Delocalization of boundary states in disordered topological insulators

Transcription

1 Journal of Physics A: Mathematical an Theoretical J. Phys. A: Math. Theor. 48 (05) FT0 (pp) oi:0.088/75-83/48//ft0 Fast Track Communication Delocalization of bounary states in isorere topological insulators Anrew M Essin an Victor Gurarie Institute for Quantum Information an Matter an Department of Physics, California Institute of Technology, Pasaena, CA 95, USA Department of Physics, CB390, University of Colorao, Bouler, CO 80309, USA victor.gurarie@colorao.eu Receive 7 October 04, revise 8 January 05 Accepte for publication 3 February 05 Publishe 3 February 05 Abstract We use the metho of bulk-bounary corresponence of topological invariants to show that isorere topological insulators have at least one elocalize state at their bounary at zero energy. Those insulators which o not have chiral (sublattice) symmetry have in aition the whole ban of elocalize states at their bounary, with the zero energy state lying in the mile of the ban. This result was previously conjecture base on the anticipate properties of the supersymmetric (or replicate) sigma moels with WZW-type terms, as well as verifie in some cases using numerical simulations an a variety of other arguments. Here we erive this result generally, in arbitrary number of imensions, an without relying on the escription in the language of sigma moels. Keywors: topological insulators, quenche isorer, Anerson localization. Introuction Topological insulators are non-interacting fermionic systems which are bulk insulators, have gapless excitations at their bounary, an which are characterize by topological invariants. As free fermion systems, they are escribe by the Hamiltonian Hˆ = αβ aˆ α aˆ β. () αβ Here α, β label points in space, spin an flavor of the fermions. A proceure evelope by Volovik in the 80s [] to relate the ege states of twoimensional (D) integer Hall systems to the bulk topological invariants in a irect way was recently generalize to all topological insulators []. In effect, that proceure showe that an ege of a topological insulator is a topological metal, characterize by its own topological 75-83/5/FT0+$ IOP Publishing Lt Printe in the UK

2 invariant whose value must be equal to the value of the bulk invariant. All of this was one in the absence of isorer. Here we show that a suitable moification of this formalism extens it to the case when the topological insulators are isorere. It immeiately follows from this formalism that the ege of topological insulators cannot be fully localize by isorer. Furthermore, it is well known that topological insulators can be split into those without chiral (sublattice) symmetry an with chiral symmetry [3]. It can then be shown that the ege of non-chiral isorere topological insulators are characterize by a ban of elocalize state spanning the energy interval between the elocalize bulk states. They are elocalize along the ege while exponentially ecaying into the bulk, as is expecte from the proper ege states. At the same time, the ege of chiral topological insulators has at least one elocalize state at zero energy while the rest of the states may be localize by isorer. These statements generally match what is expecte from the topological insulators from the stuy of sigma moels or by using other methos. In particular, the localization of the one-imensional ege of D topological insulators is very well unerstoo. There is no oubt, for example, that the ege of an integer Hall state is elocalize regarless of isorer, thanks to its chiral nature (absence of backscattering). Similar arguments can be mae in case of other D topological insulators. The three-imensional (3D) topological insulators were also stuie in recent years. Their D bounaries, in the presence of isorer, can be analyze using sigma moels. Those can belong to one of five symmetry classes. Of those, arguably the most important is the AII topological insulator, or the stanar strong time-reversal invariant topological insulators with spin orbit coupling [4, 5]. It has no sublattice symmetry. That insulator is known to have a fully elocalize ege, as confirme in a variety of stuies [6, 7]. This agrees with our claim that all non-chiral insulators have a ban of elocalize states at the bounary. The remaining four insulators in 3D are all chiral. Among them, the simplest is the insulator with sublattice symmetry only, known as AIII topological insulator. Its ege is escribe by D Dirac fermions with ranom gauge potential [8]. Surprisingly, it was shown only a few years ago that when the ranom gauge potential is zero on the average these insulators have either a fully elocalize ege or an ege with localize states with localization length which iverges as energy is taken to zero, epening on whether their topological invariant is even or o [9]. This results is obtaine by mapping the problem at finite energy into a Pruisken-type sigma moel with a topological term which correspons to exactly the point of the integer quantum Hall transition it escribes if the invariant is o integer, an to the localize quantum Hall plateau if the invariant is even integer. However, this result is not completely robust: aing a constant magnetic fiel to the ranom gauge potential shifts the coefficient of the Pruisken-type sigma moel, potentially localizing all states except those at zero energy, in all cases [0]. The next is the insulator in class DIII represente by a superflui 3 He in its phase B. By mapping it into a sigma moel, one fins that, similarly to the AIII case, this insulator has an ege which is either fully elocalize if the invariant is o (like for 3 He where it is exactly ) or with just one elocalize state at zero energy if the invariant is even. Technically this occurs because at finite energy an ege of such an insulator crosses over to the symmetry class AII which has a structure. The insulator in class CI (represente by an exotic spin-singlet superconuctor []) is known to have a fully localize ege except one state in the mile of the ban, since at finite energy its ege crosses over to class AI, time-reversal invariant spin rotation invariant systems which were known for a very long time to be localize in two-imensions [].

3 Finally, the topological insulator in class CII, the most exotic of the five topological insulators in three-imensions, has a fully localize ege except one state, as can be argue base on its mapping to the trivial (non-topological) AII insulator at finite energy. All of these examples match what follows from the arguments which are presente in this paper. However, we note that a variety of chiral insulators, at least in three-imensions, have a fully elocalize ege going beyon the preiction of at least one elocalize state given here. Whether our metho can be generalize to explain these aitional features is not known. On the other han, our metho works not only in two or three-imensions, but also for all topological insulators of arbitrarily large spacial imension where sigma moels might be ifficult to analyze. Finally we woul like to point out that there exists an alternative metho of stuying topological insulators with isorer, base on the non-commutative Chern number an its generalizations [3 7]. These methos provie yet another way to look at the bounary of topological insulators with isorer [3], which may well prove to be more powerful than the methos iscusse here.. Topological invariants of isorere insulators We start with non-chiral topological insulators in even-imensional space. When isorer is absent they are characterize by the topological invariant which can be constructe out of its Green s function G = [i ω ]. () Assuming translational invariance, the Green s function takes a form of a matrix Gab ( ω, k) which epens on the frequency ω an the -imensional lattice momentum k with inices a, b labeling bans as well as spin an flavor. The topological invariant is known to take the form N = C ϵ tr ω k G G G G. (3) α α α α 0 0 Here C is a constant which makes N an integer, known to be given by C ( )! = + (πi) ( + )!. (4) Each of the inices α 0, α,, α in equation (3) is actually implicitly summe over the values 0,,,, an a convenient shorthan notation is introuce α k, i α i where furthermore k, k,, k are the cartesian components of the -imensional vector k, with an aitional notation k 0 ω. This type of a summation will occur throughout the paper. To simplify notations, from now on to escribe it we will simply say that each of the inices α is being summe over ω, k,, k. Other notations in equation (3) inclue ϵ as the Levi Civita symbol an tr as the trace over the matrix inices of G ab. The integration over ω is taken from to, while the integration over k is over the first Brillouin zone of the lattice where equation () is efine. This invariant, if equation () is taken into account, is nothing but the Chern number of negative energy (fille) bans at =, secon Chern number at = 4, etc. 3

4 The fact that equation (3) is an integer value topological invariant relies on the existence of the homotopy group π + (GL(, )) =, even, (5) an on the fact that Green s functions given in equation () are not singular (have neither zero nor infinite eigenvalues; the latter follows from the system being an insulator an having no zero energy eigenvalues). The equivalence of the topological invariant equation (3) with the Chern number was first iscusse in [8]. More generally, the invariants of this form, inucing the precise coefficient given in equation (4) in front to make them integer, are introuce an iscusse in [, 3]. It is actually very straightforwar to check that equation (3) is an invariant as any perturbation of the form G G + δg keeps equation (3) unchange. It is a little more ifficult to check that it always prouces integer values [3]. Once the insulator is isorere, it is no longer translationally invariant an equation (3) loses any meaning. An alternative form for the topological invariant can be introuce in the following way. Following [8] (see also [9]), introuce the finite size system such that the wave function for each particle satisfies perioic bounary conitions with an aitional phases θ i, where i =,, labels irections in space. In other wors ψ ( i i) x + L = e i θ i ψ ( x) (6) for a particular coorinate x i, where L i is the size of the system in the ith irection. Then one introuces a Green s function G αβ ( θ) which is no longer Fourier transforme, but which epens on the angles θ. Here α, β label not only spin an flavor of the fermions but also the sites of the lattice. The topological invariant is given by essentially exactly the same formula, (3), but just interprete in a slightly ifferent way. The inices α 0,, α are now summe over ω, θ,, θ, an the symbol tr implies summation over all the matrix inices, while the integral is performe over ω θ N = C ϵ tr ω θ G G G G. (7) α α α α 0 0 Here the integration over each θ extens from π to π an the trace is over the matrix inices of G αβ. In case when there is no isorer, the invariant introuce in this way coincies with the invariant efine in terms of momenta. Inee, we can take avantage of translational invariance an reintrouce the momenta in equation (7). Due to the perioic bounary conitions with the phases, the momenta are restricte to the values (for each of the ( irections) ki n ) = ( πni + θi) Li with n i being integers. Integration over θ i an summation over n i together are now equivalent to an unrestricte integration over all the values of k, as in equation (3). Thus in the absence of isorer, equations (3) an (7) are equivalent. Yet unlike equation (3), the expression in terms of phases equation (7) can be use even in the presence of isorer when translational invariance is broken. If is o, then equation (3) is zero. Instea we follow [3] an consier insulators with chiral symmetry, such that there is a matrix Σ Σ Σ =. (8) As well known, only the insulators with this symmetry have invariants of the integer type in o spatial imensions. The invariant itself, without isorer, can be written as follows 4

5 Figure. A -imensional system with a omain wall separating two phases, one with a topological invariant N L at large negative coorinate s an one with a topological invariant N R at large positive coorinate s. The insulator is translationally invariant in the irection perpenicular to s so that irection can be spanne by the momenta k. N C = ϵα α Σ tr k V αv V α V. (9) Here α, α are summe over k to k each, an V = G =. (0) ω= 0 (We coul equally well use instea of V in the efinition of the topological invariant, but use V to smoothen the ifference between even an o cases.) Again, in case if there is isorer present (which preserves symmetry equation (8), we can rewrite the invariant in terms of phases N C = ϵαα α θ Σ tr V αv V αv. () Here α,, α are summe over values θ to θ each. Again, simple arguments can be given that these two expressions for N in case when there is no isorer coincie. Once the isorer is switche on, equation (9) loses any meaning, while equation () can still be use. We will not separately stuy the invariants of the type, as those can be obtaine by imensional reuction from the invariants of the type introuce above. This will be briefly iscusse at the en of the paper. 3. Bounary of topological insulators 3.. Bounary of a isorer-free topological insulator Following our prior work [], we woul like to consier a situation where a omain wall is present such that the topological insulator is characterize by one value of the topological invariant on the one sie of the omain wall an another value on another sie. We woul like to examine the nature of the ege states forming at the bounary. Let us first review the approach taken in [] in case when there is no isorer. It is possible to introuce the Green s function of the entire system G ( ω ; k,, k ; s, s ) ab. Here k,, k span the bounary separating two insulators, an s is the coorinate perpenicular to the bounary where the system is not translationally invariant, see figure. With its help it was furthermore possible to introuce the Wigner transforme Green s function 5

6 ω = r r Gab W i (, k, R) r e k r G ω; k,, k ; R + R, ab. () Here k now enotes all momenta k,, k. We can now introuce the concept of a Green s function on the far left of the bounary an the far right of the bounary R W G ( ω, k) = lim G ( ω, k, R), R GL ( ω, k) = lim GW ( ω, k, R). (3) R These can now be use to calculate the topological invariant on the right an on the left of the bounary, N R an N L respectively, accoring to equations (3) or (9), epening on whether is even or o, with G R an G L substitute for G. Since we are consiering a situation where the bounary separate two topologically istinct states, these two values are istinct, N R = N L. Furthermore, as it was iscusse in [], there is also a bounary topological invariant, which can be efine with the help of the original Green s function Gab ( ω ; k,, k; s, s ) by N = C ϵ nα0x, B α α α α α 0 Xαα α = Tr G αg α G. (4) Here we introuce convenient notations A B an Tr A, following [], where ( ) ab bc ( ) ( A B) s, s = s A ( s, s ) B s, s, ac b Tr A= s A ( s, s). (5) a aa The integral in equation (4) is over a -imensional surface in the -imensional space forme by ω, k,, k. α n 0 is a vector in this -imensional space normal to the surface which is being integrate over. This surface is close an surrouns the singularities of G in this space (those are present because the bounary is not an insulator an has gapless excitations; thus at ω = 0 the zero eigenvalues of, relate to G via equation (), make G singular). In fact, it is the presence of these singularities which makes equation (4) to be nonzero []; in their absence it can be shown that X is a ivergence-free vector an thus the bounary invariant vanishes. It was shown in [] that N R N L = N B. (6) (A much earlier work [] showe this for =.) If the space is of o-imensions, closely similar efinition of N B can be given, with equation (6) still vali. For completeness, let us give them here. We now have a system with a chiral symmetry, implying that ΣG( ωσ ) =G( ω). (7) 6

7 Figure. A -imensional system perioically repeate. Each rectangle represents a system with isorer. That isorer is ientical in every spatial replica of the system. However, a certain parameter in the Hamiltonian varies from replica to replica, as well as within replicas, along the irection of the iscrete coorinate s so that at some point the system has a bounary to a topologically istinct phase, inicate here by a think ashe line, whose irection is spanne by the phases θ to θ. Far from the bounary that parameter asymptotically reaches the two values corresponing to the left phase with the topological invariant N L an the right phase with the topological invariant N R. The bounary invariant can now be efine with Here C N = ϵ α B 3 α α n Xα α,,, Xα α = Tr ΣV α V V α V. (8) V( k, k,, k; s, s ) = G( ω, k,, k; s, s ). (9) Inices α to α are summe over k, k,, k. The integral is over a -imensional surface in the -imensional space forme by k,, k. The bulk invariants can still be R R compute using equation (9), with V = G substitute for, an similarly for ω= 0 VL. The bounary invariant N B can be useful in analyzing bounary theories of particular topological insulators. These bounary theories must have non-zero bounary invariants, which in case when the bounary is between a non-trivial insulator an an empty space (whose invariant is zero) must be equal, up to a sign, to the bulk invariant. 3.. Bounary with isorer We woul like to generalize N B to the case when isorer is present. It is clear that we nee to replace the momenta k,, k along the bounary with phases across the bounary θ,, θ. However, it is not immeiately clear what to o in the irection perpenicular to the bounary. In orer to eal with this irection, we use the following trick. Imposing phases across the system is tantamount to perioically replicating our system in space (with isorer being exactly the same in each replica), with the phase θ becoming equivalent to the usual crystalline quasi-momentum. Recall that in the absence of isorer we work with variable s conjugate to the momentum k. A variable conjugate to the quasi-momentum is the iscrete variable m = 0, ±, ±, which labels replicas of the system. Therefore, to maintain the continuity with the previous approach taken in the absence of isorer, we also perioically repeat the system in the irection perpenicular to the bounary. However, in that irection we shoul also ensure that the Hamiltonian changes close to the bounary, in such a way that once we move past the bounary the system goes into a ifferent topological class with a ifferent invariant. We can make the Hamiltonian change from a replica to a replica as well as within replicas (by varying some appropriate parameter in it) until the system goes through a transition somewhere within a particular 7 ω= 0

8 replica where the bounary between two topologically istinct states resies. Far away from the bounary on either sie of it the Hamiltonian no longer changes from replica to replica. It is even possible, if esire, to make this parameter change across just the single replica, an staying constant in the rest of the replicas (although it we shall see below, it is avantageous to work in the limit when the parameter changes very little across each replica an it takes many replicas for it to reach its asymptotic far from the bounary value). This is schematically shown in figure. Although this may appear to be a rather special kin of a setup with systems with perioically repeate isorer, it shoul be clear that this constitutes just a convenient approach which allows to probe the bounary, which really occurs within a particular replica of the system, as shown in figure. The rest of the replicate systems are there just for allowing the formalism presente here to escribe simultaneously the bounary an the insulator far away from the bounary. Assuming that there are only two phases in the replicate systems, the one in replicas on the left of the bounary, an the one in the replicas on the right of the bounary, there is only one bounary here which separates topologically istinct systems (the bounaries between replicas o not separate systems which are topologically istinct). Labeling the replicas of the system in the irection perpenicular to the bounary by the (iscrete) variable m, we can efine the Green s function of the replicate system as G ( ω; θ,, θ ; m, m ). Here θ i, i =,, phases are quasimomenta (or phases) across the -imensional bounary, an the remaining variables m, m label copies of the system in the irection perpenicular to the bounary. Given this Green s function we efine the Wigner transforme function by Fourier series (compare with equation ()) W isθ G ( ω; θ,, θ ; M) = e G ω; θ,, θ ; M + m M m,. (0) s Here M can be either integer or half-integer, an the summation over m goes over either even integers or o integers respectively. Given this function, we can again fin the far left an far right Green s function R W G ( ω; θ,, θ ) = lim G ( ω; θ,, θ, M), M GL ( ω; θ,, θ ) = lim GW ( ω; θ,, θ, M). () M These can be use to efine the left an right invariants accoring to equation (7) for even, with G LR, substitute for G, or accoring to equation () for o, with G LR, substitute ω= 0 for V. R Now the ifference N N L is again expecte to be equal to N B, the bounary invariant. This invariant is efine analogously with equations (4) an (8) for even an o respectively, with the momenta replace by the phases. If is even N = C ϵ nα0x, B α α α α α 0 Xα α = Tr G αg α G, () where the inices α i are summe over ω, an θ,, θ, an the integration over s (implicit in the efinitions of Tr an A B in equation (5)) is replace by summation over m. 8

9 Similarly for o we can use construct the bounary invariant by using C N = ϵ α B 3 α α n Xα α,,, Xα α = Tr ΣV α V V α V, (3) where V = G ω= 0, an α i are summe over θ,, θ. The integrals in equations () an (3) are taken over close surfaces in the -imensional (if even) or -imensional (if is o) space. The erivation of equation (6) for this isorere case is essentially the same as in the isorerless case presente in []. The only (minor) ifference is that here M an m are iscrete variables. The erivation of [] relies on graient expansion in powers of erivatives with respect to the variable M an it seems to be essential in this technique for M to be a continuous variable. In orer to work with M as if it were continuous variable we nee to ensure that our system changes little as M is increase by, or in other wors the parameter of the system whose change results in the topological phase transition changes just slightly as one goes from replica to replica. Therefore, at this stage of the erivation we have to assume that the system changes just slightly from replica to replica as one moves in the irection perpenicular to the bounary (in particular, a setup where the parameter controlling which phase the system is in changes only within one replica shoul not be use here). However, as always in the graient expansion of integer value topological invariants, small corrections to them coming from terms neglecte as M is mae a continuous variable must vanish to ensure that the result is still an integer. Similar issues are iscusse in a somewhat relate context in []. As a result, we expect that even as the variations in the parameters from replica to replica become larger, the approximation of continuous M remains exact. Thus we come to the conclusion that when properly efine equation (6) hols also in the presence of isorer Analysis of the bounary invariants in the presence of isorer We woul now like to analyze the bounary topological invariants. Take the invariant from the equation () (applicable when is even). The integral in equation () is compute over a close surface in the -imensional space forme by the frequency ω an phases spanning the bounary of the system. The choice of the surface is arbitrary, as long as it encloses the points or surfaces where G is singular (see [] for the iscussion concerning the surface choice). It is then natural to choose as a surface to be integrate over the two planes at two values of the phase θ =± Λ, as it is one in our prior work [] (the choice of θ is arbitrary; any of θ i can be chosen for this construction). Here Λ is such that all the singularities of G lie between these two planes in the θ-space. Then the bounary invariant can be rewritten as essentially a bulk invariant in the -imensional space forme by frequency an phases across the bounary, θ,, θ, with θ fixe. More precisely, we can efine ( ) N θ = C ϵ ω θ X, α α α α 0 0 Xα α = Tr G α G G α G. (4) 0 0 Here α 0,, α are summe over ω, θ,, θ. 9

10 Then we can rewrite N B as N = N ( Λ) N ( Λ). (5) B N Therefore, if N B is not zero, is a function of θ in such a way that as θ changes from Λ to Λ, N changes by an amount equal to N B. An as we iscusse if the bounary we stuy is the bounary of a topological insulator with non-zero bulk invariant N, N B = N = 0. Now N is a topological invariant. The only way for it to change as a function of θ is if there is some special value Λ < θ c < Λ such that at θ equal to this value, G becomes singular. Then at that value N is not well efine, an the ifference N( Λ) N( Λ) can be non-zero. G is relate to the Hamiltonian by G = [i ω ]. The only way for it to be singular if ω = 0 an has a zero eigenvalue. That means, there is a single particle energy level at the bounary whose energy is zero at θ = θ c. At the same time, when θ =± Λ, that energy level is not zero. We can now invoke a well known criterion of localization [0] which states that a localize level s energy cannot shift as a function of the phase impose across a isorere system. Therefore, in orer for N B to be non-zero, there has to be at least one energy level whose energy epens on θ. That level must be elocalize. (To be even more precise, this argument gives support to having at least one energy level which is not exponentially localize, as one can perhaps argue that some levels which are not exponentially localize are sometimes not fully elocalize either.) For this argument to work, we must also keep the lengths of the system L i, introuce in equation (6), to be much larger than the localization lengths encountere in the localize states. Furthermore, the system we stuy must have more than one elocalize energy level. Inee, zero was an arbitrary reference point for the energy. We can always consier a Hamiltonian shifte by some chemical potential μ (a position-inepenent constant) = μ. (6) We can repeat all the arguments for this shifte Hamiltonian. As long as this shift oes not change N (the bulk invariant), there shoul be a elocalize state at new zero energy, or at energy μ of the original moel. Now we can anticipate that the bulk system has states at all energies, but states in the energy interval Δ to Δ are all localize. This concept of Δ generalizes the concept of a gap in case of a isorer-free system. Then for any Δ < μ < Δ, the bulk invariant N is insensitive to μ. Then the system has a elocalize state at any energy μ which spans the interval Δ to Δ. This conclues the argument about the elocalize states at the ege of any even-imensional insulator. The situation with o-imensional insulators is somewhat more restrictive. Their bounary invariant given in equation (3) can still be rewritten in a way equivalent to equation (5), with N ( θ) given by C3 N θ = ϵ ( ) α α θ X α α, Xα α = Tr ΣV αv V α V, (7) where as before V = G ω= 0. Just as before, at the bounary of a topologic insulator N = N B = N( Λ) N ( Λ) = 0. It again follows that there is an energy level which c crosses zero as a function of θ at some value θ. That level must be elocalize. However, no other elocalize levels can be generally expecte. Inee, the Hamiltonian cannot be shifte by a chemical potential, as before, because we must ensure the symmetry 0

11 represente by equation (8). An arbitrary chemical potential ae to the Hamiltonian breaks it. Therefore, we expect only one elocalize state close to zero energy (in the limit of the infinitely large system, exactly at zero energy). All other states are generally localize. This is inee what is expecte from the chiral systems as explaine in the introuction. Some of the chiral systems have only one elocalize state at zero energy. Yet others have many elocalize states spanning some energy interval centere aroun zero, similarly to the non-chiral isorere insulators. The arguments given here cannot establish which of the chiral systems will have a fully elocalize ege. All one can establish is that at least one state at zero energy must be elocalize Ege of a topological insulator with Z invariant Finally, let us aress the rest of topological insulators escribe by a invariant taking just two values, 0 an. All topological insulators are escribe by either an integer invariant of the types iscusse here earlier or the invariant of the type as is well summarize in [3]. Systems with topological invariant can be unerstoo as a imensional reuction of the system with an invariant escribe in this paper by resiing in higher imensions [3, ]. This construction can easily be incorporate into the formalism use here, as iscusse in []. Relying on the arguments from this work, we can imagine that higher imensional system have isorer which oes not vary in the spatial irection we plan to eliminate. Those imensions can be spanne by momenta in the Green s functions, while other imensions are still spanne by phases. Setting those momenta to zero we obtain the imensionally reuce system with the invariant. The bounary states of both higher imensional parent an lower imensional escenant system must be elocalize in the same way. Thus we fin that the bounary of topological insulators have a fully elocalize ban if they o not have chiral symmetry or at least a single elocalize state at zero energy if they o have chiral symmetry. 4. Conclusions We examine the bounary states of isorere topological insulators an were able to unerstan their localization properties by irectly examining the topological invariants of these isorere systems. In oing so, we reprouce what shoul be consiere a wiely anticipate answer. Nevertheless it was only for two an some 3D insulators that this answer has been erive. Further elaborations were base on the approach of the sigma moels with WZW-type terms, an on their mostly conjecture behavior (although in some low imensional cases this can be erive). Here we erive this answer without any conjectures in an arbitrary number of imensions. Finally in view of the existence of other methos to look at bounaries of topological insulators [7], it woul be interesting to further explore the connection between these methos an the one iscusse here an see if this coul she aitional light on the structure of the bounary states in isorere topological insulators. Acknowlegments The authors are grateful to P Ostrovsky for sharing insights concerning the bounaries of three-imensional isorere topological insulators. This work was supporte by the NSF grants DMR an PHY-94 (VG), an by the Institute for Quantum Information

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