Bosonic symmetry protected topological phases with reflection symmetry
Abstract
We study twodimensional bosonic symmetry protected topological (SPT) phases which are protected by reflection symmetry and local symmetry [, , U(1), or U(1)], in the search for twodimensional bosonic analogs of topological crystalline insulators in integer spin systems with reflection and spinrotation symmetries. To classify them, we employ a ChernSimons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by symmetry are classified as for even and 0 (no SPT phase) for odd while those protected by U(1) symmetry are . We point out that the twodimensional AffleckKennedyLiebTasaki state of spins on the square lattice is a SPT phase protected by reflection and rotation symmetries.
pacs:
71.10.w, 71.27.+a, 71.10.FdCurrent address: ]Department of Physics, University of California, Berkeley, CA 94720
I Introduction
Topological insulators and superconductors are gapped phases of noninteracting fermions in which the ground state wave functions have topologically nontrivial structures in the presence of certain symmetry constraints. The topological structures in the bulk wave functions guarantee the presence of gapless excitations at the boundary which are robust against any perturbation respecting the symmetry constraints. These gapless excitations lead to novel transport properties. For example, timereversalinvariant topological insulators in three dimensions are characterized by topological invariants in the bulk.Hasan and Kane (2010); Qi and Zhang (2011) Correspondingly, stable gapless Dirac fermions emerge at the surface, which lead to the topological magnetoelectric effect.Qi et al. (2008); Essin et al. (2009, 2010) Topological insulators and superconductors of noninteracting fermions have been classified, in terms of the timereversal, particlehole, and chiral symmetries, into the “periodic table.”Schnyder et al. (2008); Kitaev (2009); Ryu et al. (2010)
Interacting bosons can also host gapped phases with gapless boundary excitations that are stable against any perturbation as far as certain symmetries are preserved. These phases are dubbed bosonic symmetry protected topological (SPT) phases. One of representative examples in one dimension is the Haldane phase in the spin1 antiferromagnetic Heisenberg chain.Haldane (1983); Affleck et al. (1987) At each end of the Haldane spin chain emerge zeroenergy spin degrees of freedom which are protected by rotation about two orthogonal axes in the spin space or by the timereversal symmetry. Pollmann et al. (2010) Bosonic SPT phases in two dimensions are also studied in twocomponent bose gas with U(1) symmetry.Senthil and Levin (2013); Wu and Jain (2013); Furukawa and Ueda (2013); Regnault and Senthil (2013) The SPT phases with local symmetries have been classified with various methods including matrix product state representation,Pollmann et al. (2010); Turner et al. (2011); Fidkowski and Kitaev (2011) group cohomology,Chen et al. (2011, 2011, 2013); Gu and Wen (2014) ChernSimons theory,Neupert et al. (2011); Lu and Vishwanath (2012); Levin and Stern (2012); Hsieh et al. (2014a) nonlinear sigma models,You and Xu (2014) and cobordisms.Kapustin (2014a, b); Kapustin et al. (2014)
Recently, the concept of topological phases of noninteracting fermions has been further extended to topological crystalline insulators by including spatial symmetries.Fu (2011) Experimental realization of topological crystalline insulator with reflection symmetry is achieved in SnTe.Tanaka et al. (2012); Hsieh et al. (2012) Surfaces of this material have an even number of Dirac cones, and the strong index for topological insulators is trivial. Thus, the timereversal symmetry does not protect these Dirac cones. Instead, reflection symmetry is responsible for the symmetry protection of Dirac cones, allowing for a nontrivial mirror Chern number defined on the reflection invariant plane in the threedimensional Brillouin zone. Now, a natural question we may ask is whether analogs of topological crystalline insulators of fermions exist in bosonic SPT phases. The bosonic SPT phases protected by spatial symmetries have not been fully explored so far, while there are several recent attempts in terms of their classification.You and Xu (2014); Hsieh et al. (2014b, a); Ware et al. (2015); Hermele and Chen (2015) In particular, few realistic models are known for such twodimensional (2D) SPT phases with spatial symmetries.
Motivated by these, we explore topological crystalline insulators in interacting bosonic systems. Specifically, focusing on integerspin systems, we study 2D bosonic SPT phases protected by reflection symmetry as well as spinrotation symmetry. In order to classify SPT phases, we employ the ChernSimons approachLu and Vishwanath (2012) which is well suited for classifying 2D SPT phases with reflection symmetry. In this approach, SPT phases are classified by analyzing the stability of gapless edge modes against any perturbation allowed under given symmetry constraints. If all gapless modes are gapped out without symmetry breaking, the system is in a trivial phase; otherwise it is an SPT phase. We apply this method to integerspin systems protected by reflection and discrete spinrotation symmetry (). Our classification results show that SPT phases form group for even , while no SPT phase is allowed for odd . Furthermore, the AffleckKennedyLiebTasaki (AKLT) state of spins on the square lattice is shown to be a bosonic SPT phase characterized by a invariant. This can be regarded as a topological crystalline insulator in spin systems.
The rest of this paper is organized as follows. Section II summarizes the classification scheme based on the ChernSimons approach. Our main results are presented in Sec. III. We focus on SPT phases protected by symmetry and discuss the AKLT state on the square lattice. Classification of 2D bosonic SPT phases under other related symmetries [, U(1), and U(1)] is discussed in Appendix.
Ii A brief review of the ChernSimons approach
In this section, we briefly review the classification scheme of 2D bosonic SPT phases using the ChernSimons approach.Lu and Vishwanath (2012) In this scheme, SPT phases are classified through stability analysis of gapless edge states. The group structure of SPT phases is obtained by studying the equivalence class of a direct sum of two phases.
ii.1 Classification scheme
In this paper we consider a class of 2D bosonic SPT phases with nonchiral gapless edge modes. We assume that SPT phases have pairs of helical edge states and that their bulk lowenergy effective theory is given by the ChernSimons action
(1a)  
(1b) 
where is the totally antisymmetric LeviCivita tensor, , , and summation is assumed over repeated indices and (we adopt this convention throughout this paper). The ChernSimons gauge fields () describe lowenergy dynamics in the gapped phase, and is a symmetric matrix in with . In bosonic systems every diagonal element of matrices is an even integer.
The gapless boundary modes along the boundary (say, at ) of the SPT phases are described by the boundary action
(2a)  
where are scalar bosonic fields satisfying the equaltime commutation relation
(3) 
The matrix in Eq. (LABEL:eq:_L^0_edge) is a nonuniversal positive definite matrix. The operator and the vertex operator are the density and the creation operators of excitations in the th edge mode. Here colons denote normal ordering.
We assume that the boundary action and the bulk action are invariant under a symmetry group . For the boundary action this means that
(4) 
for any which induces linear transformation of the bosonic fields
(5) 
where is a constant dimensional vector with () and the matrix . The transformation (5) for reflection and other local transformations will be discussed in the following section.
A ground state with gapped excitations in the bulk is in an SPT phase protected by a symmetry group , if it has gapless edge states that cannot be gapped without symmetry breaking by any perturbation allowed by the symmetry . Thus 2D SPT phases are classified by examining the stability of their edge states against perturbations of the form
(6) 
where and are real constants, and is a set of linearly independent vectors from . The cosine terms in Eq. (6) are normalordered as indicated by the colons. However, we will omit colons for normalordered vertex operators in the rest of this paper to simplify the notation.
The perturbations in are assumed to fulfill the following conditions. First, any pair of vectors, , from the set satisfies the Haldane’s null vector condition,Haldane (1995)
(7) 
so that the linearly independent combinations of the bosonic fields () can be simultaneously pinned by the cosine potentials in Eq. (6).
Second, must be invariant under the symmetry group ,
(8) 
Since , the invariance of imposes the condition
(9) 
The third condition is concerned with the absence of spontaneous symmetry breaking. The symmetry can be spontaneously broken in the ground state when linearly independent fields () are pinned, even if the interactions in respect the symmetry. To see this, let us define from the vectors a set of vectors
(10) 
Any field () takes a constant expectation value in the ground state. We then define from another set of vectors
(11) 
where gcd denotes the greatest common divisor of the integers in the parentheses. Note that . The set is a Bravais lattice, whose primitive lattice vectors are denoted by . The elementary bosonic fields
(12) 
take constant expectation values in the ground state. If the expectation values are invariant, i.e.,
(13) 
modulo for any , then the edge modes can be gapped without any symmetry breaking. Otherwise, the ground state breaks the symmetry spontaneously.
If there exists a set of vectors that satisfies the above three conditions, then the edge states can be gapped out without symmetry breaking [with strong enough even when is an irrelevant operator in the renormalizationgroup sense], and the resulting gapped phase is a trivial phase. On the other hand, if we cannot find such a set of vectors , then the edge states are stable, and the bulk ground state realizes a 2D SPT phase.
SPT phases form an Abelian group as follows.Lu and Vishwanath (2012) Elements of the Abelian group are equivalence classes of phases that are connected without a closing of a gap under adiabatic deformation of the action (Hamiltonian) while preserving the symmetry . The SPT phases described by the action (2) and the transformation (5) under the symmetry group are denoted by , while a trivial phase is denoted by “”. The summation of two phases is defined as a direct sum of the two phases,
(14) 
The bosonic fields in the direct sum of two phases, , are transformed by as
(15) 
with . The inverse element of an SPT phase is found from the relation
(16) 
which is understood by noting that the fields can be pinned without symmetry breaking by the pinning potential
(17) 
Thus, the equivalence relation between and is identical to the equivalence relation between and a trivial phase “0”.
The addition rule of SPT phases allows us to construct SPT phases of a larger number of degrees of freedom from small building blocks. Following Lu and Vishwanath,Lu and Vishwanath (2012) we consider a minimal model of SPT phases with a pair of helical edge states, described by the edge theory of two bosonic fields () with a matrix of . SPT phases with multiple pairs of helical edge states are obtained by combining minimal SPT phases, i.e., taking a direct sum of minimal models. Thus, we take
(18) 
unless otherwise noted, since matrices for bosonic systems can be reduced to . Lu and Vishwanath (2012)
Finally we note that there is a redundancy in the representation in Eq. (5). The action is unchanged by substituting the fields with and satisfying . Two representations and are therefore equivalent if they satisfy
(19) 
Iii SPT phases protected by reflection symmetry and symmetry
In this section, we classify bosonic SPT phases protected by reflection symmetry and discrete local symmetry . There are two possible group structures for these symmetries: (i) and (ii) . When the symmetry corresponds to rotation in integerspin systems, these two group structures are realized (a) when the spinrotation axis is parallel to the reflection plane and (b) when the spinrotation axis is perpendicular to the reflection plane, respectively (see Fig. 1). In this section we focus on the symmetry and show that an example of the SPT phases protected by this symmetry is given by the AffleckKennedyLiebTasaki (AKLT) state on the square lattice. The classification of SPT phases protected by (ii) symmetry, (iii) U(1) symmetry, and (iv) U(1) symmetry is discussed in Appendix. The results of the classification are summarized in Table 1.
Symmetry  Parity of  Abelian group of SPT phases 

Even  
Odd  
Even  
Odd  
–  
– 
iii.1
In this subsection, we apply the ChernSimons approach to 2D bosonic SPT phases protected by the symmetry group .
First, we determine the transformation laws [Eq. (5)] of bosonic fields ’s under and . The invariance of the Lagrangian under the reflection ,
(20) 
is guaranteed if ’s obey the transformation
(21) 
where the matrix satisfies
(22) 
This condition is satisfied by for . Since the two representations and are related by the transformation with [Eq. (19)], it suffices to take the representation
(23) 
Similarly, the Lagrangian is invariant if the bosonic fields ’s are transformed by as
(24) 
with a matrix satisfying
(25) 
Any choice from fulfills this condition. However, we discard the representations , since they are not compatible with Eq. (9) for . Here we take the representation
(26) 
since it is realized in spin models in which the symmetry corresponds to spinrotation symmetry (see Sec. III.2). We do not consider the other case in this paper. Incidentally, this case with can be relevant to bosonic systems with charge conjugation symmetry, where creation operators of quasiparticles are transformed to annihilation operators.
The representation of symmetry operation is constrained by the group structure of the symmetry group . For the symmetry group , the generators of the group satisfy the relation , where denotes the identity element of . Accordingly, the representation must satisfy the conditions
(27)  
(28) 
where is a unit matrix. Furthermore, the algebraic relation obeyed by the generators of the symmetry group leads to the additional condition
(29) 
In the following we discuss cases where is even and odd separately.
iii.1.1 Even
Given the representation
(30) 
we deduce from Eqs. (27)–(29) the transformation laws for bosonic fields
(31a)  
(31b)  
where  
(31c) 
In Eq. (31b) the phase shift caused by the reflection is set equal to zero by the basis transformation in Eq. (19) with and chosen appropriately.
Let us label by a set of integers a topological phase in which bosonic fields are transformed as in Eqs. (31). We will show that the SPT phases form an Abelian group by proving the following three properties:

Phases and are trivial ().

Any phase is generated from and , which satisfy .

The two phases and are independent generators of SPT phases.
Proof of (a): The null vector condition [Eq. (7)] with allows only pinning potentials of the form or with and . When , the pinning potential
(32) 
is invariant under the transformations in Eq. (31) and can pin the field at or depending on the sign of . No symmetry is broken by the pinning. Thus, the phase is reduced to a trivial insulator. When , the pinning potential
(33) 
is invariant under the transformations in Eq. (31) and can pin the field at or without symmetry breaking. Thus, the phase is a trivial insulator.
Proof of (b): We first show the following addition relations of SPT phases:
(34a)  
(34b) 
The composition of two phases and has bosonic fields and a matrix . The fields obey the commutation relations
(35) 
and the transformation laws
(36a)  
(36b)  
with  
(36c) 
Here, () denotes the th unit vector, . We now make a basis transformation and define a new set of bosonic fields
(37) 
which have the same matrix and commutators
(38) 
Without pinning potentials, there are two pairs of gapless helical edge modes: and . A potential of the form
(39) 
can pin the field and gap out the sector without symmetry breaking. The helical edge states in the sector remains gapless and correspond to the phase . Equation (34a) follows.
In a similar way, we obtain Eq. (34b) by making basis transformation
(40) 
and adding a potential of the form
(41) 
In this case the sector is a trivial gapped state and can be discarded. The edge states in the remaining sector corresponds to the phase , and thus we obtain Eq. (34b).
We find from Eqs. (34) that
(42a)  
(42b) 
since phase shifts are defined modulo . Furthermore, using Eqs. (34) successively, we can reduce any phase to four phases:
(43) 
where “e” and “o” stand for “even” and “odd”, respectively.
Proof of (c): We show that the two phases and are neither equivalent to each other nor connected to the trivial phase 0. To this end, we show that edge modes of the phase with cannot be gapped out, unless or . It follows from Eq. (16) that the phase has edge modes described by the bosonic fields with a matrix . The bosonic fields and obey the transformation laws of and , respectively.
Gapping the bosonic fields requires two pinning potentials and , whose integer vectors and must satisfy Eqs. (7) and (9), or equivalently,
(44a)  
(44b) 
for . Solutions to these equations are given by
(45a)  
(45b) 
with . ^{1}^{1}1 This is obtained as follows. The generic solution to the equation with is given by where . Thus the two linearly independent vectors and satisfying Eq. (44a) have the form where , , , and are integers with . Here we have two possible choices for the vectors and : or It is straightforward to check that the latter set of vectors is compatible with the condition posed by the reflection symmetry [Eq. (44b)], while the former is not. If and , then the elementary bosonic fields defined in Eq. (12) are given by or . If , we can assume and obtain the elementary bosonic fields and . In either case the fields are transformed as
(46a)  
(46b)  
(46c)  
(46d) 
where we assume or if . When , the phase shifts in Eqs. (46a), (46c), and (46d) are equal to zero (mod ) only when . This means that the edge modes cannot be gapped out without symmetry breaking unless . Similarly, when , the edge modes can be gapped out without symmetry breaking only if , i.e., . Thus, the two phases and are inequivalent, and both of them are distinct from the trivial phase.
From (a), (b), and (c), we conclude that the Abelian group of the SPT phases protected by is generated by and .
Finally, we note that the SPT phase is stable even in the absence of the reflection symmetry, while the other two SPT phases, and , are SPT phases that are stable only in the presence of both the reflection symmetry and the symmetry.
iii.1.2 Odd
When is odd, the bosonic fields are transformed under symmetry operations as
(47a)  
(47b)  
with  
(47c) 
We note that the above transformation rules are obtained from Eqs. (31) by setting . The vanishing phase shift of under the transformation is a consequence of the conditions (28) and (29) for odd .
There is no SPT phase when is odd. This conclusion is obtained by using the classification for even discussed above. Let us label phases by a set of integers . Imposing in Eq. (43), we find
(48) 
Next we prove that two phases and are equivalent by showing that the edge modes of the phase can be gapped without symmetry breaking (i.e., ). The phase has edge modes described by bosonic fields with a matrix . The bosonic fields and obey the transformation laws of and , respectively. These fields are gapped by the pinning potential of the form
(49) 
where and . These integer vectors and satisfy the conditions in Eq. (44). The corresponding elementary bosonic variables,