Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
Abstract
Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and nonAbelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix , whose entries are the fusionspace dimensions , to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2manifolds with gapped domain walls, including closed 2manifolds and open 2manifolds with gapped boundaries.
pacs:
05.30.Pr, 11.25.Hf, 71.10.Pm, 11.15.YcIntroduction – Insulator has a finite energy gap, which is rather trivial at low energy. Nonetheless, domain walls, separating different symmetrybreaking insulating regions, can enrich the physics of a trivial insulator, such as some paramagnet Ashcroft and Mermin (1976). Topological order Wen (1989); Wen and Niu (1990); Wen (1990), on the other hand, as a new kind of manybody quantum ordering, has a gapped bulk with exotic properties: some have (i) gapless edge modes, (ii) anyonic excitations with fractional or nonAbelian statistics Nayak et al. (2008), such as fractional quantum Hall states, and (iii) longrange entanglement Kitaev and Preskill (2006); Levin and Wen (2006); Chen et al. (2010). In this Letter, we would like to investigate the gapped domain walls of topological orders, and how gapped domain walls further enrich their physics.
It was conjectured that the 2+1D topological orders are completely classified by the gauge connection on the moduli space of the degenerate ground states Wen (1990); KeskiVakkuri and Wen (1993). The nonAbelian part of the gauge connection is the nonAbelian geometric phase Wilczek and Zee (1984) characterized by the matrices, which also encode the anyon statistics. The Abelian part is related to the gravitational ChernSimons term in the effective theory and is described by the chiral central charge of the edge state. Nonzero implies robust gapless edge modes.
By now we understand how to label a 2D
topological order
by a set of “topological order parameters” (),
analogous to “symmetrybreaking order parameters” for spontaneous symmetry breaking systems Ginzburg and Landau (1950); Landau and Lifschitz (1958).
However, it is less known how different topological orders are related.
To this end, it is important to investigate the following circumstance: there are several
domains in the system and each domain contains a topological order, while
the whole system is gapped. In this case, different topological orders are
connected by gapped domain walls. Our work addresses two primary questions:
(Q1) “Under what criteria can two topological orders be
connected by a gapped domain wall, and how many different types of gapped domain walls are there?”
Since a gapped boundary is a
gapped domain wall between a nontrivial topological order and the vacuum,
we also address that
“under what criteria can topological orders allow gapped boundaries?”
(Q2) “When a topologically ordered system has a gapped bulk, gapped domain walls and gapped boundaries, how to calculate its ground state degeneracy (GSD) Wen (1989); Wen and Niu (1990); Wang and Wen (2012); Kapustin (2014); Wang and Levin (2013), on any orientable manifold?”
Main result – Consider two topological orders, Phases and , described by and . Suppose there are and types of anyons in Phase and Phase , then the ranks of their modular matrices are and respectively. If and are connected by a gapped domain wall, firstly their central charges must be the same . Next, we find that the domain wall can be labeled by a tunneling matrix whose entries are fusionspace dimensions satisfying the commuting condition (2), and the stable condition (3):
(1)  
(2)  
(3) 
denotes the set of nonnegative integers. and are anyon indices for Phases . and are fusion tensors Verlinde (1988); Nayak et al. (2008) of Phases .
(1)(2)(3) is a set of necessary conditions a gapped domain wall must satisfy, i.e., if there is no nonzero solution of , the domain wall must be gapless. We conjecture that they are also sufficient for a gapped domain wall to exist. In the examples studied in Supplemental Material, are in onetoone correspondence with gapped domain walls. However, for some complicated examples Davydov (2014), a matrix may correspond to more than one type of gapped domain wall. This indicates that some additional data are needed to completely classify gapped domain walls.
As a first application of our result, we give a general method to compute the GSD in the presence of gapped domain walls on any orientable 2D surface. A simple case is the GSD on a disk drilled with two holes (equivalently a sphere with 3 circular boundaries, see Fig. 3(c)). The gapped boundaries are labeled by three vectors (onerow or onecolumn matrices) . The GSD is .
For gapped boundaries, our criteria can be understood via dimension reduction, i.e., shrinking a 1D gapped boundary to a (composite ^{1}^{1}1The concepts of trapping anyons, composite anyon types and fusion spaces are discussed in Lan and Wen (2014)) anyon . If the system is on a 2D surface drilled with gapped boundaries , then the GSD is the dimension of the fusion space Note1 with anyons ,
Since gapped domain walls talk to each other through longrange entanglement, the GSD with domain walls reveals more physics than that without domain walls. We foresee its practicality in experiments, since we can read even more physics by putting the system on open surfaces with gapped domain walls. Below we shall properly introduce and matrices.
Modular matrices – and are unitary matrices indexed by anyon types . 1 labels the trivial anyon type. The antiquasiparticle of is denoted by .
describes the self statistics. It is diagonal , where is the phase factor when exchanging two anyons . For the trivial type, . describes the mutual statistics. is the amplitude of the following process with proper normalization factors: first create a pair of and a pair of , then braid around , and finally annihilate the two pairs. is symmetric, . If , the process is just creation and annihilation, and . and form a projective representation of the modular group: , where denotes the identity matrix.
The antiquasiparticle can be read from , . The fusion tensor can be calculated via the Verlinde formula Verlinde (1988):
(4) 
Gapped domain walls – Below we demonstrate the physical meanings of the gapped domain wall conditions (1)(2)(3). First we put Phase and Phase on a sphere , separated by a gapped domain wall. Note that there can be many types of domain walls separating the same pair of phases and . What data characterize those different types of domain walls? We fix the domain wall type, labeled by , and trap Note1 an anyon in Phase , an anyon in Phase and. This configuration is denoted by . The states with such a configuration may be degenerate and the degenerate subspace is the fusion space . Here we propose using the fusionspace dimensions to characterize the gapped domain wall .
There are nonlocal operators that create a pair in Phase , and then tunnel through the domain wall to an anyon in Phase , , where is the ground state. Since we care about the fusion states rather than the operators themselves, we would take the equivalent class . We call as tunneling channels, which correspond to fusion states in . Therefore, the fusion space dimension is the number of linearly independent tunneling channels. So, we also refer to as the “tunneling matrix.”
The commuting condition (2) dictates the consistency of anyon statistics in presence of gapped domain walls. Since modular matrices encode the anyon statistics, we require that should commute with them as (2): , .
We may as well create a pair in Phase and tunnel to . describes such tunneling in the opposite direction (i.e., ). and contains the same physical data. To be consistent, tunneling to should give the same fusionspace dimension, . This is guaranteed by and .
The fusion spaces with four anyons further provide us consistence conditions of . To see this, first notice that there are generalised tunneling channels, , which, in addition to tunneling to , also create quasiparticle on the domain wall. If we combine the tunneling channels and , we can create fusion states with a domain wall and four anyons , as Fig. 1(a). In other words, form a basis of the fusion space . Let denote the number of tunneling channels , and we know that . However, the tunneling process as Fig. 1(b), i.e., fusing to , using to tunnel to and splitting to , forms another basis of the fusion space. The number of such fusion/tunneling/splitting channels is . Therefore, we must have
(5) 
We are interested in classifying stable gapped domain walls, i.e., the GSD cannot be reduced no matter what small perturbations are added near the domain wall. For stable gapped domain walls we have . Unstable gapped domain walls split as the sum of stable ones , and , for .
Now, if a gapped domain wall is stable, (5) becomes . We know that (3) is necessary for a gapped domain wall to be stable. Furthermore, setting we know that and (2) requires that , thus and cannot be the sum of more than one stable tunneling matrix; it must be stable itself. Therefore, (3) with (2) is also sufficient for a gapped domain wall to be stable.
Stability of composite domain walls – Let us consider two stable domain walls, between Phases and , and between Phases and , as in Fig. 1(c). When the two domain walls are far separated, they are both stable. Any small perturbations added near , or near , cannot reduce the GSD.
We then shrink the size of the middle Phase , such that the two domain walls are near enough to be regarded as a single domain wall. This way we obtain a composite domain wall, whose tunneling matrix is the composition , as Fig. 1(d). However, this composite domain wall may no longer be stable. Unless Phase is vacuum, we allow more perturbations to than when and are far separated. Some operators simultaneously acting on both and may reduce the GSD, in which case, the composite domain wall is not stable.
In the special case when Phase is vacuum, the composite remains stable. One can explicitly check this with (3).
GSD in the presence of gapped domain walls –
Below we derive the GSD, for a 2D system containing several topological orders separated by looplike gapped domain walls. Domain walls cut a whole 2D system into several segments. Without losing generality, let us consider an example in Fig. 2 with topological orders, Phases , and four nontrivial domain walls, , on a manifold Fig. 2(e). We first add extra trivial domain walls , so that all segments between domain walls are reduced to simpler topologies: caps, cylinders or pants. We also add oriented skeletons to the manifold, and put anyon indices on both sides of the domain walls, shown in Fig. 2(e). Next, see Fig. 2(a)(b)(c)(d), for the segments with oriented skeletons and anyon indices, we associate certain tensors: caps with , cylinders with , pants with in the corresponding topological order, and domain walls with their tunneling matrices . We may reverse the orientation and at the same time replace the index with . Finally, we multiply these tensors together and contract all the anyon indices. Physically, such tensor contraction computes the total number of winding channels of anyons, which exactly counts the number of ground states, thus the GSD.
Systems with gapped boundaries are included in our method; just imagine that there are vacuum on caps connected to the boundaries, e.g., Phases in Fig. 2(e) can be vacuum. Dimensions of generic fusion spaces can also be calculated, by putting the anyon on the cap and associating the tensor instead of .
We derive GSD on exemplary manifolds:
We apply our formalism to several topological orders. Details of our examples are organized in Supplemental Material. Part of our result is listed in Table 1 (the number of gapped domain walls types) and Table 2 (GSD).
GSD(# punctures)  1  2  3  4 

toric code  1  1, 2  2, 4  2, 4, 8 
doublesemion  1  2  4  8 
doubled Fibonacci  1  2  5  15 
doubled Ising  1  3  10  36 
Conclusion – Given matrices of topological orders with the same central charge, we have provided simple criteria (1)(2)(3) to check the existence of gapped domain walls. We want to mention that, a gapped domain wall can be related to a gapped boundary by the folding trick Kitaev and Kong (2012). By studying gapped boundaries, we can also obtain all the information of gapped domain walls. But, to compute the GSD, gapped domain walls allow more configurations on 2D surfaces than gapped boundaries.
The gapped domain walls and boundaries can be explicitly realized in lattice models Beigi et al. (2011); Kitaev and Kong (2012); Lan and Wen (2014). LevinWen stringnet models Levin and Wen (2005) are exactly solvable models for topological orders. Recently it was found that a topological order can be realized by a LevinWen model iff it has gapped boundaries Kitaev and Kong (2012); Lan and Wen (2014). Thus, our work provides the criteria whether a topological order has a LevinWen realization.
2D Abelian topological orders can be described by ChernSimons field theories. The boundary of a ChernSimons theory is gappable, iff there exists a Lagrangian subgroup Kapustin and Saulina (2011); Wang and Wen (2012); Levin (2013); Barkeshli et al. (2013a, b); Kapustin (2014). Our tunneling matrix criteria (1)(2)(3) are equivalent to the Lagrangian subgroup criteria for Abelian topological orders (a detailed proof is given in Supplemental Material), but are more general and also apply to nonAbelian topological orders.
One can also use the anyon condensation approach Bais and Slingerland (2009); Bais et al. (2009); Fuchs et al. (2013); Kong (2014); Eliëns et al. (2013); Hung and Wan (2013, 2014) to determine the gapped boundaries of (nonAbelian) topological orders, by searching for the Lagrangian condensable anyons (mathematically, Lagrangian algebras Fuchs et al. (2013); Kong (2014)), whose condensation will break the topological order to vacuum. However, we use only an integer vector to determine the anyon , while in the anyon condensation approach, besides the multiplicity , there are many additional data satisfying a series of formulas. These formulas put certain constraints on the condensable anyon, but not in a simple and explicit manner. Our claim that (1)(2)(3) are necessary and sufficient for a gapped domain wall to exist means that, Lagrangian condensable anyons must satisfy (1)(2)(3), and, for the anyon satisfying (1)(2)(3), there must exist solutions to the additional data in the anyon condensation approach.
We know that the effective 1+1D edge theory of a 2+1D topological order has a gravitational anomaly. The gravitational anomalies are classified by the bulk topological order Wen (2013); Kong and Wen (2014). When , the edge effective theory has a perturbative gravitational anomaly which leads to topological gapless edge (i.e., the gaplessness of the edge is robust against any change of the edge Hamiltonian). Even in the absence of perturbative gravitational anomaly, , certain global gravitational anomalies Witten (1985) (characterized by ) can also lead to topological gapless edge Wang and Wen (2012); Levin (2013). Our work points out that such global gravitational anomalies are described by which do not allow any nonzero solution of (1)(2)(3). The corresponding 2D topological order will have topological gapless edge.
Since a domain wall sits on the border between two topological orders, our study on domain walls can also guide us to better understand the phase transitions of topological orders.
Acknowledgements.
Acknowledgements – After posting the arXiv preprint version 1, we are grateful receiving very helpful comments from Liang Kong, John Preskill, Anton Kapustin and Yidun Wan. This research is supported by NSF Grant No. DMR1005541, NSFC 11074140, NSFC 11274192, the BMO Financial Group and the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research. Note added – During the preparation of this manuscript, we become aware that a recent work Ref. Hung and Wan (2014) has independently obtained part of our results using a different approach: anyon condensation. The comparison between our new approach and anyon condensation is explained in Conclusion.References
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Supplemental Material
Appendix A Sec. 1. Equivalence between tunneling matrices and Lagrangian subgroups for Abelian topological orders
If the fusion of anyons has a group structure, , the topological order is called Abelian. We denote the fusion group by . Since is an Abelian group, we label the trivial type by instead of . The rank of matrices is , and , . We rescale the matrix as
The Verlinde formula then implies
which means that for each , form a 1D linear representation of the fusion group . matrix is the character table of .
The gapped boundaries of Abelian topological orders
are classified by Lagrangian subgroups.
We introduce a physical definition of Lagrangian subgroup,
in terms of anyon statistics ( matrices). Levin (2013)
A Lagrangian subgroup is a subset of anyons, , such that
(i) If , then .
(ii) If , then .
(iii) If , then .
(iii) is equivalent to
(iii)’ If , then .
Note that (ii) and (iii)’ implies that is a subgroup.
First, , thus, the identity is in .
Second, if and , then .
Thus, and .
Finally, so if we also have .
Next, we want to show that, for gapped boundaries of Abelian topological orders, Lagrangian subgroups are in onetoone correspondence with tunneling matrices. First consider the stable condition. It reduces to . (We omit the anyon index of the vacuum.) In particular, , which implies that . Thus, we relate a Lagrangian subgroup and a tunneling matrix via
It is easy to see (i) is equivalent to . We will focus on the proof of (ii)(iii) .
(ii)(iii) is easier. Consider the first row
We have and
Now, if , i.e., , in the above equation must all be 1. (Note that are all phase factors ). This is (ii). If , i.e., , there must be at least one in the above equation. This is (iii).
The other direction (ii)(iii) is a bit involving. First, note the following relation
This motivates us to consider the quotient group . Each element in gives rise to a 1D representation of , i.e., . Different elements in gives different 1D representations. Since the Abelian group has in total different 1D representations, we have . On the other hand, each gives rise to a 1D representation of , i.e., . Since the matrix is invertible, different gives different 1D representations of . Again, has in total different 1D representations. We also have . Thus, we know that , i.e., .
Now, if , we have
If , then . In other words, is a nontrivial 1D representation of , and we know that . Thus,
We have proved that (ii)(iii) .
To conclude, for gapped boundaries (and gapped domain walls by the folding trick) of Abelian topological orders, our tunneling matrix criteria is equivalent to the Lagrangian subgroup criteria.
Appendix B Sec. 2. Examples
We provide explicit data of gapped boundaries and gapped domain walls of 2D topological orders, computed by our formalism developed in the main text.
A list of topological orders we consider contains (with their notations of twisted quantum double model
for a gauge group with a 3cocycle twist , and implies the number of pairs of in its matrix.):
(i). toric code (),
(ii). doublesemion (),
(iii). doubled Fibonacci phase (Fibonacci ),
(iv). doubled Ising phase (Ising ),
(v). as the quantum doubled model of the permutation group of order 6,
(vi). as the quantum doubled model of the dihedral group of order 8,
(vii). as the quantum doubled model of the quaternion group of order 8,
(viii). as a twisted quantum doubled model of the group of order 8 with a 3cocylce twist ,
(ix). as a twisted quantum doubled model of the group of order 8 with a 3cocylce twist ,
(x). as a twisted quantum doubled model of the group of order 8 with a 3cocylce twist .
One may refer to Refs. de Wild Propitius (1995); Hu et al. (2013) for an introduction to twisted quantum double models.
Here means a 3cocycle whose model generates 3 pairs of in its matrix and their generators are linear independent (). means a 3cocycle whose model generates 3 pairs of in its matrix and their generators are linear dependent (). More detail are explained in Ref. Wang and Wen (2014) and reference therein.
Below we will provide matrices, tunneling matrices of gapped boundaries and gapped domain walls of these topological orders (i)(x). We will count the number of types of gapped boundaries and gapped domain walls. We will also count some examples of their ground state degeneracy (GSD) on various manifolds with gapped boundaries on the punctures.
The matrices of all five kinds of nonAbelian twisted quantum double models are explicitly adopted from the calculation of Ref. Wang and Wen (2014).
b.1 I. Gapped boundaries of toric code phase: 2 types
The matrices of toric code phase are:
There are two types of gapped boundaries:
Conventionally, we label the 4 types of anyons (quasiparticles) as . The boundary corresponds to condensing and corresponds to condensing .
We compute GSD on a cylinder with two gapped boundaries. Note that the GSD is also the number of types of 0D defects between the two gapped boundaries. In particular, it is the number of boundary quasiparticle types if the two gapped boundaries are the same.
This agrees with Wang and Wen (2012); Bravyi and Kitaev (1998).
b.2 II. Gapped domain walls between two toric codes: 6 types
b.3 III. Gapped boundary of doublesemion phase: 1 type
The matrices of doublesemion phase are:
There is only one type of gapped boundary,
Since there is only one gapped boundary type, we would like to compute the GSD on spheres with more punctures:

2 punctures (a cylinder): GSD=2,

3 punctures (a pair of pants): GSD=4,

4 punctures: GSD=8,

5 punctures: GSD=16,

punctures: GSD=.
b.4 IV. Gapped domain walls between doublesemion and toric code phases: 2 types
Gapped domain walls between doublesemion and toric code phases only have two types. They are the compositions of gapped boundaries, and .
b.5 V. Gapped boundary of doubled Fibonacci phase: 1 type
Let . The matrices of doubled Fibonacci phase are:
There is only one type of gapped boundary,
We compute the GSD on spheres with more punctures:

2 punctures (cylinder): GSD=2,

3 punctures (a pair of pants): GSD=5,

4 punctures: GSD=15,
b.6 VI. Gapped domain walls between doubled Fibonacci and toric code phases: 2 types
Gapped domain walls between doubled Fibonacci and toric code phases only have two types. They are the compositions of gapped boundaries, and .
b.7 VII. Gapped boundary of doubled Ising phase: 1 type
Let . The matrices of doubled Ising phase are
There is only one type of gapped boundary,
We compute the GSD on spheres with more punctures:

2 punctures (cylinder): GSD=3,

3 punctures (a pair of pants): GSD=10,

4 punctures: GSD=36,
b.8 VIII. Gapped domain walls between doubled Ising and toric code phases: 3 types
There are 3 types of stable gapped domain walls between doubled Ising and toric code phases. The first one is
If we label the anyons in the doubled Ising phase as , this domain wall corresponds to the follow tunneling process
This agrees with a recent result obtained by anyon condensation Hung and Wan (2013).
The other two types of gapped domain walls, again, are the compositions of gapped boundaries, and .
We also like to use this example to illustrate the instability of composite domain walls. Insert a strip of doubled Ising phase to the toric code phase, together with gapped domain walls and . We then shrink the doubled Ising phase strip and compose the two domain walls. By straightforward calculation,
Thus, the composite domain wall splits to one trivial domain wall , and, one exchanging domain wall , between two toric code phases. It is an unstable gapped domain wall that does not satisfy the stable condition.
b.9 IX. Gapped boundaries of phase: 4 types
The matrices of phase are
There are 4 types of gapped boundaries.
We compute the GSD on a cylinder with two gapped boundaries (read from the above):
b.10 X. Gapped domain walls between and toric code: 12 types
Gapped domain walls between and toric code have 12 types in total. The first two types:
The third and fourth types are the first two types composed with the exchanging domain wall , i.e., . The other 8 types are the compositions of gapped boundaries (toric code has 2 types and has 4 types of gapped boundaries ).
b.11 XI. Gapped boundaries of phase: 11 types
Note that . To simplify notations, below we denote