A Lattices and Matrices

Short-Range Entangled Bosonic States with Chiral Edge Modes and $T$-duality of Heterotic Strings


We consider states of bosons in two dimensions that do not support anyons in the bulk, but nevertheless have stable chiral edge modes that are protected even without any symmetry. Such states must have edge modes with central charge for integer . While there is a single such state with , there are, naively, two such states with , corresponding to the two distinct even unimodular lattices in dimensions. However, we show that these two phases are the same in the bulk, which is a consequence of the uniqueness of signature even unimodular lattices. The bulk phases are stably equivalent, in a sense that we make precise. However, there are two different phases of the edge corresponding to these two lattices, thereby realizing a novel form of the bulk-edge correspondence. Two distinct fully chiral edge phases are associated with the same bulk phase, which is consistent with the uniqueness of the bulk since the transition between them, which is generically first-order, can occur purely at the edge. Our construction is closely related to -duality of toroidally compactified heterotic strings. We discuss generalizations of these results.

I Introduction

The last decade has seen enormous progress in the understanding of topological phases (see Ref. Nayak et al., 2008 and references therein) and of symmetry-protected topological (SPT) phases Chen et al. (2011, ); Kit (a); Lu and Vishwanath (2012). SPT phases are gapped phases of matter that do not have non-trivial excitations in the bulk; have vanishing topological entanglement entropy Kitaev and Preskill (2006); Levin and Wen (2006) or, equivalently, have short-ranged entanglement (SRE); but have gapless excitations at the edge in the presence of a symmetry. In the case of the most famous and best-understood example, ‘topological insulators’ (see Refs. Kane and Mele, 2005a, b; bhz, ; Fu et al., 2007; Moore and Balents, 2007; Roy, 2009; Qi et al., 2008; Schnyder et al., 2008; Hasan and Kane, 2010; Qi and Zhang, 2011 and references therein), the symmetry is time-reversal. Topological phases (without a modifier) are gapped phases of matter that are stable to arbitrary perturbations; support anyons in the bulk; and have non-zero topological entanglement entropy or, equivalently, have long-ranged entanglement (LRE). They may or may not (depending on the topological phase) have gapless edge excitations.1

However, there is a third possibility: phases of matter that do not support anyons but nevertheless have gapless excitations even in the absence of any symmetry. Thus, they lie somewhere between topological phases and symmetry-protected topological phases, but are neither. Integer quantum Hall states of fermions are a well-known example. Their gapless edge excitations Laughlin (1981); Halperin (1982) are stable to arbitrary weak perturbations even though they do not support anyons and only have SRE2. Although the existence and stability of SRE integer quantum Hall states might seem to be a special feature of fermions, such states also exist in purely bosonic systems, albeit with some peculiar features.

For any integer , there is an integer quantum Hall state of fermions with SRE, electrical Hall conductance , and thermal Hall conductance . Kane and Fisher (1997) In fact, there is only one such state for each : any two SRE states of fermions at the same filling fraction can be transformed into each other without encountering a phase transition. 3 (This is true in the bulk; see Section VII.2 for the situation at the edge.) Therefore, the state with filled Landau levels of non-interacting fermions is representative of an entire universality class of SRE states. As a result of its chiral Dirac fermion edge modes, this is a distinct universality class from ordinary band insulators. These edge modes, which have Virasoro central charge if all of the velocities are equal, are stable to all perturbations. If we do not require charge conservation symmetry, then some Hamiltonians in this universality class may not have , but they will all have .

Turning now to bosons, there are SRE states of bosons with similarly stable chiral edge modes, but only for central charges . As we discuss, they correspond to even, positive-definite, unimodular lattices. Moreover, while there is a unique such state with , there appear to be two with , twenty-four with , and more than ten million with . con () Thus, we are faced with the possibility that there are many SRE bosonic states with the same thermal Hall conductance , presumably distinguished by a more subtle invariant. In this paper, we show that this is not the case for . The two SRE bosonic states with edge excitations are equivalent in the bulk: their partition functions on arbitrary closed manifolds are equal. However, there are two distinct chiral edge phases of this unique bulk state. They are connected by an edge reconstruction: a phase transition must be encountered at the edge in going from one state to the other, but this transition can occur solely at the edge and the gap need not close in the bulk. Although we focus on the case, the logic of our analysis readily generalizes. Therefore, we claim that there is essentially a unique bulk bosonic phase for each given by copies of the so-called -state Kit (a); Lu and Vishwanath (2012). However, there are two distinct fully-chiral edge phases with , twenty-four with , more than ten million with , and even more for larger .

One important subtlety arises in our analysis. The two phases do not, initially, appear to be identical. However, when combined with a trivial insulating phase, the two bulk partition functions can be mapped directly into each other by a change of variables. This is a physical realization of the mathematical notion of stable equivalence. In general, an effective description of a phase of matter will neglect many gapped degrees of freedom (e.g., the electrons in inner shells). However, the sequence of gapped Hamiltonians that interpolates between two gapped Hamiltonians may involve mixing with these usually-forgotten gapped degrees of freedom. Therefore, it is natural, in considering a phase of matter, to allow an arbitrary enlargement of the Hilbert space by trivial gapped degrees of freedom (i.e., by SRE phases without gapless edge excitations). This is useful when, for instance, comparing a trivial insulating phase with bands with another trivial insulating phase with bands. They can be adiabatically connected if we are allowed to append trivial insulating bands to the latter system. This notion is also natural when connecting different phases of gapless edge excitations. The edge of a gapped bulk state will generically have gapped excitations that we ordinarily ignore. However, they can become gapless – which is a form of edge reconstruction – and interact with the other gapless degrees of freedom, driving the edge into a different phase. However, this does not require any change in the bulk. As we will see, such a purely edge phase transition connects the two seemingly different chiral gapped edges with . By combining a state with a trivial insulator, we are able to take advantage of the uniqueness of signature even unimodular lattices Ser (), from which it follows that the two phases are the same. This is closely-related to the fact that -duality exchanges toroidal compactifications of the and versions of the heterotic string, as explained by Ginsparg Ginsparg (1987).

In the remainder of this paper, we describe the equivalence of the two candidate phases at from two complementary perspectives. To set the stage, we begin in Section II with a short introduction to the -matrix formalism that we use to describe the phases of matter studied in this paper. In Section III, we provide a bulk description of the equivalence of the two candidate phases at . We then turn to the edge, where we show that there are two distinct chiral phases of the edge. We first discuss the fermionic description of the edge modes in Section IV and then turn to the bosonic description in Section V. There is an (purely) edge transition between these two phases. We discuss the phase diagram of the edge, which is rather intricate, and its relation to the bulk. In Section VI, we summarize how the phase diagram can change when some of the degrees of freedom are electromagnetically charged so that a symmetry is preserved. We then conclude in Section VII and discuss possible generalizations of this picture.

In Appendix A, we collect basic definitions and explain the notation used throughout the text. In Appendix B, we provide some technical details for an argument used in the main text.

Ii K-matrix Formalism

ii.1 Chern-Simons Theory

We will consider -dimensional phases of matter governed by bulk effective field theories of the form:


where , for and . See Refs. Wen and Zee, 1992 and Wen, 1995 for a pedagogical introduction to such phases. is a symmetric, non-degenerate integer matrix. (Repeated indices should be summed over unless otherwise specified.) We normalize the gauge fields and sources so that fluxes that are multiples of are unobservable by the Aharonov-Bohm effect. Consequently, if we take the sources to be given by prescribed non-dynamical classical trajectories that serve as sources of flux, they must take the form:


for integers . The sum over is a sum over the possible sources .

Therefore, each excitation of the system is associated with an integer vector . These integer vectors can be associated with the points of a lattice as follows. Let for be the eigenvalues of with the corresponding eigenvectors. We normalize the so that where . Now suppose that we view the as the components of a vector (i.e., of with a metric of signature ), where has positive eigenvalues and negative ones. In other words, the unit vector with a in the a-th entry and zeros otherwise is an orthonormal basis of so that . Then we can define . Thus, the eigenvectors define a lattice in according to ; this lattice determines the allowed excitations of the system Read (1990); Blok and Wen (1990).

The lattice enters directly into the computation of various physical observables. For example, consider two distinct excitations corresponding to the lattice vectors and in . If one excitation is taken fully around the other, then the resulting wavefunction differs from its original value by the exponential of the Berry’s phase . When the excitations are identical, , a half-braid is sufficient and a phase equal to is obtained.

Of course, any basis of the lattice is equally good; there is nothing special about the basis . We can change to a different basis , where . ( must have integer entries since it relates one set of lattice vectors to another. Its inverse must also be an integer matrix since either set must be able to serve as a basis. But since , and can both be integer matrices only if .) This lattice change of basis can be interpreted as the field redefinitions, and , in terms of which the Lagrangian (1) becomes


where . Therefore, two theories are physically identical if their -matrices are related by such a similarity transformation.

We note that the low energy phases described here may be further sub-divided according to their coupling to the electromagnetic field, which is determined by the -component vector :


It is possible for two theories with the same -matrix to correspond to different phases if they have different vectors since they may have different Hall conductances . (It is also possible for discrete global symmetries, such as time-reversal, to act differently on theories with the same -matrix in which case they can lead to different SPT phases if that symmetry is present.)

In this paper, we will be interested in states of matter in which all excitations have bosonic braiding properties, i.e., in which any exchange of identical particles or full braid of distinguishable particles leads to a phase that is a multiple of . Hence, we are interested in lattices for which is an integer for all and is an even integer if . Hence, is a symmetric integer matrix with even entries on the diagonal. By definition must also an integer matrix. Since both and are integer matrices, their determinant must be . Because (no summation on ) and , the lattice is said to be an even unimodular lattice.

It is convenient to introduce the (dual) vectors . If, as above, we view the as the components of a vector according to , then . Moreover, is the basis of the dual lattice defined by . Since the lattice is unimodular, it is equal to , up to an rotation, from which we see that must be equivalent to , up to an change of basis. (In fact, the required change of basis is provided by the defining relation .)

Now consider the Lagrangian (5) on the spatial torus. For convenience, we assume there are no sources so . We can rewrite the Lagrangian as


where we have defined and . Choosing the gauge , , the Lagrangian takes the form:


Therefore, and are canonically conjugate. Although we have gauge-fixed the theory for small gauge transformations, under a large gauge transformation, where are integers (so that physical observables such as the Wilson loop about the 1-cycle remains invariant). Therefore, we must identify and since they are related by a gauge transformation.

Suppose that we write a ground state wavefunction in the form . Then will act by multiplication and its canonical conjugate will act by differentiation. To display the full gauge invariance of the wavefunction, , it is instructive to expand it in the form:


where . This is an expansion in eigenstates of , with the term having the eigenvalue . However, by gauge invariance, takes values in . Therefore, we should restrict such that lies inside the unit cell of . In other words, the number of ground states on the torus is equal to the number of sites of that lie inside the unit cell of . This is simply the ratio of the volumes of the unit cells, . It may be shown that this result generalizes to a ground state degeneracy on a genus surface Wen (1990). Therefore, the theories on which we focus in this paper have non-degenerate ground states on an arbitrary surface, which is another manifestation of the trivial braiding properties of its excitations.

One further manifestation of the trivial braiding properties of such a phase’s excitations is the bipartite entanglement entropy of the ground state Kitaev and Preskill (2006); Levin and Wen (2006). If a system with action (1) with is divided into two subsystems and and the reduced density matrix for subsystem is formed by tracing out the degrees of freedom of subsystem , then the von Neumann entropy takes the form:


Here, is a non-universal constant that vanishes for the action (1), but is non-zero if we include irrelevant sub-leading terms in the action (e.g., Maxwell terms for the gauge fields). is the length of the boundary between regions and . The denote terms with sub-leading dependence. For the theories that we will consider in this paper, the second term, which is universal, vanishes. For this reason, such phases are called “short-range entangled.”

The discussion around Eq. (8), though essentially correct as far as the ground state degeneracy is concerned, swept some subtleties under the rug. A more careful treatment Belov and Moore () uses holomorphic coordinates , in terms of which the wavefunctions are -functions. Moreover, the normalization must account for the fact that the wavefunction is a function only on the space of with vanishing field strength (which the gauge constraint requires), not on arbitrary . Consequently, it depends on the modular parameter of the torus as where are the number of positive and negative eigenvalues of ; the torus is defined by the parallelogram in the complex plane with corners at , , , and opposite sides identified; and is is the Dedekind function, where . Consequently, the ground state wavefunction transforms non-trivially under the mapping class group of the torus (i.e., under diffeomorphisms of the torus that are disconnected from the identity, modulo those that can be deformed to the identity) which is equal to the modular group generated by and . Under , which cuts open the torus along its longitude, twists one end of the resulting cylinder by , and then rejoins the two ends of the cylinder to reform the torus, thereby enacting , the ground state transforms according to . Therefore, so long as , the bulk is not really trivial.

ii.2 Edge Excitations

The non-trivial nature of these states is reflected in more dramatic fashion on surfaces with a boundary, where there may be gapless edge excitations. For simplicity, consider the disk with no sources in its interior Elitzur et al. (1989); Wen (1992). The action (1) is invariant under gauge transformations , where , so long as at the boundary . In order to fully specify the theory on a disk, we must fix the boundary conditions. Under a variation of the gauge fields , the variation of the action (here, is the time direction) is


Here is the radial coordinate on the disk. The action will be extremized by (i.e. there won’t be extra boundary terms in the equations of motion) so long as we take boundary conditions such that . We can take boundary condition , where is the azimuthal coordinate. Here is a symmetric matrix that is determined by non-universal properties of the edge such as how sharp it is. The Lagrangian (1) is invariant under all transformations that are consistent with this boundary condition. Only those with at the boundary are gauge symmetries. The rest are ordinary symmetries of the theory. Therefore, although all bulk degrees of freedom on the disk are fixed by gauge invariance and the Chern-Simons constraint, there are local degrees of freedom at the boundary.

The Chern-Simons constraint can be solved by taking or, writing , , where . This gauge field is pure gauge everywhere in the interior of the disk (i.e. we can locally set it to zero in the interior with a gauge transformation), but it is non-trivial on the boundary because we can only make gauge transformations that are consistent with the boundary condition. Substituting this expression into the action (1), we see that the action is a total derivative which can be integrated to give a purely boundary action:


The Hamiltonian associated with this action will be positive semi-definite if and only if has non-negative eigenvalues. If we define or, in components, , then we can rewrite this in the form


where . We see that the velocity matrix parameterizes density-density interactions between the edge modes. Note that the fields satisfy the periodicity conditions for .

This theory has different dimension- fields . The theory also has ‘vertex operators’, or exponentials of these fields that must be consistent with their periodicity conditions: or, equivalently, or, simply, for . They have correlation functions:


In this equation, , where is an matrix that diagonalizes . Its first columns are the normalized eigenvectors corresponding to positive eigenvalues of and the next columns are the normalized eigenvectors corresponding to negative eigenvalues of . The velocities are the absolute values of the eigenvalues of . Therefore, this operator has scaling dimension


The scaling dimensions of an operator in a non-chiral theory generally depend upon the velocity matrix . For a fully chiral edge, however, , so .

If the velocities all have the same absolute value, for all , then the theory is a conformal field theory with right and left Virasoro central charges and . Consequently, we can separately rescale the right- and left-moving coordinates: and . The field has right and left scaling dimension for and dimension for . Meanwhile, has scaling dimension:


which simplifies, for the case of a fully chiral edge, to .

In a slight abuse of terminology, we will call the state of matter described by Eq. (1) in the bulk and Eq. (11) on the edge a , bosonic SRE phase. In the case of fully chiral theories that have , we will sometimes simply call them bosonic SRE phases. Strictly speaking, the gapless edge excitations are only described by a conformal field theory when the velocities are all equal. However, we will continue to use this terminology even when the velocities are not equal, and we will use it to refer to both the bulk and edge theories.

In the case of a , bosonic SRE phase, all possible perturbations of the edge effective field theory Eq. (11) – or, equivalently, Eq. (12) – are chiral. Since such perturbations cannot open a gap, completely chiral edges are stable. A non-chiral edge may have a vertex operator with equal right- and left-scaling dimensions. If its total scaling dimension is less than , it will be relevant and can open a gap at weak coupling. More generally, we expect that a bosonic SRE will have stable gapless edge excitations if . Some of the degrees of freedom of the theory (11) will be gapped out, but some will remain gapless in the infrared (IR) limit and the remaining degrees of freedom will be fully chiral with and . Therefore, even if such a phase is not, initially, fully-chiral, the degrees of freedom that remain stable to arbitrary perturbations is fully chiral. Therefore, positive-definite even unimodular lattices correspond to , bosonic SRE phases with stable chiral edge excitations, in spite of the absence of anyons in the bulk.

ii.3 The Cases

Positive-definite even unimodular lattices only exist in dimension for integer , Ser () so bosonic SRE phases with stable chiral edge excitations must have . There is a unique positive-definite even unimodular lattice in dimension , up to an overall rotation of the lattice. There are two positive-definite even unimodular lattices in dimension ; there are in dimension ; there are more than in dimension ; and even more in higher dimensions. If we relax the condition of positive definiteness, then there are even unimodular lattices in all even dimensions; there is a unique one with signature for .

In dimension-2, the unique even unimodular lattice in , which we will call , has basis vectors , , and the corresponding -matrix is:


This matrix has signature . (Within this discussion, is an arbitrary parameter. It will later develop a physical meaning and play an important role in the phase transition we describe.) The even unimodular lattice of signature has a block diagonal -matrix with copies of along the diagonal:


The unique positive definite even unimodular lattice in dimension- is the lattice generated by the roots of the Lie algebra of . We call this lattice . The basis vectors for are given in Appendix A, and the corresponding -matrix takes the form:


The two positive-definite even unimodular lattices in dimension are the lattices generated by the roots of and . (The latter means that a basis for the lattice is given by the roots of , but with the root corresponding to the vector representation replaced by the weight of one of the spinor representations.) We will call these lattices and . They are discussed further in Appendix A. The corresponding -matrices take the form:


(for later convenience, we permute the rows and columns of the second copy of in Eq. (A.2) so that it looks superficially different from the first ) and


The even unimodular lattice with signature has K-matrix:


The even unimodular lattice with signature has K-matrix:


These lattices are unique, so the matrix,


is equivalent to (22) under an basis change. This fact will play an important role in the sections that follow.

Iii Equivalence of the Two Bosonic SRE Phases

In the previous section, we saw that two theories of the form (1) with different -matrices are equivalent if the two -matrices are related by an transformation or, equivalently, if they correspond to the same lattice. But if two -matrices are not related by an transformation, is there a more general notion that may relate the theories? A more general notion might be expected if the difference in the number of positive and negative eigenvalues of the two -matrices coincide. Consider, for instance, the case of an -matrix and an -matrix with . Could there be a relation between them, even though they clearly cannot be related be related by an or similarity transformation?

The answer is yes, for the following reason. Consider the theory associated with , defined in Eq. (16). Its partition function is equal to on an arbitrary -manifold, , as was shown in Ref. Witten, :


One manifestation of the triviality of this theory in the bulk is that it transforms trivially under modular transformations, as we saw earlier. Furthermore, a state with this -matrix can be smoothly connected to a trivial insulator by local unitary transformations if no symmetries are maintained Chen et al. (2011). We shall not do so here, but it is important to note that, if we impose a symmetry on the theory, then we can guarantee the existence of gapless (non-chiral) excitations that live at the edge of the system Chen et al. (2011); Lu and Vishwanath (2012). (We emphasize that we focus, in this section, on the bulk and, in this paper, on properties that do not require symmetry.)

Therefore, we can simply replace it with a theory with no degrees of freedom. We will denote such a theory by to emphasize that it is a -matrix in a theory with fields and not a theory with a -matrix that vanishes. Similarly, the partition function for a theory with arbitrary -matrix on any -manifold is equal to the partition function of


Therefore, all of the theories corresponding to even, unimodular lattices of signature are, in fact, equivalent when there is no symmetry preserved. There is just a single completely trivial gapped phase. We may choose to describe it by a very large -matrix (which is seemingly perverse), but it is still the same phase. Moreover, any phase associated with a -matrix can equally well be described by a larger -matrix to which we have added copies of along the block diagonal. This is an expression of the physical idea that no phase transition will be encountered in going from a given state to one in which additional trivial, gapped degrees of freedom have been added. Of course, in this particular case, we have added zero local degrees of freedom to the bulk and we have not enlarged the Hilbert space at all. So it is an even more innocuous operation. However, when we turn to the structure of edge excitations, there will be more heft to this idea.

At a more mathematical level, the equivalence of these theories is related to the notion of “stable equivalence”, according to which two objects are the same if they become isomorphic after augmentation by a “trivial” object. In physics, stable equivalence has been used in the K-theoretic classification of (non-interacting) topological insulators Kit (b). In the present context, we will be comparing gapped phases and the trivial object that may be added to either phase is a topologically-trivial band insulator. Heuristically, stable equivalence says that we may add some number of topologically-trivial bands to our system in order to effectively enlarge the parameter space and, thereby, allow a continuous interpolation between two otherwise different states.

We now turn to the two bosonic SRE phases. Their bulk effective field theories are of the form of Eq. (1) with -matrices given by and . Their bulk properties are seemingly trivial. But not entirely so since, as we noted in Section II, they transform non-trivially under modular transformations.

These two non-trivial theories are, at first glance, distinct. They are associated with different lattices. For instance, is the direct sum of two -dimensional lattices while is not. The two -matrices are not related by an transformation.

Suppose, however, that we consider the -matrices and which describe ”enlarged” systems. (We use quotation marks because, although we now have theories with rather than gauge fields, the physical Hilbert space has not been enlarged.) These -matrices are, in fact, related by an transformation:


where is given by:


We will explain how is derived in Section V. Here, we focus on its implication: these two theories are equivalent on an arbitrary closed manifold. There is a unique bulk bosonic SRE phase of matter. However, there appear to be two possible distinct effective field theories for the edge of this unique bulk phase, namely the theories (11) with and . In the next section, we explain the relation between these edge theories.

Iv Fermionic Representations of the Two SRE Bosonic Phases

In Section III, we saw that there is a unique bulk bosonic SRE phase of matter. We now turn our attention to the two corresponding edge effective field theories, namely Eq. (11) with given by either or . These two edge theories are distinct, although the difference is subtle. To understand this difference, it is useful to consider fermionic representations Gross et al. (1985); Gre () of these edge theories.

Consider 32 free chiral Majorana fermions: