Bosonic symmetry protected topological phases with reflection symmetry
We study two-dimensional bosonic symmetry protected topological (SPT) phases which are protected by reflection symmetry and local symmetry [, , U(1), or U(1)], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer- spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons 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 two-dimensional Affleck-Kennedy-Lieb-Tasaki 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.Fd
Current address: ]Department of Physics, University of California, Berkeley, CA 94720
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, time-reversal-invariant 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 time-reversal, particle-hole, 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 spin-1 antiferromagnetic Heisenberg chain.Haldane (1983); Affleck et al. (1987) At each end of the Haldane spin chain emerge zero-energy spin- degrees of freedom which are protected by -rotation about two orthogonal axes in the spin space or by the time-reversal symmetry. Pollmann et al. (2010) Bosonic SPT phases in two dimensions are also studied in two-component 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) Chern-Simons 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 time-reversal 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 three-dimensional 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 two-dimensional (2D) SPT phases with spatial symmetries.
Motivated by these, we explore topological crystalline insulators in interacting bosonic systems. Specifically, focusing on integer-spin systems, we study 2D bosonic SPT phases protected by reflection symmetry as well as spin-rotation symmetry. In order to classify SPT phases, we employ the Chern-Simons 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 integer-spin systems protected by reflection and discrete spin-rotation symmetry (). Our classification results show that SPT phases form group for even , while no SPT phase is allowed for odd . Furthermore, the Affleck-Kennedy-Lieb-Tasaki (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 Chern-Simons 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 Chern-Simons approach
In this section, we briefly review the classification scheme of 2D bosonic SPT phases using the Chern-Simons 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 low-energy effective theory is given by the Chern-Simons action
where is the totally anti-symmetric Levi-Civita tensor, , , and summation is assumed over repeated indices and (we adopt this convention throughout this paper). The Chern-Simons gauge fields () describe low-energy 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
where are scalar bosonic fields satisfying the equal-time commutation relation
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
for any which induces linear transformation of the bosonic fields
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
where and are real constants, and is a set of linearly independent vectors from . The cosine terms in Eq. (6) are normal-ordered as indicated by the colons. However, we will omit colons for normal-ordered 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)
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 ,
Since , the invariance of imposes the condition
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
Any field () takes a constant expectation value in the ground state. We then define from another set of vectors
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
take constant expectation values in the ground state. If the expectation values are invariant, i.e.,
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 renormalization-group 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,
The bosonic fields in the direct sum of two phases, , are transformed by as
with . The inverse element of an SPT phase is found from the relation
which is understood by noting that the fields can be pinned without symmetry breaking by the pinning potential
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
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
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 integer-spin systems, these two group structures are realized (a) when the spin-rotation axis is parallel to the reflection plane and (b) when the spin-rotation 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 Affleck-Kennedy-Lieb-Tasaki (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|
In this subsection, we apply the Chern-Simons 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 ,
is guaranteed if ’s obey the transformation
where the matrix satisfies
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
Similarly, the Lagrangian is invariant if the bosonic fields ’s are transformed by as
with a matrix satisfying
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
since it is realized in spin models in which the symmetry corresponds to spin-rotation 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
where is a unit matrix. Furthermore, the algebraic relation obeyed by the generators of the symmetry group leads to the additional condition
In the following we discuss cases where is even and odd separately.
Given the representation
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
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
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:
The composition of two phases and has bosonic fields and a -matrix . The fields obey the commutation relations
and the transformation laws
Here, () denotes the th unit vector, . We now make a basis transformation and define a new set of bosonic fields
which have the same -matrix and commutators
Without pinning potentials, there are two pairs of gapless helical edge modes: and . A potential of the form
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
and adding a potential of the form
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
since phase shifts are defined modulo . Furthermore, using Eqs. (34) successively, we can reduce any phase to four phases:
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.
for . Solutions to these equations are given by
with . 111 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
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.
When is odd, the bosonic fields are transformed under symmetry operations as
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
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
where and . These integer vectors and satisfy the conditions in Eq. (44). The corresponding elementary bosonic variables,