wall decoration, P
the fluctuation should be realized by adding a decoratedL†transverse field U DW ∆H U DW= −λ i=1 τz 1 σix τz 1 . As the unitary transformation U DW does
i− 2

i+ 2

not change the energy spectrum, the critical point also takes place at λ = 1. The decorated†
model U DW (H0 + ∆H)U DW
at λ = 1, is the ZA2 × Z2G gSPT [14] (see also section 1.3.2 for thedefinition of gSPT)L XHgSPT = −σzi τ x 1 σzi+1 + σzi σzi+1 + τz 1 σix τz 1 .
i+ 2

i=1

i− 2

i+ 2

(9)

When λ > 1, the domain wall is fully proliferated, yielding a ZA2 × Z2G gapped SPT describedby the well-known cluster model [45–47]
HSPT = −L Xσzi τ x 1 σzi+1 + τz 1 σix τz 1 .
i=1

i+ 2

i− 2

i+ 2

(10)

See figure 3 for the phase diagram before and after decoration.
As commented in section 1.3.2, we can simplify the above construction
P L of gSPT by directly
starting with the Z2G Ising CFT (whose Hamiltonian is given by H0 − i=1 σix ), and conjugate
it by U DW . This simplification will be useful in section 3.
2.1.3

More on UDW

PL
We make a remark on the unitary operator U DW . Although HgSPT and H0 − i=1 σix are related through a unitary transformation U DW , they are actually not equivalent as the ZA2 × Z2Gsymmetric Hamiltonians. Recall that two Γ symmetric Hamiltonians H1 , H2 are considered
10

SciPost Phys. 17, 013 (2024)

Z2G
Ising CFTZA2 × Z2GgSPTDefectZ2G SSB

TrivialZ2G SSB
Decorated

Decoration

ZA2 × Z2G
Gapped SPTλλFigure 3: Phase diagram of Z2G Ising CFT (before decoration) and ZA2 ×Z2G gSPT (after
decoration). The horizontal axis represents the transverse field λ.
equivalent if there is a locally-symmetric unitary transformation U = exp(i

R t1
t0

d t V (t)) where

V (t) is a sum of local operators satisfying [V (t), Γ ] = 0, such that U H1 U = H2 [1, 36]. Since
U DW is a product of local unitary operators and each of them only acts on one or two unit
cells, U DW is a local unitary transformation. Moreover, U DW on a closed chain with the periodic boundary condition
is symmetricin
the sense that [U DW , Γ ] = 0. Nevertheless, as each†

local operator exp

πiz
4 (1 − σi )1 − τz 1
i+ 2

does not commute with UA and UG , U DW is not a

locally-symmetric unitary transformation. As an indication, U DW does not commute with thesymmetry generator Γ on P
an open chain, in contrast to the closed chain discussed above. In
Lsummary, HgSPT and H0 − i=1 σix are not related by ZA2 × Z2G locally-symmetric unitary transformation, hence they are not equivalent as ZA2 ×Z2G symmetric systems. This also justifies thatthe gSPT is protected by the ZA2 × Z2G .
It is interesting to compare U DW with the Kennedy-Tasaki (KT) transformation [48–50]
introduced for integer-spin chains. Although the KT transformation is also implemented by aunitary operator UK T , there are several differences. First, UK T is non-local, unlike U DW whichis as discussed above a product of local unitary operators. Second, the KT transformation
is useful for an open chain rather than for a closed chain, which is mapped to a non-local
Hamiltonian by UK T . Lastly, it maps a gapped SPT phase (on an open chain) to an SSB phase,
while U DW maps a gapped SPT phase to a trivially gapped phase. The KT transformation will
be relevant for the discussion in Section 4. In a later work by the same authors [35], we
uncover that the KT transformation on a closed chain is related to the UDW in the following
way, KT = KW · UDW · KW, where KW is the Kramers-Wannier transformation for both ZA2 × Z2Gsymmetries.

2.2

Trivializability upon stacking gapped SPTs

In this section, we will show that upon stacking a ZA2 × Z2G gapped SPT, the ZA2 × Z2G gSPT isequivalent to Z2G Ising criticality via a symmetric local unitary transformation.
Let us consider two decoupled systems. The first system is a ZA2 × Z2G gSPT given by (9).
The second system is a ZA2 × Z2G gapped SPT given by (10). Since two systems are decoupled,
the two Hamiltonians act on decoupled Hilbert spaces. We use the Pauli operators {σia , τa 1 }i+ 2

ea 1 } for the second system. The Hamiltonian for the entire
for the first system, and {eσia , τi+ 2

system is the sum
HgSPT + HSPT = −L Xe zi τ
ex 1 σe zi+1 + τ
ez 1 σe ix σez 1 .
σzi τ x 1 σzi+1 + σzi σzi+1 + τz 1 σix τz 1 + σ
i=1

i+ 2

i− 2

i+ 2

i+ 2

i− 2

i+ 2

(11)
eA × Ze G ), whose generators
The decoupled system has enlarged global symmetry (ZA2 × Z2G ) × (Z2
2

11SciPost Phys. 17, 013 (2024)

are
UA =

L
Y
i=1

τx 1 ,
i+ 2

UG =

L
Y

σix ,

eA =
U

i=1

L
Y
i=1

ex 1 ,
τ
i+ 2

eG =
U

L
Y

e ix .
σ

(12)

i=1

There exists a symmetric local unitary transformation11
L
Y








iπ
iπ
z z
z
z
z
z
z
z
Udiag =exp
1 − σi σ̃i+1 1 − τ 1 τ̃ 1 exp
1 − σi+1 σ̃i+1 1 − τ 1 τ̃ 3,
i+ 2 i+ 2
i+ 2 i+ 2
4
4
i=1
(13)
e
which (locally) preserves the diagonal Z2 × Z2 , where two Z2 ’s are generated by UA UA andeG respectively. It is straightforward to check that
UG U†
Udiag (HgSPT + HSPT )Udiag
=−

L 

X
ex 1 + σ
e ix ,
τ x 1 + σzi σzi+1 + σix + τ
i+ 2

i=1

i+ 2

(14)

which is simply the Hamiltonian of the Ising CFT, a.k.a. the Z2G Landau transition, stacked withsome trivially gapped degrees of freedom. In summary, we have shown that upon stacking a
ZA2 × Z2G gapped SPT, the ZA2 × Z2G gSPT (9) is related to an ordinary Z2G Landau transition
by a symmetric local unitary transformation. The above equivalence can be schematically
represented as
ZA2 × Z2G gSPT ⊕ ZA2 × Z2G gapped SPT ←→ Z2G Landau Transition.

(15)

This implies that the nontrivial topological properties of the gSPT in the bulk (such as nontrivial charge of the ground state under the twisted boundary condition, see section 2.3.2) arebasically inherited from the gapped SPT sector. However, we will find in section 2.3.3 that theboundary properties of the gSPT differ from those of the gapped SPT.

2.3

Symmetry features of Z2A × ZG2 gSPT

We discuss the symmetry features of the ZA2 × Z2G gSPT (9) that allow one to distinguish trivial
vs nontrivial gSPTs. As motivated in the introduction (see section 1.4), we will consider the
ground state degeneracy under open boundary condition (OBC), as well as the symmetry
charge of the ground state under twisted boundary condition (TBC). We summarize the main
properties in table 2.
2.3.1

Periodic boundary condition

On a finite chain with periodic boundary condition (PBC), the ground state of the ZA2 ×Z2G gSPT
is non-degenerate.
To see this, we first consider the Ising CFT described by the HamiltonianPLx
H0 − i=1 σi . It is well-known that the critical Ising model has only one ground state ona finite chain, and the first excited state is separated from the ground state by a finite size
gap decaying polynomially with respect to the system size. The non-degenerate ground state
preserves the ZA2 × Z2G global symmetry. Moreover, as noted in section 2.1.2, HgSPT and theIsing CFT have exactly the same energy eigenvalues because they are related via a unitarytransformation U DW , which implies that HgSPT also has a non-degenerate ground state on a
finite closed chain, with a finite size gap, and is Z2G × ZA2 symmetric under PBC.
11

eA as well as
Without multiplying over i, each exponent in Udiag commutes with the diagonal symmetries UA UeUG UG . As discussed in section 2.1.3, this implies that Udiag is a symmetric local unitary transformation, which
establishes the equivalence between different systems.

12

SciPost Phys. 17, 013 (2024)

Table 2: Ground state degeneracy and symmetry charges of the ground state under
PBC, TBC and OBC. ZA2 (or Z2G )-TBC means the boundary condition is twisted by ZA2(or Z2G ). We compare these properties between gSPT, Landau transition and gapped
SPT, all with the same global symmetry ZA2 ×Z2G . The 4 → 2 means that HgSPT has four
ground states under OBC, but two of them are lifted under a symmetric perturbation
localized on the boundary.

PBC:
ZA2 -TBC:
Z2G -TBC:

ZA2 × Z2G

ZA2 × Z2GgSPTLandau Transition

Gapped SPT

GSD

1

1

1

ZA2 × Z2G Charge(0, 0)(0, 0)(0, 0)

GSD

1

1

1

ZA2 × Z2G Charge(0, 1)(0, 0)(0, 1)

GSD

1

1

1

ZA2 × Z2G Charge(1, 0)(0, 0)(1, 0)

GSD

4→2

1

4

OBC:

2.3.2

ZA2 × Z2G

Twisted boundary condition

We show that on a closed chain with boundary condition twisted by ZA2 (or Z2G ), the groundstate of the ZA2 × Z2G gSPT carries nontrivial symmetry charges under Z2G (or ZA2 ) respectively.
The same idea has been widely used to characterize nontrivial gapped SPT order [41, 42, 51–57], and here we use it to characterize the gSPT (and also igSPT in section 3).
Twist by Z2A : We first twist the boundary condition using the ZA2 symmetry (labeled by ZA2 TBC), and measure the Z2G charge of the ground state. Twisting the boundary condition by ZA2means imposing a ZA2 domain wall between sites L − 21 and L + 21 by changing the sign of theterm τz 1 σ Lx τz 1 . The gSPT Hamiltonian (9) becomes
L+ 2

L− 2

ZA

2
HgSPT
=−

L−1 

X
σzi τ x 1 σzi+1 + σzi σzi+1 + τz 1 σix τz 1 − σzL τ x
i+ 2

i=1

i− 2

i+ 2

L+ 12

σ1z − σzL σ1z + τz

L− 21

σ Lx τz

L+ 21

.

(16)
It is useful to note that the twisted and untwisted gSPT Hamiltonian are related by a unitary
ZA

ZA

2
2
transformation HgSPT= σzL HgSPT σzL , hence the ground state of HgSPTis also non-degenerate.

ZA

Denote the ground state under PBC as |GS〉, and that under ZA2 -TBC as |GS〉tw2 . We have
ZA|GS〉tw2 = σzL |GS〉 .
It follows that

ZA

(17)
ZA

UG |GS〉tw2 = UG σzL UG† UG |GS〉 = −σzL |GS〉 = − |GS〉tw2 ,

(18)

ZG

which shows that |GS〉tw2 has Z2G charge 1.12
12

We used the fact that the ground state under PBC is neutral under Z2G . More precisely, (18) only shows therelative charge, i.e. the Z2G charge of the ground state under TBC minus that under PBC, is one. The relative charge
will be useful in section 3.

13

SciPost Phys. 17, 013 (2024)

Twist by ZG: We can alternatively twist the boundary condition using Z2G symmetry (labeled2
by Z2G -TBC), and measure the ZA2 charge of the ground state. Twisting the boundary condition
by Z2G means imposing a Z2G domain wall on the link between L-th and 1st sites, by changing
the sign of the terms σzL σ1z and σzL τ x 1 σ1z . The gSPT Hamiltonian (9) becomes
L+ 2

ZG

2
HgSPT
=−

L−1 

X
σzi τ x 1 σzi+1 + σzi σzi+1 + τz 1 σix τz 1 + σzL τ x
i+ 2

i=1

L+ 12

i+ 2

i− 2

σ1z + σzL σ1z − τz

L− 21

σ Lx τz

L+ 21

.

(19)

Z2G
ZG
Note that σzi τ x 1 σzi+1 commutes with every term in HgSPT, the ground state |GS〉tw2 should bei+its eigen-vector

2

ZG
ZG
†
σzi τ x 1 σzi+1 |GS〉tw2 = U DW
τ x 1 U DW |GS〉tw2 =
i+
i+
2

(

2

ZG

|GS〉tw2 ,i = 1, . . . , L − 1 ,ZG− |GS〉tw2 ,i = L.

(20)

Consequently, the ground state has ZA2 charge 1:
ZG
UA |GS〉tw2 =

L
Y
i=1

ZG
τ x 1 |GS〉tw2 = −
i+ 2

L
Y
ZG
ZG
(σzi σzi+1 ) |GS〉tw2 = − |GS〉tw2 .

(21)

i=1

A,G

In summary, we find that when we use Z2 to twist the boundary condition on a closed
G,Achain, the ground state of the twisted Hamiltonian has nontrivial Z2 charge. This is the property distinguished from the ZA2 ×Z2G Landau transition, where its ground state under the twisted
boundary conditions does not carry any nontrivial symmetry charge. This tells us that we canuse the symmetry charge of the ground state in the twisted sector as a topological invariant todistinguish the nontrivial gSPT from trivial gSPT (e.g. second order Landau transition). On
the other hand, the symmetry charges under TBC coincide with those of the gapped SPT. We
summarize the results in table 2.
2.3.3

Open boundary condition

As the nontrivial boundary modes protected by the global symmetry is a signature of gapped
SPT, we will find that same is true for the gSPT. We use the symmetry to analytically showthat the ground states of HgSPT have to be exactly degenerate under OBC, but the number ofdegeneracy differs from the gapped SPT. This phenomenon was discussed in [14, 15].
We place the spin system on an open chain. The left most spin is the σ spin, and theright most spin is the τ spin. The σ spins are supported on i = 1, . . . , L, and the τ spins aresupported on i + 21 = 23 , . . . , L + 12 . We first choose the OBC such that only the interactions
completely supported on the chain are kept. The Hamiltonian is
OBC
HgSPT
=−

L−1 
L
 X
X
z x
z
z z
σi τ 1 σi+1 + σi σi+1 −τz 1 σix τz 1 ,
i=1

i+ 2

i=2

i− 2

i+ 2

(22)

and the symmetry operators are
UA =

L
Y
i=1

τx 1 ,

UG =i+ 2

We find that the set of operators {σ1z , τz

L+ 12

L
Y

σix .

(23)

i=1

, σzL τ x

L+ 12

, UA, UG } all commute with the Hamilto-

nian, hence the ground state degeneracy must be at least the dimension of its irreducible
14

SciPost Phys. 17, 013 (2024)

representation. To find the representation, we choose the maximally commuting subset of operators as {σ1z , τz 1 }, and denote their eigenvalue of a particular ground state |ψ〉 by (a, b),
L+ 2

where a, b = ±1. It is then possible to generate other ground states with different quantum
numbers as follows:|ψ〉
UA |ψ〉
UG |ψ〉UA UG |ψ〉

σ1z

τz

a
a
−a
−a

b
−b
b
−b

L+ 21

.

(24)

OBC
This shows that there must be at least four exactly degenerate ground states of HgSPT
of fourdifferent sets of quantum numbers. Numerical exact diagonalization confirms that the groundstate degeneracy is exactly four.
However, symmetry does not forbid us to perturb (22) by adding symmetric boundary
terms. We can add a boundary interaction
OBC= −τ x∆HgSPT

L+ 12

,

(25)

which changes the original OBC to a new OBC. This interaction does not commute withOBC
OBC
τz 1 , so the set of operators commuting with the Hamiltonian HgSPT+ ∆HgSPT
reduces to
L+ 2{σ1z , σzL τ x

L+ 12, UA, UG }. As a consequence, the dimension of irreducible representation reducesfrom four to two. Indeed, numerical exact diagonalization confirms that there are only two
exactly degenerate ground states under the new OBC. This degeneracy splitting was alreadynoted in [14, 15]. Here, we provide a simple analytical argument of this splitting by finding
the representation. In appendix A, we show that arbitrary finite range perturbation does not
lift the 4-fold exact degeneracy of the ZA2 × Z2G gapped SPT.

2.4

Instability of gSPTAs noted in table 2, the symmetry properties of the ground states under the twisted boundary
condition are the same for the gSPT and gapped SPT. Is there a symmetric perturbation of thegSPT which derives gSPT to the gapped SPT? In this subsection, we confirm this by noting thatsuch a ZA2 × Z2G symmetric perturbation exists, which is
V = −hLXτz 1 σix τz 1 ,

i=1

i− 2

i+ 2

h > 0.

(26)

In other words, adding V to HgSPT simply modifies the coefficient of the last term of (9) by −h.
After undoing the domain wall decoration by conjugating the HgSPT + V by U DW , we get†
U DW (HgSPT + V )U DW
=−

L 

X
τ x 1 + σzi σzi+1 + (1 + h)σix ,
i=1

i+ 2

(27)

P
which is just a critical Ising model perturbed by −h i σix . It is well-known that under fermionization, this term is the mass term of the Majorana fermion, which is a relevant perturbation.
This shows that adding a perturbation with infinitesimal h drives the gSPT to gapped SPT
phase, which shows that gSPT is unstable under symmetric perturbation towards gapped SPT
phases.

15

SciPost Phys. 17, 013 (2024)

3

igSPT: (1 + 1)d spin chain with Z4 symmetry

In this section, we study a concrete lattice model of igSPT: (1 + 1)d spin chain with Z4 global
symmetry. We let A = Z2 , G = Z2 , and the symmetry extension (1) is now nontrivial. We still
use superscripts A and G to label the two Z2 ’s, and use superscript Γ to label Z4 .
The igSPT was first studied in [16]. Although essentially all the features of igSPT havebeen discussed in that work, the model discussed in our present work has the advantage ofbeing simpler, where many symmetry properties can be extracted exactly without numericalcomputation. Moreover, the characterization of topological features in [16] heavily uses thestring order parameter, while in our work we provide some alternative perspectives of characterization using the twisted boundary conditions. Although the charge of the string orderparameter and the charge of ground states under the TBC are known to be related in the continuum limit [15], it is nevertheless beneficial to discuss the TBC on the lattice and comparewith the string order parameter discussion on the lattice in [16].

3.1

Spin chain construction

3.1.1

Domain wall decoration and induced anomaly

Domain wall decoration: We construct the (1 + 1)d spin chain with ZΓ4 global symmetry,by applying the decorated defect construction reviewed in section 1.3.2. Concretely, we startwith a Z2G symmetry spontaneously broken phase with a nontrivial anomaly of Z2G , and thendecorate the Z2G domain wall by ZA2 SPT. We will show below that the domain wall decorationinduces a nontrivial Z2G anomaly due to the nontrivial extension (1), and two Z2G anomalies
are designed to cancel against each other. Thus the entire ZΓ4 symmetry is anomaly free. We
further proliferate the decorated Z2G domain wall, and fine tune the system to the critical point.
The resulting critical point is the ZΓ4 igSPT.
Induced anomaly: We explain why the domain wall decoration induces nontrivial Z2Ganomaly. Let us denote the background fields of Z2G and ZA2 as g and a respectively, bothof which are 1-cochains. The ZΓ4 background field is 2a − g̃, where eg is a lift of g to a ZΓ4valued cochain, i.e. g = e
g mod 2. By requiring the ZΓ4 background to be flat, we findδ(2a − eg ) = 2δa − δeg = 0 mod 4 ,(28)which implies
δa = Bock(g) :=1
δeg mod 2 ,2

δg = 0 mod 2 .

(29)

Bock(g) is the Bockstein of g, which is defined as in (29). As (6), decorating the Z2G domain
wall by a 1d ZA2 SPT means stacking a ZA2 Wilson line to the worldline of Z2G domain wall.
However, due to the nontrivial bundle constraint (29), the domain wall decoration is notgauge invariant, and equivalently it induces a nontrivial dependence on the extension to the
3d bulk M3 ,
 Z

 Z

 Z

a = exp iπ

exp iπ[g]a∪gM2= exp iπg ∪ Bock(g) .

(30)

M3

In the second equality, we applied total derivative to promote the 2d integral to the 3d integral
and used (29). A physical interpretation of (30) is that domain wall decoration induces a Z2Ganomaly. We will denote this anomaly as the induced anomaly.
However, the igSPT by definition should be free of ZΓ4 anomaly, and the system should be
independent of the extension to M3 . This demands that the Z2G spontaneously broken system
16

SciPost Phys. 17, 013 (2024)

before domain wall decoration should already exhibit an opposite anomaly of Z2G , which is
given by the same inflow action Zg ∪ Bock(g) .

exp iπ(31)

M3After domain wall decoration, the anomaly (31) from the low energy cancels against the induced anomaly (30) from the domain wall decoration, and the total system is anomaly free.
As commented at the end of section 2.1.2, one can simplify the discussion by directly
starting with a critical system with a non-degenerate ground state and a Z2G anomaly (31). A
standard candidate is the critical boundary theory of (2 + 1)d Z2G SPT, known as the Levin-Gu
model [42]. We then decorate the Z2G domain walls (via conjugating by the unitary operator
U DW in (7)). We will take this simplified strategy of domain wall decoration below.
3.1.2

The model

We still let the σ spins live on integer sites and τ spins live on half integer sites. Let us startfrom the Levin-Gu model [42]HLG = −LX

σix − σzi−1 σix σzi+1 ,

(32)

i=1

with an anomalous Z2G symmetry transformation:
UG =

L
Y

σix

i=1

L
Y




iπ
z z
exp
(1 − σi σi+1 ) .
4
i=1

(33)

The Z2G symmetry operator is realized in a non-on-site way, which is demanded by the Z2Ganomaly.
In the next step, we consider the following Hamiltonian which couples the τ and σ spins
and serves as the pre-decorated Hamiltonian:
Hpre = −L Xσix − σzi−1 τ x 1 σix τ x 1 σzi+1 + τ x 1 .
i− 2

i=1

i+ 2

i− 2

(34)

This Hamiltonian is invariant under the Z4 symmetry transformation:
pre
UΓ =

L
Y
i=1

σix

L
Y




iπ 
z x
z
exp
1 − σi τ 1 σi+1
.
i+ 2
4
i=1

(35)

The normal subgroup ZA2 is generated by an on-site operator
pre
UA = (UΓ )2 =

L
Y
i=1

τx 1 .
i+ 2(36)

Indeed, the two Hamiltonians (34) and (32) enjoy the same low energy theory. As the lastterm in (34) commutes with the rest of the terms, the ground state should be the eigenstateof τ x 1 with eigenvalue 1. See appendix C for a more detailed discussion on this point.13
i− 2

In the low energy sector, we simply substitute τ x 1 = 1 in (34), and obtain that the low
i− 2

13

In fact, all the low energy states with energy E − EGS ≪ 1 satisfy τ x 1 = 1.
i− 2

17

SciPost Phys. 17, 013 (2024)

energy effective Hamiltonian (34) is precisely the Levin-Gu Hamiltonian (32). Moreover, theZA2 normal subgroup decouples from the low energy. Only Z2G acts nontrivially on the low
energy degrees of freedom, in the same way as (33):


L
L
Y
Y
iπ
pre
x
z z
UΓ |low =σiexp(1 − σi σi+1 ) .
(37)
4
i=1
i=1
The additional τ operators as in (34) and (35) areQmotivated by the group extension. We
would like to introduce τ operators such that UA = i τ x 1 generates an anomaly free ZA2 ,i+ 2extending the Z2G to ZΓ4 . In other words, we demand a modification of UG in (33) such thatit squares to UA. This is precisely achieved by replacing σzi σzi+1 by σzi τ x 1 σzi+1 . This further
i+ 2

induces how the Levin-Gu Hamiltonian (32) should be mapped to (34).
Then, the Hamiltonian for the ZΓ4 igSPT is obtained by conjugating (34) using the unitary
operator U DW . The Hamiltonian is

L 
X
y
†
x z
z
z
x y
z
x
(38)
HigSPT = U DW Hpre U DW = −τ 1 σi τ 1 + τ 1 σi τ 1 + σi−1 τ 1 σi .
i=1

i+ 2

i− 2

i− 2

i− 2

i+ 2

The pre-decorated ZΓ4 symmetry operator becomespre†=UΓ = U DW UΓ U DW

L
Y

σix

L
Y

exp



i=1

i=1


iπ 
,
1 − τx 1
i+ 2
4

(39)

y
. ZΓ4 isi+ 12
anomaly free, which can be seen from the on-site-ness of UΓ 14 and justifies that Z2G anomaly

under which σix

y,zy,z→ σix , σi → −σi , τ x 1i+→ τx 1 , τ
i+ 2

2

y

i+ 12

→ τz 1 , τz 1 → −τ
i+ 2

i+ 2

in pre-decorated Hamiltonian is canceled by the induced anomaly of decorated defect construction. The normal subgroup ZA2 is also generated by an on-site operator Eq. (36). Here weremark that the Hamiltonian (38) is Jordan-Wigner dual to Eq.49 and Eq.50 in [44].

3.2

Symmetry features of ZΓ4 igSPT

We discuss the symmetry features of the ZΓ4 igSPT (38). An immediate fact to realize is that
there is no ZΓ4 gapped SPT in (1 + 1)d.15 Thus it is not possible to stack a gapped SPT to
unitarily connect it to another possibly more trivial igSPT. For this reason, the origin of thenontrivial SPT order at the critical point here is less obvious, in contrast to the ZA2 × Z2G gSPT.
In this subsection, we discuss its properties under various boundary conditions. We summarize
the main results in table 3.
3.2.1

Periodic boundary condition

We have motivated in section 1.2 that any igSPT should have one non-degenerate groundstate, with a finite size splitting with the first excited state. Thus we would like to check the
ground state degeneracy of (38) under PBC to be one.
As we find in section 3.1.2, the number of ground states of the ZΓ4 igSPT is identical to thatof the Levin-Gu model (32). In appendix B.1, we show, by Jordan-Wigner transformation, thatthe number of ground states of the Levin-Gu model depends on L mod 4 and is given as¨2,
L = 2 mod 4 ,GSD L =
(40)
1,otherwise.
14

However, not every anomaly free symmetry operator is on-site.
The (1 + 1)d bosonic SPT with a discrete symmetry G is classified by H 2 (G, U(1)). In our case, G = Z4 , and it
is well-known [36] that H 2 (Z4 , U(1)) = 0 is trivial, hence there is no nontrivial Z4 SPT phase in (1 + 1)d.
15

18

SciPost Phys. 17, 013 (2024)

Table 3: Ground state degeneracy and symmetry charges of the ground state under
PBC, TBC and OBC. We focus on the system size L = 0, 1, 3, 4, 5, 7 mod 8 to ensure
trivial ground state degeneracy. Relative ZA2 (or ZΓ4 ) charge means the difference between the corresponding charge under the TBC and that under the PBC. We compare
these properties between the igSPT and Landau transition, both with the same global
symmetry ZΓ4 .

PBC:
ZA2 -TBC:

ZΓ4 -TBC:
OBC:

ZΓ4

ZΓ4

igSPT

Landau Transition

GSD

1

1

GSD

1

1

Relative ZA2 Charge
Relative ZΓ4 Charge

0

0

2

0

GSD

L = odd : 2; L = even : 4

1

Relative ZA2 Charge
Relative ZΓ4 Charge

10

1 or 3

0

GSD

≥2

1

Thus the number of ground states of the ZΓ4 igSPT under periodic boundary condition is also
given by (40).
Let us further discuss the Z4 charge of the ground state. Denote the ground states of (32),(34) and the (38) as |GS〉LG , |GS〉pre and |GS〉 respectively. Suppose the Z2G charge of |GS〉LGin the Levin-Gu model (32) is qLG , then by definition we have†
U DW UΓ U DW|low |GS〉LG = (−1)qLG |GS〉LG .

(41)†
As ZA2 decouples from the low energy, we also have U DW UΓ U DW|GS〉pre = (−1)qLG |GS〉pre . SinceΓ|GS〉pre = U DW |GS〉, we can then compute the Z4 charge of |GS〉 via,
π†††UΓ |GS〉 = U DW
(U DW UΓ U DW
) |GS〉pre = (−1)qLG U DW|GS〉pre = e i 2 ·2qLG |GS〉 .

(42)

So the ZΓ4 charge q of the ground state |GS〉 is related to the Z2G charge of |GS〉LG via q = 2qLG
mod 4.
We are left to determine the symmetry charge of the Levin-Gu model, qLG . While theground-state degeneracy was obtained exactly in Eq. (40) by the Jordan-Wigner transformation as discussed in Appendix B.1, we could not find qLG from the Jordan-Wigner transformation. Nevertheless, we can utilize an alternative mapping to the XX chain as discussed inAppendix B.2, to determine qLG for even L’s. The analytical result for even L’s was confirmedby exact numerical diagonalization for small L’s, which also gives qLG for odd L’s. As a result,extending the L mod 4 dependence of the ground-state degeneracy (40), we find that thesymmetry charge of the Levin-Gu model qLG depends on α = L mod 8: qLG = 0 for α = 0, 1, 7,
while qLG = 1 for α = 3, 4, 5. As presented in Eq. (40), for α = 2, 6, the ground states are
two fold degenerate. We find that, each of the two degenerate ground states has qLG = 0 and
qLG = 1.
We conclude that the ZΓ4 charge q of ground state of (38) is
α = 0, 1, 7 ,
0 ,
iπq/2
UΓ |GS〉 = e|GS〉 ,
q = 2qLG = 2 ,(43)
α = 3, 4, 5 ,0 & 2 ,α = 2, 6 .
19

SciPost Phys. 17, 013 (2024)

This is consistent with that the ground state satisfiesZΓZΓ

τ x 1 |GS〉tw4 = σzi σzi+1 |GS〉tw4
i+ 2(1 ≤ i ≤ L) .

(44)

From the above result, it appears that the ground state degeneracy is not well defined in
the limit L → ∞. While we do not completely understand the physical mechanism behindthe periodic dependence of the ground-state degeneracy on the system size, the ground-statedegeneracy for α = 2, 6 might be interpreted as a consequence of an effective twist [58]. The
effective twist can be seen by mapping the Levin-Gu model to an XX chain. In appendix B,
we showed that under a unitary transformation, the Levin-Gu model with PBC can be mapped
to an XX chain with PBC and one ground state when L ∈ 4Z, and XX chain with the twisted
boundary condition and two degenerate ground states when L ∈ 4Z + 2. This is analogous
to the phenomenon that an antiferromagnetic chain of odd length is effectively subject toa twisted boundary condition. Here we simply consider the sequence of systems only withα ∈ {0, 1, 3, 4, 5, 7}. This would be reasonable if the ground-state degeneracy for α = 2, 6
is indeed due to an effective twist; we just consider the sequence of effectively untwisted
systems.16 Then the ground state degeneracy in the thermodynamic limit is regarded as one,consistent with our definition of igSPT.
There still remains the periodic dependence of the ZΓ4 charge in the ground state on thesystem size: for α = 0, 1, 7, the ground state is neutral under ZΓ4 , while α = 3, 4, 5, the groundstate gets a minus sign under the ZΓ4 transformation. However, this minus sign can always be
absorbed by suitably modifying the definition of UΓ in (39). In fact, in the following sections,
we will only be interested in the relative charge of the ground state between the periodic andtwisted boundary conditions, which turns out to be system-size independent.
3.2.2

Twisted boundary condition

We further discuss the charge of the ground state under the TBC. We can either twist by ZΓ4 ,or its normal subgroup ZA2 .
Twist by Z2A :is

We twist the boundary condition by ZA2 (labeled by ZA2 TBC). The HamiltonianZA

2
HigSPT
=−


L−1 
X
y
y
τz 1 σix τz 1 + τ 1 σix τ 1 + σzi τ x 1 σzi+1

i− 2
i+ 2
i− 2
i+ 2
i=1
y
y
+ τz 1 σ Lx τz1 + τ 1 σ Lx τ 1 − σzL τ x1 σ1z
L− 2
L− 2
2
2
2

i+ 2

(45)

= σzL HigSPT σzL ,where HigSPT is (38). We have already encountered the same algebra below (16). DeZA

ZA

2
note the ground state of HigSPT and HigSPT
as |GS〉 and |GS〉tw2 , respectively. Then we have

ZA|GS〉tw2 = σzL |GS〉. As UΓ σzL UΓ† = −σzL , we find
ZA

ZA

ZA

UΓ |GS〉tw2 = −e iπq/2 |GS〉tw2 = e iπ(q+2)/2 |GS〉tw2 ,(46)

where q is the Z4 charge of |GS〉 under PBC, given by (43). (46) means that the ZΓ4 charge ofthe ground state with the ZA2 twisted boundary condition differs from that with the periodic
boundary condition by two. We thus define the difference between the ZΓ4 charge under ZA2 TBC and that under PBC to be the relative ZΓ4 charge, which is two. Relative charge is more
16

See also [59] for the system size dependent ground state degeneracy in the (1 + 1)d Luttinger liquids.

20

SciPost Phys. 17, 013 (2024)

physical since there are ambiguities in defining the absolute charge as we noticed in the previous subsection. The nontrivial relative ZA2 charge shows that the igSPT we constructed in (38)is topologically nontrivial. We also note that the proof applies to all the states.
We remark that the second equality of (45) does not hold under a ZΓ4 symmetric perturbation. However, the conclusion that the relative ZΓ4 charge of the low lying states between the
ZA2 twisted and untwisted sectors being two still holds under a ZΓ4 symmetric perturbation. We
defer this to a subsequent paper [33].
Twist by ZΓ4 : We further use the ZΓ4 symmetry to twist the boundary condition (labeled byZΓ4 TBC). The Hamiltonian isZΓ4
HigSPT
=−


L−1 
X
y
z
x z
x y
z x
z
τ 1 σi τ 1 + τ 1 σi τ 1 + σi τ 1 σi+1
i+ 2

i− 2
i+ 2
i− 2
i+ 2
i=1
y
y
− τz 1 σ Lx τ 1 + τ 1 σ Lx τz1 + σzL τ x1 σ1z .
L− 2
L−
2
2
2
2

(47)

ZΓ

The ground state |GS〉tw4 satisfiesZΓZΓ

τ x 1 |GS〉tw4 = σzi σzi+1 |GS〉tw4
i+ 2(1 ≤ i ≤ L − 1) ,ZΓZΓτ x1 |GS〉tw4 = −σzL σ1z |GS〉tw4 .

(48)

2

We then measure the ZA2 charge using UA in (36),ZΓ

UA |GS〉tw4 = −

L−1
Y


ZΓ
ZΓ
σzi σzi+1 σzL σ1z |GS〉tw4 = − |GS〉tw4 ,

(49)

i=1
ZΓ

which means that the ground state carries ZA2 charge 1. This also implies that if |GS〉tw4 is aneigenstate of UΓ , then it should carry ZΓ4 charge 1 mod 4 or 3 mod 4.
In fact, by exact numerical diagonalization, we find that there are two degenerate groundstates if L is odd and four if L is even. If we organize them into eigenstates of ZΓ4 , half ofthem have charge 1 mod 4 and the other half have charge 3 mod 4. Since there are different
charges, an arbitrary linear combination of them is generically not an ZΓ4 eigenstate. However,as all of the ground states have ZA2 charge 1, an arbitrary linear combination of them also has
ZA2 charge 1.
From (43), the ZA2 charge of the ground state under PBC is always trivial, independentof the system size. Moreover, as we find in (49) the ZA2 charge of the ground state under ZΓ4
TBC is one, independent of the system size. We thus found that the relative ZA2 charge is sizeindependent, and it shows that the igSPT we constructed in (38) is topologically nontrivial.
Since (44) and (48) also hold for all low energy states with energy E−EGS ≪ 1, the above proof
of nontrivial ZA2 charge of the ground state also applies to low energy states, which should bestable under perturbations.
In summary, we have checked that using either ZA2 -TBC or ZΓ4 -TBC one can probe the topological nontriviality of the ZΓ4 igSPT.17
17
We need to show that the symmetry features shown in this section is stable under deformation. It was noticedin [15] that if the deformation leads to an accidental symmetry H which has a mixed anomaly with ZΓ4 , thenthe properties in this section can change due to the mixed anomaly. Assuming such accidental symmetry is the
only mechanism that can change the symmetry properties in this section (which is a common lore, and has been
discussed recently in [60]), we therefore need to demand that the deformation does not lead to any such accidental
symmetry.

21

SciPost Phys. 17, 013 (2024)

3.2.3Open boundary condition

We proceed to discuss the ground state degeneracy under the OBC. When placing the Z4 igSPT
on an open chain, as in section 2.3.3, we let the left most spin be σ spin, and right most spinbe τ spin. The Hamiltonian is
 X
L 
L−1
X
y
OBC
z
x z
x y
HigSPT = −τ 1 σi τ 1 + τ 1 σi τ 1 −(50)
σzi τ x 1 σzi+1 ,
i=2

i− 2

i+ 2

i− 2

i+ 2

i=1

i+ 2

and the symmetry operator is
UΓ =

L
Y

σix

i=1

L
Y

exp



i=1

We find that the set of operators {σ1z , σzL τ x

L+ 12


iπ 
1 − τx 1
.
i+ 2
4

(51)

, UΓ } commute with the Hamiltonian (50). The

irreducible representation of the above algebra is two, hence the ground states of (50) are atleast two fold degenerate. In appendix D.2, we show that the ground state degeneracy is fourfor L ∈ 2Z + 1, and two for L ∈ 2Z.

3.3

Stability of igSPTAs discussed in section 2, the ZA2 × Z2G gSPT is unstable upon perturbation towards the gapped
SPT phase. It immediately enters the ZA2 ×Z2G gapped SPT phase when transverse field λ passes
the critical value λc = 1. How about the stability of the ZΓ4 igSPT against perturbation into a
gapped phase with a unique ground state?
First of all, since ZΓ4 is non-anomalous, in principle, there is no obstruction to deform thesystem to Z4 symmetric gapped phase with a unique ground state [61]. Secondly, since there
is no ZΓ4 gapped SPT, the only gapped phase with a non-degenerate ground state is the triviallygapped phase. In this subsection, we will examine the most obvious ZΓ4 symmetric perturbationthat can drive the igSPT into a trivially gapped phase,
−h

L 

X
σix + τ x 1 ,
i+ 2

i=1

(52)

where h > 0. When h ≫ 1, as τ x 1 anticommutes with the first and second term of thei+ 2
Hamiltonian (38), and σix anticommutes with the third term, only (52) survives and it is in
the trivially gapped phase. This means that there must be at least one phase transition as hincreases from zero where either the ZΓ4 charge under PBC, or the relative ZA2 charge underZΓ4 -TBC or relative ZΓ4 charge under ZA2 -TBC jumps.
After adding (52), the proof of non-trivial (relative) charges under the twisted boundary
conditions in Section 3.2.2 no longer apply. We therefore numerically calculate the charges asa function of h, and indeed observe the jumps for finite h. We relegate the details of smallscale numerical study in Appendix F. In a subsequent work [33], by using the Kennedy-Tasaki
transformation, we analytically show that the phase transition happens at finite h, therefore
shows that igSPT (38) is stable under the perturbation (52).

4

gSPT and IgSPT in the spin-1 system

In this section, we briefly introduce a more realistic spin-1 model which hosts the igSPT andgSPT simultaneously. This model is studied in detail in [30] by one of the authors in this work
(L.L.) together with Yang, Okunishi and Katsura. We briefly review the results there, and fit
them into our framework.
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SciPost Phys. 17, 013 (2024)

gSPT
HaldanePhase

LandauTransition

igSPT
y′

Zz2 SSByZz2 ⋉ Z4 SSB

Z2 SSB

λ = −λ1

λ=0

λ = λ1λFigure 4: The phase diagram of (53) when θ =0.

4.1

The model and phase diagram

The Hamiltonian is given byH(θ , λ) = (1 − λ)HBLBQ + (1 + λ)UK T HBLBQ UK† T ,−π1< θ < arctan ,
4
2

(53)

where
HBLBQ = cos θ (S⃗i · S⃗i+1 ) + sin θ (S⃗i · S⃗i+1 )2 ,Y
UK T =
exp(iπSµz Sνx ) .

(54)
(55)

µ<ν

S⃗ is spin-1 operator. UK T is a non-local unitary operator implementing the Kennedy-Tasaki(KT) transformation [48–50]. Under the KT transformation, λ ↔ −λ, and λ = 0 is the selfdual point. For each θ and λ, the Hamiltonian (53) preserves three global symmetries:
Q iπS z1. Zz2 : π rotation in z direction, generated by j e j .
y

2. Z4 : π/2 rotation in y direction, generated byQ

i π2 S jyje.

3. ZT : translation symmetry.
The phase diagram of θ = 0 is obtained in [30], as shown in figure 4. See [30] for the full
2d phase diagram in the (λ, θ ) plane.

4.2

yZ2z ⋉ Z4 × ZT igSPTLet us start by discussing the self-dual point λ = 0 which we argue to be a igSPT. Taking thelow energy limit around this point, some degrees of freedom decouple, and the 3-dimensional
Hilbert space per site in the spin-1 model reduces to 2-dimensional Hilbert space per site,hence effectively becomes a spin- 21 model. The spin- 21 Hamiltonian turns out to be the XXZ
model [30]:
L
L
X
X
y y
H(λ ≪ 1) = −(1 + λ)σ xj σ xj+1 + (1 − λ)σ j σ j+1 .
(56)
j=1

j=1

This model also has three global symmetries:
Q′
1. Zz2 : generated by i σzi .
y′

2. Z2 : generated byQyi iσi .

3. ZT : translation symmetry.

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SciPost Phys. 17, 013 (2024)

We use the primes to distinguish the symmetries of the spin-1/2 model from those of the spin-1′y′model. Denote their background fields as A′z , A′y and A T . The symmetries Zz2 , Z2 and ZT2 ⊂ ZT
have a mixed anomaly [58, 62, 63] whose inflow action isω3d = eiπR
M3A′y A′z A T

.

(57)
′

y′

y

However, in the entire Hibert space of spin-1 system, the Zz2 × Z2 is realized as Zz2 ⋉ Z4
symmetry with the following extension:
Y ′ Z ′ = Rπy Z ′ Y ′ ,

(58)

QL

QL
QL
y
y
y
′
′
z
j=1 exp(iπS j ), Y =j=1 exp(iπS j /2) and Z =j=1 exp(iπS j ). exp(iπS j )has eigenvalues {−1, −1, 1}. In the low energy limit, the spin- 21 model only acts nontrivially onythe first two components of the spin-1 Hilbert space under the eigenbasis of exp(iπS j ), henceyexp(iπS j ) = −1 in the spin- 12 model, Y ′ , Z ′ in (58) reduces to the standard spin- 21 operatorsyyσzj = exp(iπS zj ) and iσ j = exp(iπS j /2). In terms of the background fields, (58) gives us theywhere Rπ =restriction
dAY = A′y A′z mod 2 ,

(59)

y

where AY is 1-cochain for Z2Y normal subgroup of Z4 symmetry. In summary, we can identify
′

y

′

y

Zz2 and Z2 in the spin- 12 theory with the Zz2 and Z4 /Z2Y in the spin-1 theory respectively.
yBesides, since exp(iπS j ) = −1 for each site in the low energy sector, the ground state isystacked by a weak gapped SPT phase protectedby translation and Z2 symmetry [64]. This isR
represented by the topological action e
a 3d bulk M3 ,
e

iπ

R
M2

iπ

M2

AY A T

AY A T=eand by (59), it depends on the extension to

iπ

R
M3

A′y A′z A T

.

(60)

This induced anomaly from stacking a weak gapped SPT phase cancels against the mixed
anomaly (57) in the low energy. Thus the total spin-1 system is anomaly free. This shows thatythe spin-1 system is a igSPT, protected by the total symmetry Zz2 ⋉ Z4 × ZT .
The total symmetry can be decomposed into two extensions,y′y

1 → Zz2 × Z2Y × ZT2 → Zz2 ⋉ Z4 × ZT2 → Z2 → 1 ,

(61)

and
y

y

1 → Z4 → Zz2 ⋉ Z4 × ZT2 → Zz2 × ZT2 → 1 .

(62)yNote that (62) is still a nontrivial extension.18 Comparing with (1), we see that the Zz2 ⋉Z4 ×ZT
y′igSPT can be constructed either by starting with G = Z2 SSB phase or G = Zz2 SSB phase,
which exactly correspond to the regimes λ > 0 and λ < 0 in figure 4. Moreover, from (57),′y′the anomalous symmetries in the low energy are b
Γ = Z2 × Zz2 × ZT2 . This provides an example where the SSB symmetry G is strictly smaller than the anomalous symmetry b
Γ , whichgeneralizes the construction in [16].
18

y

y

Both (61) and (62) are not central extensions, since both Zz2 × Z2Y and Z4 are not center subgroup of Zz2 ⋉ Z4 .

24

SciPost Phys. 17, 013 (2024)

4.3

yZ2z ⋉ Z4 × ZT gSPTLet us further consider the critical point at λ = −λ1 . The two phases around this critical point
are Zz2 SSB phase and a nontrivial gapped SPT protected by Zz2 ×Z2Y , a.k.a. the Haldane phase.
This fits into the phase diagram of gSPT in the left panel of figure 1.
Moreover, at λ = −λ1 , the Hamiltonian (53) has a unique ground state under periodic
boundary condition for a finite system size but has two ground states under the open boundary
condition (up to exponential splitting). There are also Q
three string order parameters withµµ µnonzero expectation value in the Haldane phase Oµ = 〈Sm m< j<n exp(iπS j )Sn 〉 (µ = x, y, z).
When the system is turned into this critical point, only O y remains nonzero but the other two
decay to zero algebraically quickly. All these evidence suggest that the critical point at λ = −λ1yis a nontrivial gSPT. As the system has total symmetry Zz2 ⋉ Z4 × ZT , we name the critical point
y
z
T
z
Y
as Z2 ⋉ Z4 ×Z gSPT, although only a subgroup Z2 ×Z2 protects the gapped SPT in the nearby
phase.

Acknowledgments
L.L. sincerely thanks Jian-da Wu’s group for helpful discussions on the Jordan-Winger transformation of the Levin-Gu model; Hong Yang for discussions on the gSPT and igSPT in theSpin-1 system; Yutan Zhang for kind help on the Julia code. Y.Z. sincerely thanks Jie Wang fordiscussions on stability of gapless systems, Qing-Rui Wang for discussions on decorated defectconstructions, and Nick Jones, Ryan Thorngren and Ruben Verresen for helpful comments anddiscussions. We also thank Atsushi Ueda, Yuan Yao, Kantaro Ohmori for useful discussions.
Funding information Y.Z. is partially supported by WPI Initiative, MEXT, Japan at IPMU,
the University of Tokyo. This work was supported in part by MEXT/JSPS KAKENHI Grants No.
JP17H06462 and No. JP19H01808, and by JST CREST Grant No. JPMJCR19T2.

A

Stability of boundary degeneracy of Z2A × ZG2 gapped SPT

We find in section 2.3.3 that if we suitably change OBC by adding boundary interactions, theground state degeneracy can be lifted from four to two. In this appendix, we would like toargue that exactly degenerate ground states of the ZA2 × Z2G gapped SPT, which is always four,
does not lift under arbitrary symmetric perturbations localized at the boundary.
Let us truncate the system in the same way as section 2.3.3. The σ spins are supported on
i = 1, . . . , L, and the τ spins are supported on i + 12 = 32 , . . . , L + 12 . Let us begin by choosing
one particular OBC such that the Hamiltonian is
OBC
HSPT
=−

L−1
X
i=1

σzi τ x 1 σzi+1 −
i+ 2

L
X
i=2

τz 1 σix τz 1 .
i− 2

i+ 2

(A.1)

Suppose the boundary perturbation at the left end is supported on 2 sites, 1, 32 . A genericsymmetric perturbation takes the form
β3

OBC
∆HSPT
= (σ1x )β1 (τ x3 ) 2 ,

(A.2)

2

OBC
where β1, 3 ∈ {0, 1}.19 Let us find the local operators that commute with both HSPT
and2

For perturbations supported on 3 sites, one also allows σ1z σ2z . But for 2 site perturbation, Pauli Z operators
are forbidden by the symmetries.
19

25

SciPost Phys. 17, 013 (2024)

OBC
OBC
∆HSPT
. Any interaction commuting with HSPT
are composed of the building blocks σ1z ,σ1x τz3 , τ x 1 σzL , τz 1 and all the terms that already exist in (A.1). Using these building blocks,
2

L+ 2

L+ 2

a generic term that might anticommute with the boundary perturbation takes the formOu1 u2 u3 u4 = (σ1z )u1 (σ1x τz3 )u2 (σ1z τ x3 σ2z )u3 (τz3 σ2x τz 5 )u4 ,
2

2

2

i+ 2

(A.3)

OBC
where u1,2,3,4 ∈ {0, 1}. Requiring [O, ∆HSPT] = 0, we find that the coefficients need to satisfy
the linear equations
β1 (u1 + u3 ) + β 3 (u2 + u4 ) = 0 mod 2 .
(A.4)
2Note that β1, 3 are given, while u’s are variables to be determined. There are 4 variables, and2
one equation, hence one is free to choose arbitrary value of u1 , u2 , such that ui ’s for i = 3, 4
are constrained by the equation. One solution would be u3 = β 3 − u1 , u4 = β1 − u2 . On the2other hand, the algebra between the operators {Ou1 u2 u3 u4 , UA, UB } are
′

′

′

′

′

′

′

′

′

′

Ou1 u2 u3 u4 Ou1 u2 u3 u4 = (−1)u1 u2 +u1 u2 Ou1 u2 u3 u4 Ou1 u2 u3 u4 ,UAOu1 u2 u3 u4 = (−1)u2 Ou1 u2 u3 u4 UA ,UG O

u1 u2 u3 u4u1= (−1) O

u1 u2 u3 u4

(A.5)

UG .

The commutation relations only depends on u1 , u2 ! Hence we are free to choose two commut′ ′ing independent operators O10u3 u4 and O01u3 u4 whose common eigenvalues (a, b) label theground states |(a, b)〉, where u3,4 and u′3,4 are arbitrary solutions of (A.4). The four orthogonal ground states are thus given by|(−a, −b)〉 = UA UG |(a, b)〉 .
(A.6)
The above discussion can easily be generalized to perturbation supported on arbitrary number
sites. We thus conclude that, for the ZA2 × Z2G gapped SPT, the exact four fold ground statedegeneracy on an open chain is stable under boundary perturbation.
|(a, b)〉 ,B|(−a, b)〉 = UG |(a, b)〉 ,|(a, −b)〉 = UA |(a, b)〉 ,Spectrum of Levin-Gu model under different boundary conditions

In this appendix, we show the energy spectrum of Levin-Gu model [42] under different boundary conditions analytically. The analytic results are confirmed by the numerical calculation.

B.1

Exact solutions under PBC by Jordan-Wigner transformation

The Hamiltonian of Levin-Gu model is
HLG = −LX

σix − σzi−1 σix σzi+1 ,

(B.1)

i=1

which respects the Z2 symmetry generated by
UG =

L
Y
i=1

σix

L
Y

exp



i=1


iπ
(1 − σzi σzi+1 ) .
4

(B.2)

We apply the Jordan-Wigner (JW) transformation which maps spin operator to fermion operator
i−1
Y
†
x
ni
z
σi = (−1) = 1 − 2 f i f i ,σi =(−1)n j ( f i† + f i ) ,
(B.3)
j=1

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SciPost Phys. 17, 013 (2024)

where ni := f i† f i is fermion density operator. Note that when i =1, we simply have σ1z = f1† + f1 .
a
We also assume PBC of the spins, i.e. σia = σi+L.
Applying the JW transformation to the Levin-Gu model, we can rewrite (B.1) in terms of
the fermions,LX†
HLG = −L +2 f i† f i + ( f i† − f i )( f i+2+ f i+2 ) ,
(B.4)
i=1

with boundary condition
f i+L = −(−1) F f i ,

F=

L
X

nj .

(B.5)

j=1

After Fourier transformation and Bogoliubov transformation, this Hamiltonian is diagonal


X
P †
1
†
HLG =
ωk ck ck −,(−1) k ck ck = (−1) F ,
(B.6)
2
k
where ωk = 4| cos k|. There are zero modes if k can be either π2 or 3π
2 , and whether they are
realizable depends on the boundary condition. It turns out that depending on L ∈ 4Z, 4Z + 2
or 2Z + 1, the boundary condition behaves differently. We discuss them separately.
Case 1: L ∈ 4Z
2π j

If (−1) F = −1, the fermion chain has PBC. This means k = L where j = 0, · · · , L − 1.
π
3πTherefore, when j = 4L and j = 3L
4 , we have two zero modes at k = 2 and k = 2 . Since(−1) F = −1, the ground states are: c †π |VAC〉PBC and c †3π |VAC〉PBC . The ground state energy is
2

PBC
= −2
EGS

L−1
X

2

cos

j=0



2π j
L



= −4 cot

π
L

.

(B.7)
(2 j+1)π.
If (−1) F = 1, the fermion chain has anti-periodic boundary condition (ABC) where k =LSince L ∈ 4Z, there is no zero mode. the ground state is |VAC〉ABC with ground state energy:
ABC
EGS
= −2

L−1
X

(2 j + 1)π
cos
L
j=0




=−

4
sin πL

.

(B.8)

ABC
PBC
As EGS< EGS, the Levin-Gu model has an unique true ground state which is vacuum of ABC
after Jordan-Wigner transformation.

Case 2: L ∈ 4Z + 2
2π j

If (−1) F = −1, the fermion chain has PBC where k = L , j = 0, · · · , L − 1. Since
L = 4m + 2 ∈ 4Z + 2, there is no zero mode. The ground states are c †2mπ |VAC〉PBC ,4m+2

c †2π(m+1) |VAC〉PBC , c †2π(3m+1) |VAC〉PBC and c †2π(3m+2) |VAC〉PBC . The ground state energy is
4m+2
4m+2

 4m+2
L−1
π
X
2π
j
mπ
4
PBC

EGS
= −2
cos
+ 4 cos(
)=−
+
4
sin
.
L
2m + 1L
sin πLj=0(2 j+1)π(B.9)

If (−1) F = 1, the fermion chain has ABC where k =. Since L = 4m + 2 ∈ 4Z + 2,
Lthere are two zero modes at j = m and j = 3m + 1. The ground states are double degenerate

27

SciPost Phys. 17, 013 (2024)|VAC〉ABC and c †π c †3π |VAC〉ABC with energy
2

2

ABC
EGS
= −2

L−1
X

cos



j=0

(2 j + 1)πL= −4 cotπ
L.

(B.10)

Since
ABC
PBC
EGS
− EGS
= −4 cot

π
L

+

4
sin

 − 4 sin
π

π
L

L

= −4 cotπL

1 − cos π 
L< 0 , (B.11)

the Levin-Gu model has double degenerate ground states which is vacuum of ABC.
Case 3: L ∈ 2Z + 1(2 j+1)πIf (−1) F = 1, the fermion chains has ABC where k =and where j = 0, · · · , L − 1. Now
Lsince L = 2m + 1 ∈ 2Z + 1, there is no zero mode. The ground states is |VAC〉ABC with energy
ABC
= −2
EGS

2m
X

(2 j + 1)πcos
2m + 1
j=0




m−1
X

(2 j + 1)π= −4
cos
2m + 1
j=0




− 2.

(B.12)

2π j

If (−1) F = −1, the fermion chain has PBC where k = L where j = 0, · · · , L. Now since
L = 2m + 1 ∈ 2Z + 1, there is also no zero mode. Here we note that the energy of |VAC〉PBC is
the same as (B.12)
PBC
= −2
EVAC

= −4

2m
X

2π j
cos
2m
+1
j=0

m
X

cos

j=1

m
X

2π j
cos
= −4
2m
+1
j=1







(2m − 2 j + 1)π2m + 1− 2 = −4

m−1
X



−2

cos



j=0

(2 j + 1)π2m + 1

− 2.

(B.13)

Since there is no zero mode, the ground state energy in (−1) F = −1 sector must be higher
PBC
than EVAC
which coincides with the ground state energy (B.12) under the ABC and the uniquetrue ground state is |VAC〉ABC .
In summary, the ground state degeneracy of the Levin-Gu model under PBC is two if
L ∈ 4Z + 2, and one otherwise. This proves (40).

B.2

Mapping to XX chain and charge of ground state

When the system size is even (L = 2m), there is a unitary transformation [7]
 


z
z

Y
m
m σz + σ x Y
m
Y
πi
1
−
σ
1
−
σ
πi y
2 j−1
2j
2j
2j
,
exp
U=
i
exp 
σ2 j
p
2
4
2
j=1
j=1
j=1

(B.14)

which maps the Levin-Gu model to a XX chain with imaginary hopping constant.
m
X
U HLG U = −(σ2z j−1 σ2x j − σ2x j σ2x j+1 − σ2x j−1 σ2z j + σ2z j σ2x j+1 )
†

j=1

=−

L
X

iσ+j σ−j+1 + h.c. ,

j=1

28

(B.15)

SciPost Phys. 17, 013 (2024)

where σ+j = σzj + iσ xj . The imaginary hopping XX chain can be further mapped to a standard
XX chain by a unitary transformation
U1 =

L
Y

exp



j=1

πi y
jσ j
2



.

(B.16)The resulting Hamiltonian is
U1 U HLG U † U1† = −LX(σzj σzj+1 + σ xj σ xj+1 ) ,(B.17)

j=1

with boundary condition
σzL+ j = i L σzj ,

x
L x
σ L+
j = i σj .

(B.18)

After taking the continuum limit [65, 66](σz + iσ x ) ∝ e iθ ,σy ∝a∂x φ ,
2π(B.19)

the low energy theory of standard XX chain is the free boson theory and the energy of eigenstate|m, n〉 is20
π(Em,n − E0,0 ) ∝(4m2 + n2 ) ,(B.20)
2L
where the integer pairs (m, n) are determined by the boundary conditions θ (x+L)=θ (x)+2πm
and φ(x + L)=φ(x) + 2πn. By combining (B.17), (B.18) and (B.20), we conclude as follows.
1. When L ∈ 4Z, the Levin-Gu model is equivalent to the XX chain with PBC where m ∈ Z
and n ∈ Z. Its energy minimizes at a unique value (m, n) = (0, 0), and the unique groundstate is |0, 0〉.
2. When L ∈ 4Z + 2, the Levin-Gu model is equivalent to the XX chain with ABC wherem ∈ Z + 1/2 and n ∈ Z. Its energy minimizes at two distinct values (m, n) = (± 12 , 0),
and there are two degenerate ground states |± 12 , 0〉.
This is consistent with the results from JW transformation in (B.1).
Moreover we can obtain the Z2 symmetry (B.2) after transformation
UG′ = U1 U UG U † U1† =

L
Y
j=1

L

y
σj


πi
x
z
x
z
exp
(2 + σ2 j−1 σ2 j − σ2 j σ2 j+1 ) .
4
j=1

2
Y



(B.21)

After taking the continuum limit (B.19), the Z2 symmetry operator in the low energy is given
by
 Z
Z

L
i
i
′
UG = i 2 exp∂ x φd x −∂x θ d x .
(B.22)
2
2
The charge of the state can be found by acting UG′ on |m, n〉,LUG′ |m, n〉 = i 2 e iπ(n−m) |m, n〉 .

(B.23)

Therefore when L ∈ 4Z, the charge of ground state |0, 0〉 is (−1) L/4 . When L ∈ 4Z + 2 theL−2charges of ground states |± 21 , 0〉 are ±(−1) 4 . This proves (42) for even L.
20

Since we are only interested in ground state degeneracy, we don’t consider excitations of the oscillator modes.

29

SciPost Phys. 17, 013 (2024)

B.3

Spectrum under open boundary condition

In this section, we use the transformations (B.14) and (B.16) to discuss spectrum of Levin-Gu
model under OBC
L−1
X

OBC
HLG
=−
σix − σzi−1 σix σzi+1 .
(B.24)
i=2z
There are two boundary operators σ1 and σzL commuting with Hamiltonian.
z
When L ∈ 2Z, the Hamiltonian (B.24) and the boundary operators σ1,Lafter the transfor-

mation are given byL

OBC † †U1 U HLGU U1 = −2 −1 
X
σ2x j−1 σ2x j + σ2z j+1 σ2z j+2 + σ2x j σ2x j+1 + σ2z j σ2z j+1 ,(B.25)

j=1

U1 Uσ1z U † U1† = −σ1x ,L

U1 UσzL U † U1† = (−1) 2 +1 σzL .

(B.26)

After taking the continuum limit, the boundary operators are − sin θ (x = 0) andL(−1) 2 +1 cos θ (x = L). As the ground state should be the eigenstate of the boundary operLL
ators −σ1x , (−1) 2 +1 σzL , − sin θ (x = 0) = ±1, (−1) 2 +1 cos θ (x = L) = ±1. They determine theboundary conditions θ (x = 0) = ± π2 and θ (x = L) = 0 or π. The ground state energy underthese four boundary conditions are exactly the same.
When L ∈ 2Z + 1, we only do the transformation (B.14) for even number of sites, say,i = 1, . . . , L − 1. We still do π/2 rotation along y direction, i.e. U1 in (B.16), on the L-th site.
The Hamiltonian (B.24) and the boundary operators after the transformation are given by
L−1

OBC † †U U1 = −U1 U HLG

2 
X

L−3



σ2x j−1 σ2x j + σ2x j σ2x j+1 +

j=1

2 
X


σ2z j+1 σ2z j+2 + σ2z j σ2z j+1 ,(B.27)

j=1

U1 UσzL U † U1† = σ Lx .

U1 Uσ1z U † U1† = −σ1x ,(B.28)

After taking the continuum limit, the boundary operators are − sin θ (x = 0) and sin θ (x = L)
which implies boundary conditions are θ (x = 0) = ± π2 and θ (x = L) = ± π2 , and the signs areuncorrelated. Unlike even size, the states with different boundary conditions have different
energies,
1E(∓ π2 ,± π2 ) − E(± π2 ,± π2 ) ∝ ,(B.29)
L
where the signs are correlated. Therefore the true ground states are double degenerate andare in the sector with boundary conditions θ (x = 0) = θ (x = L) = ± π2 .

C

Equivalence between ground sector of Z4 igSPT and Levin-Gu
model

In this section, we show the ground state of the pre-decorated model (34) of ZΓ4 igSPT is thesame as the Levin-Gu model (32) with τix = 1.
Let us begin with the pre-decorated model (34) with PBC, which we reproduce here†
U DW HigSPT U DW
=−

L 

X
σix − σzi−1 τ x 1 σix τ x 1 σzi+1 + τ x 1 .
i− 2

i=1

i+ 2

i− 2

(C.1)

Since the last term commutes with all other terms, the Hibert space can be divided into sectors
with different τ x configurations. In different sectors, the sign of term σzi−1 σix σzi+1 is decided
30

SciPost Phys. 17, 013 (2024)by τ x 1 τ x 1 . It is easy to see that the number of terms with τ x 1 τ x 1 = −1 must be even,i− 2 i+ 2
i− 2 i+ 2
QLxx
since i=1 τ 1 τ 1 = 1. We prove the splitting of ground state energy of first two terms in
i+ 2

i− 2

(C.1) with different τ configuration is order of 1/L or exactly zero. Therefore, when L is large
enough, the state in the ground state sector of (C.1) satisfies τ x 1 = 1 for each i.
i+ 2

When L ∈ 2Z + 1, we can prove the first two terms in (C.1) with any τ configuration canbe mapped to the standard Levin-Gu model by a unitary transformation.
This implies the ground state energy of any τ configuration is same as that of the standard
Levin-Gu model. To see the unitary transformation, let us assume that the sign of two terms
σzi−1 σix σzi+1 and σzj−1 σ xj σzj+1 are both −1 where 1 ≤ i < j ≤ L.21 There is always a unitarytransformation which can cancel these twoQ
−1 and preserve
terms: If i, j are both
Q sign of otherxx
odd (even), the unitary transformation is i<2k< j σ2k( i<2k+1< j σ2k+1). If i is odd (even)
Q
Q
Q
x
x
x
and j is even (odd), the unitary transformation is i<2k<L σ2k 1≤2k+1< j σ2k+1( j<2k<L σ2kQx1≤2k+1<i σ2k+1 ) which can do the job only when L ∈ 2Z+1. Since the number of terms with
−1 sign is even, we can cancel these −1s step by step and obtain the standard Levin-Gu modelat last.
When L ∈ 2Z, we apply the unitary transformation (B.14) and (B.16) on the first two
terms and then obtain XX chain with several minus coupling constants:
LX
Hµ1 ,µ2 = −(µ1j, j+1 σzj σzj+1 + µ2j, j+1 σ xj σ xj+1 ) ,

(C.2)

j=1

where µ1 and µ2 can be ±1. They are decided by the configuration of τ x but we don’t need toknow the exact relationship. We only use the fact that l + l ′ ∈ 2Z where l and l ′ are numberof −1 in µ1 and µ2 .22
We note that the spectrum of Hamiltonian (C.2) only depends on l, l ′ mod 2, and is independent of the configuration of µ1 and µ2 . The reason is as follows. The sites of −1 in µ1
can be labeled as µ1j , j +1 , µ1j , j +1 , · · · µ1j , j +1 where j1 < j2 < · · · < jl . After the unitary trans1 1
2 2
l l
Q ji+1x
formation k=
σ
,
µ
,
µ
will become 1 without changing spectrum. Similar
j
,
j
+1
j
,
j
+1
j +1 ki i
i+1 i+1i

for µ2 .
As l + l ′ are even, there are only two equivalence classes for spectrum: l = l ′ = 0
and l = l ′ = 1. The first case is XX chain with PBC. In the second case, we can chooseµ1L,1 = µ2L,1 = −1 without loss of generality. This is XX chain with the ABC. The splitting between ground state energy of these two boundary conditions is order of 1/L which completes
our proof.
Besides, one can apply this argument to the Z4 igSPT with TBC and OBC as well. Generally,
the ground state sector is Hilbert subspace which has eigenvalue 1 of the third term in the
Hamiltonian (45), (47) and (50).

DEdge Degeneracy of gSPT and igSPTIn section 2.3.3 and 3.2.3, we discussed the degeneracy of gSPT and igSPT under OBC bystudying the dimension of irreducible representation of operators commuting with the Hamiltonian. In this appendix, we rederive the degeneracy under OBC in an alternative way. We first
undecorate the domain wall which maps the gSPT and igSPT to the Ising and Levin-Gu models
under OBC respectively, and then use the results in appendix B to rederive the degeneracy.
We only focus on the “fundamental domain” where 1 ≤ i < j ≤ L and do not use periodicity i ∼ i + L here.
l + l ′ ∈ 2Z can be seen from the transformation (B.14) and (B.16), which maps σ2x j−1 → σ2z j−1 σ2z j ,x
σ2 j → σ2x j−1 σ2x j , −σ2z j−1 σ2x j σ2z j+1 → σ2x j σ2x j+1 and −σ2z j σ2x j+1 σ2z j+2 → σ2z j σ2z j+1 .
21
22

31

SciPost Phys. 17, 013 (2024)

D.1

Edge Degeneracy of Z2 × Z2 gSPTIn section 2.3.3, we studied the Z2 × Z2 gSPT under OBC, with the Hamiltonian (22),
OBC
HgSPT
=−

L
L−1 
 X
X
τz 1 σix τz 1 .
σzi τ x 1 σzi+1 + σzi σzi+1 −
i+ 2

i=1

i=2

i− 2

(D.1)

i+ 2

After U DW transformation, the Hamiltonian is given byOBC †U DW = −U DW HgSPT

L−1 
L
 X
X
τ x 1 + σzi σzi+1 −
σix .
i=1

i+ 2

(D.2)

i=2

τ L+ 1 decouples from the Hamiltonian which gives two ground state degeneracy. The σ1z com2
mutes with Hamiltonian which gives two fixed boundary conditions on the left end and theright end is free boundary condition. Therefore we have four exact ground states. But this isunstable under symmetric perturbations as noted in section 2.3.3. We can add the boundary
term (25) which becomes(D.3)−σzL τ x 1 ,L+ 2

after conjugated by U DW , i.e. domain wall undecoration. Now τ x

L+ 12

no longer decouples,

which lifts degeneracy due to free boundary condition on the right, and ground state degeneracy reduces to two.

D.2

Edge degeneracy of Z4 igSPTIn section 3.2.3, we studied the Z4 igSPT under OBC, with the Hamiltonian (50)OBC=−
HigSPT

 X
L 
L−1
X
y
z
x z
x y
τ 1 σi τ 1 + τ 1 σi τ 1 −σzi τ x 1 σzi+1 .
i− 2

i=2

i+ 2

i+ 2

i− 2

i=1

(D.4)

i+ 2

After undecorating the domain wall, we obtain the Levin-Gu model under OBC†
OBC
U DW HigSPT
U DW
=−

L−1
X

τx 1 −

i=1

i+ 2

L−1 
 
X
σix − σzi−1 τ x 1 σix τ x 1 σzi+1 − σ Lx − σzL−1 τ x
i=2

i− 2

The ground state should be the eigenstate of τ x 1
i− 2

L− 12

i+ 2

σ Lx τ x

L+ 21



.

(D.5)
(i < L + 1) with eigenvalue 1. The low

energy effective Hamiltonian is:†
OBC
U DW HigSPTU DW

Since τ x

L+ 12

low

=−

L−1
X

σix − σzi−1 σix σzi+1− σ Lx − σzL−1 σ Lx τ x

L+ 21

i=2

commute with effective Hamiltonian, we can redefine τ x

L+ 12



.

(D.6)

as σzL+1 and (D.6)

becomes (B.24) with system size L + 1. We thus conclude that when L ∈ 2Z + 1, the groundstate degeneracy is four and when L ∈ 2Z the ground state degeneracy is two.

E

ZT4 × Z2 igSPTIn this section we discuss another example of igSPT which respects the ZT4 × Z2 symmetries.
We will also discuss the PBC, TBC and OBC.

32

SciPost Phys. 17, 013 (2024)

E.1

Lattice Hamiltonian

Let us assign three spin- 21 s τ, σ and µ per unit cell and the Hamiltonian is: XXHZT ×Z2 =µzj τ x 1 µzj+1 + σzj µzj τ x 1 µzj+1 σzj+1 + σ xj µ xj + σ xj −
τz 1 µ xj τz 1 .
j+ 2

4

j

j+ 2

j− 2

j

This Hamiltonian respects the following symmetry:Y  1 + µ xj
1 − µ xjyxTσj +iσ j K ,Z 4 : UT ≡22jYZτ2 : Uτ ≡
τ xj ,

UT2 =

Y

j+ 2

µ xj ,

(E.1)

(E.2)

j

(E.3)

j

where T stands for time reversal, and K is the complex conjugation.
To see that (E.1) is a Z2 × ZT4 igSPT, we show that it can be obtained by starting with aµ
Zτ2 × ZT2 anomalous critical theory, and decorating the Zτ2 domain wall by 1d Z2 gapped SPT,µwhere Z2 is generated by UT2 . Let us apply U DW of τ and µ on both the Hamiltonian (E.1) andthe symmetry operators (E.2) and (E.3).
!
x zz
1 − µ xj τz 1 τz 1
Y 1 + µ j τ j− 1 τ j+ 1
j−
j+
y
2
2
2
2
†
U DW UT U DW =σ xj +
iσ j K ,
(E.4)
2
2
j
†
U DW Uτ U DW
= Uτ ,XX†µ xj .
U DW HZT ×Z2 U DW=
(τ x 1 + σzj τ x 1 σzj+1 + σ xj + τz 1 σ xj µ xj τz 1 ) −
4

j

j+ 2

j+ 2

j− 2

j+ 2

(E.5)
(E.6)

j

In (E.6), since the last term commutes with all other terms, the energy eigenstates are eigenstates of µ xj . Similar to the proof in the Z4 igSPT, we can consider the spectrum of first fourterms in the Hamiltonian (E.6) with different configurations of µ x . These four terms can bemapped to an XX chain by applying the unitary transformations (B.14):
H({µ xj }) =LXσzj τz 1 + τz 1 σzj+1 + σ xj τ x 1 + τ x 1 σ xj µ xj .
j+ 2

j=1

j+ 2

j+ 2

(E.7)

j− 2

According to the proof in appendix C, we know the spectrum of the (E.7) is invariant if we
flip even number of µ x . Thus, the spectrum of first four terms in (E.6) is that of XX chainxxzzwith boundary condition: σ L+
j = ±σ j and σ L+ j = σ j , where we take ± sign if there arexeven or odd number of µ = −1 respectively. After taking the continuum limit (B.19), these
two boundary conditions are PBC and ABC for θ respectively. The splitting between the corresponding ground state energy is also of order 1/L. Thus in the low energy state sector, one canfind that µ xj = 1. The effective Hamiltonian and symmetry are those of the boundary modelof 2+1d ZT2 × Z2 SPT [7]:!
zz
1 − τz 1 τz 1
Y 1 + τ j− 1 τ j+ 1
j−
j+
y
2
2
2
2
†
(E.8)
U DW UT U DW=
σ xj +
iσ j K ,
low
2
2
j

X
†
=
τ x 1 + σzj τ x 1 σzj+1 + σ xj + τz 1 σ xj τz 1 .
U DW HZT ×Z2 U DW
(E.9)
4

low

j

j+ 2

j+ 2

j− 2

j+ 2

Moreover the proof on the equivalence between ground state sector and XX chain can begeneralized to twisted boundary conditions and open boundary conditions. We conclude thatthe ground state sector of different boundary conditions is always Hibert subspace which haseigenvalue 1 of the last term in the Hamiltonian (E.10) and (E.14).
33

SciPost Phys. 17, 013 (2024)

E.2

Charge of twisted boundary condition

We show that the charge of the ground state under TBC is nontrivial, implying that (E.1) is a
nontrivial igSPT. Let us start by twisting the boundary condition using the Zτ2 symmetry, which
we denote as Zτ2 -TBC. The Hamiltonian (E.1) becomesZτHZT2 ×Z =
2

4

L 

X
µzj τ x 1 µzj+1 + σzj µzj τ x 1 µzj+1 σzj+1 + σ xj µ xj + σ xj
j+ 2

j=1
L−1
X

−

j=1

j+ 2

!
τz 1 µ xj τz 1 − τz

L− 12

j+ 2

j− 2

µ xj τz1.

2

The ground state satisfies
τz 1 µ xj τz 1 = 1
j− 2

τz

(0 < j < L) ,

j2

L− 12

µ xj τz1 = −1 .

(E.10)

2µwhich implies that the ground state has a nontrivial Z2 charge
L
Y

Zτ

Zτ

µ xj |GS〉tw2 = − |GS〉tw2 .

(E.11)

j=1µ

On the other hand, if we twist by Z2 symmetry, the SPT criticality Hamiltonian becomes
L−1 
L 
 X

X
z
z x
x
z z x
z
x z
z
z
x x
HZT ×Z =µ j τ 1 µ j+1 + σ j µ j τ 1 µ j+1 σ j+1 +σj µj + σj − τ 1 µj τ 1
µ

Z2
4

2

j+ 2

j=1

j+ 2

j− 2

j=1

j+ 2

− µzL τ x1 µz1 − σzL µzL τ x1 µz1 σ1z
2

2= τz1 HZT ×Z2 τz1 .
2

4(E.12)

2µIt is straightforward to check that |GS〉tw2 has Zτ2 charge 1:
Z

Z

µ

Z

µ

Uτ |GS〉tw2 = Uτ τz1 Uτ† Uτ |GS〉 = −τz1 |GS〉 = − |GS〉tw2 .
2

E.3(E.13)

2

Open boundary condition

To consider OBC, we truncate the spin chain so that σ-spins and µ-spins live on i = 1, . . . , L,and τ-spins live on i = 23 , . . . , L + 12 . We only keep the terms in (E.1) that are fully supported
on the spin chain. The Hamiltonian is
HZOBC
=
T
×Z
4

2

L−1
X

µzj τ x 1 µzj+1 + σzj µzj τ x 1 µzj+1 σzj+1 +

j=1

j+ 2

j+ 2

L
X

σ xj µ xj + σ xj −j=1

There are two boundary operators µ1x τz3 and τz
2

L+ 21

L
X
j=2

τz 1 µ xj τz 1 . (E.14)
j− 2

j+ 2

commuting with Hamiltonian. Since bothof them anticommute with Uτ , there must be at least two exactly degenerate ground states of(E.14).
The exact ground state degeneracy can be determined by undecorating the domain wall,by applying U DW on (E.14):†
U DW HZOBC
U DW
=
T
×Z
2
4

L−1
X
j=1

τ x 1 + σzj τ x 1 σzj+1 +
j+ 2
j+ 2

L
X
j=1

34

σ xj +

LX
j=2

τz 1 σ xj µ xj τz 1 + σ1x µ1x τz3 −
j− 2
j+ 2
2

L
X
j=2

µ xj ,

SciPost Phys. 17, 013 (2024)

Z4 Γ Charge
5
4
3
2
1
0
0.5
-1

Z4 Γ Charge
5
4
Z2 A TBC
32
1
0
0.5 1.0 1.5 2.0
-1

PBC

1.0

1.5

2.0

2.5

h

Z2 A Charge
3

1
2.5

0

h

Gap

Gap

Gap

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.5

1.0

1.5

2.0

2.5

h

Z2 A TBC

0.5

1.0

1.5

2.0

2.5

0.5

-1

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

PBC

Z4 Γ TBC

2

h

1.0

1.5

2.0

2.5

2.0

2.5

h

Z4 Γ TBC

0.5

1.0

1.5

h

Figure 5: ZΓ4 charge of the ground state under PBC, relative ZΓ4 charge of the groundstate under ZA2 TBC, relative ZA2 charges of the ground state under ZΓ4 TBC, and thegap between the ground state and first excited state under PBC and two TBC’s. The
horizontal axis is the perturbation strength (52). The system size is L = 11.
and the two boundary operators becomes µ1x and τz

L+ 12. In the ground state sector µ xj = 1 for

2 ≤ j ≤ L. The Hamiltonian in the low energy then simplifies to†
U DW HZOBC
U DW
T
×Z
4

2

low

=

L
L
L−1
X
X
X
τz 1 σ xj τz 1 +σ1x µ1x τz3 . (E.15)
σ xj +(τ x 1 +σzj τ x 1 σzj+1 )+
j=1

j+ 2

j+ 2

j=1

j=2

j− 2

j+ 2

2

Under the unitary transformation (B.14), this Hamiltonian is mapped to
†U DW
U U DW HZOBC
T
×Z
4

2


low

U† =

L−1
X
j=1

σzj τz 1 + τz 1 σzj+1 +
j+ 2

j+ 2

L
X

σ xj τ x 1 +
j+ 2

j=1

L
X
j=2

τ x 1 σ xj + σ1x µ1x ,j− 2(E.16)and the two boundary operators become µ1x and τ x 1 . The Hamiltonian (E.16) can be underL+ 2
stood as an XX chain on an open chain with size 2L and one spin- 12 per unit cell.
Similar to the Z4 igSPT, we can redefine µ1x as τ x1 . After taking the continuum limit (B.19),2
σ x and τ x are mapped to sin θ . Thus µ1x = ±1 and τ x 1 = ±1 correspond to the boundary
L+ 2
conditions sin θ (x = 0/L) = ±1 which implies θ (x = 0/L) = ± π2 . There is an energy splitting
between the ground states of two boundary conditions
E( π2 ,− π2 )/(− π2 , π2 ) − E( π2 , π2 )/(− π2 ,− π2 ) ∝1.
L(E.17)

In summary, the ground state degeneracy under OBC is two.

F

Small-scale numerical study for igSPT under perturbation

In this appendix, we perform the exact diagonalization numerically, and record the lowest
h where the charges jump in table 4. We also plot the charges and the gaps under various
boundary conditions for L = 11 (22 spin- 21 ’s) in figure 5.
From the plots in figure 5, we find that the ZΓ4 charge under PBC and both relative charges
under TBC’s are unchanged until h reaches the first critical value hc ≃ 0.28. This first transition is probed by the charge jump under PBC, where the finite size gap closes simultaneously.
35

SciPost Phys. 17, 013 (2024)

Table 4: Lowest h where the symmetry charge of the ground state under three boundary conditions jumps, for L = 4, 5, 7, 8, 9, 11.
L

ZΓ4 Charge under PBC

ZΓ4 Charge under ZA2 -TBC

ZA2 Charge under ZΓ4 -TBC

4

1.01

1.01

1.01

5

1.30

1.30

0.50

7

0.44

1.32

0.98

8

0.70

0.70

0.70

9

0.86

0.86

0.86

11

0.28

1.12

1.01

When h further passes hc , the system goes through a sequence of transitions, some are probed
by the ZA2 -TBC, some are probed by the ZΓ4 -TBC and the others are probed by PBC. When h issufficiently large (h > 2), the system enters into a trivially gapped phase, and all charges become trivial, which is consistent with the phase diagram by the Kennedy-Tasaki transformationin [33].
For different system sizes, for instance L = 5 as shown in table 4, the first transition canbe probed by the relative charge under TBC instead. Hence it is important to examine all theboundary conditions and find the minimal hc where the charge jumps. We plot the minimalhc for each L in figure 6.
The above discussion seems to suggest that igSPT is more stable than the gSPT. Let us
however make a cautionary remark. As observed in figure 6, the critical perturbation strength
hc depends on the system size L. At this point with small-scale ED study, we are unable toconclude whether the transition away from the igSPT at L → ∞ happens at immediately after
h = 0 or at a finite hc .
However, the analytical result by the Kennedy-Tasaki transformation shows that where hc
converges to a finite value in the thermodynamical limit [33]. It would also be interesting tostudy more sophisticated perturbation than (52) which can drive the system to the triviallygapped phase, and discuss the transition for small perturbation strength.
hc
1.4
1.2
1.0

{4, 1.01}{9, 0.86}

0.8{8, 0.7}

0.6{5, 0.5}{7, 0.44}

0.4{11, 0.28}

0.2
0.0

4

6

8

10

12

L

Figure 6: System size L dependence of the first transition out of the ZΓ4 igSPT.

36

SciPost Phys. 17, 013 (2024)

References[1] X. Chen, Z.-C. Gu and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82, 155138(2010), doi:10.1103/PhysRevB.82.155138.
[2] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006),
doi:10.1016/j.aop.2005.10.005.
[3] C. Xu and Y.-Z. You, Bosonic short-range entangled states beyond group cohomology classification, Phys. Rev. B 91, 054406 (2015), doi:10.1103/PhysRevB.91.054406.
[4] J. Wang, X.-G. Wen and E. Witten, A new SU(2) anomaly, J. Math. Phys. 60, 052301(2019), doi:10.1063/1.5082852.
[5] Y.-A. Chen and P.-S. Hsin, Exactly solvable lattice Hamiltonians and gravitational anomalies, SciPost Phys. 14, 089 (2023), doi:10.21468/SciPostPhys.14.5.089.
[6] S. Kravec, J. McGreevy and B. Swingle, All-fermion electrodynamics and fermion number
anomaly inflow, Phys. Rev. D 92, 085024 (2015), doi:10.1103/PhysRevD.92.085024.
[7] X. Chen, Y.-M. Lu and A. Vishwanath, Symmetry-protected topological phases from decorated domain walls, Nat. Commun. 5, 3507 (2014), doi:10.1038/ncomms4507.
[8] X. Yang, S. Jiang, A. Vishwanath and Y. Ran, Dyonic Lieb-Schultz-Mattis theorem andsymmetry protected topological phases in decorated dimer models, Phys. Rev. B 98, 125120(2018), doi:10.1103/PhysRevB.98.125120.
[9] Q.-R. Wang, S.-Q. Ning and M. Cheng, Domain wall decorations, anomalies and spectral
sequences in bosonic topological phases, (arXiv preprint) doi:10.48550/arXiv.2104.13233.
[10] A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404(2006), doi:10.1103/PhysRevLett.96.110404.
[11] M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys.
Rev. Lett. 96, 110405 (2006), doi:10.1103/physrevlett.96.110405.
[12] T. Grover, A. M. Turner and A. Vishwanath, Entanglement entropy of gapped
phases and topological order in three dimensions, Phys. Rev. B 84, 195120 (2011),
doi:10.1103/PhysRevB.84.195120.
[13] Y. Zheng, H. He, B. Bradlyn, J. Cano, T. Neupert and B. A. Bernevig, Structure of theentanglement entropy of (3 + 1)-dimensional gapped phases of matter, Phys. Rev. B 97,
195118 (2018), doi:10.1103/PhysRevB.97.195118.
[14] T. Scaffidi, D. E. Parker and R. Vasseur, Gapless symmetry-protected topological order, Phys.
Rev. X 7, 041048 (2017), doi:10.1103/PhysRevX.7.041048.
[15] R. Verresen, R. Thorngren, N. G. Jones and F. Pollmann, Gapless topologicalphases and symmetry-enriched quantum criticality, Phys. Rev. X 11, 041059 (2021),
doi:10.1103/PhysRevX.11.041059.
[16] R. Thorngren, A. Vishwanath and R. Verresen, Intrinsically gapless topological phases,
Phys. Rev. B 104, 075132 (2021), doi:10.1103/PhysRevB.104.075132.

37

SciPost Phys. 17, 013 (2024)[17] S. C. Furuya and M. Oshikawa, Symmetry protection of critical phases anda global anomaly in 1 + 1 dimensions, Phys. Rev. Lett. 118, 021601 (2017),
doi:10.1103/PhysRevLett.118.021601.
[18] Y. Yao, C.-T. Hsieh and M. Oshikawa, Anomaly matching and symmetry-protected criticalphases in SU(N) spin systems in 1 + 1 dimensions, Phys. Rev. Lett. 123, 180201 (2019),
doi:10.1103/PhysRevLett.123.180201.
[19] R. Verresen, J. Bibo and F. Pollmann, Quotient symmetry protected topological phenomena,(arXiv preprint) doi:10.48550/arXiv.2102.08967.
[20] A. Keselman and E. Berg, Gapless symmetry-protected topological phase of fermions in one
dimension, Phys. Rev. B 91, 235309 (2015), doi:10.1103/PhysRevB.91.235309.
[21] T. Grover and A. Vishwanath, Quantum criticality in topological insulators and superconductors: Emergence of strongly coupled Majoranas and supersymmetry, (arXiv preprint)
doi:10.48550/arXiv.1206.1332.
[22] W. Ji, S.-H. Shao and X.-G. Wen, Topological transition on the conformal manifold, Phys.
Rev. Res. 2, 033317 (2020), doi:10.1103/PhysRevResearch.2.033317.
[23] U. Borla, R. Verresen, J. Shah and S. Moroz, Gauging the Kitaev chain, SciPost Phys. 10,
148 (2021), doi:10.21468/SciPostPhys.10.6.148.
[24] X. Wang, L. Li and J. Wu, Stability and fine structure of symmetry-enriched quantum criticality in a spin ladder triangular model, (arXiv preprint) doi:10.48550/arXiv.2306.11446.
[25] Y. Hidaka, S. C. Furuya, A. Ueda and Y. Tada, Gapless symmetry-protected topological phaseof quantum antiferromagnets on anisotropic triangular strip, Phys. Rev. B 106, 144436(2022), doi:10.1103/PhysRevB.106.144436.
[26] X.-J. Yu, R.-Z. Huang, H.-H. Song, L. Xu, C. Ding and L. Zhang, Conformal boundary
conditions of symmetry-enriched quantum critical spin chains, Phys. Rev. Lett. 129, 210601(2022), doi:10.1103/PhysRevLett.129.210601.
[27] D. E. Parker, T. Scaffidi and R. Vasseur, Topological Luttinger liquids from decorated domain
walls, Phys. Rev. B 97, 165114 (2018), doi:10.1103/PhysRevB.97.165114.
[28] D. E. Parker, R. Vasseur and T. Scaffidi, Topologically protected long edge coherence times in symmetry-broken phases, Phys. Rev. Lett. 122, 240605 (2019),
doi:10.1103/PhysRevLett.122.240605.
[29] J. Zhao, Z. Yan, M. Cheng and Z. Y. Meng, Higher-form symmetry breaking at Ising transitions, Phys. Rev. Res. 3, 033024 (2021), doi:10.1103/PhysRevResearch.3.033024.
[30] H. Yang, L. Li, K. Okunishi and H. Katsura, Duality, criticality, anomaly,
and topology in quantum spin-1 chains, Phys. Rev. B 107, 125158 (2023),
doi:10.1103/PhysRevB.107.125158.
[31] R. Ma, L. Zou and C. Wang, Edge physics at the deconfined transition between aquantum spin Hall insulator and a superconductor, SciPost Phys. 12, 196 (2022),
doi:10.21468/SciPostPhys.12.6.196.
[32] R. Wen and A. C. Potter, Bulk-boundary correspondence for intrinsically gapless symmetryprotected topological phases from group cohomology, Phys. Rev. B 107, 245127 (2023),
doi:10.1103/PhysRevB.107.245127.
38

SciPost Phys. 17, 013 (2024)[33] L. Li, M. Oshikawa and Y. Zheng, Intrinsically/purely gapless-SPT from non-invertible duality transformations, (arXiv preprint) doi:10.48550/arXiv.2307.04788.
[34] J. Wang, X.-G. Wen and E. Witten, Symmetric gapped interfaces of SPT and SET states: Systematic constructions, Phys. Rev. X 8, 031048 (2018), doi:10.1103/PhysRevX.8.031048.
[35] L. Li, M. Oshikawa and Y. Zheng, Noninvertible duality transformation between symmetryprotected topological and spontaneous symmetry breaking phases, Phys. Rev. B 108,214429 (2023), doi:10.1103/PhysRevB.108.214429.
[36] X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders
and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013),
doi:10.1103/PhysRevB.87.155114.
[37] Q.-R. Wang, Y. Qi and Z.-C. Gu, Anomalous symmetry protected topologicalstates in interacting fermion systems, Phys. Rev. Lett. 123, 207003 (2019),
doi:10.1103/PhysRevLett.123.207003.
[38] E. Witten, The “parity” anomaly on an unorientable manifold, Phys. Rev. B 94, 195150(2016), doi:10.1103/PhysRevB.94.195150.
[39] X. Chen, Z.-C. Gu and X.-G. Wen, Classification of gapped symmetricphases in one-dimensional spin systems, Phys. Rev. B 83, 035107 (2011),
doi:10.1103/physrevb.83.035107.
[40] F. Pollmann, E. Berg, A. M. Turner and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012),
doi:10.1103/physrevb.85.075125.
[41] J. C. Wang, Z.-C. Gu and X.-G. Wen, Field-theory representation of gauge-gravity symmetryprotected topological invariants, group cohomology, and beyond, Phys. Rev. Lett. 114,
031601 (2015), doi:10.1103/PhysRevLett.114.031601.
[42] M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topologicalphases, Phys. Rev. B 86, 115109 (2012), doi:10.1103/PhysRevB.86.115109.
[43] Y. Yao and M. Oshikawa, Twisted boundary condition and Lieb-Schultz-Mattis
ingappability for discrete symmetries, Phys. Rev. Lett. 126, 217201 (2021),
doi:10.1103/PhysRevLett.126.217201.
[44] U. Borla, R. Verresen, J. Shah and S. Moroz, Gauging the Kitaev chain, SciPost Phys. 10,
148 (2021), doi:10.21468/SciPostPhys.10.6.148.
[45] M. Suzuki, Relationship among exactly soluble models of critical phenomena. I, Prog. Theor.
Phys. 46, 1337 (1971), doi:10.1143/PTP.46.1337.
[46] W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio and V. Vedral, Quantum phasetransition between cluster and antiferromagnetic states, Europhys. J. 95, 50001 (2011),
doi:10.1209/0295-5075/95/50001.
[47] B. Zeng, X. Chen, D.-L. Zhou and X.-G. Wen, Quantum information meets quantum matter,
Springer, New York, USA, ISBN 9781493990825 (2019), doi:10.1007/978-1-4939-90849.
[48] T. Kennedy and H. Tasaki, Hidden Z2 × Z2 symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B 45, 304 (1992), doi:10.1103/PhysRevB.45.304.
39

SciPost Phys. 17, 013 (2024)[49] T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in S = 1
quantum spin chains, Commun. Math. Phys. 147, 431 (1992), doi:10.1007/BF02097239.
[50] M. Oshikawa, Hidden Z2 × Z2 symmetry in quantum spin chains with arbitrary integer spin,
J. Phys.: Condens. Matter 4, 7469 (1992), doi:10.1088/0953-8984/4/36/019.
[51] X.-G. Wen, Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions, Phys. Rev. B 89, 035147 (2014),
doi:10.1103/PhysRevB.89.035147.
[52] L. H. Santos and J. Wang, Symmetry-protected many-body Aharonov-Bohm effect, Phys.
Rev. B 89, 195122 (2014), doi:10.1103/PhysRevB.89.195122.
[53] J. C. Wang and X.-G. Wen, Non-Abelian string and particle braiding in topological order:
Modular SL(3, Z) representation and (3 + 1)-dimensional twisted gauge theory, Phys. Rev.
B 91, 035134 (2015), doi:10.1103/PhysRevB.91.035134.
[54] C. Wang and M. Levin, Braiding statistics of loop excitations in three dimensions, Phys.
Rev. Lett. 113, 080403 (2014), doi:10.1103/PhysRevLett.113.080403.
[55] Z.-C. Gu, J. C. Wang and X.-G. Wen, Multikink topological terms and chargebinding domain-wall condensation induced symmetry-protected topological states:
Beyond Chern-Simons/BF field theories, Phys. Rev. B 93, 115136 (2016),
doi:10.1103/PhysRevB.93.115136.
[56] M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topologicalphases, Phys. Rev. B 86, 115109 (2012), doi:10.1103/PhysRevB.86.115109.
[57] C. Wang and M. Levin, Topological invariants for gauge theories and symmetry-protected
topological phases, Phys. Rev. B 91, 165119 (2015), doi:10.1103/PhysRevB.91.165119.
[58] E. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann.
Phys. 16, 407 (1961), doi:10.1016/0003-4916(61)90115-4.
[59] A. Seidel and D.-H. Lee, The Luther-Emery liquid: Spin gap and anomalous flux period,
Phys. Rev. B 71, 045113 (2005), doi:10.1103/PhysRevB.71.045113.
[60] T. T. Dumitrescu and P.-S. Hsin, Higgs-confinement transitions in QCD from symmetry protected topological phases, (arXiv preprint) doi:10.48550/arXiv.2312.16898.
[61] N. Seiberg, Thoughts about quantum field theory, in Strings 2019, Brussels, Belgium(2019).
[62] I. Affleck and E. H. Lieb, A proof of part of Haldane’s conjecture on spin chains, in Condensed
matter physics and exactly soluble models, Springer, Berlin, Heidelberg, Germany, ISBN9783642060939 (1986), doi:10.1007/978-3-662-06390-3_17.
[63] M. Oshikawa, M. Yamanaka and I. Affleck, Magnetization plateaus in spin
chains: “Haldane gap” for half-integer spins, Phys. Rev. Lett. 78, 1984 (1997),
doi:10.1103/PhysRevLett.78.1984.
[64] M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath and P. Bonderson, Translational symmetry and microscopic constraints on symmetry-enriched topological phases: A view fromthe surface, Phys. Rev. X 6, 041068 (2016), doi:10.1103/PhysRevX.6.041068.
[65] E. Brezin and J. Zinn-Justin (eds.), Fields, strings and critical phenomena, North Holland
Publishing, Amsterdam, Netherlands, ISBN 9780444884404 (1990).
40

SciPost Phys. 17, 013 (2024)[66] Y. Fuji, Effective field theory for one-dimensional valence-bond-solid phases and their symmetry protection, Phys. Rev. B 93, 104425 (2016), doi:10.1103/PhysRevB.93.104425.

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