Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases
We derive a bulk-boundary correspondence for three-dimensional (3D) symmetry-protected topological (SPT) phases with unitary symmetries. The correspondence consists of three equations that relate bulk properties of these phases to properties of their gapped, symmetry-preserving surfaces. Both the bulk and surface data appearing in our correspondence are defined via a procedure in which we gauge the symmetries of the system of interest and then study the braiding statistics of excitations of the resulting gauge theory. The bulk data is defined in terms of the statistics of bulk excitations, while the surface data is defined in terms of the statistics of surface excitations. An appealing property of this data is that it is plausibly complete in the sense that the bulk data uniquely distinguishes each 3D SPT phase, while the surface data uniquely distinguishes each gapped, symmetric surface. Our correspondence applies to any 3D bosonic SPT phase with finite Abelian unitary symmetry group. It applies to any surface that (1) supports only Abelian anyons and (2) has the property that the anyons are not permuted by the symmetries.
A gapped quantum many-body system is said to belong to a non-trivial symmetry-protected topological (SPT) phase if it satisfies three conditions: First, the Hamiltonian is invariant under some set of internal (on-site) symmetries, none of which are broken spontaneously. Second, the ground state is short-range entangled — that is, the ground state can be transformed into a product state or atomic insulator using a local unitary transformation. Third, it is impossible to continuously connect the ground state to a product state or atomic insulator, by varying some parameter in the Hamiltonian, without breaking one of the symmetries or closing the energy gap.Gu and Wen (2009); Pollmann et al. (2010); Fidkowski and Kitaev (2011); Chen et al. (2011a, b); Schuch et al. (2011); Chen et al. (2013) Famous examples of nontrivial SPT phases include the one dimensional Haldane spin chainHaldane (1983), which is protected by time reversal symmetry, and the two-dimensional (2D) and three-dimensional (3D) topological insulatorsHasan and Kane (2010); Qi and Zhang (2011) which are protected by time reversal and charge conservation symmetry.
Perhaps the most interesting property of nontrivial SPT phases is that they have protected boundary modes. Here, the precise meaning of “protected” depends on dimensionality. For example, in the two dimensional case, the edges of SPT phases are believed to be protected in the sense that they cannot be both gapped and symmetricKane and Mele (2005); Xu and Moore (2006); Wu et al. (2006); Chen et al. (2011c); Levin and Gu (2012); Else and Nayak (2014). On the other hand, in the three dimensional case, the surfaces of SPT phases are believed to be protected in the sense that any surface that is both gapped and symmetric must also support anyon excitationsVishwanath and Senthil (2013); Wang and Senthil (2013); Burnell et al. (2014); Chen et al. (2015); Bonderson et al. (2013); Wang et al. (2013); Chen et al. (2014); Metlitski et al. (2015); Fidkowski et al. (2013); Metlitski et al. (2014); Wang and Senthil (2014).
For some SPT phases, we can not only establish the existence of a protected boundary, but we can derive a “bulk-boundary correspondence.” Let us clarify what we mean by this term since there are at least two different types of bulk-boundary correspondences discussed in the literature. One type of bulk-boundary correspondence is a construction that provides a particular (i.e. non-unique) field theory description of the boundary for each bulk phaseWen (1995); Levin and Stern (2012); Lu and Vishwanath (2012); Vishwanath and Senthil (2013); Xu and Senthil (2013). Another type of bulk-boundary correspondence is a universal relation between measurable properties of the bulk and boundary. In this paper, we will be interested in bulk-boundary correspondences of the second kind.
The classic example of such a bulk-boundary correspondence appears in the context of 2D non-interacting fermion systems with charge conservation symmetry. For these systems, one can relate the bulk electric Hall conductivity , measured in units of , to the number of right-moving and left-moving edge modesHalperin (1982):
Less is known about such bulk-boundary correspondences for interacting SPT phases. One place where it would be particularly useful to have such a correspondence is in the context of 3D SPT phases with gapped symmetric surfaces. This case is interesting because surfaces of this kind are relatively easy to characterize due to the energy gap, but at the same time they exhibit nontrivial structure associated with surface anyon excitations. It is natural to ask: what are the general constraints that relate the bulk and surface properties of these systems?
There are several cases where this question has been answered — at least partially. In particular, in the case of 3D topological insulators, Refs. Metlitski et al., 2013; Wang and Senthil, 2013 derived constraints connecting the properties of the surface to properties of monopoles in the bulk. Similarly, it is possible to derive constraints for other 3D SPT phases with at least one symmetry and one anti-unitary symmetrySenthil (2015). Unfortunately, however, these constraints rely on a special combination of symmetries and therefore do not give insight into the more general structure of the bulk-boundary correspondence.
In this paper, we take a step towards a more general theory by deriving a bulk-boundary correspondence for a large class of 3D SPT phases. More specifically, we consider general 3D bosonic SPT phases with unitary Abelian symmetries. To simplify the discussion, we focus on gapped symmetric surfaces with the property that (1) the surface anyons are Abelian, and (2) these anyons are not permutedBombin (2010); Fidkowski et al. (); Barkeshli et al. (2014) by the symmetries. We denote the symmetry group by , and the group of surface anyons by — with the group law in corresponding to fusion of anyons. For this class of systems, we derive a bulk-boundary correspondence analogous to Eq. (1).
Before we can explain our correspondence, we need to describe the bulk and surface data that we use. The bulk data was originally introduced by Ref. Wang and Levin, 2015 and consists of three tensors
where the indices range over . These quantities are defined via a simple recipe. Suppose we are given a lattice boson model belonging to an SPT phase with symmetry group . To find the corresponding bulk data, the first step is to minimally couple the model to a dynamical lattice gauge field with gauge group .Kogut (1979) After gauging the model in this way, the second step is to study the braiding statistics of the “vortex loop” excitations of the resulting gauge theoryWang and Levin (2014); Jiang et al. (2014). The tensors are then defined in terms of the braiding statistics of these vortex loops, as reviewed in more detail in section II.
The surface data has a similar character and consists of five tensors
where the indices range over and the indices range over . Like the bulk data, the surface data is defined by gauging the lattice boson model and studying the braiding statistics of the excitations of the resulting gauge theory. The only difference is that we consider the braiding statistics of surface excitations instead of bulk excitations. In particular, the tensors are defined in terms of the braiding statistics of surface anyons and vortex lines ending at the surface.
The reason we use the above bulk and surface data is that this data has a number of appealing properties. First, the quantities in (2) and (3) are measurable in the sense that they can be extracted from a microscopic model by following a well-defined procedure. Second, these quantities are topological invariants: that is, they remain fixed under continuous, symmetry-preserving deformations of the (ungauged) Hamiltonian that do not close the bulk or surface gap, respectively.111To see this, recall that our gauging prescription maps gapped lattice boson models onto gapped gauge theories.Wang and Levin (2015); Levin and Gu (2012) Therefore, if two lattice boson models can be connected without closing the gap, the corresponding gauge theories can also be connected without closing the gap and hence must have the same braiding statistics data. Finally, there is reason to think that the bulk data and surface data are complete in the sense that the bulk data uniquely distinguishes every 3D SPT phase, while the surface data uniquely distinguishes every gapped symmetric surface (we discuss the evidence for this claim in sections II.2 and III.3).
The main result of this paper is a set of three equations (30-32) that connect the bulk data (2) to the surface data (3). We derive these equations by relating the bulk braiding processes that define (2) to the surface braiding processes that define (3) using topological invariance and other properties of braiding statistics.
Our results are closely related to a conjecture of Chen, Burnell, Vishwanath, and FidkowskiChen et al. (2015). To describe this conjecture, we need to recall two facts. The first fact is that many (possibly all) 3D SPT phases with finite unitary symmetry group can be realized by exactly soluble lattice boson models known as group cohomology modelsChen et al. (2013). These models are parameterized by elements of the cohomology group . The second fact is that each 2D anyon system with unitary symmetry group is associated with an anomaly which takes values in Etingof et al. (2010); Chen et al. (2015). (See Refs. Vishwanath and Senthil, 2013; Kapustin and Thorngren, 2014; Cho et al., 2014; Wang et al., 2015 for other related discussions of anomalies.) If this anomaly is nonzero then the corresponding anyon system cannot be realized in a strictly 2D lattice model. Given these two facts, Chen et al. conjectured that gapped symmetric surfaces of group cohomology model always have an anomaly that matches the defining the bulk cohomology model. The authors checked that this conjecture gives correct predictions for a particular lattice model.
What is the relationship between our bulk-boundary correspondence and this conjecture? To make a connection, we use our bulk-boundary formulas (30-32) to obtain constraints on the surfaces of group cohomology models. We then compare these constraints to those predicted by the conjecture and we show that the two sets of constraints are mathematically equivalent. Thus, our bulk-boundary correspondence gives a proof of the conjecture for the case where is Abelian. Conversely, the conjecture implies our bulk-boundary correspondence, if we assume that the group cohomology models realize every possible 3D SPT phase.
The rest of the paper is organized as follows. In Sec. II and Sec. III, we define the bulk data and surface data, respectively. In Sec. IV, we present the bulk-boundary correspondence (30-32) that relates the two sets of data to one another. We derive the correspondence in Sec. V. We discuss the implications of the correspondence for purely 2D systems in Sec. VI. In Sec. VII, we explain the connection between our bulk-boundary correspondence and the conjecture of Chen et alChen et al. (2015). Technical arguments and calculations are given in the Appendices.
Ii Bulk data: Review
In this section we review the definition of the bulk data :Wang and Levin (2015) that is, we explain how to compute these quantities given an arbitrary 3D lattice boson model belonging to a SPT phase with unitary Abelian symmetry group .
As discussed in the introduction, the computation/definition proceeds in two steps. The first step is to minimally couple the lattice boson model of interest to a dynamical lattice gauge field with gauge group .Kogut (1979) The details of this gauging procedure are not important for our purposes; the only requirement is that the gauge coupling constant is small so that the resulting gauge theory is gapped and deconfined. (See Refs. Wang and Levin, 2015; Levin and Gu, 2012 for a precise gauging prescription in which the coupling constant is chosen to be exactly zero). The second step is to study the excitations of the gauged model. The bulk data is defined in terms of the braiding statistics of these excitations.
In what follows, we focus the second step of this procedure. First, we discuss the excitations of the gauged models and review their braiding statistics. After this preparation, we give the precise definition of .
ii.1 Bulk excitations
We begin by reviewing the excitation spectrum of the gauged models. The gauged models support two types of excitations in the bulk: particle-like charges and loop-like vortices. Charge excitations are characterized by their gauge charge
where each component is an integer defined modulo . Similarly, vortex loop excitations are characterized by their gauge flux
where each component is a multiple of and is defined modulo . An important point is that while charge excitations are uniquely characterized by the amount of gauge charge that they carry, there are typically many topologically distinct types of vortex loops that carry the same gauge flux . These different loops can be obtained from one another by attaching charge excitations.
Some comments on notation: we will use Greek letters to denote vortex excitations, and we will use to denote the amount of gauge flux carried by . We will use to denote charge excitations, and we will use the same symbols to denote the amount of gauge charge that they carry.
Let us now discuss the braiding statistics of these excitations. There are several types of braiding processes one can consider: (i) braiding of two charges, (ii) braiding of a charge around a vortex loop, and (iii) braiding involving multiple vortex loops. The first kind of braiding process is not very interesting because the charges are all bosons: one way to see this is to note that the charges can be viewed as local excitations of the original ungauged model, and the ungauged model has only bosonic excitations since it is short-range entangled. As for the second kind of process, it is easy to see that if we braid a charge around a vortex loop , the resulting statistical phase is given by the Aharonov-Bohm law
where “” denotes the vector inner product.
All that remains is the last type of braiding process, involving multiple loops. There are several ways to braid vortex loops, but in this paper we will primarily be interested in a braiding process in which a loop is braided around a loop while both loops are linked to a “base loop” (Fig. 1)Wang and Levin (2014); Jiang et al. (2014). We will also consider more general braiding processes involving multiple loops , all of which are linked to a single base loop . The reason we focus on these kinds of braiding processes is that the associated Berry phases are not fixed by the Aharonov-Bohm law, but instead probe more interesting properties of the bulk.
One technical complication is that in some models the vortex loop excitations have non-Abelian braiding statistics even though the gauge group is Abelian. Therefore, even if we specialize to the case where is Abelian as we do here, it is still important to have a more complete theory of loop braiding statistics that includes the concepts of fusion rules, quantum dimensions, and so on.
Fortunately this formalism can be developed rather easily. The key point is that there is a direct analogy between 3D loop braiding and 2D particle braiding. This analogy can be seen by considering a 2D cross-section of the loop braiding process [Fig. 1(b)]. We can see that a braiding process involving two loops linked with a single base loop can be mapped onto a process involving two particles in two dimensions. More generally, a process involving loops linked to a single base loop can be mapped into a process involving particles in two dimensions. In addition to braiding, the analogy also carries over to fusion processes. Just as two particles can be fused together to form another particle, two loops that are linked to the same loop can be fused to form a new loop that is also linked to .
With this analogy, we can immediately generalize the notation and results of 2D anyon theoryKitaev (2006) to 3D loop braiding. In particular, we can define symbols, symbols, and quantum dimensions of loops in the same way as for particles. We will denote these quantities by , , and . Here are loops linked with a base loop , while is an integer vector that parameterizes the gauge flux carried by :
(The reader may wonder why we use ‘’ instead of ‘’ in our notation for , , etc. The reason is that these quantities only depend on the gauge flux carried by and this gauge flux is conveniently parameterized by ).
An additional quantity that we will need below is the topological spin of a loop linked to a base loop . We will denote this quantity by where . Here, is defined in the same way as for 2D anyon theories:
where the summation runs over all fusion channels of two loops, both of which are linked to .
ii.2 Definition of bulk data
With this preparation, we are now ready to define the bulk data. This data consists of three tensors , where the indices range over . To define these tensors, let be vortex loops linked to a base loop . Suppose that carries unit type- flux, that is where is the vector with the th entry being and all other entries being . Similarly, suppose that carry unit flux , respectively. Then, are defined as follows:
, where is the topological spin of when it is linked to ;
is the Berry phase associated with braiding the loop around for times, while both are linked to ;
is the Berry phase associated with the following braiding process: is first braided around , then around , then around in the opposite direction, and finally around in the opposite direction. Here are all linked to .
Above, we have used to denote the least common multiple of and . (Throughout this paper, we will use and to denote the least common multiple and greatest common divisor of integers respectively.)
These definitions deserve a few comments. First, we would like to point out that one needs to do some work to show that the above quantities are well-defined. In particular, one needs to establish two results: (1) are Abelian phases, and (2) depend only on and not on the choice of the loops . The first fact is not obvious since vortex loops can have non-Abelian braiding statistics in some cases so that the Berry phase associated with general braiding processes is actually non-Abelian. The second fact is not obvious either since there are multiple topologically distinct loop excitations that carry the same gauge flux.222These loops differ from one another by the attachment of charge excitations. The proof of these two properties is given in Ref. Wang and Levin, 2015.
We should also explain our motivation for using this data to characterize bulk SPT phases. Much of our motivation comes from a result of Ref. Wang and Levin, 2015, which showed that take different values in every group cohomology model with symmetry group . What makes this result especially interesting is that it has been conjectured that the group cohomology models realize every 3D SPT phaseChen et al. (2013). If this conjecture is true, then we can conclude that the above data is complete in the sense that it can distinguish all 3D SPT phases with finite Abelian unitary symmetry.
Finally, we would like to make a comment about the definition of . Unlike the other two quantities which are concretely defined through braiding processes, is a rather abstract quantity since does not have a simple interpretation in terms of physical processes. Fortunately there is an alternative definition of which is more concreteWang and Levin (2015):
is equal to times the phase associated with braiding around its anti-vortex for times, while both and are linked to .
Obviously, this definition fails when is odd since is not an integer. However, when is odd, there is no need to find an alternative definition for : the reason is that is known to satisfy the constraints (see Appendix G.2)
Together with another constraint , it is easy to see that these constraints completely determine in terms of when is odd. More specifically, one can derive the following relation:
Iii Surface data
Next we define the surface data . Our task is as follows: suppose we are given a 3D lattice boson model belonging to an SPT phase with symmetry group . Suppose that this model is defined in a geometry with a surface and that this surface has three properties: (1) it is gapped and symmetric, (2) it supports only Abelian anyons, and (3) it has the property that the surface anyons are not permuted by the symmetries. Given a microscopic model of this kind, we need to explain how to compute the corresponding surface data.
The computation/definition proceeds in the same way as for the bulk data. First, we couple the lattice boson model to a dynamical lattice gauge field with gauge group . Then, after gauging the model, we study the surface excitations of the resulting gauge theory. The surface data is defined in terms of the braiding statistics of these surface excitations.
In what follows, we build up to the definition of in several steps. First, we discuss the surface excitations of the original ungauged boson models. Then we discuss the excitations of the gauged models. Finally, after this preparation, we explain the definition of the surface data.
iii.1 Surface excitations of ungauged models
By assumption the surfaces of the ungauged boson models only support Abelian anyons. The purpose of this section is to introduce some notation for labeling these anyons and describing their braiding statistics.
Our notation for labeling the anyons is based on the observation that the anyons form an Abelian group under fusion. Denoting this Abelian group by , we label each anyon by a group element , or equivalently, an component integer vector
where each component is defined modulo . The vacuum anyon corresponds to the zero vector , while the fusion product of two anyons is given by .
To describe the braiding statistics of these particles, we focus on the unit anyons — that is, the anyons labeled by vectors with the th entry being and all other entries being . We denote the exchange statistics of the unit anyon by and we denote the mutual statistics of the unit anyons and by . From and , we can reconstruct the exchange statistics and mutual statistics of any anyons :
iii.2 Surface excitations of gauged models
We now move on to discuss the surface excitations in the gauged models. The gauged models support three types of excitations on or near the surface: (1) point-like charge excitations that can exist in either the bulk or the surface; (2) point-like surface anyons that are confined to the surface; (3) vortex line excitations that end on the surface. See Fig. 2 for a sketch of these excitations.
Let us discuss each of these excitations in turn. We begin with the charges and vortex lines. Like their bulk counterparts, the charges can be labeled by their gauge charge while the vortex lines can be labeled by their gauge flux . Just as in the bulk, the gauge charge uniquely characterizes the charge excitations, while the gauge flux does not uniquely characterize the vortex lines: there are multiple topologically distinct vortex lines that carry the same gauge flux , which differ from one another by the attachment of surface anyons and charges.
There are three types of braiding processes we can perform with charges and vortex lines. First, we can braid or exchange charges with one another. These processes are not very interesting since the charges are all bosons. Second, we can braid charges around vortex lines. As in the bulk, the Berry phase for such processes is given by the Aharonov-Bohm formula (6). Finally, we can braid vortex lines around one another.333To make sense of braiding processes involving multiple vortex lines, we have to keep track of the other ends of the vortex lines. The latter processes are highly non-trivial and can even give non-Abelian Berry phases. We will see some examples of these processes later when we derive the bulk-boundary correspondence in Sec. V.
Let us now turn to the surface anyons. To understand the structure of these excitations, it is helpful to think about their relationship to the surface anyons in the ungauged models. In particular, we will argue below that each surface anyon in the gauged model is naturally associated with a corresponding surface anyon in the ungauged model, which we denote by . We will think of the mapping as being analogous to the mapping between vortices and their gauge flux ; therefore we will use the terminology that is the “anyonic flux” carried by . Like the gauge flux, can be concretely represented as a vector — or more precisely, an component integer vector, as in Eq. 11.
At an intuitive level, is obtained by “ungauging” the excitation ; to define more precisely, let be an excited state of the gauged model which contains an anyon localized near some point . Let us suppose that has vanishing gauge flux through every plaquette in the lattice. (We can make this assumption without any loss of generality since is not a vortex excitation). Then we can find a gauge in which the state has a vanishing lattice gauge field on every link of the lattice. In this gauge, the state can be written as a tensor product where is a ket describing the configuration of the lattice gauge fields, and is a ket describing the configuration of the matter fields. Since only involves matter fields, we can think of it as an excited state of the ungauged model. By construction this state contains a localized excitation near the point ; we define to be the anyon type of this localized excitation. 444The reader may worry that the localized excitation in might not have a definite anyon type — that is, it might be a linear superposition of different anyons; however, one can argue that this situation does not happen if the symmetry does not permute the anyons, as we assume here.
The mapping between anyons and their anyonic flux has several important properties. The first property is that the mapping is not one-to-one: distinct anyons can carry the same anyonic flux, . The simplest example of this is given by the charge excitations: all the charges share the same (trivial) flux, . More generally, two anyons have the same anyonic flux if and only if can be obtained from by attaching a charge excitation, that is for some .
Another important property of the mapping is that the braiding statistics of anyons , etc. is identical to the braiding statistics of , etc. More specifically, it can be shown that the mutual statistics between any two anyons and is Abelian and is given by
Similarly, the exchange statistics of , or equivalently its topological spin , is given by
Given these results, one might be tempted to conclude that the surface anyons in the gauged model have Abelian statistics. However, this conclusion is incorrect: the surface anyons in the gauged model can be non-Abelian in general. This non-Abelian character is not manifest when we focus on braiding processes that only involve surface anyons but it becomes apparent when we consider processes in which a surface anyon is braided around a vortex line. Such processes will play an important role in the definition of the surface data in the next section.
The last property of the mapping involves fusion rules. Because the surface anyons can be non-Abelian, they can have complicated fusion rules of the general form
However, these rules have a special structure: all the fusion outcomes carry the same anyonic flux which is given by
iii.3 Definition of surface data
Now that we have discussed the surface excitations and their statistics, we are ready to define the surface data. This data consists of five tensors , where the indices run over while run over . We have already defined in Sec. III.1; below we define the other two quantities and . We discuss in the next section.
The two quantities and are defined in terms of braiding statistics of surface anyons and vortex lines. Let be a surface anyon with unit type- anyonic flux, i.e. , where with the th entry being . Let be vortex lines carrying unit type- and type- flux, i.e. and . Then and are defined by:
is the Berry phase associated with braiding around for times;
is the Berry phase associated with the following process: is first braided around , then around , then around in the opposite direction, and finally around in the opposite direction.
The above braiding processes are shown in Fig. 3. Similarly to Sec. II.2, we use the notation to denote the least common multiple of and , where the are the cyclic factors in the symmetry group , and the are the cyclic factors in the anyon group .
To show that the above quantities are well-defined, we need to prove that (i) are always Abelian phases and (ii) they only depend on the indices , i.e, on the gauge flux of the vortex lines and the anyonic flux of . In other words, if we choose another anyon with and another pair of vortex lines with , , we will obtain the same phases and . The proof of these two points is technical, and hence is given separately in Appendix A.
Our motivation for using the above surface data is twofold. First, there is reason to think that the surface data is complete in the sense that it can distinguish every gapped symmetric surface satisfying our two assumptions, namely that (1) the surface supports only Abelian anyons and (2) these anyons are not permuted by the symmetry. The main evidence for this is that our data is mathematically equivalent to another set of dataKitaev (2006); Essin and Hermele (2013); Fidkowski et al. (); Barkeshli et al. (2014) that characterizes 2D anyon systems with symmetry group . We discuss this equivalence in Sec. VII. The latter data is plausibly complete in the context of the above class of surfaces555Here it is important to distinguish between strictly 2D systems and surfaces of 3D systems. In the context of strictly 2D systems, the data needs to be supplemented by an additional piece of information, namely an element of the cohomology group .Kitaev (2006); Fidkowski et al. (); Barkeshli et al. (2014) However, we believe that this additional piece of data is unnecessary for characterizing surfaces of 3D systems, at least if one defines the notion of “topologically equivalent” surfaces in an appropriate manner. and if this is the case, then our data must be complete as well. Our second source of motivation for using the above surface data is that it can be naturally related to the bulk data via a bulk-boundary correspondence, as we discuss below.
iii.4 An auxiliary surface quantity
The last piece of surface data is a three index tensor where the indices run from while runs from . We define this quantity implicitly via the equation
More precisely, we define to be the unique integer tensor that satisfies the above equation and has components .
To understand the physical meaning of Eq. (18), let us imagine fixing the indices . Then reduces to an component integer vector, which we can think of as describing a surface anyon in the ungauged model. Likewise, the left hand side of Eq. (18) can be interpreted as the statistical phase associated with braiding a type- unit anyon around the anyon . In this interpretation, represents the unique surface anyon with the property that its mutual statistics with type- unit anyons is . From this point of view, it is not hard to see why exists and is unique. Indeed, the uniqueness of follows from the general principle of “braiding non-degeneracy” which guarantees that no two surface anyons have the same mutual statistics with respect to all other surface anyons. Kitaev (2006) As for the existence of , this is less obvious but can also be deduced, with some algebra, from braiding non-degeneracy together with the fact that is a multiple of . (See Eq. 24 below).
From Eq. 18 we can see that carries the same information as (assuming we know ). Therefore, the reader may wonder why we bother to define this additional quantity. One reason is that is often more convenient to work with than . Another reason is that the quantity plays an important role in the bulk-boundary correspondence. This role can be traced to the following thought experiment. Imagine a configuration of two linked vortex loops, and in the bulk (Fig.4) where and carry unit type- and type- flux, respectively, that is and . Now imagine that we pull the linked loops down to the surface, absorbing a part of each loop into the surface, leaving a pair of tangled vortex “arches” [Fig. 4(b)]. We then disentangle the two arches by unwinding one of the two ends of [the braiding path is shown in Fig. 4(b)]. The result is two separate arches [Fig. 4(c)]. We then shrink the two separated arches to the surface, leaving behind two localized surface excitations. We denote the resulting state by . In general, we know that any surface excitation can be written as a linear superposition of different states, each of which has a definite anyon type. Thus, we can write
where denotes a state with surface anyons and , while denotes a state with surface anyons and , and the coefficients , etc. are the corresponding complex amplitudes [Fig. 4(d)]. What does this thought experiment have to do with ? We show in Appendix B that all the anyons carry the same anyonic flux , while all the anyons carry anyonic flux :
Thus the quantity naturally appears when we think about absorbing linked vortex loops into the surface.
iii.5 General constraints on surface data
In this section, we discuss some of the constraints on . Two important constraints are
To derive these constraints, consider the unit type- anyon . Using the fact that the fusion product of of these anyons gives the vacuum excitation, it is not hard to show that , and . With these relations in hand, the above constraints follow immediately.
Another set of constraints are
We derive these constraints in Appendix G.1. A final set of constraints involve :
Note there are additional constraints on the surface data beyond those listed above. In particular, the requirement that the surface anyons obey “braiding non-degeneracy”Kitaev (2006) gives extra constraints on . We will not write out these constraints explicitly since they are not necessary for our purposes.
Iv Bulk-boundary correspondence
iv.1 The correspondence
Having defined the bulk data and the surface data , we are now ready to discuss the connection between the two. This connection is encapsulated by three equations, defined modulo :
These equations are the main results of this paper. We will present their derivation in section V, but before doing that, we make some comments about these formulas and their implications:
Note that the left hand side of equations (30-32) consists of the bulk data , while the right hand side is built entirely out of the surface data . Thus, these equations allow us to completely determine the bulk data from the surface data. They also provide some constraints on the surface data given the bulk data. This asymmetry between bulk and surface, which is also manifest in Eq. (1), is not surprising since we expect that a given bulk phase can support many different types of surfaces.
Equations (30-32) have an important corollary: any 3D short-range entangled bosonic model that has nonzero values for , has a protected surface, i.e. its surface cannot be both gapped and symmetric unless it supports anyon excitations. To derive this corollary, we note that if we could find gapped symmetric surface without anyon excitations, then the right hand sides of equations (30-32) would vanish for this surface since the sum over would run over the empty set. Clearly this vanishing is inconsistent with nonzero values of , so we conclude that such a surface is not possible.
It is natural to ask whether there could be additional constraints relating bulk and surface data beyond equations (30-32). The analysis in this paper is not capable of answering this question definitively. That being said, if there are additional constraints, we can always replace the bulk data appearing in these constraints with surface data, using (30-32). Hence, any additional constraints can be written entirely in terms of surface data.
The coefficient of in Eq. (30) is an integer. This is not obvious, but can be proven using one of the general constraints on , namely Eq. (27). This integrality property is important because is only defined modulo : therefore it is only because its coefficient is an integer that Eq. (30) gives a well-defined phase . In a similar fashion one can check that all the coefficients of the phase factors , in Eqs. (30-32) are integers, so all of these equations are well-defined. Finally, one can check that the coefficients of and in Eqs. (30-32) satisfy appropriate conditions so that all three equations are well-defined even if is defined modulo .
Several of the terms in these equations can only take two values: or . In particular, this is the case for the second term on the right hand side of (30) as well as the second term on the right hand of (31). This property is interesting because it means that the above equations are simpler than they appear. These results can be established using the general constraints (29), (22), (24).
How can we make use of the above bulk-boundary correspondence? We envision two types of applications. First, if we are given a surface theory and are able to extract all the surface data, we can use the bulk-boundary correspondence to constrain the bulk SPT phase. Conversely, if we are given a bulk SPT phase, we can use the bulk-boundary correspondence to constrain the possible surfaces. Below we give two examples to illustrate these two ways of applying the bulk-boundary correspondence.
iv.2 Example 1
In this section, we demonstrate the bulk-boundary correspondence by computing the bulk data corresponding to four different types of surfaces with symmetry. These surface theories were originally introduced and analyzed by Ref. Chen et al., 2015 as we explain below.
To set up our example, imagine that we have a 3D lattice spin model that realizes an SPT phase with a symmetry group . Imagine that we study the model in a geometry with a boundary and we find that the surface is gapped and symmetric and that it supports two distinct types of Abelian anyons: a semion with exchange statistics and the vacuum excitation with trivial statistics. Translating this information into our notation, this means that the surface anyons form a group while their statistics can be summarized by two quantities:
Here the index can only take one value — namely — since the group has only one generator.
Next, suppose we couple the system to a gauge field in order to probe its symmetry properties. After performing this gauging procedure, we take a surface anyon with anyonic flux and we braid it twice around a vortex line that carries gauge flux . We find that the Berry phase associated with this process is . We also braid the surface anyon twice around the vortex line and we find a Berry phase of . Finally, we braid the surface anyon around one vortex line carrying flux and one vortex line carrying flux , then we braid in the opposite direction around both vortex lines as described in Sec. III.3 and we find that the associated Berry phase is again . In our notation, this information is summarized by the surface data
Here the indices can take two values since the symmetry group has two generators.
To complete the surface data, we still need several more quantities, namely and . The first set of quantities can be completely fixed using general properties of . In particular, we know that by Eq. 26, while by Eq. 25. The remaining quantities are then completely determined by the definition of (18):
With the above surface data in hand, we can now illustrate the bulk-boundary correspondence (30-32). Let us focus on computing the bulk data and . Substituting the surface data into Eq. (30) we obtain
Similarly, we can go ahead and compute the remaining bulk data, e.g. , etc., with the result being that they all vanish as well. Alternatively, we can obtain this result using the general constraints (159)-(168) which completely determine these quantities in terms of and . In other words, and are the only independent bulk quantities.
For comparison, we now consider three other possibilities for the surface data, which we will refer to as ‘APS-X’, ‘APS-Y’ and ‘APS-Z’ (we explain this terminology below):
Here, as before, we assume that the surface anyons form a group with statistics (33). Applying the bulk-boundary formulas, we can compute and in the same way as above. The results are shown in Table 1, along with those corresponding to the surface data (34), which we will refer to as ‘CSL.’
What conclusions can we draw from these calculations? First, we can see from Table 1 that and take different values in each of the four cases. Therefore, we can conclude that the corresponding bulk spin models all belong to distinct SPT phases. Also we see that at least one of and is nonzero for each of the APS-X, APS-Y, APS-Z cases listed above, which implies that the corresponding bulk spin models belong to non-trivial SPT phases. Finally, since the bulk data vanishes for the ‘CSL’ case, we can conclude that the corresponding bulk spin model belongs to a trivial SPT phase — if we make the additional assumption that the bulk data is complete.
|Surface Data||Bulk Data|
As we mentioned above, these four types of surfaces were originally discussed by Ref. Chen et al., 2015. Our terminology for these surfaces follows that of Ref. Chen et al., 2015: the reason we refer to the first type of data as ‘CSL’ is that a variant of the 2D Kalmeyer-Laughlin chiral spin liquid (CSL) state is described by this data; likewise, the reason we refer to the other types of data as ‘APS-X’, ‘APS-Y’ and ‘APS-Z’ is because they correspond to the “anomalous projective semion” (APS) states of Ref. Chen et al., 2015. Here the word “anomalous” signifies the fact that the last three types of surface data are incompatible with a pure 2D lattice model and can only exist on the boundary of a nontrivial 3D SPT phase.
It is worth pointing out that Ref. Chen et al., 2015 used a different language to describe the surface data than what we use here. In this alternate description, the symmetry properties of the surface are described by an element of the cohomology group instead of the quantities . We explain this alternative language and its relationship with in section VII.
iv.3 Example 2
We can also use the bulk-boundary correspondence in the opposite direction: that is, we can use it to constrain the types of surfaces that are compatible with a given bulk Hamiltonian. For an example of this, imagine that we have a lattice boson model that realizes an SPT phase with symmetry group . As we mentioned in the previous section, there are only two independent pieces of bulk data for this symmetry group: and . Let us suppose that one or both of these quantities takes a nonzero value for our lattice spin model. Using this information we can constrain the possible surfaces of this system. In particular, assuming that the surface anyons are all Abelian and are not permuted by the symmetries, we can show that the group of surface anyons has the property that is even for at least one value of .
One way to see this is to examine the general constraints (27-29) on . In particular, from the constraint (29), we can see that . Also, if is odd, then the constraint (27) implies that . Hence if were odd for all then would necessarily vanish completely. But then and would also have to vanish according to the bulk-boundary formula (30). We conclude that if either or is nonzero then at least one of the ’s must be even.
V Derivation of the bulk-boundary correspondence
v.1 Step 1: deforming the braiding process
Our derivation proceeds in four steps. In the first step, we derive an equivalence between the three-loop braiding process associated with and another process which involves braiding vortex “arches” on the surface. The key to deriving this equivalence is the general principle that statistical Berry phases are invariant under “smooth” deformations of braiding processes: that is, if two braiding processes can be “smoothly” deformed into one another then the associated statistical Berry phases must be equal. Here, a “smooth” deformation is a sequence of local changes to the excitations involved in the braiding process. The local changes can be arbitrary except that the moving excitation must stay far apart from the other excitations at every step of the deformation. (Here, when we say “local changes” to the excitations we mean any changes that can be implemented by unitary operators supported in the neighborhood of the excitations).
To begin, let us imagine performing the three-loop braiding process in the bulk [Fig. 5(a)]. That is, we braid a loop around another loop for times while both are linked to a third loop . Here carry unit type-, type- and type- flux, i.e. , and . The Berry phase associated with this process is .
Next, we stretch and and absorb the bottoms of these loops into the surface.666The fundamental reason that this absorption/annihilation is possible is that the surface is symmetric by assumption. This symmetry guarantees that the bottoms of the vortex loops can be annihilated at the surface using local unitary operators. This step changes into vortex arches that terminate on the surface [Fig. 5(b)]. After this step, the deformed braiding process involves braiding vortex arches while they are both linked to . By the general principle described above, this deformed process must yield the same Berry phase, .
To proceed further, we now stretch and absorb its bottom into the surface. This step changes into another vortex arch [Fig. 5(c)]. Finally, we disentangle the three arches by unwinding one of the two ends of the arch [Fig. 5(d)]. The braiding process now involves braiding two unlinked arches around one another. Again the Berry phase must be the same as in the original process. Putting this all together, we conclude that is equal to the Berry phase associated with braiding the arch around the arch for times, as shown in Fig. 5(d).
v.2 Step 2: splitting the excitations
Our task is now to analyze the vortex arch braiding process in Fig. 5(d). Before doing this, it is useful to first consider a thought experiment in which we shrink down the arches so that all that is left are two localized surface excitations. From general considerations we know that the resulting surface excitations can be written as a linear superposition of states, each of which has a definite anyon type. We ask: what types of surface anyons appear in this linear superposition?
The answer is simple: if we shrink , we get a superposition of different surface anyons all of which have anyonic flux ; likewise, if we shrink , we get a superposition of surface anyons all of which have anyonic flux . To see this, notice that shrinking down in Fig. 5(d) is very similar to shrinking in Fig. 4. Furthermore, in the latter case, we know that shrinking gives a superposition of different surface anyons all of which have anyonic flux (see Eq. 20). Therefore the same must be true for Fig.5(d).
The most important point from this discussion is that if we shrink or to the surface, we will generally get non-trivial surface anyons. This property is inconvenient for our subsequent analysis and motivates us to split into more “elementary” excitations.
For this reason, the next step in our analysis is to split the arch into a surface anyon and a new arch . Similarly we split into a surface anyon and a new arch (Fig. 6). We choose the surface anyons to be any anyons with anyonic flux
while we choose the arches to be any arches with the property that can be written as a fusion product of and , and that can be written as a fusion product of and .
The motivation for performing this splitting is that the new arches have a nice property, by construction: if we shrink to the surface, we get a superposition of charge excitations instead of more complicated surface anyons. This property will play an important role below when we compute the Berry phase associated with braiding around (see Eq. 59 below).
v.3 Step 3: decomposing the braiding process
After splitting as described above, our braiding process reduces to one in which we braid and around and for times (Fig. 6). The next step is to decompose this process into a series of simpler processes.
First we introduce some notation. Let be the initial state at the beginning of the braiding, i.e. the state obtained from the deformation process shown in Fig. 5 followed by the splitting process in Fig. 6. Let be the subspace consisting of all states that are degenerate in energy with . Here, could have dimension or higher. Let be the unitary braiding matrix that describes the (possibly) non-Abelian Berry phase associated with braiding and around and once.
Then, with this notation, the fact that the Berry phase for our braiding process is translates to the equation
To proceed further, it is helpful to represent this braiding process using a 2D picture. One way to do this is to imagine folding the surface and straightening the vortex arches, as shown in Fig. 7. After doing this, the braiding process now involves vertical vortex lines. If we now take a top view of Fig. 7, then can be visualized as
With this picture in mind, it is easy to see how to decompose the above braiding process into simpler processes. In particular, let , , , and be the braid matrices corresponding to the following processes:
Then, it is easy to see that if we perform the above four processes sequentially, the result can be smoothly deformed into the process corresponding to . Translating this into algebra, we derive:
Substituting this into (38), we obtain:
Equation (45) is the main result of this step.
v.4 Step 4: evaluating the expression (45)
We now evaluate the expression on the left-hand side of Eq. (45). We accomplish this using several algebraic properties of the braid matrices , and that we will derive below. The first property is that these braid matrices all commute with each other, except for the pairs and which obey the commutation relations
where are given by
Another important property of these matrices is that when we raise them to the power , they simplify considerably:
where is the identity matrix in , and are given by