Braiding statistics approach to symmetry-protected topological phases
We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a “symmetry-protected topological phase.” We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a gauge field and then show that the -flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.
We now know that there are two distinct types of time reversal invariant band insulators: topological insulators and conventional insulators.(1); (2); (3); (4); (5); (6) The two families of insulators are distinguished by the fact that topological insulators have protected gapless boundary modes while trivial insulators do not. It is important to remember that time reversal and charge conservation symmetry play a crucial role in this physics: if either of these symmetries are broken (either explicitly or spontaneously), the boundary modes can be gapped out and the sharp distinction between topological insulators and conventional insulators disappears.
This observation motivates a generalization of topological insulators called “symmetry-protected topological (SPT) phases”(7); (8); (9); (10); (11); (12); (13); (14); (15). To define this concept, consider a general quantum many-body system. The system may be built out of fermions or bosons/spins, and can live in any spatial dimension. We will say that such a system belongs to a nontrivial SPT phase if it satisfies four properties. The first property is that the system has a finite energy gap to excitations in the bulk. The second property is that the Hamiltonian is invariant under some set of internal (on-site) symmetries, and none of these symmetries are broken spontaneously. The third property is that the ground state belongs to a distinct quantum phase from a “trivial state” with the same symmetry. That is, one cannot continuously connect the ground state with a “trivial state” without breaking one of the symmetries or closing the energy gap. Here, by a “trivial state”, we mean a product state (in the boson/spin case) or an atomic insulator (in the fermion case). The final property of an SPT phase is that the ground state can be continuously connected with a trivial state without closing the energy gap if one or more of the symmetries are broken during the process. We note that nontrivial SPT phases typically exhibit robust gapless boundary modes analogous to that of topological insulators, though we will not include this property in the formal definition.
Symmetry-protected topological phases have a long history in the one dimensional (1D) case. Most famously, the Haldane phase of the Heisenberg antiferromagnet(16) is known to belong to this class(7); (12); (13). More recently, a complete classification of 1D SPT phases was obtained for both boson/spin systems(8); (14); (9) and fermion systems.(15); (9)
Much less is known about higher dimensional SPT phases. In the case of fermion systems, our understanding is largely limited to non-interacting models such as topological insulators or superconductors. For these systems, an (almost) complete classification of SPT phases was obtained by Ref. (17); (18). In some cases, it is known that this classification scheme is not affected by interactions (e.g. the classification of topological insulators in two(19) and three(20); (21) dimensions). In general, however, this need not be the case(22) and consequently our understanding of interacting fermionic SPT phases in higher dimensions is incomplete.
The boson case has received even less attention, and will be our focus here. In this case, a major advance was made by the recent paper, Ref. (11). In that paper, the authors proposed a general classification scheme for bosonic SPT phases in general spatial dimension. Also, the authors constructed concrete microscopic models realizing each of these phases. This work established that the boson case is tractable even for interacting systems.
Nevertheless, a number of questions remain open. One problem is that we have not identified any physical properties that distinguish different SPT phases in the bulk. The boundary physics is also poorly understood: while Ref. (10) showed that the 2D SPT states have symmetry-protected gapless boundary modes, the problem for higher dimensions remains open.
In this work, we address these (and other) questions in the context of a simple example. Specifically, we consider the case of 2D spin systems with a Ising-like symmetry. According to Refs. (10); (11), there is exactly one nontrivial SPT phase with this symmetry. This phase can be thought of as a new kind of Ising paramagnet. Here, we construct an exactly soluble spin model that realizes this phase. We then derive three main results. Our first result is a simple argument that this model belongs to a distinct phase from a conventional Ising paramagnet. We derive this result by coupling the model to a gauge field. After following this procedure, we find that the resulting gauged spin model supports quasiparticle excitations with different braiding statistics from that of a conventional (gauged) paramagnet. More specifically, we find that in a conventional paramagnet, the -flux excitations have bosonic or fermionic statistics, while in the new paramagnet they have semionic statistics. It then follows immediately that the two paramagnets cannot be continuously connected without breaking the symmetry or closing the energy gap. Closely related to this observation, we show that the two spin models are “dual” to two previously studied lattice models – each of which realizes a different type of gauge theory. This duality establishes a connection between SPT phases and previous work(23) on the classification of topological gauge theories.
Our second result is a proof that the new paramagnet has gapless edge modes protected by Ising symmetry. Interestingly, our argument reveals that the protected edge states are deeply connected to the braiding statistics of the -fluxes. This approach to proving edge state protection is somewhat different from the original argument of Ref. (10) and may be more amenable to higher dimensional generalizations. In the final part of the paper, we analyze the protected edge modes at a more concrete level, focusing on a particular microscopic model of the edge. We derive a field theoretic description of the low energy modes, and analyze their stability to perturbations.
Although we focus our discussion on a particular SPT phase, we believe that our basic approach is more general. That is, we expect that in a large class of SPT phases, braiding statistics can be used to uniquely characterize the bulk and to derive the existence of protected boundary modes. We discuss these potential generalizations in the conclusion.
This paper is organized as follows. In section II, we describe spin models that realize both the conventional and the new kind of Ising paramagnet. In section III we show that the two spin models can be distinguished by the braiding statistics of the -flux excitations. In section IV we show that the two spin models are dual to two previously studied lattice models. In section V, we show that the -flux braiding statistics are directly connected to the existence of protected edge modes. Finally, in section VI we analyze a particular microscopic model for the edge.
Ii Two kinds of Ising paramagnets
To begin, consider the following spin- model defined on the triangular lattice (Fig. 1a):
This model describes a (conventional) Ising paramagnet. To see this, note that the system satisfies two properties. First, the Hamiltonian is invariant under the Ising symmetry . Second, the ground state is gapped and unique – implying that the symmetry is not broken spontaneously.
Surprisingly, there is another type of Ising paramagnet which is qualitatively different from and represents a distinct quantum phase. A microscopic model for this new type of paramagnet was first constructed in Ref. (10). Here we describe another model which is more convenient for our purposes. The model we consider is a spin- system on the triangular lattice. The Hamiltonian is given by (Fig. 1b):
where the product runs over the six triangles containing the site . We note that this Hamiltonian is Hermitian despite the factors of . To see this, notice that the product includes a factor of for each pair of neighboring spins that have opposite values of . In particular, since the number of such pairs is necessarily even, the product always reduces to a factor of . It is then clear that . (For readers who are curious as to how this model was constructed, see section IV).
First we show that describes a paramagnetic phase – that is, the Ising symmetry is not spontaneously broken. To establish this fact, we solve explicitly. The key point is that
as can be verified by straightforward algebra. As a result, we can simultaneously diagonalize . We will label the simultaneous eigenstates by where denotes the eigenvalues of . It is not hard to show that there is a unique state for each choice of , assuming a periodic geometry (i.e. a torus). In other words, the are a complete set of quantum numbers. We therefore have the full energy spectrum: each state is an energy eigenstate with energy
In particular, the ground state is unique and gapped – implying that the Ising symmetry is not spontaneously broken.
It is illuminating to compare the ground state wave functions of . The ground state of is the state where everywhere. Working in the basis, the wave function is given by
for all spin configurations (Fig. 2a). As for , we note that the ground state is the unique state with everywhere. It is straightforward to check that the corresponding wave function is given by
where is the total number of domain walls in the spin-configuration (Fig. 2b). We can see that the two ground states are nearly identical, differing only by some phase factors. Nevertheless, these two states belong to two different quantum phases, as we now show.
Iii Coupling the spin models to a gauge field
In this section, we show that belong to distinct quantum phases. Our strategy is as follows. Because have a symmetry, we can couple them to a gauge field which lives on the links of the triangular lattice. We then show that the resulting gauged spin models have quasiparticle excitations with different braiding statistics. More specifically, we show that the two systems differ in the statistics of the -flux excitations: while the -fluxes have bosonic or fermionic statistics in the case of , they have semionic statistics in the case of . It then follows immediately that cannot be continuously connected without breaking the symmetry or closing the energy gap.
Coupling to a gauge field requires several steps.(24) The first step is to apply the minimal coupling procedure, replacing nearest neighbor spin-spin interactions like with . Next, we multiply each term in the resulting Hamiltonian (either or ) by the operator
where the product runs over the six triangles adjacent to site . The operator is a projector which projects onto states that have vanishing flux through each of the adjoining triangles. We include this projection operator in order to ensure that our gauged Hamiltonian is Hermitian, and also to make the minimal coupling procedure unambiguous. (For more general models, we would replace with an operator that projects onto states that have vanishing flux through all the triangles in the vicinity of the spin-spin interactions). The final step is to add a term of the form to the Hamiltonian. This term ensures that the states with vanishing flux have the lowest energy. The resulting models are given by (Fig. 3):
Like all gauge theories, these models are defined on a Hilbert space consisting of gauge invariant states – that is, all states satisfying the constraint
for all sites .(24) This constraint can be thought of as a analog of Gauss’ law, .
Importantly, all the terms in commute with one another so these Hamiltonians can be solved exactly just like the ungauged spin models . In particular, it is easy to verify that both models have a finite energy gap.
The next task is to construct the quasiparticle excitations and show that they have different braiding statistics in the two systems. The quickest way to derive this fact is to note that can be exactly mapped onto the previously studied “toric code”(25); (26) and “doubled semion”(26) models. These two models have been analyzed in detail and are known to support quasiparticle excitations with different statistics.(26) A description of these models as well as the mapping to is given in section IV.
Alternatively, we can directly compute the quasiparticle statistics of and show that they are different. The first type of excitation is a “spin-flip”, which we will denote by . These excitations correspond to sites where for the case of , or for the case of . The second type of excitation is the “-flux”, . These excitations correspond to triangular plaquettes where . In fact, there are two types of -flux excitations, which differ by the addition of a spin-flip: .
It is clear that in both systems, if we braid a spin-flip excitation around either of the -flux excitations , the resulting statistical Berry phase is (in some sense this is the definition of a -flux excitation). It is also intuitively clear that the spin-flip excitation is a boson in both models. All that remains is to understand the statistics of the -fluxes. As we will now show, this is where the two models differ.
To determine the -flux statistics, we first identify operators that create these excitations. Like all quasiparticles with nontrivial braiding statistics, the -fluxes can be created using an extended string-like operator.(27) If we apply these string-like operators to the ground state, the result is a pair of -flux excitations – one at each end of the string. In the case of , the following string operator does the job:
Here is a path in the dual honeycomb lattice joining the two triangular plaquettes,
and the product runs over all links crossing (Fig. 4). We can verify that
creates -flux excitations at the two endpoints of by noting that anticommutes
with the flux through the two triangles at the ends of . At the same
time, this operator commutes with all the other terms in so it does not create any additional
In general, one of the most important aspects of string operators is the commutation relations satisfied by two intersecting strings. Let be two paths on the dual honeycomb lattice that intersect one another. Using the definition (11), we can see that the two corresponding string operators commute with one another:
This string algebra is important because we can use it to find the statistics of the quasiparticle .(25); (27); (26) One way to see this is to consider the special case where is a closed path and is an open path, as in Fig. 5. In this case, the two operators and have different physical interpretations: while the operator can be thought of describing a physical process in which two -fluxes are created and then moved to the endpoints of , the operator does not create any excitations at all. In fact, it is easy to check that exactly commutes with the Hamiltonian whenever forms a closed loop. This suggests that should be thought of as describing a three step process in which (1) two -fluxes are created, (2) one of the -fluxes moves all the way around the closed path , and then (3) the two -fluxes are annihilated. Using this interpretation, we can see that the state is the end result of a process in which two -fluxes are created at the endpoints of , and then afterwards another -flux is braided around one of the endpoints and annihilated with its partner. In contrast, the state corresponds to executing these two steps in the opposite order. Comparing these two processes, we expect that they will differ by a phase factor which is exactly the statistical Berry phase associated with braiding one -flux around another. In other words, the phase difference between these two states should be where is the exchange statistics for the particles:
In light of this relation, equation (12) implies that or . That is, is either a boson or a fermion. A similar analysis shows that the other -flux excitation, , is also either a boson or fermion. In fact, with a bit more work one can establish the more precise result that is boson and is a fermion. The difference in statistics between comes from the fact that where have mutual statistics . However, we will not need this more detailed result here. (See Refs. (26); (27) for an analogous calculation for the closely related “toric code” model).
We can repeat the same analysis for . In this case, the following string operator creates a -flux excitation:
Here, the first product runs over all links crossing . The next two products run over all triangles along the path such that are to the right of or to the left of respectively (Fig. 6). The last product runs over all triangles along . The operator is defined by
As in the previous case, one can check anticommutes with the flux through the two triangles at the ends of , but commutes with the Hamiltonian everywhere else. Hence, if we apply to the ground state, it creates -fluxes at the two endpoints of . We will again denote this -flux excitation by . (For readers who are curious, was constructed from the “doubled semion model” string operators(26) using the exact mapping of section IV).
In this case, one can check that the string operators satisfy a slightly different algebra: for any two paths intersecting one another, we have
Therefore by the same reasoning as in (13), we conclude that the statistical angle satisfies , so that . In other words, is a semion. A similar analysis shows that the other -flux excitation is also a semion. With a bit more work(26), one can show that have opposite statistics – that is in one case and in the other – but again we do not need this more detailed result here.
We have shown that the -fluxes have different statistics in the two gauged spin models: these excitations are bosons or fermions in the case of , and are semions in the case of . This result provides a simple physical distinction between the two systems. It also proves that the two spin models cannot be continuously connected with one another without breaking the symmetry or closing the energy gap. Indeed, if such a path existed, then we could construct a corresponding path connecting the gauged spin models – a contradiction. We note, however, that the above argument does not rule out the possibility of connecting if the Ising symmetry is broken during the process. Indeed, in appendix A we construct an explicit path of this kind.
Iv Duality between spin models and string models
In this section we explain the relationship between the spin Hamiltonians , and previously known models. Specifically, we show that are related via a duality map to two previously studied lattice models – the “toric code” model(25); (26) and the “doubled semion” model.(26) The latter two models are sometimes called “string models” and are special cases of the general class of “string-net” models constructed in Ref. (26). This duality provides another point of view on the braiding statistics analysis in the previous section, and also suggests a natural classification scheme for general 2D bosonic SPT phases with finite unitary symmetry groups.
We begin by defining the duality map: we note that every spin configuration on the triangular lattice defines a corresponding domain wall configuration on the honeycomb lattice. Formally, this correspondence is given by where is the link separating sites and corresponds to the presence or absence of a domain wall. We will refer to these domain walls as “strings.” An important point is that the dual string degrees of freedom always form closed loops – that is, they satisfy the condition where (Fig. 7)
These Hamiltonians are defined on a Hilbert space consisting of closed string states (i.e. states satisfying everywhere).
The dual Hamiltonians are closely related to two models studied in Ref. (26): the “toric code model”(25) and the “doubled semion” model. To understand the precise relationship, recall that the latter two models are defined on a Hilbert space consisting of all string states on the honeycomb lattice – both open and closed. The two Hamiltonians are (Fig. 7)
Here denotes the projector . This operator defines a projection onto states that satisfy the closed string constraint at all vertices of the plaquette .
Comparing (LABEL:Hdual) and (18), we see that can be obtained by restricting to the closed string () subspace. In other words, the spin models are dual to a restricted variant of the toric code and doubled semion models.
In fact, this duality can be extended to one that maps the gauged spin models onto the unrestricted toric code and doubled semion models (LABEL:Hdual). The extended duality is defined by setting where is the link separating sites . Substituting these expressions into and making use of the gauge invariance constraint (10) it is easy to check that the result is exactly . We note that this duality maps local operators onto local (gauge invariant) operators and should therefore be thought of as an exact equivalence between two quantum systems. Thus the gauged spin models are physically identical to the toric code and doubled semion models.
The above dualities are variants of the well-known correspondence between the 2D Ising model and 2D gauge theory.(29); (24) To see this, note that the closed string models are simply gauge theory Hamiltonians, phrased in the language of strings. The Hamiltonian is the conventional(29); (24) gauge theory Hamiltonian (in the zero coupling limit where there is no electric energy term ), while is another kind(23) of gauge theory. From this point of view, the correspondence between and is a duality between two types of 2D Ising paramagnets, and two types of 2D gauge theory.
We can understand the duality between and in a similar way. We note that the first two models can be thought of as two types of Ising paramagnets coupled to (conventional) gauge theory, while the latter two models can be thought of as two types of gauge theory coupled to a (conventional) Ising paramagnet. Hence the duality between and is a variant of the well-known self-duality of 2D gauge theory coupled to Ising matter.(24)
We expect that these dualities can be generalized from to any finite unitary symmetry group : each SPT phase with symmetry group is dual to a corresponding gauge theory with gauge group . This correspondence immediately suggests a classification scheme for 2D bosonic SPT phases with finite unitary symmetry groups: it is known that the different types of 2D gauge theories with group (or equivalently, different string-net models corresponding to ) are in one-to-one correspondence with elements of . (For a derivation of this result, see Ref. (23), and also section 10.1.E.3 of Ref. (30)). Hence, the duality map suggests that different SPT phases associated with symmetry group can also be classified by . This classification scheme is identical to the proposal of Ref. (11).
Another application of these dualities is that they give a simple method for constructing exactly soluble
models for bosonic SPT phases with finite unitary symmetry group . The first step is to construct the different “string-net”
models(26) corresponding to the group . These are models with string types given by the group elements , and branching
rules given by group multiplication: is an allowed branching if . In general, there will be a
finite number of different models with these branching rules – each one corresponding to a different solution of the self-consistency
equations of Ref. (26).
V Protected edge modes and braiding statistics
The most dramatic distinction between the two types of paramagnets is that has protected gapless edge modes, while does not. In other words, if we define in a geometry with a boundary, then the energy spectrum always contains gapless excitations. These gapless excitations are guaranteed to be present as long as the Ising symmetry is not broken (explicitly or spontaneously). In this section, we give a general argument proving this fact. Our argument reveals that these edge modes are closely connected to the semionic braiding statistics of the -flux excitations in the gauged spin model, . We note that the existence of protected edge modes was previously established in Ref. (10) using a different approach.
The statement we prove is as follows. We consider a disk geometry with a Hamiltonian of the form
where is defined as in (2) and the sum runs over all sites lying strictly in the interior of the disk. We take the edge Hamiltonian to be any Hamiltonian with local interactions which acts on the spins on or near the boundary of the disk. In this setup, it is clear that the ground state of satisfies when is far from the edge; in fact, in order to simplify the discussion, we will assume that for all lying strictly in the interior of the disk. Given these assumptions, we will show that cannot be both Ising symmetric and short-range entangled. Here, a state is “short-range entangled” if it can be transformed into a product state by a local unitary transformation – a unitary operator generated from the time evolution of a local Hamiltonian over a finite time .(8)
To understand what this result means, recall that is always Ising symmetric and short-range entangled in the bulk (see appendix A). Thus, the implication of the above theorem is that the edge either breaks the Ising symmetry or is not a short-range state. In the latter case, the edge is presumably gapless, so in this way we see that the edge is protected.
v.1 Physical argument
The argument is a proof by contradiction: we assume that is both Ising symmetric and short-range entangled and we show that these assumptions lead to a contradiction. The first step is to consider a thought experiment in which we create a pair of -fluxes in the bulk and move them along some path to two points at the boundary (Fig. 8a). This process can be implemented by applying an appropriate unitary operator to the state . We will denote this operator by . By construction contains two fluxes located near points on the boundary.
We next assert that the -fluxes at the boundary can be annihilated by local operators. In other words, there exist local operators acting near such that . To see this, note that the effect of bringing the -flux excitations to the edge is to create two Ising domain walls at points . Given that is Ising symmetric and short-range entangled, these domain walls are local excitations – that is, the two states, have identical expectation values far from . It then follows that these two states can be connected by local operators acting near these points. We emphasize that this conclusion depends crucially on the Ising symmetry of : if instead broke the Ising symmetry, the domain walls at would be nonlocal excitations, and there would be no way to annihilate them with local operators.
We now use the fact that -fluxes can be annihilated at the boundary to derive a contradiction. Consider a three step process in which two -fluxes are (1) created in the bulk, (2) moved to the boundary along the path , and (3) annihilated. Let be a unitary operator describing this process. (Formally, is given by ). Consider a second path with the geometry shown in Fig. 8b, and define in the same way. By construction, we have . Hence,
At the same time, it follows from general principles that satisfy the commutation relation
where is the exchange statistics for the -fluxes. (This result can be derived in the same way as Eq. (13)). To complete the argument, we note that the -fluxes have semionic statistics so . Equations (21),(22) are therefore in contradiction, implying that our assumption must be false and the ground state cannot be both Ising symmetric and short-range entangled.
In this analysis, we have skated over an important subtlety. The issue is that we do not know whether are even or odd under the Ising symmetry. In other words, we do not know whether the flux annihilation process involves flipping an even or odd number of spins. To understand what this means, recall that there are actually two types of -flux excitations which differ from one another by the addition of a spin-flip excitation . Thus, the operators could describe the annihilation of either one of the two types of -fluxes, depending on their parity. Since this parity is ambiguous, the existence of only shows that at least one of the two types of -fluxes can be annihilated at the boundary.
This subtlety becomes important in the last part of the argument where we derive a contradiction between equations (21),(22). In particular, since we can only guarantee that one of the two types of -fluxes can be annihilated at the boundary, the proof is only valid if we show that these equations are inconsistent for both types of fluxes. Fortunately, this is not a problem: the two types of -fluxes have exchange statistics , so in both cases.
v.2 Discussion and generalizations
The above argument does not use any properties of except the braiding statistics of the -fluxes. Therefore, it actually proves a more general statement: any SPT phase in which neither of the -fluxes is a boson or a fermion is guaranteed to have a protected edge mode. Indeed, as long as for both types of fluxes, the argument goes through unchanged. On the other hand, if either of the -fluxes is a boson or a fermion – as in a conventional paramagnet – there is no contradiction between equations (21), (22) and the argument breaks down completely. From this point of view, the key reason that has a protected edge mode and doesn’t, is the difference in their -flux braiding statistics.
It is not hard to generalize the argument to arbitrary bosonic SPT phases with unitary abelian symmetry groups . For example, consider the case of . Just as spin models support -flux excitations, models with symmetry support flux excitations with flux and . These -fluxes and -fluxes each come in three different types – just like the two types of -fluxes in the case. Using the same arguments as above, one can see that a SPT phase must have a protected edge unless there exists a set of two fluxes – consisting of one -flux and one -flux – such that (1) the fluxes in this set are bosons or fermions and (2) the fluxes in this set have trivial mutual statistics with respect to one another. Similarly to the case, this result can be derived by considering thought experiments where we annihilate and -fluxes at the boundary, and making use of the string commutation algebra (22). In fact, by using the statistical hopping algebra(27) in place of (22), we believe that this result can be strengthened even further: one can show the existence of a protected edge mode unless the above set of fluxes are all bosons. We expect that similar generalizations exist for the non-abelian case although we will not discuss them here.
v.3 Mathematical argument
Like the physical argument sketched above, the mathematical argument is a proof by contradiction. We assume that is both Ising symmetric and short range entangled (i.e. it can be turned into a product state by a local unitary transformation) and we show that these assumptions lead to a contradiction.
To begin, let be a path on the dual (honeycomb) lattice that joins two points on the edge. We define an associated unitary operator by
Here, the first product runs over all sites in the interior of the the path , while the last two products run over all triangles along the path such that are to the right of or to the left of respectively (Fig. 9). The operator is defined by
As an aside, we note that the operator is closely related to the string operator (14). Indeed the two operators are identical except for the fact that is written in terms of the formalism of the gauged spin model, while is written in terms of the original “ungauged” spin model. This similarity suggests a simple physical interpretation for : this operator describes a process in which two -fluxes are created in the bulk and then moved along the path to points at the boundary. Much of what follows can be understood using this physical picture, as discussed in section V.1.
Returning to the main argument, we note that the unitary operator has several important properties:
transforms local operators into local operators. That is, is local if and only if is local.
Let be a local operator which acts on spins within some convex region not containing either of the endpoints of . Then has the same expectation value in the two states and .
Property 1 follows from the fact that can be decomposed into a product
of two sets of commuting local unitary operators. As for property 2, there are three cases to consider:
the region of support may be contained entirely in the exterior of , it may be contained entirely in the
interior, or it may overlap the path itself. In the first case, commutes with , immediately
implying the desired equality . In the second case,
, since acts like in the interior of . Then, since
is invariant under (by the Ising symmetry assumption), we again have
. The only case where the expectation value of could be different in
the two states is if overlaps the path . However, one can check that
for any two paths with the same endpoints.
The first step is to observe that has short-range correlations (i.e., for any well separated local operators , we have up to corrections which are exponentially small in the distance between ). To see this, note that has short-range correlations since(33) it can be transformed into a product state by a local unitary transformation (by the short-range entanglement assumption). It then follows that also has short range correlations since transforms local operators into local operators (property 1).
Next we recall that share the same local expectation values away from the endpoints (property 2). Putting these facts together, we can immediately deduce the existence of the desired . To see this, consider the analogous question for the conventional paramagnet : suppose that some short-range correlated state has the same local expectation values as except near two points . In this case, the state must have far from , so it is clear that we can find local operators acting near such that . Having established this property for , it follows that the same property must also hold for since are equivalent up to a local unitary transformation (by the short-range entanglement assumption).
A key question is to understand understand how transform under the Ising symmetry . In appendix C, we show that can always be chosen so that they are either both even or both odd under . Furthermore, this even or odd parity must be the same for all pairs of endpoints . In other words, either all the operators are even under , or all of them are odd under .
We now use (25) to derive a contradiction. To this end, we consider a second path that connects two other points on the edge. We choose so that they intersect each other, and so that their endpoints are well separated (see Fig. 8b). As above, we have for some local operators acting near . Now, define
By construction, and . Hence,
At the same time, anti-commute, as we now show. To see this, we first note that anti-commute:
This relation can be checked using the explicit formula for (similarly to eq. (16)). Next, we recall that looks like in the interior of and the identity map in the exterior of so that
where the sign is determined by the parity of under . Similarly, we have
where the sign is determined by the parity of under . Importantly, these two signs are the same since the operators all share the same parity. Hence, the two pairs and either both commute or both anti-commute. In either case, the anti-commutation relation (28) implies that anti-commute:
Vi Microscopic edge analysis
In this section, we investigate the protected edge modes of at a more concrete level. We analyze a particular example of a gapless edge for , derive a field theoretic description of the low energy modes, and investigate the effect of perturbations. As in section V, we consider a disk geometry, with a Hamiltonian of the form . The bulk Hamiltonian is defined by where the sum runs over all sites that are strictly in the interior of the disk. The edge Hamiltonian can be any Ising symmetric Hamiltonian with local interactions which acts on the spins on or near the boundary.
vi.1 Zero energy edge states
We begin with the case where – that is, the edge Hamiltonian vanishes. In this case, we can compute the energy spectrum in the same way as we did for the periodic (torus) geometry. First, we simultaneously diagonalize the operators for all sites that are strictly in the interior of the disk. Next, we note that each of these simultaneous eigenstates is an energy eigenstate with energy where is the eigenvalue under . The final step is to determine the degeneracy of these simultaneous eigenspaces. A natural guess, based on dimension counting, is that each simultaneous eigenspace has a degeneracy of , where is the number of spins along the boundary of the disk. In particular, we expect that there are degenerate ground states.
We can verify this counting by constructing explicit wave functions for these degenerate ground states. Specifically, we define a wave function for each boundary spin configuration , where (Fig. 10a). This wave function is a function of the spins lying strictly in the interior of the disk, and is given by
where is the the total number of domain walls in the system. Here, we define using a particular convention where we close up all the domain walls that end at the boundary by assuming that there is a “ghost” spin in the exterior of the disk, pointing in the direction (Fig. 10b). We will denote these states by . As is apparent from this parameterization, we can think of these degenerate ground states as zero energy edge states.
It is useful to define operators that act on just like the usual Pauli spin operators. We note that the operators should not be confused with the physical boundary spin operators which act on the full Hilbert space of the spin system. In the case, the two types of operators are closely related – for example, where is the projection operator onto the dimensional edge state subspace. However, this simple relation does not hold for the or operators, or for more complicated products of spin operators.
An important question is to understand how the symmetry acts on the edge states. Using the definition (32), one finds that the Ising symmetry acts as
where the sign depends on the configuration of as follows: the sign is if the total number of domain walls between the ’s is divisible by and otherwise. In other words, the action of the Ising symmetry on the above basis states is described by the operator
In order to gain some intuition about , we note that the operators transform under the symmetry according to
vi.2 An example of an edge Hamiltonian
We now imagine adding a nonvanishing edge Hamiltonian . If is small, then we can analyze its effect using degenerate perturbation theory. The first order splitting of the degenerate ground states can be obtained by diagonalizing where is the projection onto the zero energy edge state subspace. In general, can be expressed as a function of the operators. We can therefore find the edge state spectrum by solving a spin chain with an unusual Ising symmetry (34).(10)