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4

Bulk-Edge Correspondence in2 + 1-Dimensional Abelian Topological Phases

Jennifer Cano,1 Meng Cheng,2 Michael Mulligan,2 Chetan Nayak,2, 1 Eugeniu Plamadeala,1 and Jon Yard21Department of Physics, University of California, Santa Barbara, California 93106, USA

2Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, California 93106-6105, USA

The same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. Weshow that this can occur in the integer quantum Hall states atν = 8 and 12, with experimentally-testableconsequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplestexamples being atν = 8/7, 12/11, 8/15, 16/5. We give a general criterion for the existence of multiple distinctchiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspondto lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in agenus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edgephases corresponding to the same bulk. We explain these correspondences using the theory of integral quadraticforms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations anddiscuss a fermionic generalization of the relation betweenbulk topological spins and the central charge. Thelatter follows from our demonstration that every fermionictopological phase can be represented as a bosonictopological phase, together with some number of filled Landau levels. Our analysis shows that every Abeliantopological phase can be decomposed into a tensor product oftheories associated with prime numbersp inwhich every quasiparticle has a topological spin that is apn-th root of unity for somen. It also leads to asimple demonstration that all Abelian topological phases can be represented byU(1)N Chern-Simons theoryparameterized by a K-matrix.

I. INTRODUCTION

In the limit of vanishing electron-electron interactions,theedge excitations of an integer quantum Hall state form a multi-channel chiral Fermi liquid. These excitations are stable withrespect to weak interactions by their chirality1. However, theCoulomb energy in observed integer quantum Hall states islarger than the energy of the lowest gapped edge excitation.Therefore, interactions are not weak in these experiments,andwe must consider whether interactions with gapped unpro-tected non-chiral excitations can alter the nature of the gap-less protected chiral edge excitations of an integer quantumHall state even when the bulk is unaffected.58

In this paper, we show that sufficiently strong interactionscan drive the edge of an integer quantum Hall state withν ≥ 8into a different phase in which the edge excitations form amulti-channel chiralLuttinger liquid while the bulk remainsadiabatically connected to an integer quantum Hall state ofnon-interacting electrons. This chiral Luttinger liquid is alsostable against all weak perturbations, but it is not adiabaticallyconnected to the edge of an integer quantum Hall state of non-interacting electrons even though the bulk of the system is.Forν ≥ 12, there are several possible such stable chiral edgephases corresponding to the same bulk phase. The edge ex-citations of many fractional quantum Hall states, such as theprincipal Jain series withν = n

2pn+1 form a multi-channelchiral Luttinger liquid, which is stable against weak perturba-tions due to its chirality. We show that such edges can also besubject to reconstruction into a different chiral Luttinger liq-uid as a result of strong interactions with gapped unprotectedexcitations at the edge. The new chiral Luttinger liquid is alsostable against all weak perturbations.

A similar phenomenon was recently analyzed in the contextof bosonic analogues of integer quantum Hall states2. With-out symmetry, integer quantum Hall states of bosons that onlysupport bosonic excitations in the bulk, not anyons, occur only

when the chiral central charge,c− = cR − cL, the differencebetween the number of right- and left-moving edge modes, isa multiple of eight (or, equivalently, when the thermal Hall

conductance isκxy = c−π2k2

BT3h with c− = 8k for integers

k).3 There is a unique4,5 bulk state for each possible value ofc− = 8k, but there are many possible chiral edge phases whenthe chiral central charge is greater than8: there are two chiraledge phases forc− = 16, twenty-four chiral edge phases forc− = 24, more than one billion forc− = 32, and larger num-bers of such edge phases forc− > 32. The transition betweenthe two possible chiral edge phases was studied in detail in thec− = 16 case.2,6

These fermionic and bosonic quantum Hall states illustratethe fact that the boundary-bulk correspondence in topologicalstates is not one-to-one. There can be multiple possible edgephases corresponding to the same bulk phase. This can hap-pen in a trivial way: two edge phases may differ by unstablegapless degrees of freedom, so that one of the edge theories ismore stable than the other.7–11 (One interesting refinement ofthis scenario is that the additional gapless degrees of freedomcan be protected by a symmetry so that, in the presence ofthis symmetry, both edge phases are stable12.) However, ourfocus here is the situation in which there are multiple edgephases, each of which is stable to weak perturbations with-out any symmetry considerations and none of which is more“minimal” than the others. In other words, in the integer andfractional quantum Hall states that we discuss here – whichhave the additional property that they are all chiral – all ofthe edge phases are on the same footing. Although they canbound the same bulk, such edge phases generically have dif-ferent exponents and scaling functions for transport throughpoint contacts and tunneling in from external leads. In somecases, the differences only show up in three-point and higheredge correlation functions.

In Sections VII A, VII B of this paper, we discuss fermionicinteger quantum Hall states atν = 8 andν = 12, their possi-

2

ble stable chiral edge phases, and the experimental signaturesthat could distinguish these phases. In Section VII C, we dis-cuss the simplest fractional quantum Hall states with multi-ple chiral edge phases, which occur atν = 8/7, 8/15, 16/5(fermions) andν = 12/23 (bosons). Some of the edge phasesthat we construct do not support gapless excitations with thequantum numbers of an electron. When the Hall conductanceis non-zero, the edge must have gapless excitations; in a sys-tem of electrons, there must be a finite-energy excitation ev-erywhere in the system with an electron’s quantum numbers.However, it is not necessary that the electron be among thegapless edge excitations of an electronic quantum Hall state;it may be a gapped excitation at the edge, above the gaplessexcitations that are responsible for carrying the Hall current.

Given the above statement that the same bulk phase canhave multiple distinct chiral edge phases, we should ask whatbreaks down in the usual relation between bulk topologi-cal phases and their associated edge spectra. By the usualrelationship, we mean the “integration by parts” of a bulkAbelian Chern-Simons action that gives an edge theory of chi-ral bosons with the same K-matrix13,14. The answer is simplythat the usual relation focuses only upon the lowest energyexcitations of a system and ignores higher-energy excitations.These higher-energy excitations are necessarily adiabaticallyconnected to a topologically-trivial band insulator in thebulkand, generically, gapped excitations at the edge. Surprisingly,interactions between these “trivial” modes and the degreesoffreedom responsible for the topologically non-trivial state candrive an edge phase transition that leads to a distinct edgephase without closing the bulk gap. We refer to the relation-ship between these two distinct edge theories associated withthe same bulk asstable equivalence. At the level of the gap-less edge modes, this manifests itself in the form of an edgereconstruction. While the interpolation at the edge necessar-ily involves strong interactions, these can be understood usingstandard Luttinger liquid techniques.

The relationship between the edge and the bulk can also beviewed in the following manner. Each quasiparticle in the bulkhas a topological twist factorθa = e2πiha , with 0 < ha < 1.If the edge is fully chiral, each such quasiparticle correspondsto a tower of excitations. The minimum scaling dimensionfor creating an excitation in this tower is min∆a = ha + na

for some integerna. The other excitations in the tower areobtained by creating additional bosonic excitations on topofthis minimal one; their scaling dimensions are larger than theminimal one by integers. But if the edge has a different phase,the minimal scaling dimension operator in this tower may bemin∆a = ha + na. Therefore, the spectrum of edge op-erators can be different, even though the fractional parts oftheir scaling dimensions must be the same. (In the case of afermionic topological phase, we must compare scaling dimen-sions modulo1/2, rather than modulo1. By fermionic topo-logical phase, we mean one which can only occur in a systemin which some of the microscopic consitutents are fermions.At a more formal level, this translates into the existence ofafermionic particle which braids trivially with all other parti-cles.)

The purpose of this paper is to describe the precise condi-

tions under which two different edge phases can terminate thesame bulk state, i.e. are stably equivalent. These conditionsare intuitive: the braiding statistics of the quasiparticle excita-tions of the bulk states must be identical and the chiral centralcharges of the respective states must be equal.

Let us summarize the general relation between bulkAbelian topological states and their associated edge phasesin slightly more mathematical terms. Edge phases are de-scribed by latticesΛ equipped with an integer-valued bilin-ear symmetric formB.15–20 We collectively write this data asE = (Λ, B). The signature ofB is simply the chiral cen-tral chargec− of the edge theory. Given a basiseI for Λ,the bilinear form determines a K-matrixKIJ = B(eI , eJ).In a bosonic system, the latticeΛ must be even while in afermionic system, the latticeΛ is odd. (An odd lattice is onein which at least one basis vector has(length)2 equal to anodd integer. The corresponding physical system will have afermionic particle that braids trivially with all other particles.This particle can be identified with an electron. An even lat-tice has no such vectors and, therefore, no fermionic particlesthat braid trivially with all other particles. Hence, it canoccurin a system in which none of the microscopic constituents arefermions. Of course, a system, such as the toric code, mayhave fermionic quasiparticles that braid non-trivially with atleast some other particles.) Given the latticeΛ, vertex opera-tors of the edge theory are associated with elements in the duallatticeΛ∗. For integer quantum Hall states,Λ∗ = Λ, however,for fractional statesΛ ⊂ Λ∗. The operator product expansionof vertex operators is simply given by addition inΛ∗.

Each bulk phase is characterized by the following dataconcisely written asB = (A, q, c− mod24):16,18–22 a finiteAbelian groupA encoding the fusion rules for the distinctquasiparticle types, a finite quadratic formq onA that givesthe topological spin to each particle type, and the chiral centralcharge modulo 24. As we will discuss at length, since the mapE → B associating edge dataE to a given bulkB is not one-to-one, several different edge phases may correspond to thesame bulk phase. We will provide an in-depth mathematicaldescription of the above formalism in order to precisely deter-mine when two distinct edge phases correspond to the samebulk phase. To determine all of the edge phases that can boundthe same bulk, one can perform a brute force search throughall lattices of a given dimension and determinant. (For low-dimensional cases, the results of such enumeration is in tablesin Ref. 23 and in, for instance, G. Nebe’s online Catalogue ofLattices.) Moreover, one can use a mass formula described inSection V to check if a list of edge phases is complete.

We will exemplify the many-to-one nature of the mapE →B through various examples. The most primitive example oc-curs for integer quantum Hall states. For such states, the lat-tice is self-dual,Λ∗ = Λ so there are no non-trivial quasi-particles. Forc− < 8, there is a unique edge theory for thefermionic integer quantum Hall state, however, atc− = 8,there are two distinct lattices: the hypercubic lattticeI8 andtheE8 root lattice. Therefore, the associated gapless edgetheories corresponding to each lattice may bound the samebulk state; there exists an edge reconstruction connectingthetwo edge phases. Fractional states for whichA is non-trivial

3

enrich this general structure.A rather remarkable corollary of our analysis is the follow-

ing: all rational Abelian topological phases in 2+1 dimensionscan be described by Abelian Chern-Simons theory. By ratio-nal, we mean that there is a finite number of bulk quasiparti-cle types, i.e., the groupA has finite order. As may be seenby giving a physical interpretion to a theorem of Nikulin24 theparticle types, fusion rules, and topological twist factors deter-mine a genus of lattices, from which we can define an AbelianChern-Simons theory. A second result that follows from a the-orem of Nikulin24 is that any fermionic Abelian topologicalphase can be mapped to a bosonic topological phase, togetherwith some number of filled Landau levels.

The remainder of this paper is organized as follows. We be-gin in Section II by reviewing the formalism used to describethe bulk and boundary excitations of Abelian Hall states. Asameans to both motivate the general mathematical structure andbecause of their intrinsic interest, we provide two examples ofstable equivalence in the fractional quantum Hall setting inSection III and summarize their physically distinct signatures.In Section IV, we abstract from these two examples the gen-eral method for understanding how distinct edge phases of asingle bulk are related via an edge phase transition. In Sec-tion V, we explain the bulk-edge correspondence through theconcepts of stable equivalence and genera of lattices. In Sec-tion VI, we explain how fermionic topological phases can berepresented by bosonic topological phases together with somenumber of filled Landau levels. In Section VII, we analyzeobserved integer and fractional quantum Hall states that ad-mit multiple stable, fully chiral edge phases. In Section VIII,we explain how a number of theorems due to Nikulin, that weuse throughout the text, apply to the description ofall Abeliantopological field theories in (2+1)-D. We conclude in SectionIX. We have three appendices that collect ideas used withinthe text.

II. PRELIMINARIES

A. Edge Theories

In this section, we review the formalism that describes theedges of conventional integer and Abelian fractional quantumHall states. We begin with the edges of fermionic integerquantum Hall states. We assume that the bulks of these statesare the conventional states that are adiabatically connected tothe corresponding states of non-interacting fermions. As wewill see in later sections, the edge structure is not uniquelydetermined, even if we focus solely on chiral edge phases thatare stable against all weak perturbations.

All integer quantum Hall states have one edge phase that isadiabatically connected to the edge of the corresponding non-interacting fermionic integer quantum Hall state. This edgephase has effective actionS0 + S1, where

S0 =

∫

dxdt ψ†J (i∂t +At + vJ (i∂x +Ax))ψJ (1)

andJ = 1, 2, . . . , N . We shall later study two interesting

examples that occur whenN = 8 or N = 12. The operatorψ†J creates an electron at the edge in theJ th Landau level;vJ is the edge velocity of an electron in theJ th Landau level.Inter-edge interactions take the form

S1 =

∫

dx dt(tJK(x) ei(k

JF−kK

F )x ψ†JψK + h.c.

+ vJKψ†JψJψ

†KψK + . . .

). (2)

The . . . in Eq. (2) represent higher-order tunneling and inter-action terms that are irrelevant by power counting. We neglectthese terms and focus on the first two terms. Electrons in dif-ferent Landau levels will generically have different Fermimo-menta. When this is the case, the tunneling term (the first termin Eq. (2)) will average to zero in a translationally-invariantsystem. In the presence of disorder, however,tIJ(x) will berandom and relevant (e.g. in a replicated action which is av-eraged overtIJ (x)). Moreover, it is possible for the Fermimomenta to be equal; for instance, in anN -layer system inwhich each layer has a single filled Landau level, the Fermimomenta will be the same if the electron density is the samein each layer. Fortunately, we can make the change of vari-ables:

ψJ(x)→(

P exp

(

i

∫ x

−∞dx′M(x′)

))

JK

ψK(x),

where M(x) is the matrix with entriesMJK =

tJK(x′) ei(kJF−kK

F )x′

/v, v =∑

J vJ/N , andP denotes anti-path-ordering. When this is substituted into Eq. (1), the firstterm in Eq. (2) is eliminated from the actionS0 + S1. Thisis essentially a U(N) gauge transformation that gauges awayinter-mode scattering. An extra random kinetic term propor-tional to(vJ − v)δIJ is generated, but this is irrelevant in theinfrared when disorder-averaged.

The second term in Eq. (2) is an inter-edge density-densityinteraction;vJK is the interaction between edge electrons intheJ th andKth Landau levels. This interaction term can besolved by bosonization. The actionS0+S1 from Eqs. (1) and(2) can be equivalently represented by the bosonic action

S =

∫

dx dt

(1

4πδIJ∂tφ

I∂xφJ − 1

4πVIJ∂xφ

I∂xφJ

+1

2π

∑

I

ǫµν∂µφIAν

)

, (3)

whereVII ≡ vI + vII (no summation) andVIJ ≡ vIJ forI 6= J . The electron annihilation operator is bosonized ac-cording toψJ ∼ ηJe

iφJ

. HereηJ is a “Klein factor” sat-isfying ηJηK = −ηKηJ for J 6= K, which ensures thatψJψK = −ψKψJ . Products of even numbers of Klein fac-tors can be diagonalized and set to one of their eigenvalues,±1, if all terms in the Hamiltonian commute with them. Theycan then be safely ignored. This is the case in all of the mod-els studied in this paper. This action can be brought into thefollowing diagonal form (setting the external electromagnetic

4

field to zero for simplicity):

S =

∫

dx dt

(1

4πδIJ∂tφ

I∂xφJ − 1

4πvIδIJ∂xφ

I∂xφJ

)

(4)with an orthogonal transformationφI = OI

J φJ that diago-

nalizesVIJ according toOILVIJO

JK = vLδLK . Two-point

correlation functions take the form

⟨

eimIφI

e−imKφK⟩

=

N∏

J=1

1

(x− vJ t)mImKOIJOK

J

. (5)

There is no sum overJ in the exponent on the right-hand-sideof Eq. (5). The electron Green function in theIth Landaulevel is a special case of this withmK = δIK .

It is now straightforward to generalize the preceding discus-sion to the case of an arbitrary Abelian integer or fractionalquantum Hall state14. For simplicity, we will focus on thecase of fully chiral phases in which all edge modes move inthe same direction. Such phases do not, in general, have afree fermion representation and can only be described by achiral Luttinger liquid. They are characterized by equivalenceclasses of positive-definite symmetric integerK-matricesK,and integercharge vectorst that enter the chiral Luttinger liq-uid action according to

SLL =

∫

dx dt

(1

4πKIJ∂tφ

I∂xφJ − 1

4πVIJ∂xφ

I∂xφJ

+1

2πtIǫµν∂µφ

IAν

)

. (6)

The fields in this action satisfy the periodicity conditionφI ≡φI + 2πnI for nI ∈ Z. Two phases, characterized by thepairs(K1, t1) and(K2, t2), are equivalent ifK1 =WTK2Wand t1 = t2W , whereW ∈ GL(N,Z) since the first andthird terms in the two theories can be transformed into eachother by the change of variablesφI = W I

J φJ . So long as

W ∈ GL(N,Z), the periodicity condition satisfied byφJ isprecisely the same as the periodicity condition satisfied byφI .The matrixVIJ consists of marginal deformations that do notchange the phase of the edge but affect the propagation veloci-ties. (If we wish, we can think of each phase as a fixed surfaceunder RG flow, and theVIJs are marginal deformations thatparametrize the fixed surface.) All such chiral edge theoriesare stable to all weak perturbations by the same reasoning bywhich we analyzed integer quantum Hall edges. The simplestfermionic fractional quantum Hall edge theory is that of theLaughlinν = 1/3 state, for whichK = (3) andt = (1) (a1× 1 matrix and a1-component vector, respectively). Integerquantum Hall edges are the special case,KIJ = δIJ or, al-lowing for basis changes,K =WTW with W ∈ GL(N,Z).

It is useful to characterize these phases by latticesΛ ratherthan equivalence classes of K-matrices. LeteaI be the eigen-vector ofK corresponding to eigenvalueλa: KIJe

aJ = λaeaI .

We normalizeeaJ so thateaJebJ = δab and define a metric

gab = λaδab. Then,KIJ = gabeaIe

bJ or, using vector nota-

tion,KIJ = eI · eJ . We will be focusing mostly on positive-definite lattices, so thatgab has signature(N, 0) but we will

occasionally deal with Lorentzian lattices, for which we takegab has signature(p,N−p). The metricgab defines abilinearformB on the latticeΛ (and its dualΛ∗) – this just means wecan multiply two lattice vectorseI , eJ together using the met-ric, eI · eJ = eaIgabe

bJ = B(eI , eJ). TheN vectorseI define

a latticeΛ = {mIeI |mI ∈ Z}. TheGL(N,Z) transforma-tionsK → WTKW are simply basis changes of this lattice,so we can equally well describe edge phases by equivalenceclasses ofK-matrices or by latticesΛ. The conventional edgephases of integer quantum Hall states described above corre-spond to hypercubic latticesZN , which we will often denoteby the correspondingK matrix in its canonical basis,IN . Theν = 1/3 Laughlin state corresponds to the latticeΛ = Z

with dualΛ∗ = 13Z.59 The connection of quantum Hall edge

phases to lattices can be exploited more easily if we make thefollowing change of variables,Xa = eaIφ

I , in terms of whichthe action takes the form

S =1

4π

∫

dx dt

(

gab∂tXa∂xX

b−vab∂xXa∂xXb.

)

(7)

The variablesXa satisfy the periodicity conditionX ≡X + 2πy for y ∈ Λ and vab ≡ VIJf

Iaf

Jb , where f I

a

are basis vectors for the dual latticeΛ∗, satisfyingf Iae

aJ =

eLa(K−1)LIeaJ = δIJ .

Different edge phases (which may correspond to differentbulks or the same bulk; the latter is the focus of this paper) aredistinguished by their correlation functions. The periodicityconditions on the fieldsXa dictate that the allowed exponen-tial operators are of the formeiv·X, wherev ∈ Λ∗. Theseoperators have scaling dimensions

dim[eiv·X

]=

1

2|v|2. (8)

They obey the operator algebra

: eiv1·X :: eiv2·X :∼: ei(v1+v2)·X :, (9)

where: · : denotes normal ordering. Thus, the operator spec-trum and algebra is entirely determined by the underlying duallatticeΛ∗.

In a quantum Hall state, there are two complementary waysof measuring some of the scaling exponents. The first is aquantum point contact (QPC) at which two edges of a quan-tum Hall fluid are brought together at a point so that quasipar-ticles can tunnel across the bulk from one edge to the other.Even though a single edge is completely stable against allweak perturbations, a pair of oppositely-directed edges will,in general, be coupled by relevant perturbations

S = ST + SB +

∫

dt∑

v∈Λ∗

vv eiv·[XT−XB ]. (10)

Here,T,B are the two edges, e.g., the top and bottom edgesof a Hall bar; we will use this notation throughout whenever itis necessary to distinguish the two edges. The renormalizationgroup (RG) equation forvv is

dvvdℓ

=(

1− |v|2)

vv. (11)

5

If v · f ItI 6= 0, the above coupling transfersv · f ItI units ofcharge across the junction and this perturbation will contributeto the backscattered current according to

Ib ∝ |vv|2 V 2|v|2−1. (12)

A second probe is the tunneling current from a metallic lead:

S = Sedge + Slead

+

∫

dt∑

v∈Λ

tv

[

ψ†lead∂ψ

†lead∂

2ψ†lead . . .

]

eiv·X.

The term in square brackets[...] containsn factors ofψ†lead

andn(n − 1)/2 derivatives, wheren = v · f ItI must be aninteger. The RG equation fortv

dtvdℓ

=

(

1− n2

2− 1

2|v|2

)

tv. (13)

The contribution to the tunneling current fromtv (assumingn 6= 0) is

Itun ∝ |tv|2 V |v|2+n2−1. (14)

Here, we have assumed that the spins at the edge of the quan-tum Hall state are fully spin-polarized and that tunneling fromthe lead conservesSz. If, however, either of these conditionsis violated, then other terms are possible in the action. Forinstance, charge-2e tunneling can take the form

tpair

∫

dt ψ†lead,↑ψ

†lead,↓ e

iv·X, (15)

wherev · f I tI = 2. Then, we have tunneling current

Itun ∝ |tv|2 V |v|2+1. (16)

Generically, two latticesΛ1 andΛ2 can be distinguished bythe possible squared lengths|v|2 for v ∈ Λ∗

1. In many cases ofinterest, the shortest length, which will dominate the backscat-tered current discussed above, is enough to distinguish twoedge phases of the same bulk. However, sometimes, as inthe case of the two bosonic integer quantum Hall states withc = 16 discussed in Ref. 2 the spectrum of operator scalingdimensions (not just the shortest length, but all lengths alongwith degeneracies at each length level) is precisely the samein the two theories, so they could only be distinguished bycomparing three-point correlation functions. In either case,different edge phases can be distinguished by their correlationfunctions.

B. Bulk Theories

In a later section, we will explain how bulk phases corre-spond to the mathematical notion of agenusof lattices, whiletheir associated edge theories are given by lattices withinagenus (or in the case of fermionic theories, a pair of genera,

one odd and one even). In order to explain the relation be-tween the genus of a lattice and a bulk Abelian phase, werecall some facts about Abelian topological phases.

Suppose that we have a2 + 1d Abelian topological phaseassociated to a latticeΛ. Choosing a basiseI for the latticeΛ,we defineKIJ = eI · eJ and write a bulk effective action

S =

∫

d3x( 1

4πǫµνρKIJa

Iµ∂νa

Jρ +

1

2πjµI a

Iµ

)

. (17)

A particle in this theory carrying chargemI under the gaugefieldaI can be associated with a vectorv ≡ mIf

I , wherefI isthe basis vector ofΛ∗ dual toeI and satisfying(K−1)IJeJ =f I . Recall that becauseΛ ⊂ Λ∗, any element inΛ can beexpressed in terms of the basis forΛ∗, however, the converseis only true for integer Hall states for whichΛ = Λ∗. Parti-clesv, v′ ∈ Λ∗ satisfy the fusion rulev × v′ = v + v′ andtheir braiding results in the multiplication of the wave func-tion describing the state by an overall phasee2πiv·v

′

. Sincethis phase is invariant under shiftsv → v + λ for λ ∈ Λ, thetopologically-distinct particles are associated with elementsof the so-called discriminant groupA = Λ∗/Λ. The many-to-one nature of the edge-bulk correspondence is a reflection ofthe many-to-one correspondence between latticesΛ and theirdiscriminant groupsA. Equivalent bulk phases necessarilyhave identical discriminant groups so our initial choice oflat-tice is merely a representative in an equivalence class of bulktheories.

We now define a few terms. Abilinear symmetric formona finite Abelian groupA is a functionb : A×A→ Q/Z suchthat for everya, a′, a′′ ∈ A,

b(a+ a′, a′′) = b(a, a′′) + b(a′, a′′)

andb(a, a′) = b(a′, a). As all bilinear forms considered inthis paper will be symmetric, we will simply call thembi-linear formswith symmetric being understood. Aquadraticformq on a finite Abelian groupA is a functionq : A→ Q/Zsuch thatq(na) = n2q(a) for everyn ∈ Z, and such that

q(a+ a′)− q(a)− q(a′) = b(a, a′)

for some bilinear formb : A × A → Q/Z. In this case,we say thatq refinesb, or is a quadratic refinementof b.A bilinear b or quadratic formq is degenerateif there ex-ists a non-trivial subgroupS ⊂ A such thatb(s, s′) = 0 orq(s) = 0 for everys, s′ ∈ S. Throughout this paper, all bilin-ear and quadratic forms will be assumed nondegnerate. EachK-matrix K determines a symmetric bilinear formB on Rn

viaB(x,y) = xTKy that takes integer values on the latticeZn ⊂ Rn. Every other latticeΛ ⊂ Rn on whichB is inte-gral can be obtained by acting onZn by the orthogonal group{g ∈ GL(N,R) : gKgT = K} of K. On the other hand,an integral symmetric bilinear form is equivalent to a latticeaccording to the construction before Eq. (7) in Section II A.We are therefore justified in using the terminology “lattice”and “K-matrix” in place of “integral symmetric bilinear form”throughout this paper. Every diagonal entry of a K-matrixKis even iff the(length)2 of every element in the latticeZN iseven. We callK evenif this is the case, and otherwise it isodd.

6

Even K-matrices determine integral quadratic forms onZN

viaQ(x) = 12x

TKx, while for odd K-matrices they are half-integral. When we simply write bilinear or quadratic form or,sometimes,finite bilinear formor finite quadratic form, wewill mean a nondegenerate symmetric bilinear form, or non-degenerate quadratic form, whose domain is a finite Abeliangroup. Throughout, we abbreviate the ringZ/NZ of integersmoduloN asZ/N .

The S-matrix of the theory can be given in terms of theelements of the discriminant group:

S[v],[v′] =1

√

|A|e−2πiv·v′

=1

√

|A|e−2πimI (K

−1)IJm′

J ,

(18)wherev = mIf

I ,v′ = m′J f

J ∈ Λ∗ and|A| is the dimensionof the discriminant group. The bracketed notation[v] indi-cates an equivalence class of elements[v] ∈ Λ∗/Λ = A. Ournormalization convention is to represent elements in the duallatticeΛ∗ with integer vectorsmI . The bilinear formB onΛ∗ reduces moduloΛ to define a finite bilinear form on thediscriminant groupΛ∗/Λ via

b([mIfI ], [m′

J fJ ]) = B(mIf

I ,m′J f

J ) = mI(K−1)IJm′

J .

The topological twistsθ[v], which are the eigenvalues of theT matrix, are defined by

T[v],[v′] = e−2πi24

c− θ[v] δ[v],[v′] (19)

where

θ[v] = eπiv·v. (20)

Note that Eq. (19) implies that the theory is invariant undershifts ofc− by 24 so long as the topological twistsθ[v] are in-variant, but its modular transformation properties, whichde-termine the partition function on3-manifolds via surgery25, issensitive to shifts byc− 6= 0 (mod24).

If the topological twists are well-defined on the set of quasi-particlesA, then they must be invariant underv 7→ v + λ,whereλ ∈ Λ, under which

θ[v] 7→ θ[v+λ] = θ[v] eπiλ·λ. (21)

If the K-matrix is even, so that we are dealing with a bosonictheory,λ · λ is even for allλ ∈ Λ. If the K-matrix is odd,however – i.e. if the system is fermionic – then there are someλ ∈ Λ for which λ · λ is odd. In this case, the topologicaltwists are not quite well-defined, and more care must be taken,as we describe in Section VI. Given the above definition, onlyT 2 is well-defined.

In a bosonic Abelian topological phase, we can define a fi-nite quadratic formq on the discriminant group, usually calledthediscriminant form, according to

q([v]) =1

2v2 =

1

2mI(K

−1)IJmJ modZ, (22)

wherev = mIfI . In a topological phase of fermions, we

will have to defineq with more care, as we discuss in Section

VI. Thus, we postpone its definition until then and will onlydiscuss Abelian bosonic topological phases in the remainderof this section. In terms of the discriminant formq, theT -matrix takes the form

θa = e2πiq(a), (23)

and theS-matrix takes the form

Sa,a′ =1

√

|A|e2πi(q(a−a′)−q(a)−q(−a′)) (24)

=1

√

|A|e−2πi(q(a+a′)−q(a)−q(a′)) (25)

The equation for theS-matrix makes use of the fact that the fi-nite bilinear formb can be recovered from the finite quadraticform according tob(a, a′) = q(a+a′)−q(a)−q(a′). (It is sat-isfying to observe that the relation between the bilinear form band the discriminant formq coincides exactly with the phaseobtained by a wave function when two particles are twistedabout one another.) While the introduction of the discrimi-nant form may appear perverse in the bosonic context, we willfind it to be an essential ingredient when discussing fermionictopological phases.

In any bosonic topological phase, the chiral central chargeis related to the bulk topological twists by the followingrelation26:

1

D∑

a

d2aθa = e2πic−/8. (26)

HereD =√∑

a d2a is the total quantum dimension,da is the

quantum dimension of the quasiparticle typea, andθa is thecorresponding topological twist/spin.c− = c− c is the chiralcentral charge. In an Abelian bosonic phase described by aneven matrixK, the formula simplifies to

1√

|A|∑

a∈A

e2πiq(a) = e2πic−/8, (27)

since da = 1 for all quasiparticle types. Here|A| =√

| detK| andc− = r+ − r− is the signature of the matrix,the difference between the number of positive and negativeeigenvalues. (We will sometimes, as we have done here, usethe term signature to refer to the differencer+ − r−, ratherthan the pair(r+, r−); the meaning will be clear from con-text.) Notice thate2πiq(a) is just the topological twist of thequasiparticle represented bya ∈ Λ∗/Λ. This is known as theGauss-Milgram sum in the theory of integral lattices.

Let us pause momentarily to illustrate these definitions in asimple example: namely, the semion theory described by theK-matrix,K = (2). This theory has discriminant groupA =Z/2Z = Z2 and, therefore, two particle types, the vacuumdenoted by the lattice vector[0] and the semions = [1]. Recallthat our normalization convention is to take the bilinear formon A to beb([x], [y]) = x · 1

2 · y; the associated quadraticform is thenq([x]) = 1

2b([x], [x]). The discriminant form,evaluated on the semion particle, is given byq([1]) = 1

2 ·

7

12

2 . The T matrix equalsexp(−2πi/24)diag(1, i), and the S-matrix,S = 1√

2

(1 11 −1

). Evaluating the Gauss-Milgram sum

confirms thatc− = 1.In order to determine the discriminant group from a given

K-matrix, we can use the following procedure. First, we com-pute the Gauss-Smith normal form of theK-matrix, whichcan be found using a standard algorithm27. GivenK, this al-gorithm produces integer matricesP ,Q,D such that

K = PDQ. (28)

Here bothP andQ are unimodular|detP | = |detQ| = 1,andD is diagonal. The diagonal entries ofD give the ordersof a minimal cyclic decomposition of the discriminant group

A ≃∏

J

Z/DJJ ,

with the fewest possible cyclic factors, giving yet anothersetof generators for the quasiparticles. Although more compact,this form does not directly lend itself towards checking theequivalence of discriminant forms.

Now recall that the bases ofΛ andΛ∗ are related byK:

eI = KIJ fJ (29)

Substituting the Gauss-Smith normal form, this can be rewrit-ten

(P−1)ILeL = DIKQKJ fJ . (30)

The left-hand side is just a basis change of the original lattice.On the right-hand side, the row vectors ofQ that correspondto entries ofD greater than1 give the generators of the cyclicsubgroups of the discriminant group. A non-trivial exampleisgiven in Appendix A.

III. TWO ILLUSTRATIVE EXAMPLES OF BULKTOPOLOGICAL PHASES WITH TWO DISTINCT EDGE

PHASES

The chiral Luttinger liquid action is stable against all smallperturbations involving only the gapless fields in the actionin Eq. (6) (or, equivalently in the integer case, the action inEq. (1)). This essentially follows from the chirality of thethe-ory, but it is instructive to see how this plays out explicitly.1

However, this does not mean that a given bulk will have onlya single edge phase.28 A quantum Hall system will have addi-tional gapped excitations which we can ignore only if the in-teractions between them and the gapless excitations in Eq. (6)are weak. If they are not weak, however, we cannot ignorethem and interactions with these degrees of freedom can leadto an edge phase transition2.

We will generally describe the gapped excitations with a K-matrix equal toσz =

(1 00 −1

). We may imagine this K-matrix

arising from a thin strip ofν = 1 fluid living around theperimeter of our starting Hall state.28 For edge phase transi-tions between bosonic edges theories, we should instead take

the gapped modes to be described by a K-matrix equal toσx =

(0 11 0

). It is important to realize that the existence of the

localized (gapped) edge modes described by either of theseK-matrices implies the appropriate modification to the Chern-Simons theory describing the bulk topological order. This ad-dition does not affect the bulk topological order29; withoutsymmetry, such a gapped state is adiabatically connected toatrivial band insulator.

We will illustrate this with two concrete examples. We be-gin with the general edge action

S =

∫

dx dt

(1

4πKIJ∂tφ

I∂xφJ

− 1

4πVIJ∂xφ

I∂xφJ +

1

2πtIǫµν∂µφ

IAν

)

. (31)

The first example is described by the K-matrix

K1 =

(1 00 11

)

, (32)

with t = (1,−1)T . This is not an example that is particularlyrelevant to quantum Hall states observed in experiments – wewill discuss several examples of those in Section VII – but itissimple and serves as a paradigm for the more general structurethat we discuss in Sections V and VI.

Let us suppose that we have an additional left-movingand additional right-moving fermion which, together, formagapped unprotected excitation. The action now takes the form

S =

∫

dx dt

(1

4π(K1 ⊕ σz)IJ ∂tφI∂xφJ

− 1

4πVIJ∂xφ

I∂xφJ +

1

2πtIǫµν∂µφ

IAν

)

, (33)

where we have now extendedt = (1,−1, 1, 1)T . The K-matrix for the two additional modes is taken to beσz. Wewill comment on the relation to theσx case in Sections IVand V.

If the matrixVIJ is such that the perturbation

S′ =

∫

dx dt u′ cos(φ3 + φ4), (34)

is relevant, and if this is the only perturbation added toEq. (33), then the two additional modes become gapped andthe system is in the phase (32). Suppose, instead, that the onlyperturbation is

S′′ =

∫

dx dt u′′ cos(φ1 − 11φ2 + 2φ3 + 4φ4). (35)

This perturbation is charge-conserving and spin-zero (i.e., itsleft and right scaling dimensions are equal). If it is relevant,then the edge is in a different phase. To find this phase, it ishelpful to make the basis change:

WT (K1 ⊕ σz)W = K2 ⊕ σz , (36)

8

where

K2 =

(3 11 4

)

, (37)

and

W =

0 0 1 00 −2 0 1−2 3 0 −21 −7 0 4

. (38)

Making the basis changeφ =Wφ′, we see that

φ1 − 11φ2 + 2φ3 + 4φ4 = φ′3 + φ′4. (39)

Therefore, the resulting phase is described by (37).To see that these are, indeed, different phases, we can com-

pute basis-independent quantities, such as the lowest scalingdimension of any operator in the two theories. In theK1

theory, it is1/22 while in theK2 theory, it is3/22. Mea-surements that probe the edge structure in detail can, thereby,distinguish these two phases of the edge. Consider, first,transport through a QPC that allows tunneling between thetwo edges of the Hall bar, as described in Sec II A. In thestate governed byK1, the most relevant backscattering termis cos(φT2 −φB2 ). Applying Eq (12), the backscattered currentwill depend on the voltage according to

Ib1 ∝ V −9/11. (40)

An alternative probe is given by tunneling into the edge froma metallic lead. The most relevant term in theK1 edge phasethat tunnels one electron into the lead isψ†

leadeiφT

1 . ApplyingEq (14) yields the familiar current-voltage relation,

Itun1 ∝ V. (41)

In contrast, in the phase governed byK2, the most relevantbackscattering term across a QPC is given bycos(φ′T2 −φ′B2 ),which from Eq (12) yields the current-voltage relation

Ib2 ∝ V −5/11, (42)

while the most relevant single-electron tunneling term is givenbyψ†

leade−3iφ′T

1 −iφ′T2 , which yields the scaling from Eq (14)

Itun2 ∝ V 3. (43)

Since the two edge theories given byK1 andK2 are con-nected by a phase transition just on the edge, we may expectthey bound the same bulk Chern-Simons theory. Indeed, thebulk quasiparticles can be identified up to ambiguous signsdue to their fermionic nature. First, the discriminant groupof theK1 theory isZ/11. We define a quasiparticle basis forthis theory asψj ≡ (−j,−6j)T , j = 0, 1, . . . , 10. (Whilethe cyclic nature of the groupZ/11 implies the identification(a, b) ≡ (a′, b′) mod (1, 11) for a, b, a′, b′ ∈ Z, we choosethe above basis in order to ensure charge conservation.) TheS matrix is given bySjj′ = 1√

11e−

72πi11

jj′ . For the other

theory given byK2, the discriminant group obviously hasthe same structure with the generator being(0, 1)T and thequasiparticles are denoted byψ′

j . TheS matrix is given by

S′jj′ =

1√11e−

6πi11

jj′ . Now we make the following identifica-tion:

ψ′j ←→ ψj . (44)

This identification preserves theU(1) charge carried by eachquasiparticle. TheS matrices are also identified:

Sj,j′ =1√11e−

72πi11

jj′ =1√11e−

6πi11

jj′ = S′jj′ . (45)

Since the diagonal elements ofS are basicallyT 2, it followsthat the topological spins are also identified up to±1.

Our second example is

K ′1 =

(1 00 7

)

, (46)

with t = (1, 1)T . As before, we suppose that a non-chiral pairof modes comes down in energy and interacts strongly withthe two right-moving modes described by (46). The actionnow takes the form

S =

∫

dx dt

(1

4π(K ′

1 ⊕ σz)IJ ∂tφI∂xφJ

− 1

4πVIJ∂xφ

I∂xφJ +

1

2πtIǫµν∂µφ

IAν

)

. (47)

If the matrixVIJ is such that the perturbation

S′ =

∫

dx dt u′ cos(φ3 + φ4) (48)

is relevant and this is the only perturbation added to Eq. (47),then the two additional modes become gapped and the sys-tem is in the phase in Eq. (46). Suppose, instead, the onlyperturbation is the following:

S′′ =

∫

dx dt u′′ cos(φ1 + 7φ2 + φ3 + 3φ4). (49)

This perturbation is charge-conserving and spin-zero. If it isrelevant, then the edge is in a different phase. To find thisphase, it is helpful to make the basis change

W ′T (K ′1 ⊕ σz)W ′ = K ′

2 ⊕ σz, (50)

where

K ′2 =

(2 11 4

)

(51)

and

W ′ =

2 1 0 −11 −1 0 −10 0 −1 0−3 2 0 3

. (52)

9

Making the basis changeφ =W ′φ′, we see that

φ1 + 7φ2 + φ3 + 3φ4 = φ′4 − φ′3. (53)

Therefore, the resulting phase is described by (51). This isa different phase, as may be seen by noting that the latticecorresponding to Eq. (51) is an even lattice while the latticecorresponding to Eq. (46) is odd.

The difference between the two edge phases is even moredramatic than in the previous example. One edge phasehas gapless fermionic excitations while the other one doesnot! This example shows that an edge reconstruction canrelate a theory with fermionic topological order to one withbosonic topological order. Again, these two edge phases oftheν = 8/7 can be distinguished by the voltage dependenceof the current backscattered at a quantum point contact and thetunneling current from a metallic lead. In theK ′

1 edge phase(46), the backscattered current at a QPC is dominated by thetunneling termcos(φT2 − φB2 ); using Eq (12) this yields thecurrent-voltage relation

Ib1 ∝ V −5/7, (54)

while the single-electron tunneling into a metallic lead isdom-inated by the tunneling termψ†

leadeiφT

1 , which, using Eq (14),yields the familiar linear current-voltage scaling

Itun1 ∝ V. (55)

In theK ′2 edge phase (51), the backscattered current at a QPC

is dominated by the backscattering termcos(φ′T2 −φ′B2 ), yield-ing:

Ib2 ∝ V −3/7. (56)

The tunneling current from a metallic lead is due to thetunneling of charge-2e objects created by the edge operatoreiφ

′

1+4iφ′

2 . If we assume that the electrons are fully spin-polarized andSz is conserved, then the most relevant term thattunnels2e into the metallic lead isψ†

lead∂ψ†leade

iφ′T1 +4iφ′T

2 .Using Eq (14) the tunneling current is proportional to a veryhigh power of the voltage:

Itun2 ∝ V 7. (57)

Again, although the theories look drastically different, wecan show that the bulkS matrices are isomorphic. First, thediscriminant group of theK ′

1 theory isZ/7 whose generatorwe can take to be the(0, 4) quasiparticle. We label all quasi-particles in this theory asψj ≡ (0, 4j), j = 0, 1, . . . , 6. TheSmatrix is given bySjj′ = 1√

7e−

32πi7

jj′ . For the other theory

given byK ′2, the discriminant group is generated by(0, 1)T

and we denote the quasiparticles byψ′j . TheS matrix is given

by S′jj′ =

1√7e−

4πi7

jj′ . Now we make the following identifi-cation:

ψ′j ←→ ψj . (58)

TheS matrices are then seen to be identical:

Sj,j′ =1√7e−

32πi7

jj′ =1√7e−

4πi7

jj′ = S′jj′ . (59)

IV. EDGE PHASE TRANSITIONS

In the previous section, we gave two simple examples ofedge phase transitions that can occur between two distinct chi-ral theories. In this section, we discuss how edge transitionscan occur in full generality.

The chiral Luttinger liquid action is stable against all per-turbations involving only the gapless fields in the action inEq. (6) (or, equivalently in the integer case, the action inEq. (1)). However, as we have seen in the previous section,strong interactions with gapped excitations can drive a phasetransition that occurs purely at the edge. While the bulk iscompletely unaffected, the edge undergoes a transition intoanother phase.

On the way to understanding this in more generality, wefirst consider an integer quantum Hall state. At the edge ofsuch a state, we expect additional gapped excitations that weordinarily ignore. However, they can interact with gaplessexcitations. (Under some circumstances, they can even be-come gapless.28) Let us suppose that we have an additionalleft-moving and and additional right-moving fermion which,together, form a gapped unprotected excitation. Then addi-tional terms must be considered in the action. Let us first con-sider the case of an integer quantum Hall edge. The action inEqs. (1) and (2) becomesS0 + S1 + Su with

Su =

∫

dx dt(

ψ†N+1 (i∂t + vN+1i∂x)ψN+1

+ ψ†N+2 (i∂t − vN+2i∂x)ψN+2

+ uψ†N+1ψN+2 + h.c.

+ vI,N+1ψ†IψIψ

†N+1ψN+1 + vI,N+2ψ

†IψIψ

†N+2ψN+2

+ LN,L

)

, (60)

whereψN+1, ψN+2 annihilate right- and left-moving excita-tions which have an energy gapu for vI,N+1 = vI,N+2 = 0.So long asvI,N+1 andvI,N+2 are small, this energy gap sur-vives, and we can integrate outψN+1, ψN+2, thereby recov-ering the actionS0 + S1 in Eqs. (1) and (2), but with the cou-plings renormalized. However, ifvI,N+1 andvI,N+2 are suffi-ciently large, then some of the other terms in the action, whichwe have denoted byLN,L in Eq. (60) may become more rele-vant thanu. These include terms such as

LN,L = uIψ†IψN+2 + h.c.+ . . . . (61)

In order to understand these terms better, it is helpful toswitch to the bosonic representation, where there is no addi-tional overhead involved in considering the general case ofachiral Abelian state, integer or fractional:

S =

∫

dx dt

(1

4π(K ⊕ σz)IJ ∂tφI∂xφJ−

1

4πVIJ∂xφ

I∂xφJ

+∑

mI

umIcos(mIφ

I)+

1

2π

∑

I

ǫµν∂µφIAν

)

. (62)

Here,I = 1, 2, . . . , N + 2; and(K ⊕ σz)IJ is the direct sumof K andσz : (K ⊕ σz)IJ = KIJ for I = J = 1, 2, . . . , N ,

10

(K ⊕ σz)IJ = 1 for I = J = N + 1, (K ⊕ σz)IJ = −1 forI = J = N + 2, and(K ⊕ σz)IJ = 0 if I ∈ {1, 2, . . . , N},J ∈ {N + 1, N + 2} or vice-versa. The interaction ma-trix has VI,N+1 ≡ vI,N+1, VI,N+2 ≡ vI,N+2. ThemIsmust be integers because theφIs are periodic. For instance,mI = (0, 0, . . . , 0, 1,−1) corresponds to the mass termu(ψ†

N+1ψN+2+h.c.) in Eq. (60), soumI= u. In the last term,

we are coupling all modes equally to the electromagnetic field,i.e. this term can be written in the formtIǫµν∂µφIAν withtI = 1 for all I. This is the natural choice, since we expectadditional fermionic excitations to carry electrical chargee.

In general, most of the couplingsumIwill be irrelevant at

the Gaussian fixed point. An irrelevant coupling cannot openagap if it is small enough to remain in the basin of attraction ofthe Gaussian fixed point. However, if we make the couplinglarge enough, it may be in the basin of attraction of anotherfixed point and it may open a gap. We will not comment moreon this possibility here. However, we can imagine tuning theVIJs so that any givenumI

is relevant. To analyze this possi-bility, it is helpful to change to the variablesXa = eaIφ

I , interms of which the action takes the form

S =

∫

dx dt

(1

4πηab∂tX

a∂xXb − 1

4πvab∂xX

a∂xXb

+∑

mI

umIcos(mIf

IaX

a)+

1

2π

∑

I

fJa ǫµν∂µX

aAν .

)

(63)

eaI andf Ia are bases for the latticeΛN+2 and its dualΛ∗

N+2,where the latticeΛN+2 corresponds toK ⊕ σz . The vari-ablesXa satisfy the periodicity conditionX ≡ X + 2πy fory ∈ ΛN+2. Note that, since one of the modes is left-moving,the Lorentzian metricηab = diag(1N−1,−1) appears in Eq.(63).

Sincef Ia is a basis of the dual latticeΛ∗

N+2, the cosine termcan also be written in the form

∑

v∈Λ∗

N+2

uv cos (v ·X) .

The velocity/interaction matrix is given byvab = VIJfIaf

Jb .

Now suppose that the velocity/interaction matrix takes theform

vab = v OcaδcdO

db, (64)

whereO ∈ SO(N + 1, 1). Then we can make a change ofvariables toXa ≡ Oa

bXb. We specialize to the case of a

single cosine perturbation associated with a particular vectorin the dual latticev0 ≡ pIf

I which we will make relevant(we have also setAν = 0 since it is inessential to the presentdiscussion). Now Eq. (63) takes the form

S =1

4π

∫

dx dt

(

ηab∂tXa∂xX

b − vδab∂xXa∂xXb

+ uv0cos(

pIfIa (O

−1)abXb))

. (65)

If this perturbation has equal right and left scaling dimensions(i.e., is spin-zero), then its scaling dimension is simply twiceits left scaling dimension with corresponding beta function

duv0

dℓ=(2− q2N+2

)uv0

, (66)

whereqb ≡ pIfIa (O

−1)ab. The transformationO−1 can bechosen to be a particular boost in the(N + 2)-dimensionalspaceRN+1,1. Becauseqa is a null vector (i.e., a light-likevector) in this space, by taking the boost in the opposite direc-tion of the “spatial” components ofqa, we can “Lorentz con-tract” them, thereby makingqN+2 as small as desired. Thus,by takingvab of the form (64) and choosingO ∈ SO(N+1, 1)so thatq2N+2 < 2, we can make this coupling relevant.

When this occurs, two modes, one right-moving and oneleft-moving, will acquire a gap. We will then be left over witha theory withN gapless right-moving modes. The gaplessexcitationsexp(iv ·X) of the system must commute withv0 ·X and, since the cosine fixesv0 ·X, any two excitations thatdiffer by v0 ·X should be identified. Thus, the resulting low-energy theory will be associated with the latticeΓ defined byΓ ≡ Λ⊥/Λ‖, whereΛ⊥,Λ‖ ⊂ ΛN+2 are defined byΛ⊥ ≡{v ∈ ΛN+2 |v · v0 = 0} andΛ‖ ≡ {nv0 |n ∈ Z}. If gI is abasis forΓ, then we can define a K-matrix in this basis,KIJ =gI ·gJ . The low-energy effective theory for the gapless modesis

S =

∫

dx dt

(1

4πKIJ∂tφ

I∂xφJ − 1

4πVIJ∂xφ

I∂xφJ

+1

2πtIǫµν∂µφ

IAν

)

. (67)

Whenv0 = (0, 0, . . . , 0, 1,−1) is the only relevant opera-tor, φN+1 andφN+2 are gapped out. Therefore,Γ = Λ andKIJ = KIJ . However, when other operators are present,Γcould be a different latticeΓ ≇ Λ, from which it follows thatKIJ 6= KIJ (and,K 6=WTKW for anyW ).

We motivated the enlargement of the theory fromK toK⊕σz by assuming that an additional pair of gapped counter-propagating fermionic modes comes down in energy andinteracts strongly with the gapless edge excitations. Thiscounter-propagating pair of modes can be viewed as a thinstrip of ν = 1 integer quantum Hall fluid or, simply, as afermionic Luttinger liquid. Of course, more than one suchpair of modes may interact strongly with the gapless edge ex-citations, so we should also consider enlarging the K-matrixtoK⊕σz⊕σz . . .⊕σz . We can generalize this by imaginingthat we can add any one-dimensional system to the edge of aquantum Hall state. (This may not be experimentally-relevantto presently observed quantum Hall states, but as a matter ofprinciple, this is something that could be done without affect-ing the bulk, so we should allow ourselves this freedom.) Anyclean, gapless 1D system of fermions is in a Luttinger liq-uid phase (possibly with some degrees of freedom gapped).Therefore,K ⊕ σz ⊕ σz . . .⊕ σz is actually the most generalpossible form for the edge theory.

One might wonder about the possibility of attaching a thinstrip of a fractional quantum Hall state to the edge of the sys-

11

tem. Naively, this would seem to be a generalization of ourputative most general formK⊕σz⊕σz . . .⊕σz. To illustratethe issue, let us consider a bulkν = 1 IQH state and place athin strip of ν = 1/9 FQH state at its edge. The two edgesthat are in close proximity can be described by the followingK-matrix:

K =

(1 00 −9

)

. (68)

As discussed in Ref. 9, this edge theory can become fullygapped with charge-non-conserving backscattering. Then weare left with the outer chiral edge of the thin strip, which isdescribed byK = (9), which can only bound a topologicallyorderedν = 1/9 Laughlin state. The subtlety here is thata thin strip of the fractional quantum Hall state has no two-dimensional bulk and should be considered as a purely one-dimensional system. Fractionalized excitations, characterizedby fractional conformal spins only make sense when a true 2Dbulk exists. If the width of the strip is small, so that there isno well-defined bulk between them, then we can only allowoperators that add an integer number of electrons to the twoedges. We cannot add fractional charge since there is no bulkwhich can absorb compensating charge. Thus the minimalconformal spin of any operator is1/2. In other words, startingfrom an one-dimensional interacting electronic system, onecannot change the conformal spin of the electron operators.So attaching a thin strip of FQH state is no different fromattaching a trivial pair of modes.

In a bosonic system, we cannot even enlarge our theory bya pair of counter-propagating fermionic modes. We can onlyenlarge our theory by a Luttinger liquid of bosons or, equiva-lently, a thin strip ofσxy = 2e2

h bosonic integer quantum hallfluid9,12,30. Such a system has K-matrix equal toσx, whichonly has bosonic excitations. Equivalently, bosonic systemsmust have evenK-matrices – matrices with only even num-bers along the diagonal – because all particles that braid triv-ially with every other particle must be a boson. Since the en-larged matrix must have the same determinant as the originalone because the determinant is the ground state degeneracy ofthe bulk phase on the torus17, we can only enlarge the theoryby σx, the minimal even unimodular matrix. Therefore, in thebosonic case, we must enlarge our theory byK → K ⊕ σx.

In the fermionic case, we must allow such an enlargementbyσx as well. We can imagine the fermions forming pairs andthese pairs forming a bosonic Luttinger liquid which enlargesK by σx. In fact, it is redundant to consider bothσz andσx:for an odd matrixK,W (K ⊕ σz)WT = K ⊕ σx, where

W =

1 0 · · · 0 y1 −y10 1 · · · 0 y2 −y2...

......

......

...0 0 · · · 1 yN −yN0 0 · · · 0 1 −1x1 x2 · · · xN s 1− s

(69)

Here the vectorx has an odd length squared, i.e.xTKx isodd; by definition ofK odd, such anx must exist. The vector

y is defined asy = −Kx and the integers by s = 12 (1 −

xTKx). ThusK⊕σx isGL(N +2,Z)-equivalent toK⊕σzand our previous discussion for fermionic systems could beredone entirely with extra modes described byσx. However,if K is even, thenK ⊕ σx is notGL(N + 2,Z)-equivalent toK ⊕ σz.

We remark that althoughσz enlargement andσx enlarge-ment are equivalent for fermionic states when topologicalproperties are concerned, they do make a difference in chargevectors: the appropriate charge vector for theσz block shouldbe odd and typically taken to be(1, 1)T . However the chargevector for theσx block must be even and needs to be deter-mined from the similarity transformation.

To summarize, a quantum Hall edge phase described bymatrixK1 can undergo a purely edge phase transition to an-other edge phase withGL(N,Z)-inequivalentK2 (with iden-tical bulk) if there existsW ∈ GL(N + 2k,Z) such that

K2 ⊕ σx ⊕ . . .⊕ σx = WT (K1 ⊕ σx ⊕ . . .⊕ σx) W . (70)

for some numberk of σxs on each side of the equation. In afermionic system withK1 odd, an edge phase transition canalso occur to an even matrixK2 if

Keven2 ⊕ σz ⊕ . . .⊕ σx = WT

(Kodd

1 ⊕ σx ⊕ . . .⊕ σx)W .(71)

V. STABLE EQUIVALENCE, GENERA OF LATTICES,AND THE BULK-EDGE CORRESPONDENCE FOR

ABELIAN TOPOLOGICAL PHASES

A. Stable Equivalence and Genera of Lattices

In the previous section, we saw that a bulk Abelian quan-tum Hall state associated withK1 has more than one differentstable chiral edge phase if there existsGL(N,Z)-inequivalentK2 andW ∈ GL(N + 2k,Z) such that

K2 ⊕ σx ⊕ . . .⊕ σx = WT (K1 ⊕ σx ⊕ . . .⊕ σx) W . (72)

This is an example of a stable equivalence; we say thatK1

andK2 arestably equivalentif, for somen, there exist sig-nature(n, n) unimodular matricesLi such thatK1 ⊕ L1 andK2 ⊕ L2 areintegrally equivalent, i.e. areGL(N + 2n,Z)-equivalent. If there is a choice ofLis such that both are even,we will say thatK1 andK2 are “σx-stably equivalent” sincetheLis can be written as direct sums ofσxs. We also sawin Eq. 71 that whenK1 is odd andK2 is even, we will needL2 to be an odd matrix. We will call this “σz-stable equiva-lence” sinceL2 must contain aσz block. We will useU todenote the signature(1, 1) even Lorentzian lattice associatedwith σx. Thenσx-stable equivalence can be restated in thelanguage of lattices as follows. Two latticesΛ1, Λ2 areσx-stably equivalent ifΛ1 ⊕ U · · · ⊕ U , andΛ2 ⊕ U · · · ⊕ U areisomorphic lattices. Similarly,Uz will denote the Lorentzianlattice associated withσz. Occasionally, we will abuse nota-tion and useσx andσz to refer to the corresponding latticesU , Uz.

12

Stable equivalence means that the twoK-matrices areequivalent after adding “trivial” degrees of freedom – i.e.purely 1D degrees of freedom that do not require any changeto the bulk. This is analogous to the notion of stable equiva-lence of vector bundles, according to which two vector bun-dles are stably equivalent if and only if isomorphic bundlesare obtained upon joining them with trivial bundles.

We now introduce the concept of thegenusof a lattice orintegral quadratic form. Two integral quadratic forms are inthe samegenus23,31 when they have the same signature andare equivalent over thep-adic integersZp for every primep.Loosely speaking, equivalence overZp can be thought of asequivalence modulo arbitrarily high powers ofp, i.e. inZ/pn

for everyn. The importance of genus in the present contextstems from the following statement of Conway and Sloane23:

Two integral quadratic formsK1 and K2 are in the samegenus if and only ifK1⊕σx andK2⊕σx are integrally equiv-alent.

Proofs of this statement are, however, difficult to pin downin the literature. It follows, for instance, from results inRef.31 about a refinement of the genus called the spinor genus.Below, we show how it follows in the even case from re-sults stated by Nikulin24. This characterization of the genusis nearly the same as the definition ofσx-stable equivalencegiven in (72), except that Eq. (72) allows multiple copieswhich is natural since a physical system may have access tomultiple copies of trivial degrees of freedom. Its relevanceto our situation follows from the following theorem that wedemonstrate below:

TwoK-matricesK1 andK2 of the same dimension, signatureand type are stably equivalent if and only ifK1 ⊕ σx andK2 ⊕ σx are integrally equivalent, i.e. only a single copy ofσx is needed in Eq. (72).

Thus any edge phase that can be reached via a phase tran-sition involving multiple sets of trivial 1D bosonic degreesof freedom (described by K-matrixσx) can also be reachedthrough a phase transition involving only a single such set.Wedemonstrate this by appealing to the following result stated byNikulin24 (which we paraphrase but identify by his number-ing):

Corollary 1.16.3:The genus of a lattice is determined by itsdiscriminant groupA, parity, signature(r+, r−), and bilinearform b on the discriminant group.

Since taking the direct sum with multiple copies ofσx doesnot change the parity, or bilinear form on the discriminantgroup, anyK1 andK2 that areσx-stably equivalent are in thesame genus. The theorem then follows from the statement23

above that only a single copy ofσx is needed.In the even case, the theorem follows directly from two

other results found in Nikulin24:

Corollary 1.13.4: For any even latticeΛ with signature(r+, r−) and discriminant quadratic formq, the latticeΛ⊕Uis the only lattice with signature(r++1, r−+1) and quadraticform q.

Theorem 1.11.3:Two quadratic forms on the discriminantgroup are isomorphic if and only if their bilinear forms areisomorphic and they have the same signature(mod 8).

If latticesΛ1 andΛ2 are in the same genus, they must havethe same(r+, r−) and bilinear formb. According to Theo-rem 1.11.3, they must have the same quadratic form, namelyq([x]) = 1

2b([x], [x]), which is well-defined in the case of aneven lattice. Then, Corollary 1.13.4 tells us thatΛ1⊕U is theunique lattice with signature(r+ + 1, r− + 1) and quadraticform q. SinceΛ2⊕U has the same signature(r++1, r−+1)and quadratic formq, Λ1 ⊕ U ∼= Λ2 ⊕ U . Thus, we see thatany two evenK-matrices in the same genus are integrally-equivalent after taking the direct sum with a single copy ofσx.Of course, our previous arguments that used Nikulin’s Corol-lary 1.16.3 and the characterization of genus from Conwayand Sloane23 are stronger since they apply to odd matrices.

B. Bulk-Edge Correspondence

Since the quadratic formq([u]) gives theT andS matri-ces according to Eqs. (23) and (25), we can equally-well saythat the genus of a lattice is completely determined by theparticle types,T -matrix,S-matrix, and right- and left-centralcharges. For a bosonic system, the genus completely deter-mines a bulk phase. Conversely, a bulk topological phasealmost completely determines a genus: the bulk phase de-termines(c+ − c−) mod 24 while a genus is specified by(c+, c−). However, if the topological phase is fully chiral,so that it can havec− = 0, then it fully specifies a familyof genera that differ only by adding central charges that areamultiple of24, i.e. 3k copies of theE8 state for some integerk (see Section VII A for a discussion of this state). Thus, up toinnocuous shifts of the central charge by24, we can say that

A bulk bosonic topological phase corresponds to a genus ofeven lattices while its edge phases correspond to the differentlattices in this genus.

The problem of detemining the different stable edge phasesthat can occur for the same bosonic bulk is then the problemof determining how many distinct lattices there are in a genus.

In the fermionic case, the situation is more complicated.A fermionic topological phase is determined by its particletypes, itsS-matrix, and its central charge (mod 24). It does nothave a well-definedT -matrix because we can always changethe topological twist factor of a particle by−1 simply byadding an electron to it. According to the following resultof Nikulin, these quantities determine an odd lattice:

Corollary 1.16.6:Given a finite Abelian groupA, a bilinearform b : A × A → Q/Z, and two positive numbers(r+, r−),then, for sufficiently larger+, r−, there exists an odd latticefor whichA is its discriminant group.b is the bilinear formon the discriminant group, and(r+, r−) is its signature.

Since theS-matrix defines a bilinear form on the Abeliangroup of particle types, this theorem means that the quanti-ties that specify a fermionic Abelian topological phase are

13

compatible with an odd lattice. Clearly, they are also com-patible with an entire genus of odd lattices sinceσx stableequivalence preserves these quantities. Moreover, by Corol-lary 1.16.3, there is only a single genus of odd lattices thatare compatible with this bulk fermionic Abelian topologicalphase. However, Corollary 1.16.3 leaves open the possibilitythat there is also a genus of even lattices that is compatiblewith this fermionic bulk phase, a possibility that was realizedin one of the examples in Section III. This possibility is dis-cussed in detail in Section VI. However, the general result thatwe can already state, up to shifts of the central charge by24 is

A bulk fermionic topological phase corresponds to a genus ofodd lattices while its edge phases correspond to the differentlattices in this genus and, in some cases (specificed in SectionVI), to the different lattices in an associated genus of evenlattices.

In principle, one can determine how many lattices there arein a given genus by using the Smith-Siegel-Minkowski massformula23 to evaluate the weighted sum

∑

Λ∈g

1

|Aut(Λ)| = m(K) (73)

over the equivalence classes of lattices in a given genusg.Each equivalence class of forms corresponds to a latticeΛ.The denominator is the order of the automorphism groupAut(Λ) of the latticeΛ. The right-hand-side is themassofthe genus ofK, which is given by a complicated but explicitformula (see Ref. 23).

Given a K-matrix for a bosonic state, one can compute thesize of its automorphism group60, which gives one term inthe sum in (73). If this equals the mass formula on the right-hand-side of Eq. (73), then it means the genus has only oneequivalence class. If not, we know there is more than oneequivalence class in the genus. Such a program shows32 that,in fact, all genera contain more than one equivalence class forN > 10, i.e. all chiral Abelian quantum Hall states with cen-tral chargec > 10 have multiple distinct stable chiral edgephases. For3 ≤ N ≤ 10, there is a finite set of generawith only a single equivalence class33; all others have multi-ple equivalence classes. The examples ofν = 16 analyzed inRef. 2 andν = 12/23 that we gave in Section VII are, in fact,the rule. Bosonic chiral Abelian quantum Hall states with asingle stable chiral edge phase are the exception, they can onlyexist forc ≤ 10 and they have been completely enumerated33.

This does not tell us how, given one equivalence class,to find other equivalence classes ofK-matrices in the samegenus. However, one can use the Gauss reduced form23 tofind all quadratic forms of given rank and determinant by bruteforce. Then we can use the results at the end of previous Sec-tion to determine if the resulting forms are in the same genus.

C. Primary Decomposition of Abelian Topological Phases

According to the preceding discussion, two distinct edgephases can terminate the same bulk phase if they are both in

the same genus (but not necessarily only if they are in thesame genus in the fermionic case). It may be intuitively clearwhat this means, but it is useful to be more precise about whatwe mean by “the same bulk phase”. In more physical terms,we would like to be more precise about what it means fortwo theories to have the same particle types andS- andT -matrices. In more formal terms, we would like to be moreprecise about what is meant in Nikulin’s Theorem 1.11.3 byisomorphic quadratic forms and bilinear forms. In order to dothis, it helps to view an Abelian topological phase in a some-what more abstract light. When viewed from the perspectiveof an edge phase or, equivalently, a K-matrix, the bulk phaseis determined by the signature(r+, r−), together with the bi-linear form on the discriminant groupΛ∗/Λ induced by thebilinear form on the dual latticeΛ∗ determined byK. Aswe have seen, this data uniquely specifies a nondegeneratequadratic formq : Λ∗/Λ → Q/Z on the discriminant group.Therefore, we may view the genus more abstractly in termsof an arbitrary finite Abelian groupA and a quadratic formq : A → Q/Z, making no direct reference to an underlyinglattice. We will sometimes call such a quadratic form afinitequadratic form to emphasize that its domain is a finite Abeliangroup. The elements of the groupA are the particle types inthe bulk Abelian topological phase.

Now suppose we have two bulk theories associated withAbelian groupsA, A′, quadratic formsq : A → Q/Z,q′ : A′ → Q/Z and chiral central chargesc−, c′−. These the-ories are the same precisely when the chiral central chargessatisfyc− ≡ c′− mod 24, and when the associated quadraticforms areisomorphic. This latter condition means that thereexists a group isomorphismf : A′ → A such thatq′ = q ◦ f .Note that if the quadratic forms are isomorphic then the chiralcentral charges must be equal (mod 8) according to the Gauss-Milgram sum. However, the bulk theories are the same only ifthey satisfy the stricter condition that their central charges areequal modulo 24.

The implications of this become more apparent after ob-serving that any Abelian group factors as a direct sumA ≃⊕pAp over primes dividing|A|, whereAp ⊂ A is the p-primary subgroup of elements with order a power ofp. Anyisomorphismf : A′ → A must respect this factorization bydecomposing asf = ⊕pfp, with eachfp : A′

p → Ap. Fur-thermore, every finite quadratic form decomposes into a directsumq = ⊕pqp of p-primary forms; we callqp thep-part of q.This ultimately leads to a physical interpretation forp-adic in-tegral equivalence: ifp is odd, two K-matrices arep-adicallyintegrally equivalent precisely when thep-parts of their asso-ciated quadratic forms are isomorphic. Additional subtletiesarise whenp = 2 but, as we will see, these are the reason forthe distinction betweenσx- andσz-equivalence.

The image of a given finite quadratic formq is a finite cyclicsubgroupN−1

q Z/Z ⊂ Q/Z isomorphic toZ/Nq, whereNq isthe levelof the finite quadratic formq. The level is the small-est integerN such thatq factors throughZ/N , implying thatthe topological spins of particles inAq areNqth roots of unity.Because the level of the direct sum of finite quadratic formsis the least common multiple of the levels of the summands,the level ofq = ⊕pqp is equal to the productNq =

∏

pNqp

14

of the levels of theqp. If p is odd, the level ofqp is the or-der of the largest cyclic subgroup ofAp, while it is typicallytwice as big forq2. Physically, this means that the entire the-ory uniquely factors into a tensor product of anyon theoriessuch that the topological spins of the anyons in thepth theoryarepth-power roots of unity. This decomposition lets us ex-press alocal-to-global principlefor finite quadratic forms:qandq′ are isomorphic iffqp andq′p are for everyp. Indeed,if one views prime numbers as “points” in an abstract topo-logical space61, this principle says thatq andq′ aregloballyequivalent(at all primes) iff they are locally equivalent at eachprime dividing|A|.

Further information about the prime theories is obtained bydecomposing eachAp into a product

Ap ≃mp∏

m=0

(Z/pm)dpm (74)

of cyclic groups, wheredp0 , . . . , dpmp−1 ≥ 0 anddpmp > 0.Whenp is odd, there is a 1-1 correspondence between bilinearand quadratic forms onAp because multiplication by 2 is in-vertible in everyZ/pm. Furthermore, given a quadratic formqp onAp for oddp, we claim there always exists an automor-phismg ∈ Aut(Ap) that fully diagonalizesqp relative to afixed decomposition (74) such that

qp ◦ g =⊕

m

(q+pm ⊕ . . .⊕ q+pm ⊕ q±pm

︸ ︷︷ ︸

dpm terms

), (75)

where

q+pm(x) =1

pm2−1x2 modZ,

q−pm(x) =1

pmup2

−1x2 modZ

andup is some fixed non-square modulopn. A dual perspec-tive is that, givenqp, it is always possible to choose a decom-position (74) ofAp relative to whichqp has the form of theright-hand-side of (75). However, not every decompositionwill work for a givenqp becauseAut(Ap) can mix the differ-ent cyclic factors. For example,Aut((Z/p)d) ≃ GL(d,Z/p)mixes the cyclic factors of orderp. There will also be auto-morphisms mixing lower-order generators with ones of higherorder, such as the automorphism ofZ/3 ⊕ Z/9 = 〈α3, α9〉defined on generators byα3 7→ α3 andα9 7→ α3 + α9. Phys-ically, this means that the anyon theory associated toAp fur-ther decomposes into a tensor product of “cyclic” theories,al-though now such decompositions are not unique because onecan always redefine the particle types via automorphisms ofAp.

D. p-adic Symbols

Two K-matrices arep-adically integrally equivalent iffthe diagonalizations of thep-parts of their associated finite

quadratic forms coincide. The numbersdpm and the sign ofthe last form in themth block thus form a complete set of in-variants forp-adic integral equivalence of K-matrices. Thisdata is encoded into thep-adic symbol, which is written as1±d

p0p±dp1 (p2)±d

p2 · · · (terms withdpm = 0 are omitted)and can be computed using Sage34. Two K-matrices arep-adically integrally equivalent iff theirp-adic symbols coin-cide.

Thep-adic symbol can be computed more directly by not-ing that K-matrices are equivalent over thep-adic integerswhen they are equivalent by arational transformation whosedeterminant and matrix entries do not involve dividing byp.Such transformations can be reduced modulo arbitrary pow-ers ofp and give rise to automorphisms of thep-partAp of thediscriminant group. Given a K-matrixK, there always exists ap-adically integral transformationg puttingK into p-adicallyblock diagonalized23 form

gKgT = Kp0 ⊕ pKp1 ⊕ p2Kp2 ⊕ · · · , (76)

wheredet(Kpm) is prime top for everym.A more direct characterization of the genus can now be

given: Two K-matrices are in the same genus iff they are re-lated by arational transformation whose determinant and ma-trix entries are relatively prime to twice the determinant,orrather, to the levelN of the associated discriminant forms.Such a transformation suffices to simultaneouslyp-adicallyblock-diagonalizeK over thep-adic integers for everyp di-viding twice the determinant, and a similar reduction yieldsthe entire quadratic form on the discriminant group, with someextra complications whenp = 2. Such a non-integral trans-formation mapping two edge theories asg(Λ1) = Λ2 doesnot, however induce fractionalization in the bulk since it re-duces to an isomorphism between the discriminant groupsΛ∗1/Λ1 → Λ∗

2/Λ2. For example, theν = 12/11 K-matrices(32) and (37) are related by the following rational transforma-tion that divides by 3:

(1 0−1/3 1

)(3 11 4

)(1 −1/30 1

)

=

(1 00 11

)

.

One might be tempted to look at this transformation and con-clude that one of the particle types on the left-hand-side hasundergone fractionalization and divided into3 partons (due tothe−1/3 entries in the matrix), thereby leading to the phaseon the right-hand-side. But in mod 11 arithmetic, the number3 is invertible, so no fractionalization has actually occurred.

When p 6= 2, the p-adic symbol can be directly com-puted from any suchp-adic block diagonalization, as the term(pm)±dpm records thedimensiondpm = dim(Kpm) andsign± of det(Kpm), the latter being given by theLegendre symbol

(det(Kpm)

p

)

=

{

+1 if p is a square modp−1 if p is not a square modp.

In this case, it is further possible top-adically diagonalize allof the blocksKpm , in which case there exists ap-adically in-tegral transformationg that diagonalizes the formQ(x) =12x

TK−1x on the dual latticeΛ∗ such that its reduction mod-uloΛ takes the form (75).

15

K-matrix p-adic symbols quadratic form(

1 00 7

)

1+2

0 1+17+1

q+7(

2 11 4

)

1+2even 1+17+1

(

1 00 11

)

1−2

4 1+111+1 q+11(

3 11 4

)

(

3 00 5

)

1+2

0 1−13+1 1−15+1

q+3 ⊕ q+5(

2 11 8

)

1+2even 1−13+1 1−15+1

(

2 33 16

)

1+2even 1+123+1 q+23(

4 11 6

)

KA41−4

even 1+35+1

q+55⊕ I3 1−4

0 1+35+1

KE81+8

even 0I8 1+8

0

KE8⊕ I4

1+12

40I12

KD

+12

(

22

)

2+2even q+2,2

KD41−2

even2−2even q−2,2

(

4 22 4

)

2−2even 1+13+1 q−2,2 ⊕ q+3

TABLE I: Here we list thep-adic symbols and discriminant quadraticforms for various K-matrices appearing in this paper, beginning withthe canonical 2-adic symbol in every case, followed by the sym-bols for each prime dividing the determinant. Each block con-tains inequivalent-but-stably-equivalent matrices. Thelast few rowscontain K-matrices giving rise to some of the exceptional 2-adicquadratic forms mentioned in the text.

Whenp = 2, it is possible that only some of the blocksK2m

in the decomposition (76) can be 2-adically diagonalized23

(we call these blocksodd). The remainingevenblocks canonly be block diagonalized into2 × 2 blocks of the form(2a bb 2c

)with b odd, or rather, some number of copies of

σx and(2 11 2

). As with odd p, the 2-adic symbol associ-

ated to such a block diagonalization records the dimensionsd2m of the blocks, together with thesignsof the determinantsdet(K2m), which are given by theJacobi symbols

(2

det(K2m)

)

=

{

+1 if det(K2m) ≡ ±1 mod8

−1 if det(K2m) ≡ ±3 mod8

and record whether or notdet(Kpm) is a square mod 8. Inaddition to this data, the 2-adic symbol also records the par-ities as well as the tracesTrK2m mod 8 of the odd blocks.An additional complication is that a given K-matrix can be2-adically diagonalized in more than one way, and while thedimensions and parities of the blocks will be the same, thesigns and traces of the odd blocks – and thus the 2-adic sym-bols – can be different. While this makes checking 2-adicequivalence more difficult, it is nonetheless possible to defineacanonical2-adic symbol23 thatis a complete invariant for 2-adic equivalence. We record these canonical 2-adic symbolsfor many of the K-matrices considered in this paper in Table I.

The reason for the additional complexity whenp = 2 is be-cause multiplication by 2 is not invertible on the 2-primary

part (Q/Z)2 of Q/Z. This implies that ifq refines a bi-linear form on a 2-group then so doesq + 1

2 modZ, andsometimes these refinements are not isomorphic. For exam-ple, there is only one nondegenerate bilinear formb2(x, y) =xy2 modZ onZ/2, with two non-isomorphic quadratic refine-

mentsq±2 (x) = ±x4 modZ. Each of these refinements has

level 4 and corresponds respectively to the semionK = (2)and its conjugateK = (−2). These give theS andT matrices

S2 =1√2

(

1 1

1 −1

)

, T±2 = e∓

2πi24

(

1

±i

)

.

OnZ/2 × Z/2, there are two isomorphism classes of non-degenerate bilinear forms. The first class is represented by

(b2 ⊕ b2)(x, y) = 12 (x1y1 + x2y2) modZ

and has theS-matrix

S2 ⊗ S2 =1

2

1 1 1 1

1 −1 1 −11 1 −1 −11 −1 −1 1

.

All the refinements in this case have level 4 and are given bytensor products of semions. Up to isomorphism, this givesthree refinementsq+2 ⊕ q+2 , q+2 ⊕ q−2 andq−2 ⊕ q−2 , determinedby the K-matrices

(22

),(2−2

)and

(−2−2

)with c− =

2, 0,−2 respectively.The second class of bilinear forms onZ/2 × Z/2 contains

the single form

b2,2(x, y) =12 (x1y2 + x2y1) modZ

and gives theS-matrix

S2,2 =1

2

1 1 1 1

1 1 −1 −11 −1 1 −11 −1 −1 1

.

It is refined by two isomorphism classesq±2,2 of quadraticforms with T-matricesT±

2,2 = diag(1,±1,±1,−1) (thesehave level 2, the exception to the rule), up to the usual phaseof −2πic−/24. The formq+2,2 is given by the K-matrix

(2

2

)

and corresponds to the toric code. The formq−2,2 is given bythe K-matrix

KD4=

2 0 1 0

0 2 −1 0

1 −1 2 −10 0 −1 2

of SO(8)1, or equivalently, by the restriction of the quadraticform associated to the K-matrix

(4 22 4

)to the 2-part of its dis-

criminant groupZ/2 × Z/2 × Z/3. Again, these are distin-guished by their signatures, which are 0 and 4 mod 8. The

16

2-adic diagonalizations of these K-matrices contain examplesof even blocks, as illustrated in to even blocks in Table I.

Further complexity arises for higher powers of 2: There aretwo bilinear formsb±4 onZ/4, and fourb1,3,5,72m on eachZ/2m

whenm ≥ 3. There are also four quadratic formsq1,3,5,72m onZ/2m for everym ≥ 2, all with level 2m+1. Therefore, thebilinear formsb±4 have two refinements each, while the resthave unique refinements. On top of all this, even more com-plexity arises from the fact that factorizations of such formsis not typically unique. It is therefore less straightforwardto check equivalence of 2-adic forms. It is nonetheless stillpossible to define a canonical 2-adic symbol23 that is a com-plete invariant for 2-adic equivalence of K-matrices. However,this symbol carries strictly more information than the isomor-phism class of the 2-part of the discriminant form because itknows the parity ofK. To characterize the even-odd equiva-lences that we investigate in the next section, the usual 2-adicequivalence is replaced with equivalence of the 2-parts of dis-criminant forms as in the oddp case above.

The 2-adic symbol contains slightly more information thanjust the equivalence class of a quadratic form on the discrim-inant group. This is evident in our even-odd examples, forwhich allp-adic symbols for oddp coincide, with the only dif-ference occurring in the 2-adic symbol. It is however clear thattwo K-matricesKevenandKodd of different parities are stablyequivalent precisely when eitherKeven⊕1 andKodd⊕1 are inthe same genus, or otherwise, whenKeven⊕σz andKodd⊕σzare in the same genus. A detailed study of the 2-adic symbolsin this context will appear elsewhere.

VI. STABLE EQUIVALENCE BETWEEN ODD AND EVENMATRICES: FERMIONIC BULK STATES WITH BOSONIC

EDGES PHASES

We now focus on the case of fermionic systems, which aredescribed by oddK-matrices (i.e., matrices that have at leastone odd number on the diagonal). We ask: Under what cir-cumstances is such a K-matrix equivalent, upon enlargementby σz (or σx, since it makes no difference for an odd matrix),to an even K-matrix enlarged byσz:

Kodd ⊕ σz =WT (Keven ⊕ σz)W? (77)

This question can be answered using the theory of quadraticrefinements.21,22

As we have alluded to earlier, the naive definition of aquadratic form on the discriminant group breaks down for oddmatrices. To be more concrete,1

2u2 (mod 1) is no longer

well-defined on the discriminant group. In order to be well-defined on the discriminant group, shiftingu by a lattice vec-tor λ ∈ Λ must leaveq(u) invariant modulo integers, so thate2πiq(u) in Eq. (23) is independent of which representative inΛ∗ we take for an equivalence class inA = Λ∗/Λ. WhenKis odd, there are some vectorsλ in the original latticeΛ suchthat

q(u+ λ) ≡ q(u) + 1

2mod1. (78)

Physically, such a vector is just an electron (λ·λ is an odd inte-ger). One can attach an odd number of electrons to any quasi-particle and change the exchange statistics by−1. In a sense,the discriminant group should be enlarged toA⊕ (A+λodd):quasiparticles come in doublets composed of particles withopposite fermion parity, and therefore opposite topologicaltwists. The Gauss-Milgram sum over this enlarged set ofquasiparticles is identically zero, which is a clear signaturethat the Abelian topological phase defined by an odd K-matrixis not a TQFT in the usual sense.

While theT matrix is not well-defined for a fermionic the-ory, theS matrix, which is determined by the discriminant bi-linear formb([v], [v′]), makes perfect sense. This is becausea full braid of one electron around any other particle does notgenerate a non-trivial phase.

Given a bilinear formb, a systematic approach for defin-ing a quadratic form that is well-defined on the discrimi-nant group comes from the theory of quadratic refinements.The crucial result is that a given bilinear form canalwaysbe lifted to a quadratic formq on the discriminant group.The precise meaning of “lifting” is that there exists a well-defined discriminant quadratic form such thatb([v], [v′]) =q([v + v′]) − q([v]) − q([v′]).21,22 With q, the topologicaltwists are well-defined:e2πiq(u) = e2πiq(u+λ) for all u ∈ Λ∗

andλ ∈ Λ. We will give a constructive proof for the existenceof such aq, given any odd K-matrix.

Once the existence of such a quadratic formq([v]) is es-tablished, we can evaluate the Gauss-Milgram sum (27) anddeterminec− mod8. We then appeal to the following resultof Nikulin24:

Corollary 1.10.2:Given an Abelian groupA, a quadratic formq onA, and positive integers(r+, r−) that satisfy the Gauss-Milgram sum forq, there exists an even lattice with discrim-inant groupA, quadratic formq on the discriminant group,and signature(r+, r−), providedr++r− is sufficiently-large.

Using Corollary 1.10.2, we immediately see that an evenlattice characterized by(A, q, c− mod8) exists, whose Grammatrix is denoted byKeven. Recall that the chiral centralchargec− is equal to the signatureσ = r+ − r− of the lat-tice. Next we show thatKeven is σz-stably equivalent to theodd matrix we started with: namely, (77) holds for thisKeven.SinceKeven andK share the same discriminant group andSmatrix, they are stably equivalent upon adding unimodular lat-tices, according to Theorem 1. 1. 9. In other words, there existunimodular matricesU andU ′ such that

K ⊕ U ≃ Keven⊕ U ′. (79)

ApparentlyU ′ must be odd. We now add to both sides of theequation the conjugate ofU ′ denoted byU ′:

K ⊕ (U ⊕ U ′) ≃ Keven⊕ (U ′ ⊕ U ′). (80)

On the right-hand side,U ′ ⊗ U ′ is equivalent toσz ⊕ σz ⊕ · · ·σz . On the left-hand side,U ⊕ U ′

can be transformed to the direct sum ofIn wheren = σ(U) − σ(U ′) = σ(Keven) − σ(K) and several

17

σz/x’s. HereIn is the|n| × |n| identity matrix and whenn isnegative we take it to be−I|n|. If n 6= 0 mod8, thenKeven

has a different chiral central charge asK. Therefore we havearrived at the following theorem:

For any oddK matrix, K ⊕ In is σz-stably equivalentto an even K-matrix for an appropriaten.

The physical implication is that by adding a certain numberof Landau levels the edge phase of a fermionic Abeliantopological phase is always stably equivalent to a purelybosonic edge phase which has no electron excitations in itslow-energy spectrum.

The possible central charges of the bosonic edge theory arecferm + n + 8m for m ∈ Z. We can consider a fermionicsystem with an additional8m + n Landau levels, wheremis the smallest positive integer such that8m + n > 0. Sucha fermionic theory has precisely the same discriminant groupas the original fermionic theory and, consequently, is associ-ated with precisely the same bosonic system defined by therefinementq([u]). So even if the original fermionic theorydoes not have a stable chiral edge phase with only bosonicexcitations, there is a closely-related fermionic theory withsome extra filled Landau levels which does have a chiral edgephase whose gapless excitations are all bosonic. A simpleexample of this is given by theν = 1/5 Laughlin state,which hasK = 5. The corresponding bosonic state hasc = 4, so theν = 1/5 Laughlin state does not have a chiraledge phase whose gapless excitations are all bosonic. How-ever, the central charges do match if, instead, we consider theν = 3 + 1

5 = 16/5 state. This statedoeshave a bosonic edgephase, with K-matrix

KA4=

2 1 0 0

1 2 1 0

0 1 2 1

0 0 1 2

(81)

corresponding toSU(5)1. (Ordinarily, the Cartan matrix forSU(5) is written with−1s off-diagonal, but by a change ofbasis we can make them equal to+1.)

In the following we demonstrate concretely how to obtaina particular discriminant quadratic formq, starting from theodd lattice given byK. We already know that the naive defini-tion 1

2u2(mod1) does not qualify as a discriminant quadratic

form. In order to define a quadratic form on the discriminantgroup, we first define a quadratic functionQw(u) accordingto:

Qw(u) =1

2u2 − 1

2u ·w, (82)

for w ∈ Λ∗. Such a linear shift preserves the relation be-tween the quadratic function (T matrix) and the bilinear form(S matrix):

Qw(u+ v) −Qw(u)−Qw(v) = u · v. (83)

(Notice thatu · v is the symmetric bilinear formb(u,v) inStirling’s thesis22). Notice that at this stageQw is not yet aquadratic form onA, being just a quadratic function.

If, for anyλ ∈ Λ,Qw satisfiesQw(u+λ) ≡ Qw(u)mod1or, in other words,

λ · λ ≡ λ ·wmod2. (84)

then we can define the following quadratic form on the dis-criminant group:

q([u]) = Qw(u).

Expandingw in the basis of the dual latticew = wIfI and

expandingλIeI , we find that this condition is satisfied if wetakewI ≡ KII mod2. Thus, for a Hall state expressed inthe symmetric basis, we may identifyw with twice the spinvectorsI = KII/2.35,36

A central result of Ref. 21 is that such aw leads to a gener-alized Gauss-Milgram sum:

1√

|A|e

2πi8

w2∑

u

e2πiQw(u) = e2πiσ/8, (85)

where, in order for the notation to coincide, we have replacedthe chiral central charge with the signatureσ on the right-hand-side of the above equation. Note that the choice ofw

here is not unique. We can check that the modified Gauss-Milgram sum holds forw + 2λ∗ whereλ∗ ∈ Λ∗. First notethat

Qw+2λ∗(u) =1

2u2 − 1

2u ·w− u · λ∗

= Qw(u− λ∗)− 1

2λ∗2 − 1

2λ∗ ·w, (86)

while at the same time

(w + 2λ∗)2 = w2 + 4λ∗ ·w + 4λ∗2. (87)

Therefore,

e2πi8

(w+2λ∗)2∑

u

e2πiQw+2λ∗ (u)

= e2πi8

w2∑

u

e2πiQw(u−λ∗) = e2πiσ/8. (88)

One can freely shiftw by 2λ∗. Consequently,w is really anequivalence class inΛ∗/2Λ∗.

In Appendix B, we further prove that such a representativew can always be chosen to lie in the original latticeΛ. Wedenote such aw by w0. The advantage of such a choice canbe seen from the expression

e2πiQw0(u) = eπiu

2

eπiu·w0

the topological twists. Sincew0 now lives inΛ, we haveu ·w0 ∈ Z andeπiu·w0 = ±1. This corroborates our intuitionthat one can salvage the Gauss-Milgram sum in the case ofodd matrices by inserting appropriate signs in the sum.

In addition, we can prove that our quadratic functionnow defines a finite quadratic form becauseQw0

(nu) ≡

18

n2Qw0(u) modZ. To see why this is true, we use the defi-

nition of q:

Qw0(nu) =

n2

2u2 − n

2u ·w0

≡(n2

2u2 − n2

2u ·w0

)

modZ. (89)

The second equality follows from the elementary fact thatn2 ≡ n (mod2) together withu · w0 ∈ Z. Therefore thedefinitionq([u]) = Qw0

(u) modZ is well-defined.Having found the discriminant quadratic formq(u), the

generalized Gauss-Milgram sum now can be re-interpreted asthe ordinary Gauss-Milgram sum of a bosonic Abelian topo-logical phase. As aforementioned, there exists a lifting toaneven lattice with the signatureσ′ ≡ (σ −w2

0)mod8 whereσis the signature of the odd matrixK and thus the number ofLandau levels we need to add isn = −w2

0 mod8.Hence, we have the sufficient condition for the existence of

an even lattice that is stably equivalent to a given odd lattice:σ′ = σ, orw2

0 ≡ 0mod8.An obvious drawback of this discussion is that it is not con-

structive (which stems from the non-constructive nature oftheproof of Nikulin’s theorem24): we do not know how to con-struct uniquely the even matrix corresponding to a given dis-criminant group, quadratic formq, and central chargec. Thedistinct ways of lifting usually result in lattices with differentsignatures.

VII. NOVEL CHIRAL EDGE PHASES OF THECONVENTIONAL BULK FERMIONIC ν = 8, 12, 8

15, 16

5

STATES

Now that the general framework has been established, inthis section we consider a few experimentally relevant exam-ples and their tunneling signatures.

A. ν = 8

The integer quantum Hall states are the easiest to produce inexperiment and are considered to be well understood theoret-ically. But surprisingly, integer fillings, too, can exhibit edgephase transitions. The smallest integer filling for which thiscan occur is atν = 8, because eight is the smallest dimensionfor which there exist two equivalence classes of unimodularmatrices. One class contains the identity matrix,I8, and theother containsKE8

, defined in Appendix C, which is gener-ated by the roots of the Lie algebra ofE8. KE8

is an evenmatrix and hence describes a system whose gapless excita-tions are all bosonic2,12 (although if we consider the bosonsto be paired fermions, it must contain gapped fermionic exci-tations.) Yet, counterintuitively, it is stably equivalent to thefermionicI8; for W8 defined in Appendix C,

WT8 (KE8

⊕ σz)W8 = I8 ⊕ σz , (90)

This is an example of the general theory explained in SectionVI, but it is an extreme case in which both phases have only

a single particle type – the trivial particle. The chiral cen-tral charges of both phases are equal and so Nikulin’s theoremguarantees that the two bulk phases are equivalent (when thebosonicE8 state is understood to be ultimately built out ofelectrons) and that there is a corresponding edge phase transi-tion between the two chiral theories.

The action describing theI8 state with an additional left-and right-moving mode is

S =

∫

dx dt

(1

4π(I8 ⊕ σz)IJ ∂tφI∂xφJ

− 1

4πVIJ∂xφ

I∂xφJ +

1

2π

∑

I

ǫµν∂µφIAν

)

. (91)

The charge vector is implicitlytI = 1 for all I. As we haveshown in previous sections, the basis changeφ′ =W8φmakesit straightforward to see that if the perturbation

S′ =

∫

dxdtu′ cos (φ′9 ± φ′10) (92)

is the only relevant term, then the two modesφ′9 and φ′10would be gapped and the system would effectively be de-scribed byKE8

.As in the previous examples, measurements that probe the

edge structure can distinguish the two phases of the edge.Consider, first, transport through a QPC that allows tunnel-ing between the two edges of the Hall bar. In theν = 8 statewith K = I8, the backscattered current will be proportionalto the voltage

IbI8∝ V (93)

because the most relevant backscattering operators,cos(φTI −φBI ), correspond to the tunneling of electrons. In contrast,whenK = KE8

, there is no single-electron backscatteringterm. Instead, the most relevant operator is the backscatteringof charge-2e bosons – i.e. of pairs of electrons – from termslike cos(φ′T1 −φ′T4 −φ′B1 +φ′B4 ), which yields different current-voltage relation

IbE8∝ V 3. (94)

An alternative probe is given by tunneling into the edgefrom a metallic lead. In theK = I8 case, the leading con-tribution is due to electrons tunneling between the lead andthe Hall bar from the termsψ†

leadeiφT

I , yielding

ItunI8∝ V. (95)

However, in theKE8case there are no fermionic charge-e op-

erators to couple to the electrons tunneling from the lead. In-stead, the leading term must involve two electrons from thelead tunneling together into the Hall bar. The amplitude forthis event may be so small that there is no detectable current.If the amplitude is detectable, then we consider two cases: ifthe quantum Hall state is not spin-polarized or if spin is notconserved (e.g. due to spin-orbit interaction), then the lead-ing contribution to the tunneling current is from terms like

19

ψ†lead,↓ψ

†lead,↑e

iφ′T1 −iφ′T

4 , which represents two electrons ofopposite spin tunneling together into the Hall bar, yielding

ItunE8∝ V 3. (96)

If the quantum Hall state is spin-polarized, and tunneling fromthe lead is spin-conserving, then the pair of electrons thattun-nels from the lead must be a spin-polarizedp-wave pair, cor-responding to a tunneling term likeψ†

lead,↓∂ψ†lead,↓e

iφ′T1 −iφ′T

4

in the Lagrangian, and we instead expect

ItunE8∝ V 5. (97)

Another important distinction between the two edge phasesis the minimal value of electric charge in the low-energy sec-tor, which can be probed by a shot-noice measurement37,38, aswas done in theν = 1/3 fractional quantum Hall state39,40.The I8 phase has gapless electrons, so the minimal charge isjust the unit chargee. However, theE8 edge phase is bosonicand consequently the minimal charge is at least2e (i.e. apair of electrons). (Electrons are gapped and, therefore, donot contribute to transport at low temperatures and voltages.)Quantum shot noise, generated by weak-backscattering at theQPC is proportional to the minimal current-carrying chargeand the average current. So we expect a shot-noise measure-ment can also distinguish the two edge phases unambiguously.

B. ν = 12

In dimensions-9, -10, and -11, there exist two unique pos-itive definite unimodular lattices, whoseK-matrices are (inthe usual canonical bases)I9,10,11 or KE8

⊕ I1,2,3. In eachdimension, the two lattices, when enlarged by direct sum withσz , are related by the similarity transformation of the pre-vious section. However in dimension-12, a new lattice ap-pears,D+

12, defined in Appendix C. One salient feature ofthis matrix is that it has an odd element along the diagonal,but it is not equal to1, which is a symptom of the fact thatthere are vectors in this lattice that have odd(length)2 butnone of them have(length)2=1. The minimum(length)2 is2. Upon taking the direct sum withσz, the resulting matrixis equivalent toI12 ⊕ σz – and hence toKE8

⊕ I4 ⊕ σz us-ing the transformation of the previous section – by the relationWT

12(KD+

12

⊕ σz)W12 = I12 ⊕ σz, whereW12 is defined inAppendix C.

Consider the action of theν = 12 state with two addi-tional counter propagating gapless modes and with the im-plicit charge vectortI = 1:

S =

∫

dx dt

(1

4π(I12 ⊕ σz)IJ ∂tφI∂xφJ

− 1

4πVIJ∂xφ

I∂xφJ +

1

2π

∑

I

ǫµν∂µφIAν

)

. (98)

The matrixW12 suggests a natural basis changeφ′ = W12φin which the perturbation

S′ =

∫

dxdtu′ cos (φ′9 ± φ′10) (99)

can open a gap, leaving behind an effective theory describedbyKD+

12

.It is difficult to distinguish theI12 edge phase from the

E8 ⊕ I4 phase because both phases have charge-e fermionswith scaling dimension-1/2. However, both of these edgephases can be distinguished from theD+

12 phase in the man-ner described for theν = 8 phases in the previous subsection.At a QPC, the most relevant backscattering terms will havescaling dimension 1; one example is the termcos(φ′T11−φ′B11 ),which yields the current-voltage relation

IbD+

12

∝ V 3. (100)

This is the same as in theE8 edge phase atν = 8 be-cause the most-relevant backscattering operator is a charge-2e bosonic operator with scaling dimension2. There is acharge-e fermionic operatorexp(i(φ′T2 + 2φ′T12)), but it hasscaling dimension3/2. Its contribution to the backscat-tered current is∝ V 5, which is sub-leading compared tothe contribution above, although its bare coefficient may belarger. However, if we couple the edge to a metallic leadviaψ†

lead exp(i(φ′T2 + 2φ′T12)), single-electron tunneling is the

dominant contribution for a spin-polarized edge, yielding

ItunD+

12

∝ V 3, (101)

while pair tunneling via the couplingψ†lead∂ψ

†leade

iφ′T11

gives a sub-leading contribution∝ V 5. If the edgeis spin-unpolarized, pair tunneling via the couplingψ†lead,↑ψ

†lead,↓e

iφ′T11 gives a contribution with the same

V dependence as single-electron tunneling.

C. Fractional Quantum Hall States with Multiple Edge Phases

In Section III, we discussed theν = 8/7 state, which hastwo possible edge phases. Our second fermionic fractionalquantum Hall example is

K1 =

(

3 0

0 5

)

(102)

with t = (1, 1)T . We again assume that a pair of gappedmodes interacts with these two modes, and we assume thatthey are modes of oppositely-charged particles (e.g. holes),so thatt = (1, 1,−1,−1)T . Upon enlarging byσz, we findthatK1 ⊕ σz =WT (K2 ⊕ σz)W , where

K2 =

(

2 1

1 8

)

(103)

and

W =

1 3 0 1

0 3 0 1

0 0 1 0

1 8 0 3

. (104)

20

If the following perturbation is relevant, it gaps out a pairofmodes:

S′ =

∫

dx dt u′ cos(−3φ1 − 5φ2 + φ3 + 3φ4). (105)

Under the basis change (104),−3φ1 − 5φ2 + φ3 + 3φ4 =φ′3 + φ′4, so the remaining theory has K-matrix (103).

In theK1 edge phase (102), the backscattered current at aQPC is dominated by the tunneling termcos(φT2 −φB2 ), whichyields

Ib1 ∝ V −3/5, (106)

while the tunneling current from a metallic lead is dominatedby the single-electron tunneling termψ†

leade3iφT

1 , which yields

Itun1 ∝ V 3. (107)

In theK2 edge phase (103), the backscattered current at aQPC is dominated by the tunneling termcos(φ′T2 − φ′B2 ),yielding

Ib1 ∝ V −11/15, (108)

while the tunneling current from a metallic lead is dominatedby the pair-tunneling termψ†

lead∂ψ†leade

iφ′T1 −7iφ′T

2 , which as-sumes a spin-polarized edge, and yields

I tun2 ∝ V 11. (109)

As we discussed in Section VI, theν = 16/5 state can havetwo possible edge phases, one with

K1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 5

, (110)

which is essentially the edge of theν = 1/5 state, togetherwith 3 integer quantum Hall edges. The other possible phasehas

K2 =

2 1 0 0

1 2 1 0

0 1 2 1

0 0 1 2

. (111)

Upon enlarging by a pair of gapped modes, the two matricesare related byK1 ⊕ σz =WT (K2 ⊕ σz)W , where

W =

1 0 0 2 0 −1−1 1 0 −4 0 2

1 −1 1 6 0 −3−1 1 −1 −8 1 4

0 0 0 5 0 −2−1 1 −1 −10 1 5

(112)

If the gapped modes are oppositely charged holes, then thefollowing perturbation carries no charge:

S′ =

∫

dxdtu′ cos(−φ1+φ2−φ3−5φ4+φ5+3φ6) (113)

If this perturbation is relevant, it will gap out a pair of modesand leave behind an effective theory describe by the K-matrix(111),

The two edge phases of theν = 16/5 state can be distin-guished by the voltage dependence of the current backscat-tered at a quantum point contact and the tunneling currentfrom a metallic lead. In theK1 edge phase, the backscatteredcurrent at a QPC is dominated by the quasiparticle backscat-tering termcos(φT4 − φB4 ), yielding the current-voltage rela-tion

Ib1 ∝ V −3/5. (114)

In theK2 edge phase, there are several terms that are equallymost-relevant, including, for examplecos(φ′T1 − φ′B1 ), whichyield the current-voltage relation

Ib2 ∝ V 3/5. (115)

Meanwhile, in theK1 edge phase, single-electron tunnelingfrom a metallic lead given by, for example,ψ†

leadeiφT

1 , yieldsthe dependence

Itun1 ∝ V, (116)

while in the K2 edge phase there are only pair-tunnelingterms; one such term for a spin-polarized edge isψ†lead∂ψ

†leade

iφ′T1 +iφ′T

4 , which yields

Itun2 ∝ V 5. (117)

We now consider an example of a bosonic fractional quan-tum Hall state withν = 12/23,

Kb1 =

(

2 3

3 16

)

(118)

and t = (1, 1)T . (This is a natural choice of charge vectorfor bosonic atoms in a rotating trap. For paired electrons inamagnetic field, it would be more natural to havet = (2, 2)T )By a construction similar to the one discussed in the fermioniccases ofν = 8, 12, 8/7, 8/15and the bosonic integer quantumHall cases ofν = 8, 16, this state has another edge phasedescribed by

Kb2 =

(

4 1

1 6

)

(119)

and t = (1,−1)T . As in the previous cases, the two edgephases can be distinguished by transport through a QPC ortunneling from a metallic lead.

21

VIII. SOME REMARKS ON GENERA OF LATTICES ANDBULK TOPOLOGICAL PHASES

The focus in this paper is on the multiple possible gaplessedge phases associated with a given bulk topological phase.However, having established that the former correspond tolattices while the latter correspond to genera of lattices (or,possibly, pairs of genera of lattices), we note here that someresults on genera of lattices published by Nikulin in Ref. 24have direct implications for bulk topological phases. We hopeto explore these relations more thoroughly in the future.

We begin by noting that the data that determine a genus oflattices is precisely the data that determine a2 + 1-D Abeliantopological phase. Recall that the elements of the discrim-inant groupA of a lattice form the particle content of anAbelian topological phase. We can turn this around by not-ing that the particle content and fusion rules of any Abeliantopological phase can be summarized by an Abelian groupA whose elements are the particle types in the theory andwhose multiplication rules give the fusion rules of the the-ory. The fusion rules take the form of the multiplication rulesof an Abelian group because only one term can appear onthe right-hand-side of the fusion rules in an Abelian topolog-ical phase. Meanwhile, specifying theS-matrix for the topo-logical phase is equivalent to giving a bilinear form on theAbelian groupA according toS[v],[v′] =

1√|A|e−2πib([v],[v′]).

A quadratic formq on the Abelian groupA determines thetopological twist factors or, equivalently, theT -matrix of anAbelian topological phase according toθ[v] = e2πiq([v]). Fi-nally, the signature of the form, the number of positive andnegative eigenvaluesr+ andr− of the quadratic formq, deter-mines the right and left central charges, according tocR = r+and cL = r−. The chiral central chargec− = cR − cL isgiven byc− = r+ − r− which, in turn, determines the mod-ular transformation properties of states and, consequently, thepartition functions of the bulk theory on closed3-manifolds(e.g. obtained by cutting a torus out ofS3, performing a Dehntwist, and gluing it back in). The signature is determined (mod8) by the quadratic formq, according to the Gauss-Milgramsum:

1√

|A|∑

a∈A

e2πiq(a) = e2πic−/8

We now consider Nikulin’s Theorem 1.11.3, given in SectionV and also his result

Proposition 1.11.4:There are at most4 possible values forthe signature(mod 8)for the quadratic forms associated witha given bilinear form on the discriminant group.

Theorem 1.11.3 (given in Section V) states that theS-matrix andr+ − r− (mod 8) completely and uniquely deter-mine theT -matrix, up to relabellings of the particles that leavethe theory invariant. In Section VI we show constructivelythat such aT -matrix exists in the fermionic case. Proposition1.11.4 tells us that, for a givenS-matrix, there are at most4possible values for the signaturer+ − r− (mod 8) and, there-fore, at most4 possibleT -matrices. One way to interpret this

is that the elements of theT -matrix are the square roots of thediagonal elements of theS-matrix; therefore, they can be de-termined, up to signs from theS-matrix. There are, at most,four consistent ways of doing this, corresponding to, at most,four possible values of the Gauss-Milgram sum.

Then, Theorem 1.10.2, stated in Section VI, tells us thatthe quadratic form defines an even lattice. Thus, to anyfermionic Abelian topological phase, we can associate abosonic Abelian topological phase with the same particletypes, fusion rules, andS-matrix. The bosonic phase has awell-definedT -matrix, unlike the fermionic phase. In addi-tion, we have:

Theorem 1.3.1:Two latticesS1 and S2 have isomorphicbilinear forms on their discriminant groups if and only if thereexist unimodular latticesL1,L2 such thatS1⊕L1

∼= S2⊕L2.

In other words, two lattices have isomorphic bilinear formsif they are stably equivalent under direct sum with arbitraryunimodular lattices, i.e. if we are allowed to take direct sumswith arbitrary direct sums ofσx, σz , 1, andKE8

. One ex-ample of this is two lattices in the same genus. They havethe same parity, signature, and bilinear form and are stablyequivalent under direct sum withσx, as required by the the-orem. However, we can also consider lattices that are not inthe same genus. The example that is relevant to the presentdiscussion is a pair of theories, one of which is fermionic andthe other bosonic. They have the sameS-matrix but may nothave the same chiral central charges. The theorem tells us thatthe difference can be made up with unimodular theories. Butsinceσx andσz do not change the chiral central charge, theunimodular lattices given by the theorem must be hypercubiclattices. (In the fermionic context, theE8 lattice isσz-stablyequivalent to the8-dimensional hypercubic lattice.) In otherwords, every fermionic Abelian topological phase is equiv-alent to a bosonic Abelian topological phase, together withsome number of filled Landau levels.

Finally, we consider Nikulin’s Corollary 1.16.3, given inSection V, which states that the genus of a lattice is deter-mined by its parity, signature, and bilinear form on the dis-criminant group. Recall that the parity of a lattice is even orodd according whether its K-matrix is even or odd. The evencase can occur in a purely bosonic system while the odd casenecessarily requires “fundamental” fermions, i.e. fermionsthat braid trivially with respect to all other particles. There-fore, specifying the parity, signature, and bilinear form onan Abelian groupA is equivalent to specifying (1) whetheror not the phase can occur in a system in which the micro-scopic constituents are all bosons, (2) theS-matrix, and (3)the chiral central charge. (According to the previous theorem,the T -matrix is determined by the latter two.) This is suf-ficient to specify any Abelian topological phase. Accordingto Corollary 1.16.3, these quantities specify a genus of lat-tices. Thus, given any Abelian topological phase, there is anassociated genus of lattices. We can take any lattice in thisgenus, compute the associated K-matrix (in some basis) anddefine aU(1)r++r− Chern-Simons theory. A change of ba-sis of the lattice corresponds to a change of variables in theChern-Simons theory. Different lattices in the same genus

22

correspond to different equivalentU(1)r++r− Chern-Simonstheories for the same topological phase. Therefore, it followsfrom Corollary 1.16.3 thatevery Abelian topological phasecan be represented as aU(1)N Chern-Simons theory.

IX. DISCUSSION

A theoretical construction of a bulk quantum Hall state typ-ically suggests a particular edge phase, which we will callK1.The simplest example of this is given by integer quantum Hallstates, as we discussed in Sections II and VII. However, thereis no reason to believe that the state observed in experimentsis in this particular edge phaseK1. This is particularly im-portant because the exponents associated with gapless edgeexcitations, as measured through quantum point contacts, forinstance, are among the few ways to identify the topologi-cal order of the state41,42. In fact, such experiments are vir-tually the only way to probe the state in the absence of in-terferometry experiments43–49 that could measure quasiparti-cle braiding properties. Thus, given an edge theoryK2 thatis deduced from experiments, we need to know if a purelyedge phase transition can take the system fromK1 to K2 –in other words, whether the edge theoryK2 is consistent withthe proposed theoretical construction of the bulk state. Wewould also like to predict, given an edge theoryK2 deducedfrom experiments, what other edge phasesK3,K4, . . . mightbe reached by tuning parameters at the edge, such as the steep-ness of the confining potential. In this paper, we have givenanswers to these two questions.

The exotic edge phases atν = 8, 12 discussed in this pa-per may be realized in experiments in a number of materialswhich display the integer quantum Hall effect. These includeSi-MOSFETs50, GaAs heterojunctions and quantum wells(see, e.g. Refs. 51, 52 and references therein), InAs quanutmwells53, graphene54, polar ZnO/MgxZn1−xO interfaces55. Inall of these systems, edge excitations can interact strongly andcould be in anE8 phase atν = 8 or theD+

12 phase or theE8 ⊕ I4 phase atν = 12. To the best of our knowledge, thereare no published studies of the detailed properties of edge ex-citations at these integer quantum Hall states.

The novel edge phase that we have predicted atν = 16/5could occur at theν = 3+ 1/5 state that has been observed56

in a 31 million cm2/Vs mobility GaAs quantum well. Thisedge phase is dramatically different than the edge of theν =1/5 Laughlin state weakly-coupled to3 filled Landau levels.Meanwhile, aν = 8/15 state could occur in an unbalanceddouble-layer system (or, possibly, in a single wide quantumwell) with ν = 1/3 and1/5 fractional quantum Hall states inthe two layers. Even if the bulks of the two layers are veryweakly-correlated, the edges may interact strongly, therebyleading to the alternative edge phase that we predict. Finally,if an ν = 8/7 state is observed, then, as in the two casesmentioned above, it could have an edge phase without gaplessfermionic excitations.

We have focussed on the relationship between theK-matrices of different edge phases of the same bulk. However,in a quantum Hall state, there is also at-vector, which spec-

ifies how the topological phase is coupled to the electromag-netic field. An Abelian topological phase specified by aK-matrix splits into several phases with inequivalentt-vectors.Therefore, two differentK-matrices that are stably equiva-lent may still belong to different phases if the correspond-ing t-vectors are are not related by the appropriate similar-ity transformation. However, in all of the examples that wehave studied, given a(K, t) pair, and aK ′ stably equiva-lent toK, we were always able to find at′ related tot bythe appropriate similarity transformation. Said differently, wewere always able to find an edge phase transition driven by acharge-conserving perturbation. It would be interesting to seeif there are cases in which there is no charge-conserving phasetransition between stably-equivalentK, K ′ so that charge-conservation symmetry presents an obstruction to an edgephase transition betweenK,K ′.

When a bulk topological phase has two different edgephases, one that supports gapless fermionic excitations andone that doesn’t, as is the case in theν = 8 integer quan-tum Hall state and the fractional states mentioned in the pre-vious paragraph, then a domain wall at the edge must supporta fermionic zero mode. For the sake of concreteness, let usconsider theν = 8 IQH edge. Suppose that the edge of thesystem lies along thex-axis and the edge is in the conven-tional phase withK = I8 for x < 0 and theKE8

phase forx > 0. The gapless excitations of the edge are fully chiral; letus take their chirality to be such that they are all right-moving.A low-energy fermionic excitation propagating along the edgecannot pass the origin since there are no gapless fermionic ex-citations in theE8 phase. But since the edge is chiral, it cannotbe reflected either. Therefore, there must be a fermionic zeromode at the origin that absorbs it.

We discussed how the quadratic refinement allows us torelate a given fermionic theory to a bosonic one. One ex-

ample that we considered in detail relatedK1 =

(

1 0

0 7

)

to

K2 =

(

2 1

1 4

)

. Both of these states are purely chiral. How-

ever, we noted that we are not restricted to relating purely chi-ral theories; we could have instead considered a transitionbe-tween theν = 1/7 Laughlin edge and the non-chiral theory

described byK =

2 1 0

1 4 0

0 0 −1

. This transition does not pre-

serve chirality, but the chiral central charges of the two edgetheories are the same. It can be shown that there exist re-gions in parameter space where the non-chiral theory is stable– for example, if the interaction matrix, that we often writeasV , is diagonal, then the lowest dimension backscatteringoperator has dimension equal to4. Even more tantalizingly,it is also possible to consider theν = 1/3 Laughlin edgewhich admits an edge transition to the theory described by

K ′ =

(

−2 −1−1 −2

)

⊕ I3×3. The upper left block is simply the

conjugate or(−1) times the Cartan matrix forSU(3)1. Aboutthe diagonalV matrix point, the lowest dimension backscat-

23

tering term is marginal; it would be interesting to know if sta-ble regions exist.

The theory of quadratic refinements implies that anyfermionic TQFT can be realized as a bosonic one, togetherwith some filled Landau levels, as we discussed as the end ofSec. VIII. In particular, it suggests the following picture: asystem of fermions forms a weakly-paired state in which thephase of the complex pairing function winds2N times aroundthe Fermi surface. The pairs then condense in a bosonic topo-logical phase. The winding of the pairing function gives theadditional central charge (and, if the fermions are charged,the same Hall conductance) asN filled Landau levels. Theremarkable result that follows from the theory of quadraticre-finements is that all Abelian fermionic topological phases canbe realized in this way.

In this paper, we have focused exclusively on fully chiralstates. However, there are many quantum Hall states that arenot fully chiral, such as theν = 2/3 states. The stable edgephases of such states correspond to lattices of indefinite signa-ture. Once again, bulk phases of bosonic systems correspondto genera of lattices while bulk phases of fermionic systemscorrespond either to genera of lattices or to pairs of genera–one even and one odd. Single-lattice genera are much morecommon in the indefinite case than in the definite case23. If ann-dimensional genus has more than one lattice in it then4[

n2]d

is divisible byk(n

2) for some non-square natural numberk sat-isfying k ≡ 0 or 1 (mod 4), whered is the determinant ofthe associated Gram matrix (i.e. the K-matrix). In particular,genera containing multiple equivalence classes ofK-matricesmust have determinant greater than or equal to17 if their rank

is 2; greater than or equal to128 if their rank is3; and5(n

2) or

2 · 5(n

2) for, respectively, even or odd rankn ≥ 4.

Quantum Hall states are just one realization of topologi-cal phases. Our results apply to other realizations of Abeliantopological states as well. In those physical realizationswhichdo not have a conservedU(1) charge (which is electric chargein the quantum Hall case), there will be additionalU(1)-violating operators which could tune the edge of a system be-tween different phases.

Although we have, in this paper, focussed on Abelian quan-tum Hall states, we believe that non-Abelian states can alsohave multiple chiral edge phases. This will occur when twodifferent edge conformal field theories with the same chiralcentral charge are associated with the same modular tensorcategory of the bulk. The physical mechanism underlying thetransitions between different edge phases associated withthesame bulk is likely to be the same as the one discussed here. Inthis general case, we will not be able to use results on latticesand quadratic forms to find such one-to-many bulk-edge cor-respondances. Finding analogous criteria would be useful forinterpreting experiments on theν = 5/2 fractional quantumHall state.

Acknowledgments

We would like to thank Maissam Barkeshli, AndreiBernevig, Parsa Bonderson, Michael Freedman, TaylorHughes, Chenjie Wang, and Zhenghan Wang for helpful dis-cussions. We thank David Clarke for discussions and for shar-ing unpublished work57 with us. C.N. and E.P. have been par-tially supported by the DARPA QuEST program. C.N. hasbeen partially supported by the AFOSR under grant FA9550-10-1-0524. J.Y. has been partially supported by the NSF un-der grant 116143. J.C. acknowledges the support of the Na-tional Science Foundation Graduate Research Fellowship un-der Grant No. DGE1144085.

Appendix A: A Non-Trivial Example of using the Gauss-SmithNormal Form to find the Discriminant Group

We now apply the method described in Section V to theSO(8)1 theory,which is given by the followingK matrix:

K =

2 0 1 0

0 2 −1 0

1 −1 2 −10 0 −1 2

(A1)

It is not clear, simply by inspection, what vectors correspondto generators of the fusion group.

The Gauss-Smith normal form is

D =

1 0 0 0

0 1 0 0

0 0 2 0

0 0 0 2

(A2)

Hence, the fusion group of the theory isZ/2× Z/2.and the Q matrix

Q =

2 0 1 0

3 1 0 1

2 0 0 1

1 0 0 0

(A3)

So the fusion group is generated by the two quasiparticlescorresponding to(2, 0, 0, 1) and (1, 0, 0, 0). We can thencompute theS, T matrices and the result agrees with whatis known (all nontrivial quasiparticles are fermions and theyhave semionic mutual braiding statistics with each other).

Another useful piece of information from the Smith normalform is that the discriminant group for a2× 2 K-matrix

K =

(

a b

b c

)

(A4)

with gcd(a, b, c) = 1 andd = |ac − b2| is Z/d. More gener-ally, it is Z/f × Z/(d/f) whengcd(a, b, c) = f .

24

Appendix B: Proof that w ∈ Λ exists such thatλ · λ ≡ λ ·w mod 2 for all λ ∈ Λ

We begin by showing that for any K-matrix, there exists aset of integerswJ such that

KII ≡N∑

J=1

KIJwJ mod2, for all I (B1)

whereN is the dimension of the K-matrix.Assume the K-matrix hasM ≤ N rows that are linearly

independent mod 2; denote these rowsR1, ...RM and definethe setR = {Ri}. The linear independence of theRi impliesthat Eq (B1) is satisfied for these rows, i.e., there exists a setof integers(w0)J satisfying

KII ≡N∑

J=1

KIJ(w0)J mod2, for all I ∈ R (B2)

For a rowI 6∈ R, the elements of theI th row in K can bewritten as a linear combination of the rows inR:

KIJ ≡∑

Ri∈R

cIRiKRiJ mod2, for I 6∈ B (B3)

where thecIRi∈ {0, 1} are coefficients. It follows that for

I 6∈ R:

KII ≡∑

Ri∈R

cIRiKRiI ≡

∑

Ri∈R

cIRiKIRi

≡∑

Ri,Rj∈R

cIRicIRj

KRiRj≡∑

Ri∈R

c2IRiKRiRi

≡∑

Ri∈R

cIRiKRiRi

mod2 (B4)

Furthermore, forI 6∈ R

N∑

J=1

KIJ(w0)J ≡N∑

J=1

∑

Ri∈R

cIRiKRiJ(w0)J

≡∑

Ri∈R

cIRiKRiRi

mod2 (B5)

Hence, forI 6∈ R, KII ≡∑N

J=1KIJ(w0)J mod2. Sincethis equation already holds forI ∈ R, we have shown thatw0

is a solution to Eq (B1).

It follows that for any choice ofλ = λJeJ ∈ Λ,

λ · λ =

N∑

I,J=1

λIλJKIJ ≡N∑

I=1

λIKII

≡N∑

I=1

λI

N∑

J=1

KIJ(w0)J ≡ λ ·w0 mod2 (B6)

wherew0 = (w0)JeJ is a vector inΛ.

Appendix C: Relevant large matrices

Here we define matrices referred to in VII:

KE8=

2 −1 0 0 0 0 0 0

−1 2 −1 0 0 0 −1 0

0 −1 2 −1 0 0 0 0

0 0 −1 2 −1 0 0 0

0 0 0 −1 2 −1 0 0

0 0 0 0 −1 2 0 0

0 −1 0 0 0 0 2 −10 0 0 0 0 0 −1 2

(C1)

25

W8 =

−5 −5 −5 5 5 5 5 5 8 16

−10 −10 −10 9 9 9 9 9 15 30

−8 −8 −8 8 7 7 7 7 12 24

−6 −6 −6 6 6 5 5 5 9 18

−4 −4 −4 4 4 4 3 3 6 12

−2 −2 −2 2 2 2 2 1 3 6

−7 −7 −6 6 6 6 6 6 10 20

−4 −3 −3 3 3 3 3 3 5 10

1 1 1 −1 −1 −1 −1 −1 −3 −4−2 −2 −2 2 2 2 2 2 4 7

(C2)

KD+

12

=

2 0 1 0 0 0 0 0 0 0 0 −10 2 −1 0 0 0 0 0 0 0 0 0

1 −1 2 −1 0 0 0 0 0 0 0 0

0 0 −1 2 −1 0 0 0 0 0 0 0

0 0 0 −1 2 −1 0 0 0 0 0 0

0 0 0 0 −1 2 −1 0 0 0 0 0

0 0 0 0 0 −1 2 −1 0 0 0 0

0 0 0 0 0 0 −1 2 −1 0 0 0

0 0 0 0 0 0 0 −1 2 −1 0 0

0 0 0 0 0 0 0 0 −1 2 −1 0

0 0 0 0 0 0 0 0 0 −1 2 0

−1 0 0 0 0 0 0 0 0 0 0 3

(C3)

W12 =

11 6 6 −6 −6 −6 −6 −6 −6 −6 −6 −6 0 22

−9 −4 −5 5 5 5 5 5 5 5 5 5 0 18

−18 −9 −9 10 10 10 10 10 10 10 10 10 0 36

−16 −8 −8 8 9 9 9 9 9 9 9 9 0 32

−14 −7 −7 7 7 8 8 8 8 8 8 8 0 28

−12 −6 −6 6 6 6 7 7 7 7 7 7 0 24

−10 −5 −5 5 5 5 5 6 6 6 6 6 0 20

−8 −4 −4 4 4 4 4 4 5 5 5 5 0 16

−6 −3 −3 3 3 3 3 3 3 4 4 4 0 12

−4 −2 −2 2 2 2 2 2 2 2 3 3 0 8

−2 −1 −1 1 1 1 1 1 1 1 1 2 0 4

3 2 2 −2 −2 −2 −2 −2 −2 −2 −2 −2 0 −70 0 0 0 0 0 0 0 0 0 0 0 1 0

2 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 0 −4

(C4)

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59 This statement assumes the periodicity convention,φ ≡ φ+2πn,for n ∈ Z.

60 For generic K-matrices without any symmetries, the automor-phism group often only consists of two elements:W = ±IN×N .

61 This space is known asSpec(Z). Rational numbers are identifiedwith functions on this space according to their prime factoriza-tions.