Exactly soluble models for fractional topological insulators in 2 and 3 dimensions
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Exactly soluble models for fractional topological insulators in 2 and 3 dimensions
Michael Levin, F. J. Burnell,
2, 3
Maciej Koch-Janusz, and Ady Stern Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA Rudolf Peierls Centre for Theoretical Physics, 1 Keble Rd, Oxford OX1 3NP, UK All Souls College, Oxford, OX1 4AL, UK Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
We construct exactly soluble lattice models for fractionalized, time reversal invariant electronicinsulators in 2 and 3 dimensions. The low energy physics of these models is exactly equivalent to anon-interacting topological insulator built out of fractionally charged fermionic quasiparticles. Weshow that some of our models have protected edge modes (in 2D) and surface modes (in 3D), andare thus fractionalized analogues of topological insulators. We also find that some of the 2D modelsdo not have protected edge modes – that is, the edge modes can be gapped out by appropriate timereversal invariant, charge conserving perturbations. (A similar state of affairs may also exist in 3D).We show that all of our models are topologically ordered, exhibiting fractional statistics as well asground state degeneracy on a torus. In the 3D case, we find that the models exhibit a fractionalmagnetoelectric effect.
I. INTRODUCTION
One of the more surprising discoveries of the pastdecade has been that time-reversal invariant band insula-tors come in two kinds: topological insulators and trivialinsulators. These two families of insulators exist in bothtwo and three dimensional systems. They are dis-tinguished by the fact that the interface of a topologicalinsulator with the vacuum always carries a gapless edgemode (in two dimensions) or surface mode (in three di-mensions), while no such protected boundary modes existfor the trivial insulator.Though much of our current understanding of topolog-ical insulators has focused on non-interacting or weaklyinteracting systems, it is natural to consider the fateof this physics in the presence of strong interactions.Strongly interacting insulators can be divided into twoclasses: systems that can be adiabatically connectedto (non-interacting) band insulators without closing thebulk gap, and those that cannot. In the former case it hasbeen shown (explicitly in 2D , and implicitly in 3D ) thatthe gapless boundary modes of a topological insulator arestable to strong interactions as long as time reversal sym-metry and charge conservation are not broken (explicitlyor spontaneously). Therefore there is a well-defined no-tion of interacting topological insulators in systems thatare adiabatically connected to band insulators.Here we will consider the second possibility: stronglyinteracting, time reversal invariant electron systemswhose ground state cannot be adiabatically connectedto a band insulator. The same question can be posed:do some of these systems have protected gapless edgemodes? This question is particularly interesting in lightof the fact that such phases can be fractionalized , leadingto a great diversity of possibilities. That is, such phasesneed not have excitations that resemble electrons; in gen-eral, the quasiparticles may have fractional charge andstatistics.Our understanding of these fractionalized insulators is, however, limited. Focusing on the two dimensionalcase, Ref. 10 analyzed a class of strongly interact-ing toy models where spin-up and spin-down electronseach form fractional quantum Hall states with oppo-site chiralities. Ref. 10 concluded that some of thesestrongly interacting, time reversal invariant insulatorshave protected edge modes, while some do not. Thetwo kinds of insulators were dubbed “fractional topolog-ical insulators” and “fractional trivial insulators” sincethey are analogous to non-interacting topological andtrivial insulators, but they contain quasiparticle exci-tations with fractional charge and fractional statistics.These models demonstrate that fractionalized analoguesof topological insulators are possible in principle, butthey do not exhaust all the possibilities for these phases.In the three dimensional case, even less is known. Refs.12 and 13 used a parton construction to build time rever-sal invariant insulators with a fractional magnetoelectriceffect. However, this work did not prove that these stateshave protected surface modes (though Ref. 12 did con-jecture that this is the case). Also, Refs. 12 and 13 didnot construct a microscopic Hamiltonian realizing thesephases–a standard limitation of parton or slave particleapproaches.In this paper, we address both of these issues. First, weconstruct a set of exactly soluble lattice electron models– in both two and three dimensions – that realize timereversal invariant insulators with fractionally charged ex-citations. Second, we prove that some of these fractional-ized electronic insulators have protected edge or surfacemodes (that is, we show that perturbations cannot gapout the boundary modes without breaking time reversalsymmetry or charge conservation symmetry, explicitly orspontaneously). In this sense, these models provide con-crete examples of “fractional topological insulators” inboth two and three dimensions. An added bonus of ouranalysis is that the argument we use to establish the ro-bustness of the edge or surface modes is not specific toour exactly soluble models, and can be applied equallywell to more general fractionalized (or unfractionalized)insulators.The low energy physics of our models is exactly equiv-alent to a non-interacting topological insulator built outof fractionally charged fermions. Surprisingly, however,some of the models do not have protected boundarymodes. Specifically, we find that in some 2D models(namely those for which our protected-edge argumentbreaks down) the edge modes can be gapped out byadding an appropriate time reversal invariant, chargeconserving perturbation. In the 3D case, our understand-ing is more limited: we can prove that some of the modelshave protected surface modes, but we do not know thefate of the surface modes in the remaining models.The models that we construct and solve are rather farfrom describing systems that are presently accessible toexperiments. Nevertheless, they are of value for two rea-sons. First, they allow for a study of matters of principle,such as the existence of fractional topological insulators,their properties and their stability. And second, sincethe phases we consider are robust to arbitrary deforma-tions of the Hamiltonian that do not close the bulk gap(or break time reversal or charge conservation symme-try), these phases may also be realized by models thatare significantly different from the ones discussed here.The paper is organized as follows. In section II wedescribe the basic physical picture and summarize ourresults. In section III, we construct models in the 2Dcase. In section IV, we analyze the physical properties ofthese models, including the structure of the edge modesand the topological order in the bulk. In section V, weconsider the same models in the 3D case. We then an-alyze the physical properties of the 3D models in sec-tion VI, including the structure of the surface modes,the topological order in the bulk, and the nature of themagnetoelectric effect. In the final part of the paper,section VII, we investigate whether the boundary modesin our models are robust to arbitrary time reversal in-variant, charge conserving perturbations. The Appendixcontains some of the more technical calculations. TableI lists the various symbols that we use in the text.
II. SUMMARY OF RESULTS
This section is aimed at introducing the reader to ourexactly soluble models, and to the main results we find byanalyzing these models. We will emphasize the physicalpicture, leaving the detailed calculations to the followingsections.
A. Constructing exactly soluble models forfractional topological insulators
To obtain candidate fractional topological insulators,we build models with two important properties: (1) frac-tionally charged fermionic quasiparticles and (2) a topo-
Symbol Description Section N site number of sites s in the lattice II N link number of links h ss ′ i in the lattice II N plaq number of plaquettes P in the lattice II m integer parameter in boson model III A b † s boson creation operator on site s III A b † ss ′ boson creation operator on link h ss ′ i III A n s boson occupation number on site s III A n ss ′ boson occupation number on link h ss ′ i III A Q s cluster charge on site s in boson model III A B P ring exchange term on plaquette P III A α s sublattice weighting factor for site s III A U ss ′ hopping term on link h ss ′ i III A q s eigenvalue of Q s III A 1 b P eigenvalue of B P III A 1 | q s , b P i simultaneous eigenstate of Q s , B P III A 1 | q s i simultaneous eigenstate | q s , b P = 1 i III A 2 q ch electric charge of charge excitation III A 2 c † sσ electron creation operator III B n sσ electron occupation number III B n s,e total number of electrons on site s III B˜ Q s cluster charge in electron model III B k integer parameter in electron model III B˜ q s eigenvalue of ˜ Q s III B 1 | ˜ q s , b P , n sσ i simultaneous eigenstate of ˜ Q s , B P , n sσ III B 1 | n sσ , elec i electron state with occupation { n sσ } III B 2 | n sσ i eigenstate | ˜ q s = 0 , b P = 1 , n sσ i III B 2 q f electric charge of fermion excitation III B 2 H hop hopping term for fermion excitations III C t ss ′ ,σσ ′ hopping amplitudes for fermions III C d † sσ creation operator for fermions III C θ ch,fl Mutual statistics of charges and fluxes IV B θ f,fl Statistics of fermions and fluxes IV B e ∗ smallest charged excitation VI CTABLE I. List of symbols, their description, and the sectionwhere they are defined. logical insulator band structure for these excitations.Our construction has three steps. In the first step weconstruct lattice boson models with fractionally chargedbosonic excitations. In the second step we add elec-trons to the lattice, and define an electron-boson inter-action that binds each electron to fractionally chargedexcitations of the bosonic model, thus creating a frac-tionally charged fermion. In the third step we constructa hopping term on the lattice that allows the fraction-ally charged fermion to hop between lattice sites withoutexciting other degrees of freedom. We then choose thehopping terms so that these fermions have a topologicalinsulator band structure.The boson models we construct are similar in spiritto the “toric code” model and its Z m generalizations ,but are built out of bosonic charged particles whose to-tal charge is conserved. In this sense, these models area hybrid between the toric code model (which is exactlysoluble but not charge conserving) and the fractionalizedbosonic insulators of Refs. 15–19 (which are charge con-serving, but not exactly soluble).The construction of the models is based on the follow-ing recipe. We consider a system of bosons that live onthe sites s and links h ss ′ i of a bipartite square (or in3D, cubic) lattice. We construct a bosonic Hamiltoniancomposed of two parts: H B = H + H . Each term hasan associated energy scale whose magnitude is of minorsignificance to our discussion. Both, however, depend onan integer parameter m which plays a crucial role, as itdetermines the fractional charge carried by the quasipar-ticle excitations.The first term H is the “charging” Hamiltonian. Thisterm depends only on the number of bosons on each site n s and on each link n ss ′ . Each boson is made of twoelectrons, of a charge e each. The charging Hamilto-nian assigns different energies to different charge config-urations { n s , n ss ′ } , by coupling the charge on a site tothe charges on the four links neighboring the site. Thespectrum of H is discrete, as expected from a chargingHamiltonian. The spectrum is also highly degenerate,since many charge configurations have the same energycost. In fact, the number of degenerate eigenstates ofthe lowest eigenvalue of H is m N link − N site +1 with N link being the number of links in the lattice and N site beingthe number of sites.The second term H is the “hopping” Hamiltonian.This term makes bosons hop between neighboring latticesites and links. A crucial aspect of our model is that thetwo parts are mutually commuting: [ H , H ] = 0. Thus,the hopping Hamiltonian H only has matrix elementsbetween degenerate states of the charging Hamiltonian H , and splits the degeneracy for the ground state.As we want to build an insulator, we need the groundstate of H B to be separated from the excited states bya finite energy gap. Furthermore, because we want frac-tionally charged excitations, H B must be topologicallyordered (in gapped systems, fractional charge impliesthe existence of topological order). The presence of topo-logical order means that the degeneracy of the groundstate must depend on the topology of the system. More specifically, we need the degeneracy of the groundstate to be independent of the system size, and to bedifferent for a system with open and periodic boundaryconditions.The first condition – existence of an energy gap – isguaranteed in our model by having the spectra of thecharging Hamiltonian H and the hopping Hamiltonian H discrete. Note that this is not a common featureto hopping Hamiltonians. The continuous spectrum ofthe Josephson Hamiltonian is a representative exampleto the contrary. To make the spectrum discrete, we needto choose a carefully tailored hopping operator. While aconventional hopping Hamiltonian allows a single particle to hop between two neighboring sites, the hopping termwe introduce allows only for a simultaneous correlatedhopping of several particles around a single plaquette.The second condition – a ground state degeneracy thatdepends on the topology of the system – is a consequenceof the way that the hopping Hamiltonian splits the de-generacy of the ground state of the charging Hamiltonian.For example, consider the case of the 2D system definedon a torus. Each of the terms in H describes hoppingaround one of the N plaq plaquettes of the lattice and onlyone out of m N plaq − ground states of H is also a groundstate of H . Thus, the ground state degeneracy of theHamiltonian H B on a torus is m N link − N site − N plaq +2 . ByEuler’s theorem, this number is exactly m . A similarcalculation in a 2D open geometry yields a ground statedegeneracy of 1. In the 3D case, the analysis is similar.One finds that the ground state degeneracy in a 3D opengeometry is again 1, while on a 3D torus it is m .This counting agrees with the generalized “toric code”model with gauge group G = Z m . The quasiparticleexcitations of the boson model are also similar to the Z m toric code: there are two types of quasiparticle excita-tions – “charge” particles and “flux” particles – which areindividually bosons but have fractional mutual statistics.Also, like the toric code model, the boson model does nothave gapless edge modes. The main difference from the Z m toric code model is that the “charge” quasiparticlescarry a fractional electric charge, 2 e/m .After constructing the bosonic models, we next intro-duce single electron degrees of freedom that live on thelattice sites. The electrons couple to the bosons throughthe charging energy, and the electron-boson coupling ischaracterized by a second integer parameter k . We designthis coupling so that it energetically binds an electron toa composite of k fractionally charged bosonic quasiparti-cles, each carrying charge 2 e/m . The resulting compositeparticle then has a fractional charge of q f = e (1 + 2 k/m ),and follows fermionic statistics. We denote the Hamilto-nian of this modified lattice model by H e .In order for the composite particle to be a stable de-gree of freedom, it must be able to hop between latticesites “in one piece”, i.e. without affecting the other typesof excitations. In the final step of the construction, wefind a hopping term H hop that does just that. We thenadd H hop to the Hamiltonian H e , choosing the hoppingamplitudes so that the composite particles have a bandstructure of a topological insulator. The energy gap be-tween the bands is a parameter of H hop , and we assumeit to be much smaller than the energy gap of the bosonicexcitations.This construction results in a system of non-interactingfermions of spin-1 / q f = e (1 + 2 k/m ) ina topological insulator band structure in either two orthree dimensions. The smallest charged excitation in thesystem carries a charge e ∗ = 2 e/m when m is even, anda charge of e ∗ = e/m when m is odd. In the formercase, this is a bosonic excitation. In the latter, it is acomposite of fermionic and bosonic excitations. B. Properties of the models
1. The two-dimensional case
In two dimensions our models realize quantized spinHall states, with a pair of gapless edge modes and a spin-Hall conductivity of e π (cid:0) km (cid:1) . The topological or-der characterizing the states originates from the bosonicmodels underlying them. The ground state degeneracyon a torus is m . In the bulk there are three types ofexcitations: the bosonic charge excitation with electriccharge 2 e/m , the bosonic flux excitation which is neutral,and the fermion excitation with charge q f = e (1 + 2 k/m ).We find the flux excitation to have a non-trivial mutualstatistics with the other two types of excitations. Whena bosonic charge excitation of charge 2 e/m winds arounda flux excitation, it accumulates a phase of 2 π/m . Con-sequently, when a fermion, which is a composite of anelectron and k bosonic charge excitations, winds arounda flux particle, it accumulates a phase of 2 πk/m .In certain limits the only active degrees of freedom arethose of the fermions at the edge, where a gapless modeexists. The system can then be described as a topo-logical insulator built out of non-interacting fermions offractional charge q f = e (1+2 k/m ). In particular, this de-scription holds when the system is driven at low frequen-cies and long wave lengths by a weak electromagneticfield or when thermodynamical properties are probed atlow temperatures. Under these conditions, the systemwould show a two-terminal conductance of 2 q /h , theshot noise associated with tunneling between edges wouldcorrespond to a charge of q f , and the heat capacity wouldbe linear in temperature and proportional to the system’scircumference, as expected from a 2D topological insula-tor of non-interacting charge q f fermions.When deviating from these conditions, the bosonic de-grees of freedom can become active. Examples includethe application of a magnetic flux of the order of a fluxquantum, Φ = hc/e , per plaquette, the application ofbias charges of order of e/ m to particular sites and theapplication of an electromagnetic field at frequencies thatcorrespond to the gap to bosonic excitations.
2. The three dimensional case
In three dimensions our models are strong topologicalinsulators built out of charge q f = e (1 + 2 k/m ) fermions,with a gapless Dirac cone on each surface. When timereversal symmetry is broken on the surface, the modelsexhibit a surface Hall effect with a fractional Hall con-ductivity of q / h . As in the two dimensional case, thetopological order in the 3D model originates from thetopological order of the underlying bosonic system. Thecharged excitations carry electric charges of 2 e/m and e (1+2 k/m ) and are identical to those in the 2D case, butthe flux excitation becomes a flux loop rather than thepoint particle it is in 2D. The ground state degeneracy on a 3D torus is m . Again, in certain limits the bosonicdegrees of freedom may be neglected and the only activedegrees of freedom are the fermionic ones. The condi-tions for these limits to hold are similar to those of thetwo dimensional case.The bosonic degrees of freedom are active in severalcases, one of which is of particular interest. In a 3D topo-logical insulator of non-interacting electrons, a magneticmonopole in the bulk of the insulator binds a half integerelectric charge. Hence, a monopole/anti-monopolepair – which may be created by a finite-length solenoidcarrying a flux quantum Φ and positioned within thebulk – creates an electric dipole with a half-integer elec-tric charge at its ends. In our model we find that sucha solenoid leads to a dipole with a charge which is ahalf-integer multiple of q /e . Unlike the non-interactingcase, however, the energy involved in creating the dipoleis proportional to its length – indicating that the twoends of the dipole cannot be effectively separated fromone another. The two ends of the dipole can be sepa-rated only when the flux carried by the solenoid is e/e ∗ flux quanta. Furthermore, because we could presumablytrap any number of additional charge e ∗ quasiparticlesnear the ends of the solenoid by adding an appropriatelocal potential, the only quantity which is independentof microscopic details is the monopole charge modulo e ∗ .Calculating this quantity, we find that the charge at theend of the ( e/e ∗ )Φ solenoid is a half-integer multipleof e ∗ for the models where q f /e ∗ is odd, and an integermultiple of e ∗ for the models where q f /e ∗ is even. C. The stability of the edge or surface modes
In conventional topological insulators, the edge or sur-face modes are protected as long as time reversal symme-try and charge conservation are not broken. If either ofthese symmetries is broken, e.g. by a Zeeman magneticfield that couples to the electron spin or by a proximity-coupling to a superconductor that allows for Cooper pairsto tunnel into and out of the edge or surface modes, thesemodes may be gapped. The breaking of time reversalsymmetry may be spontaneous rather than explicit, in-duced for example by the Fock term of electron-electroninteraction. The stability of the edge or surface modesto perturbations that do not break these symmetries isthe distinguishing feature of topological insulators in 2Dand strong topological insulators in 3D.An important question is whether the phases we studyhere have protected edge or surface modes similar to con-ventional topological insulators. We find that some ofthe models do indeed have edge or surface modes pro-tected by time reversal symmetry and charge conserva-tion, while some do not. (Independent of this difference,all the models are topologically ordered, as demonstratedby their topological ground state degeneracy).
1. The two dimensional case
In the 2D case, we find that our models conform to thegeneral rule derived in Ref. 10: that is, the edge modesare protected if and only if the ratio σ sH /e ∗ is odd, where σ sH is the spin-Hall conductivity in units of e/ π and e ∗ is the elementary charge in units of e . In our models,this criterion is equivalent to the condition that the ratio q f /e ∗ is odd.We establish the stability of the edge modes for themodels with odd q f /e ∗ by a general flux insertion argu-ment similar to the used in Ref. 10, and establish the in-stability in the case of even q f /e ∗ by explicitly construct-ing the perturbations whose combination gaps the edge.This combination is rather interesting. As defined, themodels have two fermionic edge modes of opposite chiral-ities – the bosonic excitations are gapped at the edge. Inorder to gap the fermionic edge modes, we introduce oneperturbation whose role is to close the gap of the bosonicexcitations at the edge, and then two additional pertur-bations that couple the bosonic and fermionic modes,gapping them both.For the closure of the bosonic gap at the edge we applya perturbation aimed at turning the edge of the bosonicsystem from an insulator into a superfluid. The naturalway of doing that is by introducing a hopping Hamilto-nian that allows fractionally charged bosonic excitationsat the edge to hop from one site to another. When thehopping term is strong enough it can overcome the charg-ing term described by H , thereby closing the gap at theedge. As for the perturbations that couple the bosonsand the fermions at the edge, the first such perturbationbreaks a spinless boson of charge 2 e into two electronsof opposite spin directions on the same lattice site. Thesecond of these perturbations flips the direction of aninteger number of electrons’ spins, while simultaneouslyoperating on the flux degrees of freedom on the edge.Both of these perturbations make use of the bosonic de-grees of freedom and therefore do not have analogues innon-interacting electron systems.
2. The three dimensional case
Just as in the 2D case, we find that the 3D models withodd q f /e ∗ have protected surface modes. We establishthis result using a 3D generalization of the flux insertionargument of Ref. 10. We note that this argument isof interest beyond the particular models discussed here,and can be applied to more general fractionalized andconventional insulators. Unlike the 2D case, we are notable to determine the stability of the surface modes forthe models with even q f /e ∗ . Addressing this questionrequires either the construction of specific perturbationsthat gap out the surface, or an argument proving thatthe surface modes are protected. Q s
41 2334 U P FIG. 1. In the lattice boson model, bosons live on both thesites s and links h ss ′ i of the square lattice. The Hamiltonian H B (1) is a sum of a Q s term (2), which acts on four links h ss ′ i and one site s , and a B P term (4), which acts on thefour sites and links adjacent to a plaquette P . The B P termis a product of four link operators U ss ′ (5) which each act onthe sites s, s ′ and the link h ss ′ i . III. LATTICE MODELS FOR 2D FRACTIONALTOPOLOGICAL INSULATORSA. Step 1: 2D lattice boson models with fractionalcharge
In this section we describe a collection of exactlysoluble lattice boson models with fractionally chargedexcitations—one for each integer m ≥
2. The modelscan be defined on any bipartite lattice in 2 or higher di-mensions. Here, for simplicity, we will focus on the caseof the square lattice. Later, when we construct 3D mod-els, we will consider the cubic lattice case.The basic degrees of freedom in these models are charge2 e spinless bosons which live on the sites s and links h ss ′ i of the square lattice. We denote the boson creation oper-ators on the sites and links by b † s and b † ss ′ and the corre-sponding boson occupation numbers by n s and n ss ′ . TheHamiltonian H B can be written as a sum of two terms,one associated with sites s , and the other associated withplaquettes P of the square lattice (Fig. 1): H B = H + H = V X s Q s − u X P ( B P + B † P ) (1)We will take u, V > Q s term is a “cluster charge” term which measures the totalcharge on the site s and the four neighboring links h ss ′ i with appropriate weighting factors. It is defined as thesum Q s = α s X s ′ n ss ′ + m · n s (2)where α s = ( s ∈ Am − s ∈ B (3)and A and B are the two sublattices of the square lat-tice. Since V is positive, V Q s describes a short rangerepulsive interaction between the bosons. This interac-tion breaks the sublattice symmetry between the A and B sublattices, except in the case m = 2. The B P termcan be thought of as a ring exchange term. It is definedas the product B P = U U U U (4)where U ss ′ is a boson hopping term on the link h ss ′ i : U ss ′ = (cid:0) b † s (cid:1) α s − b † s ′ b α s ss ′ + b α s ′ − s ′ b s (cid:16) b † ss ′ (cid:17) α s ′ (5)The hopping term U ss ′ describes processes where bosonshop from the sites s, s ′ to the link h ss ′ i and vice versa.It is designed so that it has two special properties. First, U ss ′ changes the number of bosons on the site at thecenter of the link h ss ′ i by ± s ∈ B and the − sign when s ∈ A . This change iscompensated by a corresponding increase or decrease inthe number of bosons in the two neighboring sites s, s ′ so that the total number of bosons is conserved. Second, U ss ′ decreases the cluster charge Q s by 1 and increasesthe cluster charge Q s ′ by 1 and doesn’t affect the chargeon any other site:[ Q r , U ss ′ ] = ( δ rs ′ − δ rs ) U ss ′ (6)An important consequence of this relation is that Q s commutes with the product of U ss ′ around any set ofclosed loops, and in particular,[ Q s , B P ] = 0 . (7)Equation (7) is at the root of why our system is an in-sulator: the B P operator has no effect on the clustercharges Q s and hence does not provide for the long-distance transport of electric charge.The final component of the model is our definition ofthe boson creation operators b † s , b † ss ′ . For the site bosons b † s , we use a rotor representation, letting b † s = e iθ s with[ θ s , n s ] = i . The boson occupation number on the sitescan therefore be any integer, n s ∈ ( −∞ , ∞ ). On theother hand, we take the link bosons b † ss ′ to be a kind ofgeneralized hard-core boson, restricting the boson occu-pation number to n ss ′ ∈ { , , ..., m − } , and defining b † ss ′ to be the m × m matrix b † ss ′ = · · ·
00 0 1 · · · · · ·
10 0 0 · · · (8)when written in the normalized number basis {| m − i , ..., | i} on the link h ss ′ i . We note that while thesegeneralized hard-core bosons are unconventional, they can arise as an effective description of a conventionalboson system in an appropriate limit. For example, ifthe number of bosons on the link h ss ′ i is very large butis restricted to a set of m contiguous values {N , N +1 , ..., N + m − } by appropriate energetics, then the abovedefinition of b † ss ′ becomes a good approximation to con-ventional bosons (up to an overall normalization factor).To summarize, the full Hilbert space for our model isspanned by the occupation number states | n s , n ss ′ i with n s ∈ ( −∞ , ∞ ) and 0 ≤ n ss ′ ≤ m −
1. Solving the boson model
We will now show that the Hamiltonian (1) is exactlysoluble, and compute its exact energy spectrum. We arealready part way there, having established that Q s , B P commute with each other (7). Next, we note that[ U ss ′ , U rr ′ ] = 0 , U † ss ′ = U − ss ′ = U s ′ s (9)from which it follows that[ B P , B P ′ ] = [ B P , B † P ′ ] = 0 . (10)Combining these results with the obvious relation[ Q s , Q s ′ ] = 0 , (11)we conclude that { Q s , B P , B † P } all commute, and there-fore can be diagonalized simultaneously.The simultaneous eigenstates of these operators can belabeled as | q s , b P i , where Q s | q s , b P i = q s | q s , b P i B P | q s , b P i = b P | q s , b P i B † P | q s , b P i = b ∗ P | q s , b P i (12)The corresponding energies are E = V X s q s − u X P ( b P + b ∗ P ) . (13)It is clear from the definition (2) that Q s has integereigenvalues so q s is an integer. Similarly, using thefact that U ss ′ changes the occupation number n ss ′ by ± B mP = 1 (14)so b P must be a m th root of unity. This relation guaran-tees what we promised in section II: the spectrum of H is discrete .The only remaining question is to determine the degen-eracy of each of the q s , b P eigenspaces. This degeneracydepends on the geometry we consider. We first considerthe case of a rectangular piece of square lattice with openboundary conditions (Fig. 2(a)). We show in Appendix Q s Q s (b)(a) FIG. 2. We consider the lattice boson model H B (1) in twogeometries: (a) a rectangular geometry with open boundaryconditions, and (b) a periodic (torus) geometry. In the rect-angular geometry (a), the Q s operators at the corners act ontwo links h ss ′ i , while those at the edge act on three links. A 1 that in this geometry, there is a unique eigenstate foreach collection of { q s , b P } satisfying the global constraint X s q s ≡ { q s , b P } are independent and completequantum numbers, except for this single global con-straint. A similar result holds for a (periodic) torus geom-etry (Fig. 2(b)). In this case, we find that there are m states for each collection of { q s , b P } satisfying (15) as wellas the additional constraint Q P b P = 1 (see AppendixA 2). This degeneracy is a consequence of the fact thatthe system is topologically ordered (see Section IV B).Below we will focus exclusively on the open boundarycondition geometry, unless otherwise indicated.Putting this all together, we conclude that the groundstate of (1) is the unique state with q s = 0 , b P = +1 ev-erywhere. There are two types of elementary excitations:“charge” excitations where q s = 1 for some site s , and“flux” excitations where b P = e πi/m for some plaquette P . The total number of charge excitations, in the bulkand edge together, must sum up to 0 modulo m . Thecharge excitations cost an energy of V , while the fluxloop excitations cost an energy of u (1 − cos(2 π/m )). Inparticular, as long as u, V >
0, then the ground state isgapped.
2. Fractional charge in the boson model
An important property of the boson model (1) is thatthe charge excitations carry fractional electric charge q ch = 2 e/m . We can derive this result by explicitly cal-culating the electric charge distribution in these states.Consider an eigenstate | q s i ≡ | q s , b P = 1 i with some ar-bitrary configuration of charges { q s } , and with no fluxes.It is straightforward to show that the (un-normalized)microscopic wave function for this state in the occupa-tion number basis is given by h n ss ′ , n s | q s i = ( α s P s ′ n ss ′ + mn s = q s for all s n ss ′ on any link h ss ′ i , then we would findeach of the m possible values, n ss ′ = 0 , , ...m −
1, withequal probability, 1 /m . This is true on every link ss ′ , in-dependent of the configuration of charges { q s } . It followsthat the expectation value of n ss ′ in the state | q s i is h n ss ′ i q s = 0 + 1 + ... + m − m = m −
12 (17)Using this result, together with the constraint α s P s ′ n ss ′ + mn s = q s , we deduce h n s i q s = q s − α s P s ′ h n ss ′ i m = q s − zα s ( m − / m (18)where z is the coordination number of lattice ( z = 4 forthe square lattice).We are now in a position to compute the fractionalcharge q ch . Consider the case of an isolated chargeexcitation—that is, a state where q s = 1 at some site s and q s = 0 at all nearby sites. Denote this state by | q s = 1 i . Similarly, consider an eigenstate | q s = 0 i where q s = 0 both at s and at all nearby sites. Thenthe electric charge q ch carried by the excitation is givenby the difference in expectation values q ch = 2 e [ h N S i − h N S i ] (19)where N S is an operator which measures the total num-ber of bosons in some large area S containing s : N S = X s ∈S n s + X s,s ′ ∈S n ss ′ (20)Using the above results (17-18) we derive h n ss ′ i − h n ss ′ i = 0 h n s i − h n s i = δ s s m (21)implying that q ch = em . Furthermore, we can see fromthese expressions that this charge is perfectly localizedto the site s . This perfect localization is specific to theexactly soluble model: in a generic gapped system weexpect excitations to have a finite size of order the cor-relation length.Alternatively, we can derive the fractional charge usinga simple identity: for any set of sites S in the squarelattice, we have the relation X s ∈S Q s = m X s ∈S n s + X s,s ′ ∈S n ss ′ + X s ∈S ,s ′ ∈S c α s n ss ′ = mN S + X s ∈S ,s ′ ∈S c α s n ss ′ (22)Taking expectation values of both sides in the two states | q s = 1 i , | q s = 0 i , and subtracting gives h N S i −h N S i = 1 m − X s ∈S ,s ′ ∈S c α s [ h n ss ′ i −h n ss ′ i ] (23)To complete the calculation, we note that the secondterm on the right hand side vanishes in the limit that S becomes large, since in that case, the sum only in-volves links ss ′ that are far from s , and the excess charge( h n ss ′ i − h n ss ′ i ) must vanish at large distances from s .It follows that q ch = 2 e/m , as claimed. B. Step 2: 2D lattice electron models withfractional charge
We are now ready to construct a model with fraction-ally charged spin-1 / / s . In addition, we now think of the lattice bosonsas being charge 2 e , spin-singlet pairs of electrons. Thismicroscopic picture for the bosons is important conceptu-ally because ultimately we want a model for a fractionaltopological insulator which is constructed out of electrondegrees of freedom.We will denote the creation and annihilation operatorfor the (unpaired) electrons by c † sσ , c sσ , and their occu-pation number by n sσ . We will use n s,e to denote thetotal number of the (unpaired) electrons on site s : n s,e = X σ n sσ = X σ c † sσ c sσ (24)Later, we will be interested in models with multiple or-bitals on each lattice site s . In that case, we will let σ include both the spin and orbital degrees of freedom.In this notation, the Hamiltonian H e for the electronmodel is a sum of three terms: H e = V X s ˜ Q s − u X P ( B P + B † P ) − µ X sσ n sσ (25)where ˜ Q s = Q s − k · n s,e (26)and B P , Q s are defined as in Eq. (2-5). We will take V, u >
0, but will consider both positive and negative µ ,and arbitrary integer k .
1. Solving the electron model
The electron model (25) can be solved in the same wayas the original boson model. Just as before,[ ˜ Q s , ˜ Q s ′ ] = [ B P , B P ′ ] = [ B P , B † P ′ ] = [ ˜ Q s , B P ] = 0 (27) Also, it is clear that[ ˜ Q s ′ , n sσ ] = [ B P , n sσ ] = 0 (28)Hence, we can simultaneously diagonalize { ˜ Q s , B P , B † P , n sσ } . Let | ˜ q s , b P , n sσ i denote the si-multaneous eigenstates, where q s is an integer, b P is a m th root of unity, and n sσ = 0 ,
1. The correspondingenergies are: E = V X s ˜ q s − u X P ( b P + b ∗ P ) − µ X sσ n sσ (29)By our analysis of the lattice boson model, we know thatthere is a unique state for each choice of ˜ q s , b P , n sσ sat-isfying the global constraint X s ˜ q s + k X sσ n sσ ≡ µ < q s = 0 , b P = 1 , n sσ = 0. The system hasthree types of elementary excitations: “charge” excita-tions with ˜ q s = 1 for some site s , “flux” excitationswhere b p = e πi/m on some plaquette p , and spin-1 / n sσ = 1 for some site s andspin σ = ↑ , ↓ . The charge excitations cost an energy of V ,the flux excitations cost an energy of u (1 − cos(2 π/m )),and the fermion excitations cost energy − µ . In particu-lar, as long as u, V > µ <
0, the ground state isgapped.
2. Fractional charge in the electron model
Our next task is to show that the fermion excitationscarry fractional charge. To do this, we first need to intro-duce some notation. Let | n sσ , elec i denote the electronoccupation basis state | n sσ , elec i = Y s ( c † sσ ) n sσ | i (31)where | i is the empty state. Also, let | n ss ′ , n s i denotethe boson occupation state defined in section III A. Acomplete basis for the Hilbert space of the electron modelis given by tensor product states | n sσ , elec i ⊗ | n s , n ss ′ i .In addition to these general basis states, we will alsofind it useful to think about the set of eigenstates | n sσ i ≡| ˜ q s = 0 , b P = 1 , n sσ i made up of some arbitrary config-uration of fermions { n sσ } with no charge or flux exci-tations. (To construct these states we must impose theglobal constraint P sσ n sσ ≡ | n sσ i = | n sσ , elec i ⊗ | q s = kn s,e i (32)where | q s i are the boson eigenstates defined in (16).In order to compute the fractional charge carried bythe fermion, it suffices to consider a state with an isolated fermion excitation— that is, suppose n sσ = 1 at some site s and n sσ vanishes at all nearby sites. Denote this stateby | n s σ = 1 i . According to the definition (32), this stateis a tensor product of an electron state and a boson state: | n s σ = 1 i = | n s σ = 1 , elec i ⊗ | q s = k i (33)We can see that | n s σ = 1 , elec i consists of a single spin- σ electron at site s , while | q s = k i corresponds to k “charge” excitations at site s . Thus, the fermion exci-tation is a composite particle made of an electron and k charge excitations. To compute the total charge ofthe fermion, we need to add together the contributionscoming from these two pieces. By our analysis of thefractional charge in the bosonic model, we know thateach charge excitation carries charge 2 e/m . On the otherhand, the electron clearly has charge e . Adding togetherthese two contributions, we conclude that the fermionexcitation has charge q f = e (1 + 2 k/m ) (34)
3. Time reversal symmetry and the electron model
To construct candidate fractional topological insula-tors, it will be important to understand how the fermionicexcitations in our model transform under time reversal.We use the usual convention for T , where the electroncreation operators transform according to: T : c † s ↑ → c † s ↓ , c † s ↓ → − c † s ↑ (35)In this convention, the bosons in (25) transform trivially,since they are spin singlet pairs of electrons: b † ss ′ ⇒ b † ss ′ , b † s → b † s (36)Applying these transformation laws, we see that thefermion excitations transform like spin-1 / C. Step 3: Building candidate 2D fractionaltopological insulators
In the last two sections, we have shown that thefermionic excitations of the electron model (25) carryspin-1 / q f (34). Given these properties, it is easy to build acandidate fractional topological insulator: we simply putthe fractionally charged fermions into a non-interactingtopological insulator band structure. We accomplish thisby adding a new term to the electron Hamiltonian H e (25): H = H e + H hop = V X s ˜ Q s − u X P ( B P + B † P ) − µ X sσ n sσ ! + H hop (37)where H hop = − X h ss ′ i ( t ss ′ σσ ′ c † s ′ σ ′ c sσ U kss ′ + h.c. ) (38)and ˜ Q s , B P , U ss ′ are defined as before. The new term H hop gives an amplitude for the fermion excitations tohop from site to site without affecting any of the otherdegrees of freedom, as we now show. We will assumethat t ss ′ σσ ′ ≪ u, V so that the bandwidth of the fermionexcitations is much smaller than the gap to the bosonicexcitations.We can understand the effect of H hop by computingthe matrix elements of this operator between differenteigenstates of H e , h ˜ q ′ s , b ′ P , n ′ sσ | H hop | ˜ q s , b P , n sσ i (39)This computation is considerably simplified by the factthat [ ˜ Q s , H hop ] = [ B P , H hop ] = 0 (40)implying that the matrix elements are only nonzero when˜ q ′ s = ˜ q s , and b ′ P = b P . In what follows, we specialize tothe ˜ q s = ˜ q ′ s = 0 , b P = b ′ P = 1 case, since these arethe lowest energy states and this is all we will need tounderstand the low energy physics. These states containonly fermions and no other excitations. As in (32), wewill denote a state with some arbitrary configuration offermions { n rτ } using the abbreviated notation | n rτ i ≡| ˜ q s = 0 , b P = 1 , n rτ i (Here r labels the sites of the lattice,while τ = ↑ , ↓ labels the two possible spin states). To findthe matrix elements h n ′ rτ | H hop | n rτ i , we write c † s ′ σ c sσ U kss ′ | n rτ i = c † s ′ σ ′ c sσ | n rτ , elec i ⊗ U kss ′ | q r = kn r,e i (41)and then analyze each of these two pieces in turn. Usingthe explicit form of | q r i (16), we find U kss ′ | q r i = | q ′ r i (42)where q ′ r = q r − kδ rs + kδ rs ′ (43)Similarly, we have c † s ′ σ ′ c sσ | n rτ , elec i = ±| n ′ rτ , elec i (44)where n ′ rτ = n rτ − δ rs δ στ + δ rs ′ δ σ ′ τ (45)0and the ± sign depends on the ordering of the electroncreation and annihilation operators in Eq. (31). Com-bining these two results with (41), we derive c † s ′ σ c sσ U kss ′ | n rτ i = ±| n ′ rτ , elec i ⊗ | q ′ r i ≡ ±| n ′ rτ i (46)where n ′ rτ is defined as in (45). This relation establisheswhat we promised earlier: H hop gives an amplitude forthe fermion excitations to hop from site to site, but doesnot affect the other types of excitations. (We should nottake these properties for granted: for example, if we hadnot included the operator U kss ′ in the definition of H hop (38) then H hop would have affected the bosonic chargeexcitations).Denoting the creation operators for the fermion excita-tions by d † sσ , we conclude that the matrix elements of H within the low energy ˜ q s = 0 , b P = 1 subspace are givenby the free fermion Hamiltonian H eff = − X h ss ′ i ( t ss ′ σσ ′ d † s ′ σ ′ d sσ + h.c. ) − µ X sσ d † sσ d sσ (47)To complete the construction, we choose the hopping am-plitudes t ss ′ σσ ′ and the chemical potential µ so that H eff is a non-interacting topological insulator. The low en-ergy physics is then described by a topological insulatorbuilt out of fractionally charged fermions.There are many possible choices for t ss ′ σσ ′ , µ , but tobe concrete we will focus our discussion on the followingtight binding model on the square lattice . We con-sider a model with two orbitals, whose hopping matrixelements are related by time reversal symmetry: letting( σ, σ ′ ) denote the fermion spin, and (1 ,
2) denote the or-bital index, we have: t ss ′ σσ ′ , = t ∗ ss ′ σσ ′ , (48)and there is no hopping matrix element connecting thetwo orbitals. For each spin, the hopping matrix elementsfor orbital 1 are: t ss ′ σσ ′ , = X ˆ e i =ˆ x, ˆ y δ s − s ′ , ˆ e i ( tσ z + iλσ i ) + δ s − s ′ , t ( κ − σ z µ = 0 (49)where σ x,y,z are Pauli matrices acting on the spin in-dices. Here t parametrizes the spin-independent hoppingterms, and λ is a spin-orbit coupling. The constant κ sets the band gap at the momenta (0 , , ( ± π, , (0 , ± π );in the regime 0 < κ <
2, the model is in a topologicallyinsulating phase.In addition to being a topological insulator, the model(49) conserves the total z component of the spin. Theground state consists of a filled spin-up band with Chernnumber ν = +1, and a filled spin-down band with Chernnumber ν = −
1. As a result, the model exhibits a spin-Hall conductivity of σ sH = q f π (50) D. Effect of an electromagnetic field
In this section, we investigate the response of the frac-tionalized insulator (37) to an applied electromagneticfield. We show that for weak, slowly varying fields, themodel behaves like a non-interacting system of fraction-ally charged fermions. On the other hand, for strongerfields, we find that other, non-fermionic, degrees of free-dom contribute to the response.The first step is to understand how to incorporate avector potential A into the Hamiltonian. As the model(37) contains charged bosons that live on links of thelattice in addition to those that live on the sites, we de-fine a lattice vector potential A ss ′ , , A ss ′ , for each of thetwo halves of each link h ss ′ i (see Fig. 3). The hoppingoperator U ss ′ is then U ss ′ = (cid:0) b † s (cid:1) α s − b † s ′ b α s ss ′ e ie (1 − α s ) A e ieA (51)+ b α s ′ − s ′ b s (cid:16) b † ss ′ (cid:17) α s ′ e ie (1 − α s ′ ) A e ieA We can simplify this expression with the help of the uni-tary transformation W A = exp − iem · X h ss ′ i n ss ′ · ( α s A ss ′ , − α s ′ A ss ′ , ) A little algebra shows that W A U ss ′ W − A = h(cid:0) b † s (cid:1) α s − b † s ′ b α s ss ′ + b α s ′ − s ′ b s (cid:16) b † ss ′ (cid:17) α s ′ i e i em A ss ′ (52)where A ss ′ = A ss ′ , + A ss ′ , is the total vector potentialon the link h ss ′ i . In retrospect, this expression is to beexpected as U ss ′ hops a charge 2 e/m from s to s ′ , and assuch, should be multiplied by a phase factor e ieA ss ′ /m in the presence of an electromagnetic vector potential.Substituting this expression into H (37), we find thatthe Hamiltonian can be written as W A HW − A = V X s ˜ Q s − u X P ( B P e ieφ P /m + h.c. ) − µ X sσ n sσ + H hop (53)where φ P = A + A + A + A , H hop = − X h ss ′ i ( t ss ′ σσ ′ e iq f A ss ′ c † s ′ σ ′ c sσ U kss ′ + h.c. ) (54)and ˜ Q s , B P , U ss ′ are defined as in the A = 0 case.We now analyze this Hamiltonian in several cases.First, we consider the case where A is time independentand the flux φ P through each plaquette is small com-pared with a unit flux quantum. We proceed by simul-taneously diagonalizing { ˜ Q s , B P , B † P , n sσ } , denoting theeigenstates by | ˜ q s , b P , n sσ i . We then define rotated states | ˜ q s , b P , n sσ , A ss ′ i ≡ W A | ˜ q s , b P , n sσ i (55)1 s s’ A ss’,1 A ss’,2 FIG. 3. In order to include an electromagnetic vector poten-tial in H (37), we need to define a vector potential for eachof the two halves of each link h ss ′ i . We denote these vectorpotentials by A ss ′ , , A ss ′ , . We note that in the absence of the fermion hopping term H hop , these states are exact eigenstates of H with ener-gies E = V X s ˜ q s − u X P ( b P e ieφ P /m + b ∗ P e − ieφ P /m ) − µ X sσ n sσ (56)When the flux φ P is small (specifically, | φ P | < Φ / q s = 0 , b P = 1.We will denote these (purely fermionic) states using theabbreviated notation | n rτ , A ss ′ i ≡ W A | ˜ q s = 0 , b P = 1 , n rτ i (57)To obtain the low energy effective Hamiltonian H eff , weproject H (53) to the subspace spanned by these states(57). By the calculation in the previous section, the ma-trix elements of (53) within this subspace are given by H eff = − X h ss ′ i ( t ss ′ σσ ′ e iq f A ss ′ d † s ′ σ ′ d sσ + h.c. ) − µ X sσ d † sσ d sσ (58)This result proves that the model behaves like a non-interacting system of fractionally charged fermions, evenin the presence of a weak, time independent vector po-tential A .In fact, the low energy effective Hamiltonian (58) isalso valid for a weak, time dependent A , as long as A varies slowly compared with the energy gap u, V of thebosonic excitations. One way to derive this result is tonote that when A is slowly varying, we can make an adi-abatic approximation and can assume that the systemalways remains in the instantaneous low energy subspacespanned by the fermion states | n rτ , A ss ′ i . The time evo-lution is therefore described by projecting the Hamilto-nian (53) to the instantaneous low energy subspace, andwe again obtain the low energy effective Hamiltonian H eff (58). As for Berry phase effects, one can check thatnon-abelian Berry connection h n rτ , A ss ′ | i∂ A | n r ′ τ ′ , A ss ′ i reduces to a overall c -number phase factor and hence canbe neglected for our purposes. (In other words, Berryphase effects only contribute a global phase factor to thetime evolution of the wave function). We emphasize, however, that the above low energy ef-fective theory (58) is only valid when the flux througheach plaquette is small compared to Φ . Examining (56),we see that when the flux becomes comparable to Φ ,the states with b P = 1 become energetically favorable.Hence, in this regime, we cannot describe the low energyphysics in terms of the fermions alone: we also need tokeep track of the other types of excitations.One case where these additional degrees of freedomare particularly relevant – and which will play an im-portant role in our analysis in section VII – is in fluxinsertion thought experiments. Imagine we take a ge-ometry with open boundary conditions, and we adia-batically increase the flux through a plaquette P from φ P = 0 to φ P = Φ , while keeping the flux throughall other plaquettes constant at φ P = 0. For simplicity,let us assume that the system is initialized in the state | ˜ q s = 0 , b P = 1 , n sσ = 0 , A ss ′ i when φ P = 0, and letus neglect the fermion hopping term H hop . From (56),we can see that there are two level crossings during theflux insertion process – one at φ P = Φ / φ P = 3Φ /
4. At the first level crossing, the state with b P = e − πi/m becomes lower in energy than the statewith b P = 1, while at the second level crossing, thestate with b P = e − πi/m becomes lower in energy than b P = e − πi/m . For a finite system size, we expect thatthese crossings will be avoided crossings, at least if we adda generic local perturbation to the Hamiltonian. There-fore, if we insert flux sufficiently slowly, the system willfollow the lowest energy state at all times, finally evolv-ing into a state with b P = e − πi/m and φ P = Φ . Onecan check using the definition (55) that this final stateis the same as the initial state. Hence, the insertion ofa unit flux quantum returns us to our original startingpoint – as expected from general considerations (e.g. theByers-Yang theorem).An important point, however, is that the gap at theseavoided level crossings vanishes in the limit that the pla-quette P is far from the edge of the system, since thestates involved in the level crossings have different val-ues of b P and therefore only couple to one another via aprocess where a flux quasiparticle tunnels from the edgeto P . As a result, the above picture is only valid at ex-tremely long time scales. If we insert the flux at a ratewhich is fast compared with the gap at the level crossing(but still slow compared with the bulk gap) the outcomeis different. In this case, the system passes through thelevel crossings unaffected, leading to a final state with b P = 1 at Φ P = Φ . One can check using (55) that thisfinal state corresponds to having 2 flux quasiparticles onthe plaquette P . This result shows that other degrees offreedom beyond the fermions come into play in flux in-sertion experiments. In addition, it implies that we needto insert m flux quanta (if m is odd) or m/ m is even) for the system to return to its original con-figuration. In this sense, our models have a reduced fluxperiodicity – like fractional quantum Hall states. IV. PHYSICAL PROPERTIES OF THE 2DMODELS
In the previous section, we introduced a Hamiltonian H (37) whose low energy physics is exactly described bya non-interacting 2D topological insulator of fractionallycharged fermions. We now describe the physical proper-ties of this fractionalized insulator, and derive an effectiveChern-Simons field theory which summarizes them. A. Edge states
One of the distinguishing features of non-interacting2D topological insulators is that they have gapless edgestates which are protected by time reversal symmetry andcharge conservation.
Here we discuss the analogousedge states for the 2D fractionalized insulator H (37).Fortunately, the exact mapping between the low en-ergy physics of H and the non-interacting topologicalinsulator H eff (47) holds for any lattice, including ge-ometries with an edge. Hence, our problem reduces tounderstanding the edge structure of the insulator H eff .We focus on the specific band structure (49), for simplic-ity. This insulator has two filled bands: a spin-up bandwith Chern number ν = 1, and a spin-down band withChern number ν = −
1. Therefore, by the usual bulk-boundary correspondence, H eff must have one spin-upedge mode and one spin-down edge mode with oppositechiralities. Translating this result over to the fractional-ized insulator H , we conclude that H also has a pair ofcounterpropagating free fermion edge modes. The onlydifference from the non-interacting case is that the lowenergy fermions carry charge q f instead of charge e . Thelow energy edge Hamiltonian for H is thus of the form H edge = vd †↑ ( i∂ x + q f A x ) d ↑ − vd †↓ ( i∂ x + q f A x ) d ↓ (59)where v is the velocity of the two edge modes. We leavethe discussion of the stability of these edge modes tosection VII. B. Topological order
To fully characterize our model, we must also ana-lyze the topological order of the fractionalized insulator H (37). We first recall the concept of topological or-der in 2D systems. Topologically ordered systems intwo dimensions have three important physical properties.First, these systems have a finite energy gap separatingthe ground state(s) from excited states. Second, when atopologically ordered system is defined in a torus geome-try (periodic boundary conditions in both directions), theground state is typically multiply degenerate. This de-generacy is not a consequence of any symmetry and is ro-bust to arbitrary perturbations.
Third, and perhaps P U s s ss s s ss s U U s s
FIG. 4. We can move a charge at position s around a fluxat position P by applying a string of U ss ′ operators along aclosed path C = s s ...s k s encircling P . most importantly, these systems have quasiparticle exci-tations with fractional statistics: when one such quasi-particle is braided around another, it acquires a nonvan-ishing Berry phase.Before analyzing the fractionalized insulator H (37),it is useful to first understand the topological order inthe lattice boson model H B (1). As we discussed earlier,this model contains two types of elementary excitations:“charge” excitations where q s = 1 at some site s , and“flux” excitations where B P = e πi/m on some plaquette P . We will now show that these excitations have non-trivial mutual statistics: when a charge moves around aflux, it acquires a Berry phase of θ ch,fl = 2 π/m .Imagine we have a flux on plaquette P and a chargeon some site s . We can denote this state by | s , P i ≡ | n s σ = 1 , b P = 1 i . (60)We need to adiabatically move the charge around the fluxand compute the resulting Berry phase. To this end, wenote that the operator U s s moves the charge from site s to a neighboring site s : U s s | s , P i = | s , P i (61)Indeed, this follows from the commutation relations be-tween Q t , B P and U ss ′ . Therefore, we can move thecharge around the flux by applying a string of U ss ′ oper-ators along some closed path C = s s ...s k s encircling P (see Fig. 4): U s k s ...U s s U s s | s , P i (62)To find the accumulated Berry phase, we use an operatoridentity: U s k s ...U s s U s s | s , P i = Y P ∈ C B P | s , P i = e πi/m | s , P i (63)Here, the second equality follows from the fact that b P =1 everywhere except for P = P , where b P = e πi/m .To isolate the statistical Berry phase from other geo-metric phases, we need to compare this phase with the3phase accumulated when the flux is not enclosed by thepath C . In that case, the same operator identity gives U s k s ...U s s U s s | s , P i = | s , P i (64)Comparing (63),(64), we conclude that the statisticalBerry phase is precisely θ ch,fl = 2 π/m , as we claimed.In a similar way, we can check that there is no Berryphase associated with exchanging a pair of charge or fluxexcitations. In other words, the charge and flux excita-tions are bosons. To complete our analysis of the topo-logical order, we need to find the ground state degeneracyon a torus. We describe this calculation in Appendix A 2.There, we show that the lattice boson model has exactly D = m degenerate ground states in a torus geometry.The above ground state degeneracy and quasiparticlestatistics are identical to the statistics and ground statedegeneracy of the generalized “toric code” model withgauge group G = Z m . Thus, the fractionalized bosonicinsulator H B (1) has the same topological order as the Z m toric code, or equivalently 2D Z m gauge theory cou-pled to bosonic matter.Given these results, we can now easily analyze thetopological order of the fractionalized insulator H (37).This model contains three types of elementary excitations– charges, fluxes, and fermions. The charge and flux exci-tations are identical to the excitations in the lattice bosonmodel and obey the same statistics. On the other handthe fermion excitation is a composite of an electron and k “charge” excitations, as we argued in equation (33).This decomposition implies that the fermion excitationhas mutual statistics θ f,fl = 2 πk/m with respect to theflux excitations, but no mutual statistics with respect tothe charges. As for the ground state degeneracy on atorus, it is easy to check that D = m just as in thelattice boson model. C. Chern-Simons field theory description
In this section, we derive a field theoretical descriptionof the fractionalized insulator H (37). This field theoryis useful as it captures all of the (universal) properties ofthe fermionic phase–including both the topological orderin the bulk, and the edge modes at the boundary (seesection VII C 1).We begin by writing down a field theory description ofthe lattice boson model H B (1). As we discussed in sec-tion IV B, the topological order in this model is the sameas Z m gauge theory coupled to bosonic matter. Previouswork has shown that this kind of topological order isdescribed by the Chern-Simons theory, L B = m π ǫ λµν ( α λ ∂ µ β ν + β λ ∂ µ α ν ) − e π ǫ λµν A λ ∂ µ β ν (65)where A λ is the electromagnetic gauge field. (This par-ticular form of U (1) × U (1) Chern-Simons theory is some- times referred to as 2D BF theory ). In this descrip-tion, the boson number current is given by j λboson = 12 π ǫ λµν ∂ µ β ν (66)Also, the two types of quasiparticle excitations, “charges”( q s = 1) and “fluxes” ( b p = e πi/m ), are described bycoupling (65) to bosonic particles carrying unit α µ and β µ charge respectively.This field theory correctly describes the fractionalcharges and statistics of the quasiparticle excitations inthe lattice boson model, as well as the ground state de-generacy on a torus. We can verify this using the “ K -matrix” formalism for abelian Chern-Simons the-ory and abelian fractional quantum Hall states. First wewrite L B in K -matrix notation: L B = K IJ π ǫ λµν a Iλ ∂ µ a Jν − π t I ǫ λµν A λ ∂ µ a Iν (67)where K IJ = mm ! , t I = e ! , a I = αβ ! (68)In this notation, the gauge charge carried by the twotypes of particles (charges and fluxes) can be representedin terms of the vectors l ch = ! , l fl = ! (69)According to the K -matrix formalism, the physical elec-tric charge of each excitation is given by q l = l T K − t (70)while the mutual statistics associated with braiding oneparticle around another is given by θ ll ′ = 2 πl T K − l ′ (71)(The statistical phase associated with exchanging twoidentical particles is θ l = θ ll / l ch and l fl , we see that q ch = 2 em , q fl = 0 , θ ch,fl = 2 πm , θ ch = θ fl = 0 (72)in agreement with the properties of the lattice bosonmodel. Furthermore, according to the K -matrix formal-ism, the ground state degeneracy on a torus is given by D = | det ( K ) | = m (73)Again, this is in agreement with the properties of thelattice boson model. We conclude that the Chern-Simonstheory (65) does indeed describe the topological order inthe lattice boson model.4To obtain a field theory description of the fractional-ized insulator H (37), we think of this system as a non-interacting topological insulator built out of fractionallycharged fermions. These fermions are composite particlescomposed of k “charge” excitations of the lattice bosonmodel together with one electron. In terms of the fieldtheory (65) we can describe these composite particles bycoupling (65) to fermionic particles, each of which carries k units of α µ charge and e units of electric charge. Weconclude that the Lagrangian for H is of the form L = L B + L TI [ kα µ + eA µ ] (74)where L TI [ kα µ + eA µ ] is the Lagrangian of a non-interacting topological insulator coupled to an externalgauge field kα µ + eA µ .To complete our derivation, we need to construct afield theory L T I for the non-interacting topological insu-lator coupled to an external gauge field. We can writedown such a field theory explicitly in the case of the bandstructure (49). In this case, the spin-up fermions form aband insulator with Chern number ν = +1, while thespin-down fermions form an insulator with Chern num-ber ν = −
1, so the appropriate field theory is a sum oftwo decoupled Chern-Simons theories: L TI [ kα µ + eA µ ] = 14 π ǫ λµν ( γ ↑ λ ∂ µ γ ↑ ν − γ ↓ λ ∂ µ γ ↓ ν ) − π ǫ λµν ( kα λ + eA λ ) ∂ µ ( γ ↑ ν + γ ↓ ν ) (75)In this description, the spin-up and spin-down fermioncurrents are given by j λ ↑ = 12 π ǫ λµν ∂ µ γ ↑ ν (76) j λ ↓ = 12 π ǫ λµν ∂ µ γ ↓ ν (77)while the spin-up and spin-down fermion excitations canbe described by coupling (75) to bosonic sources carryingunit γ ↑ µ and γ ↓ µ charge respectively.Combining (65), (74), (75), we see that the fractional-ized electronic insulator H is described by the 4 compo-nent Chern-Simons theory L = K IJ π ǫ λµν a Iλ ∂ µ a Jν − π t I ǫ λµν A λ ∂ µ a Iν (78)where K IJ = m − k − km − k − k − , t I = eee , a I = αβγ ↑ γ ↓ (79)As in the bosonic lattice model case (67), this Chern-Simons theory contains all the information about thefractional charge and fractional statistics of the three types of excitations, as well as the ground state degen-eracy on the torus. The “charge” excitations correspondto particles with unit a charge, the “flux” excitationscorrespond to particles with unit a charge, and the spin-up/spin-down fermions correspond to particles with unit a or a charge. We represent these particles with thevectors l ch = , l fl = , l ↑ = , l ↓ = − (80)Applying the formulas (70-71), we can check that thequasiparticle charges and statistics are q ch = 2 em , q fl = 0 , q f = e (2 k + m ) m (81) θ ch,fl = 2 πm , θ f,fl = 2 πkm , θ f = π , θ ch = θ fl = 0in agreement with our previous calculations. Also, theground state degeneracy on a torus is given by D = | det ( K ) | = m (82)as expected. V. LATTICE MODELS FOR 3D FRACTIONALTOPOLOGICAL INSULATORS
In this section we construct exactly soluble latticemodels for 3D fractionalized insulators. The 3D con-struction is a simple generalization of the 2D case, andmuch of our analysis carries over without any change.As before, we proceed in three steps, beginning by con-structing lattice boson models with fractional charge.
A. Step 1: 3D lattice boson models with fractionalcharge
The 3D lattice boson models are identical to the 2Dmodels (1), except that we consider these systems on thecubic lattice instead of the square lattice. In other words,the Hamiltonian is still H B = V X s Q s − u X P ( B P + B † P ) (83)but now the sums over s and P run over the sites s andplaquettes P of the cubic lattice (Fig. 5(a)).This model is exactly soluble just as in the 2D case,since Q s , B P , B † P all commute with one another. Again,we can choose simultaneous eigenstates | q s , b P i where q s is an integer and b P is a m th root of unity, and thesestates have energies E = V X s q s − u X P ( b P + b ∗ P ) (84)5 (a) (b) Q s B P FIG. 5. (a) The lattice boson model defined on the cubiclattice. As in the square lattice case, the Hamiltonian H B (83) is a sum of Q s operators and B P operators, but now the Q s operators act on six links h ss ′ i and one site s , and the B P operators act on the plaquettes of the cubic lattice. (b)The B P operators obey the identity Q P ∈ C B P = 1 where theproduct runs over the six plaquettes P adjacent to a “cube” C , and where we choose appropriate orientations on theseplaquettes. Furthermore, as before, we can find the degeneracy of the q s , b P eigenspaces for particular system geometries – saya rectangular slab of cubic lattice with open boundaryconditions. The only difference from the 2D case is thatthe B P operators satisfy the identity Y P ∈ C B P = 1 (85)where the product runs over the six plaquettes P adja-cent to a “cube” C in the cubic lattice, and where wechoose appropriate orientations on these plaquettes (Fig.5(b)). Therefore, the b P quantum numbers obey the localconstraint Y P ∈ C b P = 1 (86)for every “cube” C in the cubic lattice, in addition to theglobal constraint X s q s ≡ { q s , b P } satisfying theseconstraints, there is a unique eigenstate | q s , b P i (see Ap-pendix B 1).As in the square lattice case, the ground state of the3D cubic lattice model (with open boundary conditions)is the unique state with q s = 0 , b P = 1, and there are twotypes of elementary excitations. The first kind of excita-tion are “charges” where q s = 1 at some site s . These ex-citations are very similar to the charge excitations in the2D model. In particular, they carry the same fractionalcharge 2 e/m , as can be verified using the arguments insection III A 2. On the other hand, the second kind ofexcitation is slightly different from the 2D case: insteadof particle-like “flux” excitations, the 3D model has “fluxloop” excitations where b P = e πi/m along some closedloop in the dual cubic lattice. The reason that these ex-citations are loop-like in this case is the constraint (86)which ensures that the flux is divergence free and hencemust form closed loops. B. Step 2: 3D lattice electron models withfractional charge
The next step, as in the 2D case, is to modify the3D boson model (83) by including additional spin-1 / s . The resulting3D electron model is identical to the square lattice case(25). The Hamiltonian is still H e = V X s ˜ Q s − u X P ( B P + B † P ) − µ X sσ n sσ (88)except now the sums over s , P run over sites and plaque-ttes of the cubic lattice.As in the 2D case, we can choose simultaneous eigen-states | ˜ q s , b P , n sσ i with corresponding energies E = V X s ˜ q s − u X P ( b P + b ∗ P ) − µ X sσ n sσ (89)There is a unique state for each choice of ˜ q s , b P , n sσ sat-isfying the two constraints Y P ∈ C b P = 1 , X s ˜ q s + k X sσ n sσ ≡ µ < q s = 0 , b P = 1 , n sσ = 0. Thereare three types of elementary excitations, just as in the2D case: “charge” excitations with ˜ q s = 1, “flux loop”excitations with b P = e πi/m along some loop in the duallattice, and spin-1 / n sσ =1. The spin-1 / q f = e (1 + 2 k/m )(34), and transform under time reversal in the same way. C. Step 3: Building candidate 3D fractionaltopological insulators
The final step is to modify the lattice electron model(88) so that the fractionally charged fermions form anon-interacting topological insulator. As in the 2D case,we accomplish this by adding a new term, H hop to theHamiltonian: H = H e + H hop (91)where H hop is defined just as in the 2D case (38): H hop = − X h ss ′ i ( t ss ′ σσ ′ c † s ′ σ ′ c sσ U kss ′ + h.c. ) (92)Again, we assume that t ss ′ σσ ′ ≪ u, V so that the band-width of the fermion excitations is small compared withthe gap to the bosonic excitations. Just as in the 2D6case, we can show that the low energy physics of H isdescribed by the non-interacting fermion model H eff = − X h ss ′ i ( t ss ′ σσ ′ d † s ′ σ ′ d sσ + h.c. ) − µ X sσ d † sσ d sσ (93)We can then complete the construction, just as in the2D case: we simply choose the hopping amplitudes t ss ′ σσ ′ and chemical potential µ so that H eff is a non-interactingtopological insulator. The low energy physics is then de-scribed by a topological insulator built out of fraction-ally charged fermions. There are many possible choicesfor t ss ′ σσ ′ , but to be concrete, we consider a 2-orbitalmodel on the cubic lattice, introduced by Ref. 25. The 4fermions on each site can be expressed as a single vector( d s ↑ , d s ↓ , d s ↑ , d s ↓ ). In this basis the hopping matrixelements can be expressed as a tensor product of matri-ces τ i acting on the orbital indices (1 ,
2) and σ i acting onthe spin indices ( ↑ , ↓ ). We take t ss ′ σσ ′ = X ˆ e i =ˆ x, ˆ y, ˆ z δ s − s ′ , ˆ e i [ iλ ( τ z ⊗ σ i ) σσ ′ + t ( τ x ⊗ ) σσ ′ + h.c. ] + mδ s − s ′ , ( τ x ⊗ ) σσ ′ (94)where σ, σ ′ ∈ { (1 , ↑ ) , (2 , ↑ ) , (1 , ↓ ) , (2 , ↓ ) } specify both thespin and orbital indices. The first term is a spin-orbitcoupling whose sign differs for the two orbitals; the sec-ond term is a nearest-neighbor hopping from one orbitalto the other, and the third term is an on-site hoppingbetween the two orbitals. Taking µ = 0, together with t > λ >
0, this yields a topologically insulatingband structure, provided that 2 t < | m | < t . VI. PHYSICAL PROPERTIES OF THE 3DMODELS
In the previous section, we introduced an electronHamiltonian H (91) whose low energy physics is exactlydescribed by a non-interacting topological insulator offractionally charged fermions. We now describe the phys-ical properties of the resulting fractionalized insulator. A. Surface states
One of the most important properties of 3D non-interacting topological insulators is that they have gap-less surface states which are protected by time reversalsymmetry and charge conservation. In this section, wediscuss the analogue of these surface states for the frac-tionalized insulator realized by H (91).To begin, we review the structure of the surface statesof the non-interacting topological insulator with bandstructure (94). To be concrete, we consider this modelin an L x × L y × L z slab of cubic lattice which is periodicin the x, y directions and has open boundary conditionsat z = 0 and z = L z . (This geometry is sometimes referred to as a “Corbino donut”). This system is trans-lationally invariant in the x, y directions, so we can stilldefine a 2D band structure as a function of k x , k y . Ithas been shown that the surface band structure for thetight binding model (94) has two Dirac points—one foreach surface. The surface states have a particularly sim-ple structure if the chemical potential is tuned near theseDirac points. In that case, the low energy surface stateson each surface are described by the Dirac Hamiltonian H TIsurf = vc † α [ σ αβ · ( i ∇ + e A )] c β − µc † α c α (95)where α, β = ↑ , ↓ , and σ = ( σ x , σ y , σ z ).The surface states for the fractionalized insulator H (91) are almost identical to the non-interacting case,since the low energy physics of H is exactly describedby the free fermion model H eff . The only difference isthat the fermions excitations carry fractional charge q f .Therefore, the low energy surface Hamiltonian for H nearthe Dirac points is given by exactly (95), except with e replaced by q f (and c replaced by d ): H surf = vd † α [ σ αβ · ( i ∇ + q f A )] d β − µd † α d α (96)We leave the discussion of the stability of these surfacemodes to section VII. B. Surface quantum Hall effect
The presence of a single Dirac cone in the surface modespectrum has an interesting consequence if we break timereversal on this surface, for example by attaching to it athin magnetic film. Let us first review what this does to anon-interacting topological insulator with band structure(94). Assuming that the magnetic film is only weaklycoupled to topological insulator, we can model its effectby including a Zeeman term H Z = Bd † α σ zαβ d β in thesurface Dirac Hamiltonian (95). This term opens up agap in the Dirac spectrum of size 2 | B | . If the chemicalpotential lies within this gap, then the surface states arecompletely gapped and the system exhibits a half-integersurface Hall conductivity σ TI xy = e h (97)The total Hall conductivity on the two surfaces is e /h or 0 depending on whether the magnetic field B has thesame or opposite signs at z = 0 and z = L z .The behavior of the fractionalized insulator H (91) issimilar. As in the non-interacting case, the magnetic filmintroduces a Zeeman term, thereby gapping the spectrumof the surface Dirac Hamiltonian (96). If the chemicalpotential lies within this gap, then the system exhibits aquantized surface Hall conductivity. The only differenceis that the fermionic excitations carry charge q f so theHall conductivity is a half-integer in units of q /h : σ xy = q h (98)7The total Hall conductivity on both surfaces is q /h or 0 depending on whether the magnetic field B has thesame or opposite signs at z = 0 and z = L z . The first caseis particularly interesting, since in this case the total Hallconductivity is fractional. Hence, if we view our geometryas a quasi-2D system, taking the thermodynamic limit L x , L y → ∞ while keeping L z finite, then H realizes afractional quantum Hall state. The properties of thisstate can be derived using the same approach as sectionIV C. C. Charge acquired by a magnetic monopole
Another important property of non-interacting topo-logical insulators is their response in the presence of amagnetic monopole. Imagine threading a microscopicsolenoid into a topological insulator which is so small thatit fits within a single plaquette of the lattice. If we inserta full flux quantum Φ = hc/e , we can effectively con-struct a monopole/anti-monopole pair at the two ends ofthe solenoid. An interesting property of non-interactingtopological insulators is that this process causes a half-integer charge q TI = ( n + 1 / e (99)to be localized at each end of the solenoid . In otherwords, a magnetic monopole (or anti-monopole) in atopological insulator carries half-integer charge. (Herethe integer n depends on the microscopic form of theHamiltonian near the ends of the solenoid. As such, itcan be varied without encountering a phase transition inthe bulk).We consider the analogous question for the fractional-ized insulators (91). In this case, we have to be carefulbecause the Dirac quantization condition for monopolesis modified due to the presence of fractionally charged ex-citations. Rather than requiring monopoles to carry fluxwhich is an integer multiple of Φ , the Dirac argumentimplies that monopoles carry flux which is a multiple of( e/e ∗ )Φ where e ∗ is the smallest charged quasiparticleexcitation in the system. In our case, the minimal charge e ∗ is given by e ∗ = ( em if m is odd em/ if m is even (100)When m is even, the bosonic charge excitations carrythe minimal charge e ∗ ; when m is odd, an excitationwith minimal charge e ∗ can be constructed by forming acomposite of an electron and (1 − m ) / e/e ∗ )Φ , the result is that the monopole isjoined by a physically observable Dirac string to theanti-monopole partner. One way to see this is to cal-culate the ground state energy in the presence of the monopole/anti-monopole pair using the 3D analogue ofthe eigenvalue spectrum (56). Examining the b P term,we can see that a Dirac string with flux j Φ will havean energy cost ∼ uL (1 − cos(4 jπ/m )) proportional toits length L unless 2 j/m is an integer – that is, unless j is a multiple of e/e ∗ (100). (We cannot eliminate thisenergy cost by changing the value of b P along these pla-quettes due to the constraint Q P ∈ C b P .) Intuitively, thepoint is that, unless this condition is satisfied, a charge e ∗ quasiparticle excitation will acquire a nonvanishing Berryphase when it travels around the Dirac string.Hence, the natural quantity in the fractional case is theelectric charge carried by a flux ( e/e ∗ ) φ monopole. Tocompute this charge, we again use the fact that the lowenergy physics of H (91) is described by the free fermionmodel (93). Because the fermions carry charge q f , thissystem behaves exactly like a non-interacting topologicalinsulator with a flux ( q f /e ∗ ) φ monopole. In this case, weknow that ( n + 1 / q f /e ∗ fermions are localized near themonopole, as we discussed above. We conclude that theflux ( e/e ∗ ) φ monopole in the fractionalized insulator H carries electric charge q = ( n + 1 / q /e ∗ (101)On the other hand, we could presumably trap anynumber of additional charge e ∗ quasiparticles near themonopole, by adding an appropriate local potential.Therefore, the only quantity which is independent of mi-croscopic details is the charge of the monopole modulo e ∗ . This quantity is given by q = ( e ∗ (mod e ∗ ) if q f /e ∗ is odd0 (mod e ∗ ) if q f /e ∗ is even (102)More explicitly, if we use the expressions for q f (34) and e ∗ (100), we find that q f /e ∗ is even if and only if 2 k + m is divisible by 4. D. Topological order
In the previous two sections, we showed that the 3Dmodel (91) exhibits a fractional surface Hall conductiv-ity (98) and a fractional charge bound to a magneticmonopole (102). These phenomena are manifestationsof a fractional magnetoelectric effect . On the otherhand, it was argued in Ref. 12 that the only way a frac-tional magnetoelectric effect can be consistent with timereversal symmetry is if the ground state on a three di-mensional torus is multiply degenerate. In this section,we show that our model does indeed have multiply de-generate ground states on a 3D torus. This ground statedegeneracy originates from the fact that the model istopologically ordered, as we now discuss.We begin by analyzing the topological order of the 3Dlattice boson model (83). To derive the quasiparticlestatistics in this model, we recall that (83) contains two8types of elementary excitations: point-like “charge” ex-citations where q s = 1 at some site s , and “flux-loop” ex-citations where B p = e πi/m along some loop in the duallattice. Using the same approach as in the 2D case (sec-tion IV B), one can show that when one braids a chargearound a flux-loop, it acquires a Berry phase of e ± πi/m where the sign depends on the orientation of the braidingtrajectory. We can think of this as the 3D analogue ofmutual statistics. It is also straightforward to show thatthere is no Berry phase associated with exchanging twocharges, so the “charge” excitations are bosons.In addition to nontrivial statistics, the lattice bosonmodel (83) also displays the second signature of topologi-cal order, namely ground state degeneracy. In particular,if we define this model in a 3D torus geometry, we find m degenerate ground states (see Appendix B 2).The quasiparticle statistics and ground state degener-acy of (83) is identical to the the statistics and groundstate degeneracy of the 3D toric code with gaugegroup G = Z m . Thus, the lattice boson model (83) hasthe same topological order as the Z m toric code in threedimensions, or equivalently 3D Z m gauge theory coupledto bosonic matter.Given these results, it is easy to analyze the topolog-ical order in the fractionalized insulator H (91). Thismodel contains three types of elementary excitations –charges, flux loops, and fermions. The charge and fluxloop excitations are identical to the excitations in the lat-tice boson model and obey the same statistics. On theother hand the fermion excitation is a composite of anelectron and k “charge” excitations, just as in the 2Dcase (33). This decomposition implies that the fermionexcitation has mutual statistics e πik/m with respect tothe flux loop excitations. As for the ground state degen-eracy on a 3D torus, we can show that this is m , just asfor the lattice boson model. VII. GAPLESS BOUNDARY MODES:PROTECTED OR UNPROTECTED?
In the previous sections we have constructed fractional-ized insulators (37,91) whose low energy physics is equiv-alent to a non-interacting topological insulator built outof fractionally charged fermions. We have also seen thatthese models have gapless boundary modes. Howeverthese two properties are not by themselves sufficient todeclare these models “fractional topological insulators”:we must also show that the boundary modes are pro-tected . That is, we must show that the modes are robustto arbitrary time reversal invariant, charge conservingperturbations.In this section we investigate this robustness, and findtwo key results. First, we show that in both 2 and 3dimensions, the models for which the ratio q f /e ∗ is oddhave protected boundary modes. Thus, these models areindeed authentic “fractional topological insulators.” Sec-ond – and perhaps more unexpectedly – we show that in the 2D case, the models with even q f /e ∗ do not haveprotected edge modes, and are thus “fractional trivial in-sulators.” We do not have an analogous result in the 3Dcase. That is, we do not establish either the stability orinstability of the surface modes when q f /e ∗ is even.In the 2D case, our analysis of edge modes closely fol-lows the arguments of Ref. 10. Our conclusion is alsovery similar to Ref. 10: in that paper, the authors foundthat a certain class of s z conserving models have pro-tected edge modes if and only if σ sH /e ∗ is odd, where σ sH is the spin-Hall conductivity in units of e/ π , and e ∗ is the elementary charge in units of e . Here, we find thatour exactly soluble models have protected edge modes ifand only if q f /e ∗ is odd. The two criteria agree exactly,since as one can easily check, q f /e ∗ = σ sH /e ∗ (50) for themodels discussed here. A. 2D flux insertion argument
In this section we present a general argument thatshows that when q f /e ∗ is odd, the 2D fractionalized in-sulator H (37) has protected gapless edge modes. Theargument is very similar to the one given in Ref. 10,which is in turn a generalization of the flux insertion ar-gument of Ref. 8.To be precise, we prove the following statement: weconsider the exactly soluble model H in a cylindrical ge-ometry with an even number of electrons and zero fluxthrough the cylinder. We assume that the ground stateis time reversal invariant and does not have a Kramersdegeneracy on either of the two edges. (We discuss themeaning of this Kramers degeneracy assumption in sec-tion VII A 1 below). Given these assumptions, we showthat if q f /e ∗ is odd, then there is always at least one ex-cited state whose energy gap vanishes in the thermody-namic limit. Furthermore, this excited state is robust ifwe add an arbitrary time reversal invariant, charge con-serving, local perturbation to the Hamiltonian, as longas we do not close the bulk gap. Finally, this excitedstate has the important property that it is in the same“topological sector” as the ground state, as we explainbelow. We interpret this low lying excited state as evi-dence for a protected gapless edge mode – that is, a modewhich cannot be gapped out without breaking time rever-sal symmetry or charge conservation, explicitly or spon-taneously.
1. Non-interacting case
We first review the argument for the case of non-interacting topological insulators. The proof that thesystem always has a low lying state begins as follows:we consider the system on a cylinder, and start in themany-body ground state Ψ at zero flux. We imagineadiabatically inserting Φ / hc/ e flux through thecylinder. We call the resulting state Ψ π . Similarly, we9 /2 −Φ Φ /2 ΦΨΨ Ψ ex E π ∆ Φ Ψ −π FIG. 6. The flux insertion argument in the 2D non-interacting case: we start in the ground state Ψ and adi-abatically insert ± Φ / ± π . The state Ψ π has a Kramers degeneracy atthe two ends of the cylinder, and is thus degenerate in energywith three other states, one of which is Ψ − π . If we start inone of these three degenerate states and then adiabaticallyreduce the flux to 0, we obtain an excited state Ψ ex whoseenergy gap vanishes in the thermodynamic limit. let Ψ − π be the state obtained by adiabatically inserting − Φ / π and Ψ − π aretime reversed partners.To proceed further, we recall that 2D topological insu-lators are characterized by a Z invariant ν = −
1. Thisvalue of the invariant means that Ψ π has a Kramers de-generacy at the two ends of the cylinder if and only ifΨ does not. (At an intuitive level, when we say that amany-body state has a Kramers degeneracy at the twoends of the cylinder, we mean that it can be broughtto a state which is time reversal invariant by removingan odd number of electrons from each of the two ends.Assuming the ends are far apart, we can treat them asseparate systems, each of which must have a two-fold de-generacy according to Kramers theorem. A precise def-inition of this notion of “local Kramers degeneracy” isgiven in Ref. 37). By our assumption above, Ψ has noKramers degeneracy so Ψ π must have a Kramers degen-eracy at the two ends of the cylinder. Hence, as long asthe two ends are well separated, Kramers theorem guar-antees that Ψ π is part of a multiplet of four states (twoassociated with each end) which are nearly degenerate inenergy. More precisely, these four states are separated byan energy splitting which vanishes exponentially as thedistance between the two ends grows. We note that Ψ − π is one of these degenerate states, being the time reversedpartner of Ψ π .Next, we imagine starting with the system at Φ / π , then adiabatic flux removaltakes us to the ground state Ψ . However, if we startwith Ψ − π (or one of the other two degenerate states), theresult is a eigenstate Ψ ex of the zero flux Hamiltonian,which is distinct from Ψ (see Fig. 6). At the same time,it necessarily has low energy since the energy change ∆ E associated with an adiabatic insertion of flux througha cylinder must vanish in the thermodynamic limit aslong as charge conservation is not broken (in a gapless system with linear dispersion, we expect a scaling like∆ E ∼ /L , while in a gapped system, ∆ E ∼ e − const. · L ).To complete the argument, we now imagine adding anarbitrary time reversal invariant, charge conserving, localperturbation to the system (for example, we could addshort-ranged interactions between electrons). As long asthe perturbation does not close the bulk gap, the abovepicture must stay the same: the Kramers degeneracy be-tween Ψ π and Ψ − π must remain intact, and hence Ψ ex must continue to be low in energy. We conclude thatthe system always contains at least one low-lying excitedstate Ψ ex whose energy gap vanishes in the thermody-namic limit.
2. Fractionalized case
We now explain the analogous argument for our frac-tionalized insulator H (37). The crucial difference fromthe non-interacting case is that this model has excita-tions with fractional charge e ∗ . As a result, we need tomodify the flux insertion argument, inserting ± ee ∗ · Φ flux through the cylinder instead of ± Φ flux. To understand why this is so, imagine we insert ± Φ / π , Ψ − π differ bythe insertion of a single flux quantum. But the inser-tion of a single flux quantum changes the Berry phase θ associated with moving a charge e ∗ particle around thecylinder by ∆ θ = 2 π e ∗ e . As ∆ θ is not a multiple of 2 π ,Ψ π and Ψ − π are in different “topological sectors.” Thatis, to get from one state to the other, we need to transfera fractionalized excitation (in this case, two flux quasi-particles – see section III D) from one end of the cylinderto the other. If we then try to construct a low ly-ing excited state at zero flux in the usual way (i.e. bystarting from Ψ − π and adiabatically reducing the flux tozero), the resulting state Ψ ex is in a different topologi-cal sector from Ψ . Unfortunately, this property meansthat we cannot conclude much from the existence of Ψ ex .The reason is that topologically ordered systems like H always have a finite number of low lying states that lie indifferent topological sectors. These states have noth-ing to do with time reversal symmetry and continue toexist even if time reversal symmetry is broken and theedge is gapped. Instead, these states are “topologicallyprotected” and are analogous to the degenerate groundstates in a torus geometry. Therefore, it is importantthat we construct a low-lying state in the same topolog-ical sector as the ground state. This will establish theexistence of an “unexpected” low lying state, which canplausibly be taken as evidence for a time reversal pro-tected gapless edge mode.For this reason, we insert ± N Φ flux through the cylin-der where N = e/e ∗ : when we insert this amount of flux,the states Ψ Nπ , Ψ − Nπ lie in the same topological sector,so we do not run into the issue discussed above. Otherthan this simple modification, the argument proceeds asbefore: we note that the low energy physics of H is de-0scribed by a topological insulator built out of charge q f fermions, so our insertion of ± ee ∗ · Φ / ± q f e ∗ · Φ flux for a non-interacting topologicalinsulator. If q f /e ∗ is odd, then this process changes thetime reversal polarization at the two ends of the cylin-der, implying that Ψ Nπ , Ψ − Nπ are Kramers degenerate.As before, we can then construct a protected low lyingstate Ψ ex at zero flux by starting with Ψ − Nπ and adia-batically inserting − N Φ flux quanta. Furthermore, Ψ ex is in the same topological sector as the ground state Ψ since Ψ Nπ , Ψ − Nπ are in the same topological sector. Weconclude that when q f /e ∗ is odd, the fractionalized in-sulator H has a protected low lying excited state in thesame topological sector as the ground state. This is whatwe wanted to show. B. 3D flux insertion argument
Next, we show that a similar flux insertion argumentcan be used to establish the existence of protected low-lying surface states in the 3D models (91) with q f /e ∗ odd.The precise statement we will prove is this: we considerthe 3D models in a “Corbino donut” geometry, periodicin the x and y directions, and with open boundary condi-tions in the z direction (see Fig. 7a). We assume that thenumber of electrons is even and that the ground state istime reversal invariant when there is zero flux through thetwo holes of the Corbino donut. In particular, we assumethat the ground state does not have a Kramers degener-acy on the z = 0, z = L z surfaces. We then show thatif q f /e ∗ is odd, then there is always at least one excitedstate whose energy gap vanishes in the thermodynamiclimit. Furthermore, this excited state is robust if we addan arbitrary time reversal invariant, charge conserving,local perturbation to the Hamiltonian, as long as we donot close the bulk gap. Finally, this excited state is inthe same topological sector as the ground state. Just asin the 2D case, we interpret this low lying excited stateas evidence for a protected gapless surface mode–thatis, a mode which cannot be gapped out without breakingtime reversal symmetry or charge conservation, explicitlyor spontaneously.
1. Non-interacting case
It is useful to first give the argument in the case of thenon-interacting 3D topological insulator. (This argumentwas alluded to in Ref. 9, though not explicitly discussed).The proof that the system has a protected low lying statebegins similarly to the 2D case: we start in the many-body ground state at zero flux, which we call Ψ (0 , , andthen imagine adiabatically inserting Φ / ( π, ) , Ψ (0 ,π ) , Ψ ( π,π ) .We next recall that 3D topological insulators are char-acterized by a Z invariant ν = −
1. Similar to the situa- φ y φ x ππ φ y0 φ x ππ φ y0 φ x ππ φ y0 φ x ππ φ y0 φ x (a)(b) FIG. 7. (a) The 3D flux insertion argument makes use of a“Corbino donut” geometry, with flux φ x , φ y through the twoholes of the Corbino donut. (b) In the non-interacting case,we start in the ground state Ψ (0 , and adiabatically insertΦ / ( π, , Ψ (0 ,π ) , Ψ ( π,π ) . An odd number of these statesmust have Kramers degeneracies at the two surfaces of theCorbino donut, leading to four possible degeneracy patterns(filled circles represent states with Kramers degeneracies). Inparticular, at least one state has a Kramers degeneracy, andhence can be used to construct a protected low lying excitedstate Ψ ex , as in the 2D case. tion in 2D, this value of the invariant implies that an odd number of the 4 states Ψ (0 , , Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) havea Kramers degeneracy on the z = 0 and z = L z surfacesof the Corbino donut. (We give a simple derivation ofthis result at the end of this section). By our assump-tion above, Ψ (0 , has no Kramers degeneracy, so eitherall three of the other states have a degeneracy, or ex-actly one of them does (see Fig. 7b). In particular, atleast one of the three other states has a Kramers degen-eracy. Let us denote this state by Ψ ( α,β ) . Then, as longas L z is large so that the two surfaces of the Corbinodonut are well separated, Kramers theorem guaranteesthat Ψ ( α,β ) is part of a multiplet of four states (two as-sociated with each surface) which are nearly degeneratein energy. More precisely, these four states are separatedby an energy splitting that vanishes exponentially as thedistance between the two surfaces grows.Just as in the 2D case, we can use this Kramers degen-erate state to construct a low lying excited state at zeroflux: as in that case, we start in one of the three stateswhich are degenerate with Ψ ( α,β ) , and then adiabaticallyreduce the flux to (0 , ex of the zero flux Hamiltonian. This state is distinct fromthe ground state Ψ (0 , and its energy gap vanishes inthe thermodynamic limit. Furthermore, time reversal in-variant perturbations cannot change this picture, sincethey cannot split the degeneracy between Ψ ( α,β ) and itsKramers partners. Hence, even in the presence of arbi-trary time reversal invariant, charge conserving pertur-bations, the system always has at least one low lyingexcited state Ψ ex .For completeness we now explain why non-interactingtopological insulators have the property that an oddnumber of the 4 states Ψ (0 , , Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) havea Kramers degeneracy on the z = 0 and z = L z surfacesof the Corbino donut. This result was first established1 k x ε k Ψ (0,0) (a) k x ε k Ψ ( π ,0) (b)k x ε k Ψ (0, π ) (c) k x ε k Ψ ( π , π ) (d) FIG. 8. Surface state configurations ofΨ (0 , , Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) for a system with a singleDirac cone on each surface, and a chemical potential justabove the Dirac point. Here we plot the surface states as afunction of k x ; the bands indicate different discrete valuesof k y . The band that crosses the tip of the Dirac cone,corresponding to k y = 0, is singly degenerate; all otherbands, for which k y = 0, are doubly degenerate. The 4panels show the arrangement of the discrete surface momentarelative to the tip of the Dirac cone in each of the 4 fluxsectors. The filled circles indicate occupied states. in Ref. 9; here we give an alternative derivation usingthe fact that these systems have an odd number of Diraccones on each surface.For simplicity we assume a band structure with a single Dirac cone on each surface. We also assume that thechemical potential is tuned to lie just above the Diracpoint. We begin with the system at zero flux: ( φ x , φ y ) =(0 , L x and L y be the dimensions in the twoperiodic directions, the momenta k x , k y are quantized as: k x = 2 πm x L x k y = 2 πm y L y (103)where m x , m y are arbitrary integers.Let us focus on one of the two surfaces, say the z = 0surface. Suppose that the Dirac cone on that surface islocated at ( k x , k y ) = (0 , (0 , will be a Slater determinant consist-ing of all the momentum states with ǫ k ≤ µ – including both momentum states at ( k x , k y ) = (0 , x or y direc-tion, the momenta will shift by half a unit in k x or k y ,leading to the states Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) shown in Fig.8b-d. Examining these configurations, we can see that allthree have a single filled state in the upper band, imply-ing that all three of them have a Kramers degeneracy onthe z = 0 surface (these states must also have a Kramersdegeneracy on the z = L z surface since the total numberof electrons is even).On the other hand, if the Dirac cone is located at adifferent point in the Brillouin zone, say ( k x , k y ) = ( π, π ), then depending on the parity of L x , L y , the momentumstates near the Dirac point at flux ( φ x , φ y ) = (0 , (0 , will be a Slater determinant involvingall states in the lower band. In these three cases, onecan check that when one inserts flux through the twoholes of the Corbino donut, exactly one of the resultingstates Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) has an unpaired momentumstate, and therefore a Kramers degeneracy on the L z = 0surface. Combining all of these cases, we conclude thatthe system always has a Kramers degeneracy in either 1or 3 of the four possible states. This is what we wantedto show.
2. Strong vs. weak topological insulators
The reader may have noticed that, in the flux inser-tion argument, we only used the fact that at least one ofthe states Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) had Kramers degenera-cies: we did not make use of the additional informationthat the number of such states was odd . In other words,the argument would have worked equally well if 2 of thesestates had degeneracies.We can understand the physical meaning of this obser-vation as follows. Recall that, in addition to the usual3D topological insulator, there is another interesting kindof 3D time reversal invariant band insulator known as a“weak topological insulator.” (In this terminology, theusual 3D topological insulator is known as a “strong topo-logical insulator.”) Weak topological insulators can bethought of as layered systems, where each layer is a 2Dtopological insulator. The reason that weak topologicalinsulators are relevant here is that these systems give ex-amples where 2 of the three states Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) have Kramers degeneracies (assuming, as before, thatΨ (0 , is not Kramers degenerate). More precisely, thispattern of degeneracies must arise in a weak topologicalinsulator with an odd number of layers, with the layersoriented perpendicular to the z = 0 , L z surfaces.This example clarifies why the flux insertion ar-gument extends to systems where 2 of the statesΨ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) have Kramers degeneracies: just likestrong topological insulators, these odd-layer weak topo-logical insulators have protected surface modes. One wayto understand these surface modes is to note that each2D topological insulator layer contributes a one dimen-sional edge mode at the boundary. These edge modes canbe gapped out in pairs by appropriate perturbations, butif there are an odd number of layers, we will always beleft with at least one gapless mode.At the same time, this example reveals a shortcom-ing of the flux insertion argument: the argument doesnot distinguish strong topological insulators from weaktopological insulators with an odd number of layers, asboth systems have protected surface modes. Yet it isclear intuitively that the surfaces of these two systemsare topologically distinct: the surface modes of a strong2topological insulator are in some sense “more two dimen-sional” than those of a weak topological insulator withan odd number of layers.In the following, we refine the flux insertion ar-gument so that it addresses this issue. We usethe additional information that an odd number ofΨ (0 , , Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) have Kramers degeneracies toshow that our system not only has a low lying surfacemode, but this mode is delocalized in both the x and y directions. To be precise, we prove that the low lyingsurface modes in our system can never be localized toone dimensional strips. This property distinguishes oursystem from a weak topological insulator with an oddnumber of layers since in that case a carefully designedperturbation can gap out all low lying excitations ex-cept a single one dimensional edge mode. (As an aside,we note that the difference between the surface modes ofweak and strong topological insulators largely disappearsif one considers the surfaces in the presence of a random potential ).We give a proof by contradiction. Suppose that anappropriate perturbation could localize all low-lying sur-face modes to a single one dimensional strip. Let W x , W y denote the number of times this strip winds around the(toroidal) surface in the x and y directions respectively.Then, when we insert flux φ x , φ y = 0 , π through thetwo holes of the Corbino donut, the low lying excita-tions will only couple to φ x , φ y via the linear combination φ strip = W x φ x + W y φ y . In particular, it follows that thepresence or absence of a Kramers degeneracy at Ψ ( φ x ,φ y ) only depends on φ x , φ y via φ strip = W x φ x + W y φ y . Butthis kind of φ x , φ y dependence always leads to an even number of Ψ (0 , , Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) having Kramersdegeneracies, as we now show.It is useful to consider the cases where W x , W y areeven and odd separately. First suppose that W x is evenand W y is arbitrary. In this case, the effective flux φ strip takes the same value at ( φ x , φ y ) = (0 , , ( π,
0) andsimilarly at ( φ x , φ y ) = (0 , π ) , ( π, π ). Therefore, since φ strip completely determines whether the system has aKramers degeneracy, it follows that either Ψ (0 , , Ψ ( π, both have Kramers degeneracies or they both have nodegeneracy, and similarly for Ψ (0 ,π ) , Ψ ( π,π ) . But thenthe Kramers degenerate states come in pairs, implyingthat an even number of the four states have Kramersdegeneracies. The same is true if W y is even and W x is arbitrary. The only remaining possibility is if W x , W y are both odd. In this case, φ strip takes the same valueat ( φ x , φ y ) = (0 , , ( π, π ) and similarly at ( φ x , φ y ) =(0 , π ) , ( π, (0 , , Ψ ( π, , Ψ (0 ,π ) , Ψ ( π,π ) haveKramers degeneracies must have excitations which aretruly two dimensional and are not localized to a one di-mensional strip.
3. Fractionalized case
The argument for the 3D fractionalized insulator H (91) is completely analogous. For the same reason asin 2D, we insert N Φ / / N = e/e ∗ . Inserting flux through each of thetwo holes in the Corbino donut, we construct four states,Ψ (0 , , Ψ ( Nπ, , Ψ (0 ,Nπ ) , Ψ ( Nπ,Nπ ) . Since the low energyphysics is described by a topological insulator built outof charge q f fermions, this flux insertion is equivalentto inserting q f e ∗ · Φ flux for a non-interacting topologi-cal insulator. If q f /e ∗ is odd, then an odd number ofΨ (0 , , Ψ ( Nπ, , Ψ (0 ,Nπ ) , Ψ ( Nπ,Nπ ) have Kramers degen-eracies. We can then construct a protected low lyingstate Ψ ex at zero flux, by starting in one of the Kramersdegenerate states and adiabatically reducing the flux to0. Moreover, this state is in the same topological sectoras the ground state, just as in the 2D case. This is whatwe wanted to show.Finally, we note that we can refine the flux insertion ar-gument just as in the non-interacting case. To be specific,by repeating the analysis in the previous section, we canprove that if q f /e ∗ is odd, the fractionalized insulators notonly have low lying surface modes, but these modes aretruly two dimensional and cannot be localized to a onedimensional strip. Just as in the non-interacting case,this allows us to distinguish these systems from “weakfractional topological insulators” with an odd number oflayers. C. Microscopic theory of 2D edge
While the flux insertion argument shows that the edgemodes are protected for the models (37) with odd q f /e ∗ , itdoes not tell us anything about the converse statement—that is, whether the modes can be gapped out in themodels with even q f /e ∗ . In order to establish this fact wenow analyze the microscopic theory of the edge.
1. Constructing the 2D edge theory
The first step is to construct the edge theory. The freefermion theory (59) is unfortunately not a good startingpoint since it is not truly a complete edge theory. Thereason it is not complete is that it doesn’t include all thequasiparticle excitations with nontrivial statistics, suchas flux quasiparticles. As a result, this theory cannot beused to analyze the most general edge perturbations—for example, those that are large compared to the fluxquasiparticle gap.A simple way to construct a complete edge theory isto use the general bulk-edge correspondence for abelianChern-Simons theory.
According to this correspon-dence, the edge of H (37) is described by a four compo-3nent chiral boson theory L = 14 π ∂ x Φ I ( K IJ ∂ t Φ J − V IJ ∂ x Φ J )+ 12 π t I ǫ µν ∂ µ Φ I A ν (104)Here Φ is a four component vector of fields, V is the ve-locity matrix and K and t are defined as in (79). In thislanguage, the most general quasiparticle creation opera-tors can be written as e il T Φ , where l an integer valuedfour component vector. The charge carried by e il T Φ is l T K − t . The subset of these operators that are “local”(i.e. products of electron creation and annihilation op-erators) can be written as e i Θ(Λ) where Θ(Λ) ≡ Λ T K Φand Λ is an integer valued four component vector.The final component of the edge theory is the phys-ical interpretation of the e i Θ(Λ) operators in terms ofthe microscopic model H . Using our understanding ofthe corresponding bulk Chern-Simons theory, we iden-tify Λ T = (0 , , , , (0 , , ,
1) with creation operatorsfor spin-up and spin-down electrons, respectively, andΛ T = (0 , , ,
0) with a creation operator for the charge2 e spinless boson. As for (1 , , , m flux quasiparticles. The most general e i Θ(Λ) is a com-posite of these elementary operators.These identifications have the added benefit of fixingthe transformation properties of Φ under time reversal. Ifwe require that the electron creation operators transformas T : c †↑ → c †↓ , c †↓ → − c †↑ (105)and that the spinless charge 2 e boson is invariant undertime reversal, while the flux changes sign, we deduce thatΦ transforms as Φ → T Φ + πK − χ (106)where T = − ; χ = (107)Equivalently, Θ(Λ) transforms asΘ(Λ) → Θ( − T Λ) − Q (Λ) · π (108)where Q (Λ) ≡ Λ T χ . To complete the story, we needto explain the relationship between the above edge the-ory (104) and the edge (59) of the exactly soluble model H (37). An important clue is that (104) has four gap-less modes while the free fermion edge (59) has only twomodes. This mode counting suggests that (59) can be ob-tained from (104) by adding a perturbation that gaps outtwo of the modes. In appendix C, we confirm this guess. We show that a time reversal invariant perturbation ofthe form U (Λ) = U ( x )[cos(Θ(Λ) − α ( x ))+ ( − Q (Λ) cos(Θ( T Λ) − α ( x ))] (109)with Λ T = (1 , , ,
0) does the job. That is, the pertur-bation U (Λ ) gaps out two of the edge modes of (104)leaving behind exactly the free fermion edge (59).Turning this statement around, if we start with theexactly soluble model (37), we must be able to find aperturbation that closes the gap to the bosonic excita-tions at the edge, and results in the four component edgetheory (104). At a microscopic level, the following per-turbation accomplishes this task:∆ H edge = − J X h ss ′ i∈ ∂X ( U ss ′ + h.c ) (110)Here, the sum runs over links on the boundary ∂X ofthe exactly soluble system. Physically, this term givesan amplitude for the bosonic charge excitations to hop.When J ≪ V , the bosonic charge excitations are gappedin both the bulk and the edge, and the only low energyedge excitations are the two chiral fermion modes in (59).However, when J is sufficiently large, the gap to thebosonic charge excitations closes at the edge, resultingin a 1D superfluid. The resulting edge has four gaplessmodes and is described by (104).
2. Analysis of edge mode stability
In the previous section we derived an edge theory (104)for H (37), and showed it could be obtained from the ex-actly soluble edge by adding an appropriate perturbation.We now investigate whether this edge theory (104) can begapped out completely by charge conserving, time rever-sal symmetric perturbations. We find that the edge canbe gapped out if and only if q f /e ∗ is even, in agreementwith the flux insertion argument.Our analysis closely follows that of Ref. 10. We focuson scattering terms of the form (109). These perturba-tions can be divided into two different classes: perturba-tions where Λ , T Λ are linearly independent, and pertur-bations where T Λ = ± Λ. Perturbations of the first typecan gap out four edge modes, while perturbations of thesecond type can gap out two edge modes. Therefore, inorder to fully gap out the edge (104), we either need oneperturbation of the first type or two perturbations of thesecond type. Here, we will focus on the second possibility(though we would obtain the same results if we consid-ered the first kind of perturbation instead). That is, wewill look for perturbations of the form U (Λ ) + U (Λ ) (111)where T Λ = ± Λ and similarly for Λ .To begin, we note that the most general charge con-serving solution to T Λ = Λ is Λ T = (0 , x, − x, − x ),4while the most general solution to T Λ = − Λ is Λ T =( y, , z, − z ). Therefore, in order to get two linearly inde-pendent Λ’s, we can either takeΛ T = ( y , , z , − z ) , Λ T = ( y , , z , − z ) (112)or we can takeΛ T = (0 , x, − x, − x ) , Λ T = ( y, , z, − z ) (113)In the first case (112), the corresponding perturbation(111) can indeed gap out the edge, but hand in hand withthat it spontaneously breaks time reversal symmetry, aswe now explain. We note that when (111) gaps out theedge, it freezes the value of Θ(Λ ) , Θ(Λ ). It thereforealso freezes the value of Θ( y Λ − y Λ ), which is a mul-tiple of Θ(0 , , , − , , , − − α ) is oddunder time reversal (this follows from (108) or alterna-tively from the fact that cos(Θ(0 , , , − − α ) describesa Zeeman field oriented in the xy plane) so we concludethat this perturbation spontaneously breaks time rever-sal symmetry.Therefore, if we want to gap out the edge withoutbreaking time reversal symmetry, we are led to pertur-bations of the form (113). According to Haldane’s nullvector criterion such a perturbation can gap out theedge if Λ , Λ satisfyΛ T K Λ = Λ T K Λ = Λ T K Λ = 0 (114)(The origin of the criterion (114) is that this conditionguarantees that we can make a linear change of variablesfrom Φ to Φ ′ such that (a) the action for Φ ′ consists oftwo decoupled non-chiral Luttinger liquids, and (b) thetwo perturbations correspond to backscattering terms forthe two liquids. See appendix C for a related example).For the above Λ , Λ , this leads to the condition(2 k + m ) y − z = 0 (115)It is convenient to parameterize y, z by y = us , z = vs where u, v have no common factors. Then the abovecondition becomes(2 k + m ) u − v = 0 (116)We now consider two cases: 2 k + m divisible by 4, and2 k + m not divisible by 4. If 2 k + m is not divisible by4 then we must have v odd (since even v implies even u , which contradicts the fact that u, v have no commonfactors). It then follows that the perturbation U (Λ )spontaneously breaks time reversal symmetry: the per-turbation U (Λ ) freezes the values of Θ(Λ ) which is amultiple of Θ( u, , v, − v ). But cos(Θ( u, , v, − v ) − α ) isodd under time reversal symmetry (since v is odd), sothis perturbation must spontaneously break time rever-sal symmetry. We conclude that when 2 k + m is notdivisible by 4, we cannot gap out the edge using theseperturbations without breaking time reversal symmetry.We note that this agrees with the flux insertion argument since q f /e ∗ is odd precisely when 2 k + m is not divisibleby 4.On the other hand, when 2 k + m is a multiple of 4 (orequivalently, q f /e ∗ is even) the above condition suggestsa natural solution for (Λ , Λ ): we can take u = 1, v = k + m/ s = 1, x = 1, so thatΛ T = (0 , , − , − T = (1 , , k + m/ , − k − m/
2) (117)The physical meaning of the corresponding perturbationsis clear: U (Λ ) describes a process where a charge 2 e boson breaks into two electrons with opposite spin di-rections, while U (Λ ) describes a process which flips thespins of k + m/ m flux quasiparticles.To see that the perturbation U (Λ ) + U (Λ ) can gapout the edge, we note that the above Λ , Λ obey Hal-dane’s criterion (114). Moreover, this perturbation istime reversal invariant as long as α = π/ π/ a, b with no common factors, thelinear combination a Λ + b Λ , is non-primitive – that is, a Λ + b Λ = n Λ, where Λ is an integer vector, and n is an integer larger than 1. (To understand where thiscondition comes from, see the example of spontaneoussymmetry breaking given in the discussion after (113)).But it is clear from the form of Λ , Λ that all such linearcombinations a Λ + b Λ are primitive. We conclude that U (Λ ) + U (Λ ) gaps out the edge without breaking timereversal symmetry explicitly or spontaneously. VIII. CONCLUSION
In this work, we have constructed a family of exactlysoluble models for fractionalized, time reversal invariantinsulators in 2 and 3 dimensions. At low energies, thesemodels behave like topological insulators made of frac-tionally charged fermions of charge q f . As a result, the2D models have two edge modes of opposite chiralities,while the 3D models have a gapless Dirac cone on eachsurface. At high energies, the insulators possess othertypes of excitations, including quasiparticles with mini-mal charge e ∗ . As expected for systems with fractionalcharge, all of the 2D and 3D insulators are topologicallyordered, exhibiting fractional statistics as well as groundstate degeneracy in geometries with periodic boundaryconditions.An important characteristic of the 3D models is thatthey exhibit a fractional magnetoelectric effect. Morespecifically, if time reversal symmetry is broken at thesurface, the 3D models exhibit a surface Hall effect withfractional Hall conductivity σ xy = q / h . In addition, ifa magnetic monopole is inserted into the bulk, it bindsa fractional electric charge. Somewhat surprisingly, the5size of the elementary charge e ∗ plays a significant rolein this charge binding physics.A key question is whether the gapless boundary modesare robust to perturbations, so that these systems trulyqualify as “fractional topological insulators.” We havestudied this question in detail and have shown that theboundary modes are protected for the models where theratio q f /e ∗ is odd. That is, the edge or surface modes can-not be gapped out by perturbations that are time reversalsymmetric and charge conserving. In fact, these modesare immune even to large perturbations, as long as theperturbations do not close the bulk gap or spontaneouslybreak time reversal or charge conservation symmetry atthe boundary.In contrast, the gapless boundary modes are not nec-essarily stable when q f /e ∗ is even. In the 2D case, wehave demonstrated this point explicitly by constructingtime reversal invariant, charge conserving perturbationsthat open a gap at the edge. The situation in 3D is lessclear: though our argument for proving the existence ofprotected surface modes breaks down in the models witheven q f /e ∗ , we have not explicitly constructed perturba-tions that gap out the surface. Determining the robust-ness of the surface modes in these models is an interestingquestion for future research.Our findings regarding the stability of the edge modesare intriguing for several reasons. First, we have demon-strated that simply having fractionally charged fermionsin the right band structure does not guarantee a systemis a fractional topological insulator. Second, although e ∗ is a high energy property, it evidently determines the fateof the low energy excitations on the edge. And third, itis interesting that the condition that guarantees stabilitythe boundary modes turns out to be identical for boththe two and three dimensional models considered hereand those considered in Ref. 10.We have focused here on a particular set of models,which does not exhaust all the possibilities for fractionaltopological insulators. In two dimensions, the fractionalquantum spin Hall systems discussed by Refs. 3 and 10give a whole other class of examples. Similarly, in threedimensions, there may be other families of 3D fractionaltopological insulators beyond the ones discussed here.Constructing microscopic models of the other families,and finding a complete classification of the possibilities,remain tantalizing open questions. ACKNOWLEDGMENTS
ML was supported in part by an Alfred P. Sloan Re-search Fellowship. AS thanks the US-Israel BinationalScience Foundation, the Minerva foundation and Mi-crosoft Station Q for financial support.
Appendix A: Eigenstate degeneracy of 2D latticeboson model
In this section, we find the degeneracy of the | q s , b P i eigenstates of the 2D lattice boson model (1). We con-sider two geometries: a rectangular piece of square latticewith open boundary conditions (Fig. 2(a)), and a rect-angular piece of square lattice with periodic boundaryconditions—that is, a torus (Fig. 2(b)).
1. Open boundary condition geometry
The first step is to define projectors P q s , P b P , whichproject onto states with Q s = q s and B P = b P respec-tively. The degeneracy D of the q s , b P eigenspace canthen be written as the trace of the product of all theprojectors: D = T r Y s P q s Y P P b P ! (A1)We next write down an explicit expression for Q P P b P .To this end, we recall that the eigenvalues of B P are p throots of unity, so the projector can be written as P b P = 1 m m − X j =0 ¯ b jP B jP (A2)Expanding out the product, we have Y P P b P = Y P m m − X j =0 ¯ b jP B jP = 1 m N plaq X { j P } Y P ¯ b j P P Y P B j P P (A3)where N plaq is the total number of plaquettes. Using thedefinition B P = U U U U , we get an expression ofthe form Y P P b P = 1 m N plaq X { j P } Y P ¯ b j P P Y h ss ′ i U ∆ j ss ′ ss ′ (A4)Here, ∆ j ss ′ = j P − j P ′ where P, P ′ are the two plaquettesbordering the link h ss ′ i . If h ss ′ i happens to live on theboundary of the system, then ∆ j ss ′ = j P where P is theunique plaquette bordering this link. Substituting thisinto (A1) gives D = 1 m N plaq X { j P } Y P ¯ b j P P · T r Y s P q s Y h ss ′ i U ∆ j ss ′ ss ′ (A5)The next step is to note that the operator U rss ′ changesthe boson number n ss ′ by ± r (mod m). We concludethat the trace T r (cid:16)Q s P q s Q h ss ′ i U ∆ j ss ′ ss ′ (cid:17) vanishes unless6∆ j ss ′ ≡ j P = 0everywhere (since we are assuming a geometry with openboundary conditions). In this way, we see that the onlynonvanishing term in (A5) is the one with j P = 0, so that D = 1 m N plaq · T r Y s P q s ! (A6)All that remains is to compute this trace, or equivalently,count the number of states with Q s = q s . Working in thenumber basis, | n s , n ss ′ i , and using the definition (2) of Q s it is clear that there is a one-to-one correspondencebetween states with Q s = q s and configurations of n ss ′ satisfying α s X s ′ n ss ′ ≡ q s (mod m) (A7)We can count these configurations by comparing thenumber of constraints (A7) to the number of free param-eters. Examining (A7), we see that we have a constraintfor every site s of the lattice. However, these constraintsare not all independent, since if we add all the constraintequations together, we get X s q s ≡ X s α s X s ′ n ss ′ ≡ P s q s ≡ N constr = N site − N param = N link (A10)Combining these two calculations, we see that the totalnumber of configurations of n ss ′ satisfying (A7) is givenby N config = m N param − N const = m N link − N site +1 (A11)We now substitute this into our formula (A6) for thedegeneracy, obtaining D = 1 m N plaq N config = m N link − N site − N plaq +1 (A12)We then note that N link − N site − N plaq = − V − E + F = χ . We conclude that D = 1: there isa unique eigenstate | q s , b P i for each configuration satis-fying P s q s ≡
2. Periodic (torus) geometry
The calculation in the torus geometry proceeds sim-ilarly to the open boundary condition case discussedabove. The expression (A5) for the degeneracy is stillapplicable in this case, and we can still deduce that theonly nonvanishing terms in this sum are the ones where∆ j ss ′ ≡ m termssatisfying ∆ j ss ′ ≡ j P is constant. Therefore, in the torus geometry case,(A5) reduces to D = 1 m N plaq X j Y P ¯ b jP · T r Y s P q s ! = ( m N plaq − · T r ( Q s P q s ) if Q P b P = 10 otherwise (A14)Just as in the open boundary condition case, we can com-pute T r ( Q s P q s ) by counting the number of configura-tions n ss ′ satisfying (A7), and again this number is givenby (A11). Substituting (A11) into (A14), and specializ-ing to the case Q P b P = 1, we find D = 1 m N plaq N config = m N link − N site − N plaq +2 (A15)At this stage, there is again a difference from the openboundary condition case. Instead of (A13), we have themodified relation N link − N site − N plaq = 0 (A16)(again, this can be derived directly or via Euler’s formula V − E + F = χ ). We conclude that there are D = m de-generate eigenstates | q s , b P i for every configuration satis-fying Q P b P = 1 and P s q s ≡ | q s = 0 , b P = 1 i is m -fold degenerate. Appendix B: Eigenstate degeneracy of 3D latticeboson model1. Open boundary condition geometry
We use the equation (A5) in an almost unchangedform, the only difference being that the quantity ∆ j ss ′ isnow defined as j P − j P + j P − j P , where P i are the fourplaquettes meeting at the edge h ss ′ i and the labeling isclockwise as we go around the link (i.e. there is a relativeminus sign between plaquettes which are perpendicularto each other. This is a different convention from the 2Dcase.). The only nonvanishing matrix elements are thosefor which: ∆ j ss ′ ≡ { j P } that satisfy those conditions.7To count the allowed configurations of { j P } , we will con-struct an explicit mapping from choices of { j P } satisfyingEq. (B1) to configurations of Z m spins living at the cen-ters of the cubes on the lattice (i.e. Z m spins on the duallattice). One can picture j P = 0 as the absence of a do-main wall between the cubes; more generally the valueof j P assigns a label to the face separating any pair ofcubes, which we will later identify with the change in the Z m spin between these cubes. The conditions (B1) en-sure that domain walls can never end on an edge – andfurther, that the net change in the spin along any closedcurve is a multiple of m , so that an allowed configurationof { j P } does indeed specify a set of domain walls between Z m spins on adjacent cubes.The precise mapping is as follows. Let us focus onsome link h ss ′ i incident at the corner of the sample –there are only two plaquettes P, P ′ bordering that linkand ∆ j ss ′ = j P − j P ′ , so in order for it to contributeto the trace we need j P = j P ′ which can assume m val-ues. We place the corresponding value of j C = j P in thecorner cube of the lattice. We assign a value to all othercubes sequentially using the following algorithm: start inthe corner cube and construct a closed directed loop go-ing through a number of other cubes. If C, C ′ are cubesin the loop separated by a plaquette P and the value j C is already assigned, then j C ′ ≡ j C ± j P (mod m). Weadopt the convention that we always pick up a positivesign as we leave the corner cube and later it alternatesas shown in figure (9). The consistency is guaranteedby the fact that as we go around a single link h ss ′ i andreturn to the cube C we pick up a contribution of pre-cisely ± ∆ j ss ′ ≡ C we can decompose it into anumber of small loops around individual links enclosed bythe big loop, all of which contribute 0 (mod m), whichcan also be seen in figure (9). Thus the assignment ofspins is consistent. Once the corner cube has been as-signed a value the mapping is unique. Conversely, givena configuration of { j C } in the cubes we can reconstructthe corresponding assignment of { j P } . Therefore thereis a bijection between the plaquette configurations andthe cube configurations – of which there are m N cube . Weconclude that there are m N cube terms contributing to thesum in equation (A5).For each plaquette configuration { j P } , we must nowevaluate the product1 m N plaq Y P b j P P T r Y s P q s ! . (B2)The evaluation of the trace is identical to the 2D case,and again gives m N link − N site +1 . The factor Q P b j P P isalways equal to 1 – as long as Q P ∈ C b P = 1 for each cube C . (Using this relation, we can simultaneously reduce thelabels on all of the faces of a given cube by any integer j (mod m ). This preserves the condition (B1) everywhere.Since the allowed configurations constitute closed domainwalls, a series of these reductions can be used to reduce all FIG. 9. An example of mapping to spin configuration for m =3. The numbers in red correspond to a choice of { j P } . Thebase of the loop is the lower left corner, which for concretenesshas been assigned a value 0. The sign with which we pick upthe appropriate j P (or equivalently with which the plaquettecontributes to ∆ j ss ′ for a specific link h ss ′ i enclosed by theloop) has been coded in blue for + and green for − . Theinvolved links have been colored red. of the labels to 0, which proves that Q P ¯ b j P P = Q P ¯ b P = 1for the allowed configurations { j P } ). Thus: D = m N cube m N plaq · T r Y s P q s ! = m N cube − N plaq + N link − N site +1 (B3)The exponent can be evaluated by explicit counting orusing the generalized Euler formula: N cube − N plaq + N link − N site + 1 = 0 . (B4)Thus we have shown that D = 1, provided the constraint Q P ∈ C b P = 1 is satisfied for all cubes.
2. Periodic (torus) geometry
Next, we repeat the above counting argument for asystem with periodic boundary conditions in the x , y ,and z directions. It is instructive to start with the openboundary case again. We may divide the configura-tions { j P } satisfying (B1) into bulk and boundary parts: { j P } P ∈ boundary and { j P } P ∈ bulk . Assume we have fixedsome bulk configuration. In the open geometry, fixingone of the boundary plaquettes (say, in the corner) toone of the m possible values automatically fixes all otherboundary plaquettes. This is because the links h ss ′ i onthe boundary always have only two incident boundaryplaquettes (and possibly a bulk one, but it is alreadyfixed). Hence, the total number of configurations is m times the number of bulk configurations.Let us now define the periodic geometry by identifyingopposite faces of the boundary. This allows us to inheritthe notion of bulk and boundary configurations from theopen case. Note that the two cases do not differ at allin the bulk – the allowed configurations satisfying (B1)are the same. However, at the boundary there are morepossible configurations if we impose periodic boundaryconditions. Let us again fix a bulk configuration. In8the periodic case choosing a value for one of the bound-ary plaquettes only fixes the value of the plaquettes inthe same boundary plane. The two other perpendicularplanes still remain unfixed as each boundary link in thecorner has four incident boundary plaquettes. Indepen-dently choosing a value for one plaquette in the remainingplanes produces a factor of m . Hence the total numberof configurations in periodic case is m times the numberof bulk configurations or m times the total number ofconfigurations in the open case. Thus we have m N cube +2 configurations. Note also, that in the periodic case thereis an additional algebraic constraint satisfied by opera-tors B P : if we take any boundary plane i (which is a 2Dtorus embedded in the 3D one) then: Y P ∈ plane ( i ) B P = 1 . (B5)The rest of the calculation is unchanged: D = m N cube +2 − N plaq + N link − N site +1 = m , (B6)since in the periodic case we have N cube − N plaq + N link − N site = 0 either by explicit counting or using the Eulerformula. Thus there is a D = m degeneracy, providedthe constraints (86) and (B5) are satisfied. In particular,the ground state | q s = 0 , b P = 1 i has an m degeneracy. Appendix C: Relationship between the bosonic edgetheory (104) and the free fermion edge theory (59)
In this section, we show that the perturbation U (Λ )(109) where Λ = (1 , , ,
0) can gap out two of the edgemodes of (104), leaving behind exactly the free fermionedge (59). We accomplish this via a change of variables,Φ = W ˜Φ , W = /m k/m k/m (C1) Substituting these expressions into the edge action (104),we find L = 14 π ∂ x ˜Φ I ( ˜ K IJ ∂ t ˜Φ J − ˜ V IJ ∂ x ˜Φ J )+ 12 π ˜ t I ǫ µν ∂ µ ˜Φ I A ν (C2)where˜ K IJ = − , ˜ t I = e/m (2 k + m ) e/m (2 k + m ) e/m (C3)and ˜ V = W T V W . In these variables, the perturbationbecomes U (Λ ) = 2 U ( x ) cos( ˜Φ − α ( x )) (C4)We next assume that that the interactions on the edgeare tuned so that ˜ V = vδ IJ (we can make this assumptionwithout any loss of generality since the velocity matrixis non-universal and can be modified by appropriate per-turbations at the edge). In this case, the edge theorycan be written as a sum of two decoupled actions, oneinvolving ˜Φ , ˜Φ and one involving ˜Φ , ˜Φ : L = L + L (C5)where L = 14 π (cid:16) ∂ x ˜Φ ∂ t ˜Φ − v ( ∂ x ˜Φ ) − v ( ∂ x ˜Φ ) (cid:17) + emπ ǫ µν ∂ µ ˜Φ A ν (C6)and L = 14 π ∂ x ˜Φ ( ∂ t ˜Φ − v∂ x ˜Φ ) − π ∂ x ˜Φ ( ∂ t ˜Φ + v∂ x ˜Φ )+ (2 k + m ) e mπ ǫ µν ∂ µ ( ˜Φ + ˜Φ ) A ν (C7)It is now easy to analyze the effect of the perturbation U (Λ ) (C4): this term gaps out the non-chiral Luttingerliquid described by L by freezing the value of ˜Φ (atleast if U ( x ) is large). The resulting edge then has onlytwo gapless modes, and is described by L . On theother hand, it is easy to check that L is nothing butthe bosonized description of the free fermion edge theory(59), with e i ˜Φ , e − i ˜Φ corresponding to the fermion cre-ation operators d †↑ , d †↓ respectively. We conclude thatthe perturbation U (Λ ) does indeed gap out two of themodes of (104) leaving the free fermion edge (59). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005). B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. ,106802 (2006). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010). R. Roy, Phys. Rev. B , 195322 (2009). L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. ,106803 (2007). J. E. Moore and L. Balents, Phys. Rev. B , 121306(2007). L. Fu and C. L. Kane, Phys. Rev. B , 195312 (2006). L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007). M. Levin and A. Stern, Phys. Rev. Lett. , 196803(2009). M. Freedman, C. Nayak, K. Shtengel, K. Walker, andZ. Wang, Ann. Phys. , 428 (2004). B. Swingle, M. Barkeshli, J. McGreevy, and T. Senthil,Phys. Rev. B , 195139 (2011). J. Maciejko, X.-L. Qi, A. Karch, and S.-C. Zhang,Phys. Rev. Lett. , 246809 (2010). A. Y. Kitaev, Annals of Physics , 2 (2003). O. I. Motrunich and T. Senthil, Phys. Rev. Lett. ,277004 (2002). T. Senthil and O. Motrunich, Phys. Rev. B , 205104(2002). N. Read and B. Chakraborty, Phys. Rev. B , 7133(1989). R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B , 024504 (2001). C. Nayak and K. Shtengel, Phys. Rev. B , 064422(2001). X.-G. Wen, Adv. Phys. , 405 (1995). X.-G. Wen,
Quantum Field Theory of Many-body Systems:From the Origin of Sound to an Origin of Light and Elec- trons (Oxford University Press, 2007). T. Einarsson, Phys. Rev. Lett. , 1995 (1990). E. Witten, Physics Letters B , 283 (1979), ISSN 0370-2693. F. Wilczek, Phys. Rev. Lett. , 1799 (1987). G. Rosenberg and M. Franz, Phys. Rev. B , 035105(2010). B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science , 1757 (2006). J. B. Kogut, Rev. Mod. Phys. , 659 (1979). Y. Gefen and D. J. Thouless, Phys. Rev. B , 10423(1993). M. Levin and X.-G. Wen, Phys. Rev. B , 245316 (2003). S.-P. Kou, M. Levin, and X.-G. Wen, Phys. Rev. B ,155134 (2008). G. Y. Cho and J. E. Moore, arXiv:1011.3485(2010). T. H. Hansson, V. Oganesyan, and S. L. Sondhi, Annals ofPhysics , 497 (2004). X. G. Wen and A. Zee, Phys. Rev. B , 2290 (1992). X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B ,195424 (2008). A. M. Essin, A. M. Turner, J. E. Moore, and D. Vanderbilt,Phys. Rev. B , 205104 (2010). A. Hamma, P. Zanardi, and X.-G. Wen, Phys. Rev. B ,035307 (2005). M. Levin and A. Stern, in preparation. Z. Ringel, Y. E. Kraus, and A. Stern,arxiv:1105.4351(2011). F. Haldane, Phys. Rev. Lett.74