Interaction Protected Topological Insulators with Time Reversal Symmetry
IInteraction Protected Topological Insulators with Time Reversal Symmetry
Raul A. Santos , and D.B. Gutman Department of Physics, Bar-Ilan University, Ramat Gan, 52900, Israel and Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Anderson’s localization on the edge of two dimensional time reversal (TR) topological insulator(TI) is studied. For the non-interacting case the topological protection acts accordingly to the Z classification, leading to conducting and insulating phases for odd and even fillings respectively.In the presence of repulsive interaction the phase diagram is notably changed. We show that forsufficiently strong values of the interaction the zero temperature fixed point of the TI is conducting,including the case of even fillings. We compute the boundaries of the conducting phase for variousfillings and types of disorder. I. INTRODUCTION
Time reversal non-interacting TIs are realized in ma-terials with strong spin orbit interaction in two andthree dimensions. Also known as quantum spin Hallinsulators, these materials have an insulating bulk, whilehosting gapless surface states.For the non-interacting case the classification of dis-ordered TIs is complete . In two dimensions, non in-teracting TR invariant TIs are classified by a Z topo-logical invariant, according to the number of helical edgestates . The back scattering by TR invariant disor-der is possible only between states that are not Kramer’spartners. Therefore for an odd number of the helicalstates the conducting state is protected. For an evennumber of helical edge modes the electrons can be local-ized completely by scattering among non-Kramers pairs.This state is therefore equivalent to a trivial insulator inagreement with a more general Haldane criterion .Interacting integer and fractional TI’s are a subject ofactive research. Interaction may lead to strongly corre-lated ground states with fractional excitations andnon trivial statistics . The classification of TI’s in theinteracting case is yet unknown. An approach, based onthermal response and its relation to a quantum anomaly,valid beyond single particle picture, was proposed inRef. .It is commonly accepted, that the charge transport inthe ideal TI occurs via protected a single helical edgemode, with the universal quantized conductance 2 e /h .In reality the conductance differ from this value due toback scattering processes. The latter may occur via thecombination of the two electron scattering and the dis-order potential , coupling to the bulk via electronpuddles or due to the magnetic impurities. In the latercase the interaction stabilizes the conducting phase, andthe quantum phase transition as function of Luttinger liq-uid (LL) parameter is predicted for the fractional TIs .In this work we study the localization by TR disorderon the edge states of a TI in the presence of repulsiveinteraction. Although we focus on the TI’s at integerfillings ( ν ), that in the absence of the disorder possess ν helical edge states, the same analysis applies for narrowstripe of TI at ν = 1. The inclusion of TR disorder drives the non-interacting system to a state with a singleor no helical edge states. We show that the presenceof the repulsive interaction can stabilize the conductingphase.We model the disorder by a short range static poten-tial that due to the spin orbit interaction mixes differenthelical states, except those that are connected by the TRsymmetry. We consider a generic finite range interactionbetween the electrons, with all possible matrix elementsallowed by symmetry. We consider the case of a sin-gle impurity and the random disorder, with a scatteringlength shorter that the sample size. We perform one looprenormalization group (RG) analysis, analogous to Kane-Fisher and Giamarchi-Schultz study of localization inone dimensional systems.Our analysis shows that the low energy fixed pointis determined by the magnitude of the interaction andits effective radius. For interaction stronger that somecritical value the low temperature phase is conducting. II. THE MODEL
The appearance of the helical edge states can be un-derstood on the level of non interacting electrons. Inthe presence of a Rashba spin-orbit (SO) interaction thesingle particle Hamiltonian is given by H = ˆ p x + ˆ p y m e + α SO ( (cid:126)p × (cid:126)σ ) · ∇ V ( x, y ) + V ( x, y ) , (1)where m e is effective mass of an electron and α SO is thestrength of the SO coupling. For the parabolic potential V ( y ) = y / m e α SO , the Hamiltonian (1) H = ˆ p y m e + 12 m e (cid:18) ˆ p x − yα SO σ z (cid:19) (2)corresponds to two replicas of fermions subject to oppo-site magnetic fields | B | = α − SO . For the integer fillings, ν = α − SO A/ Φ ( A being an area of sample and Φ = hc/e the flux quantum) the bulk forms an incompressible statewith a gap of size (cid:126) α − SO /m e .An addition of a smooth confining potential curves theLandau levels, as shown in Fig.1, leading to ν gapless a r X i v : . [ c ond - m a t . s t r- e l ] S e p helical edge states . Assuming that the single par-ticle gap formed in the bulk is not closed the effects ofinteraction can be taken into account within the helicaledge states. This phenomenological approach can be mi-croscopically justified within the sliding Luttinger liquidmodel . However, the resulting helical edge descrip-tion is believed to be the correct low energy model, validbeyond the sliding LL approximation.To account for the interaction, it is natural topass to the bosonic description, defining bosonicfields R i /L i , related to the right/left density compo-nents by ρ R,i = ∂ x R i / π and ρ L,i = − ∂ x L i / π .These fields satisfy the canonic commutation relations[ R j ( x ) , R j (cid:48) ( y )] = − [ L j ( x ) , L j (cid:48) ( y )] = iπ sgn( x − y ) δ jj (cid:48) .The electronic operators are represented as ψ R,j = e − iR j / √ πa, and ψ L,j = e − iL i / √ πa, with a a shortdistance cut-off.In the absence of the Umklapp and 2 k F electron-electron scattering the interaction between ν modes, con-sistent with TR symmetry, is represented by the followingaction S = 12 π ˆ dxdt (cid:16) ∂ x Φ T K ∂ t Φ − ∂ x Φ T M ∂ x Φ (cid:17) , (3)where we use the compact notations Φ = ( R , L ), R =( R , R , . . . , R ν ) and similarly for L . Here the matrix K encodes the commutation relations of the fields andcan be written as K = σ z ⊗ I ν , where the σ z is a Paulimatrix that acts in the right/left movers subspace while I ν is the identity matrix in the space of ν modes. Thepositive definite matrix M accounts for interaction. Thehelical edge modes are separated in space by a distance d . The symmetry under TR requires that { T, K} =[ T, M ] = 0, where T = σ x ⊗ I ν is a time reversal symme-try operator. This restricts the interaction matrix M tothe form M = (cid:20) M fw M bw M bw M fw (cid:21) = I ⊗ M fw + σ x ⊗ M bw , (4)where ( M fw ) ij describes the forward interaction betweenthe copropagating modes ρ R,i and ρ R,j (similarly for leftmovers). ( M bw ) ij is a an element of a symmetric matrixin the channel space that describes the backward interac-tion between R i and L j . We assume that the interactionbetween helical modes i and j is translationally invariant,and depends only on the relative distance | i − j | . In theabsence of disorder the spectrum of this model is gapless.The presence of impurities may dramatically changethe states of edge modes. TR invariant disorder mixeshelical states that belong to different Kramer’s pairs andinduces backward scattering processes. We consider twocases: (a) single impurity scattering and (b) random dis-order. Single impurity scattering is the dominant processif the mean free path of the electrons is larger than thesample size. In the opposite case (the electrons’ meanfree path is smaller than the sample size) the localizationis dominated by multiple scattering. E y µ FIG. 1. (color on line) Helical edge modes. The wavy linesrepresent the back-scattering events between non-Kramerspairs (in red and blue). Insert.- Schematic band structureof a TI.
A. Single Impurity
We analyze the single impurity case first. The sin-gle impurity, located on the edge backscatter betweenstates i and j that are not connected by TR symmetry, O imp ij = µ ij ψ † R,i (0) ψ L,j (0), where µ ij is proportional tothe impurity potential at k = 2 k F . The renormalizationof the strength µ ij is a straightforward generalization ofstandard Kane-Fisher analysis dµ ij dl = (1 − ∆ ij ) µ ij . (5)Here ∆ ij is the scaling dimension of the scattering pro-cess (cid:104) ψ † R,i ( τ ) ψ L,j ( τ ) ψ † L,j (0) ψ R,i (0) (cid:105) ∼ | τ | − ij . FromEq. (5) it follows that a single impurity is an irrelevantperturbation if ∆ ij > M bw . For the simple case ν = 2only two helical modes propagate on the edge. If theseparation between the helical states is larger than theinteraction radius the effective Hamiltonian is given by H = H fw + H bw with H fw = 12 π (cid:88) i =1 ˆ dx (cid:18) v F + g π (cid:19) (( ∂ x R i ) + ( ∂ x L i ) ) ,H bw = − g π (cid:88) i =1 ˆ dx∂ x R i ∂ x L i . (6)Here v F the Fermi velocity while g and g param-eterize the forward and backward interactions respec-tively. Defining the fields ϕ i = ( R i − L i ) / √ θ i = ( R i + L i ) / √
2, the above Hamiltonian can be writtenin a Luttinger liquid (LL) form H = u π (cid:88) i =1 ˆ dx (cid:18) ( ∂ x θ i ) K + ( ∂ x ϕ i ) K (cid:19) , (7)where u = ( v F + g /π ) (cid:112) − ( λ ) is the renormalizedsound velocity, λ = g π ( v F + g /π ) and K = (cid:113) − λ λ is theLL parameter ( K <
K > O imp12 has the scaling dimension∆ = K/ / K , so the single impurity is irrelevantfor any interaction. This is in stark contrast with thenon helical LL where the scaling dimension of the disor-der operator is K , so the disorder is irrelevant only foran attractive interaction . If the LL’s are different thescaling dimension is controlled by both LL parameters∆ = ( K + 1 /K + K + 1 /K ) /
4. This result impliesthat the scattering between different edge states is ac-companied by the zero bias anomaly that suppresses theprobability of this process. This is in contrast to Kane-Fisher case , where the back scattering occurs in thesame LL and has no zero bias anomaly suppression. ForTI this processes is forbidden by the TR symmetry.For the case where the helical edge states are locatedwithin the radius of interaction (or for any long rangeinteraction potential) the interaction matrices (4) are M fw = (cid:18) v F + g g g v F + g (cid:19) , M bw = − (cid:18) g g g g (cid:19) . (8)The scaling dimension ∆ is∆ = 12 ( F + F ) , (9)where, for repulsive interactions F = (cid:115) λ − λ − λ λ + λ + λ , F ∈ [0 ,
1] (10) F = (cid:115) − λ + λ − λ − λ − λ + λ , F ∈ [0 , ∞ ] (11)with λ ba = g ba π ( v F + g /π ) . In the limit g = g = g = 0of no interactions within the helical modes the scalingdimension ∆ = (cid:113) − λ λ < . B. Random Disorder
Now we switch to the case of multiple impurities onthe edge. This perturbation is described by O dis ij = ˆ dxξ ij ( x )( ψ † R,i ψ L,j − ψ † L,i ψ L,j ) . (12)Here ξ ij ( x ) is the (random) scattering amplitude, and ξ ii = 0 due to the TR symmetry. We model the scatter-ing to be local along the edge and uncorrelated for thedifferent pairs of helical states (cid:104) ξ ij ( x ) ξ kl ( x (cid:48) ) (cid:105) = W ij δ ik δ jl δ ( x − x (cid:48) ) . (13) We now follow the steps of Giamarchi-Schultz renor-malization group analysis . For the weak disorder onefinds dW ij d(cid:96) = (3 − ij ) W ij , (14)where ∆ ij = ∆ | i − j | is the scaling dimension of scatteringprocess (12) between helical states i and j allowed byTR symmetry. In the conducting phase the disorder isan irrelevant perturbation, and all W ij flow to zero. Thisrequires ∆ | i − j | > for all pairs i, j . Let us consider twolimiting cases: (i) disorder that mixes only the nearestmodes W ij ∼ W δ i,j +1 ; (ii) the disorder that mixes themodes uniformly W ij ∼ W . All physical realizations liein between these two limits.The simplest situation is realized for ν = 2 wherethe limits (i) and (ii) coincide. In that case, the scal-ing of the disorder operator is given by Eq.(9). In theabsence of inter-mode interaction the scaling dimensionof a back scattering operator is ∆ = ( K + K − ) / K < (3 − √ /
2. In the presence of inter mode inter- λ λ λ =0 λ =0.2 λ =0.4 λ =0.5 FIG. 2. (color online) Phase diagram for ( λ , λ ). Red regioncorresponds to the conducting phase for a single impurity.Multiple impurities are irrelevant in the blue region. Thegray region is forbidden by positivity of matrix M . action the phase diagram is show in Fig. (2) as functionof interaction parameters. The symmetry between intraand inter mode forward scattering ( λ →
1) enhances theconducting phase.
C. Effect of two particle processes
For weak interactions, two particle processes are lessrelevant than single particle events. For sufficientlystrong interactions, they start to compete. We analyzehere the following processes involving two particle events O II,c = t c ˆ dxδ ( x ) ψ † R, ψ † L, ψ R, ψ L, , (15) O II,s = t s ˆ dxδ ( x ) ψ † R, ψ † R, ψ L, ψ L, , (16)which correspond to the transfer of 2 e charge and twoparticle backscattering respectively. These processesrenormalize according to dt a dl = (1 − ∆ a ) t a , (17)with a = ( s, c ). Here ∆ a is the scaling dimension of theoperator O II,a . They are∆ c = 2 F and ∆ s = 2 F . (18)In the case of 1 / < F <
1, the system is in theconducting phase for F + F >
2. The correction to theconductance G = 4 e /h scales with the temperature as δG ∼ (cid:40) − c µ (cid:0) πaTu (cid:1) F + F − , if F < F , − c µ v F (cid:0) πaTu (cid:1) F − , if F > F . (19)where c i are non-universal parameters. If F < /
2, thesecond order process O II,s becomes relevant. The con-ductance then becomes non monotonous at large temper-atures (see Fig.3) G T* T K+1/K-24K-2 T T FIG. 3. (color online). Sketch of the conductance as functionof temperature. For F < /
2, the two particle process O II,s becomes relevant, leading to a non monotonous behaviour ofthe conductance at high temperatures. In the figure we take F = K and F = 1 /K , which correspond to the simplest caseof interactions just within each helical mode, discussed in Eq.(6) For random disorder, the two particle operators are D II,c = ˆ dxξ c ( x ) ψ † R, ψ † L, ψ R, ψ L, , (20) D II,s = ˆ dxξ s ( x ) ψ † R, ψ † R, ψ L, ψ L, , (21)where the ξ ( x ) a are uncorrelated random variables with (cid:104) ξ a ( x ) ξ b ( x (cid:48) ) (cid:105) = W a δ ab δ ( x − x (cid:48) ). These processes renor-malize acording to the RG equations dW a dl = (cid:18) − ∆ a (cid:19) W a , (22)with ∆ a given by (18). For 3 / < F < F + F > F < / F + F > D II,s becomes relevant underRG and conductance becomes non-monotonous at hightemperatures (similar to the case of single impurity). D. ν (cid:29) Helical Edge Modes
We now proceed with a more general case of ν helicalstates. To calculate ∆, we consider the operators of theform Ψ m = e i m · Φ where each vector m = ( m R , m L )corresponds to a different physical process. For example( m R ) i = δ ik , ( m L ) i = − δ i,k + l , (23)describes an operator Ψ m that backscatter a right moverin the mode k to a left mover in the mode k + l . Us-ing the (quadratic) action (3), one computes the scalingdimension of Ψ m ∆[Ψ m ] = 12 m T Λ m , (24)where Λ = M − |M KM |M − . Here the absolutevalue of a matrix in the right hand side is defined asthe absolute value of its eigenvalues. In other words, if A is a diagonalizable matrix A = U DU − , then | A | = U | D | U − = U | d | · · · | d | · · · ... . . . U − . (25)Note that, for translationally invariant interaction weconsider, the interaction matrices M are of Toeplitz type,i.e. ( M fw / bw ) ij = ( M fw / bw ) | i − j | .Now on we focus on the limit where the number ofmodes is large ( ν (cid:29) M | ν − i − j | = M | i − j | , and can be easilydiagonalized .We adopt a g -ology type notations and model the in-teraction by g and g components( M fw ) | i − j | = v F δ ij + g ( | i − j | ) , (26)( M bw ) | i − j | = − g ( | i − j | ) . (27)Here the distance dependent g ( i ) accounts for the for-ward interactions between electron densities of the samechirality at distance i , while g ( i ) parameterizes the back-ward interactions of densities of opposite chiralities. Thescaling dimension ∆ (cid:96) , defined in Eq.(24) with m givenby Eq.(23) is∆ (cid:96) = 1 ν ν − (cid:88) k =0 − G ( k ) cos(2 πk(cid:96)/ν ) (cid:112) − G ( k ) . (28)The function G ( k ), is determined by the interaction pa-rameters g , G ( k ) = ˜ g ( k ) v F + ˜ g ( k ) . (29)Here ˜ g , ( k ) = (cid:80) νj =1 cos(2 πjk/ν ) g , ( j ) is the cosinetransform of g , ( r ). The condition of | G ( k ) | < M . With the scaling ofdisorder operators at hand we can analyze their behaviorunder renormalization.We now focus on a finite range interaction. One caneasily show that the scattering processes between distantmodes are less effective for the localization than backscat-tering between close ones. For the model of isotropic in-teraction g ( r ) = g ( r ) = g exp( − r /R ) the scatteringbetween the distant modes (12) is irrelevant for (cid:114) g v F > π Rd . (30)Here we assumed that | i − j | = (cid:96) (cid:29) g/v F (cid:29) (cid:96) = 1) im-poses more stringent conditions on the interaction con-stants λ ba , as shown in Fig.4. In particular, the conduct-ing phase is stable only for the nearly symmetric inter-action. For the fixed values of interaction strength thelocalization is enhanced by increasing the interaction ra-dius. In other words, strong and short range interactionmost efficiently drives the system towards the conductingphase. III. SUMMARY
To summarize, we have studied the localization of theedge modes in TIs with TR symmetry. We find thata combination of TR symmetry and zero bias anomalychanges the scaling dimensions of scattering operators.This notably affects the phase diagram. For a sufficientlystrong values of interaction the zero temperature fixedpoint is a conductor with a number of edge modes that g v π F g v π F R=0.1d R=dR=5d R=10d
FIG. 4. (color online). ( g , g ) phase diagram ν (cid:29)
1, fi-nite ranges interaction g , ( n ) = g , exp (cid:0) − ( ndR ) (cid:1) . Panelscorrespond to the different values of the interaction radius R .Color code is the same as in Fig.2 are stable against TR disorder. This holds also for theeven fillings, where the non-interacting system is equiv-alent to a trivial insulator. We have analyzed the prob-lem in several limiting cases, for the single impurity andrandom disorder, short and long range interaction, fora variety of filling fractions ν . We have computed theboundaries of the conducting phase in all these cases.For intermediate values of interaction electric conductiv-ity is a non-monotonous function of temperature, due tointerplay of single and two electron scattering processes.The authors acknowledge discussions with E. Berg, Y.Gefen, I. V.Gornyi, N. Kainaris, A.D. Mirlin, I.V. Pro-topopov, E. Sela. This work was supported by GIF andISF. I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Perge, H. Ji, R. J. Cava, B. A. Bernevig, and A. Yazdani, Nature Physics , 664 (2014); K. Suzuki, Y. Harada,K. Onomitsu, and K. Muraki, Phys. Rev. B , 235311 (2013); I. Knez, C. T. Rettner, S.-H. Yang, S. S. P. Parkin,L. J. Du, R. R. Du, and G. Sullivan, Phys. Rev. Lett. ,026602 (2014). M. K¨onig, S. Wiedmann, C. Brne, A. Roth, H. Buhmann,L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science ,766 (2007). M. K¨onig, H. Buhmann, L. W. Molenkamp, T. Hughes,C.-X. Liu, X.-L. Qi, and S.-C. Zhang, J. Phys. Soc. Jpn. , 031007 (2008). D. Hsieh , D. Qian , L. Wray , Y. Xia , Y.S. Hor , R.J.Cava, and M.Z. Hasan , Nature , 970 (2008). D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil,J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S.Hor, R. J. Cava, and M. Z. Hasan, Science , 919 (2009). Y. Xia, D. Qian , D. Hsieh , L. Wray , A. Pal , H. Lin, A.Bansil, D. Grauer, Y.S. Hor, R. J. Cava, and M.Z. Hasan,Nat Phys , 398 (2009). D. Hsieh, Y. Xia, D. Qian, L. Wray, J.H. Dil , F. Meier,J. Osterwalder, L. Patthey, J.G. Checkelsky, N.P. Ong ,A.V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y.S. Hor ,R.J. Cava, and M.Z. Hasan, Nature , 1101 (2009). G. Zhang, H. Qin, J. Teng, J. Guo, Q. Guo, X. Dai,Z. Fang, and K. Wu, Applied Physics Letters , 053114(2009). A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-wig, Phys. Rev. B , 195125 (2008). A. Kitaev, AIP Conference Proceedings , 22 (2009). S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Lud-wig, New Journal of Physics , 065010 (2010). S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Lud-wig, New Journal of Physics , 065010 (2010). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005). B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. ,106802 (2006). F. D. M. Haldane, Phys. Rev. Lett. , 2090 (1995). M. Levin and A. Stern, Phys. Rev. Lett. , 196803(2009). T. Neupert, C. Chamon, C. Mudry, and R. Thomale, Phys.Rev. B , 205101 (2014). J. Klinovaja and Y. Tserkovnyak, Phys. Rev. B , 115426(2014). L. Santos, T. Neupert, S. Ryu, C. Chamon, and C. Mudry,Phys. Rev. B , 165138 (2011). R. A. Santos, C.W. Huang, Y. Gefen, and D. B. Gutman,to appear in Phys. Rev. B, arXiv:1502.00236. T. Neupert, L. Santos, S. Ryu, C. Chamon, and C. Mudry,Phys. Rev. B , 165107 (2011). B. Scharfenberger, R. Thomale, and M. Greiter, Phys. Rev.B , 140404 (2011). M. Levin and A. Stern, Phys. Rev. B , 115131 (2012). S. Ryu, J. E. Moore, and A. W. W. Ludwig, Phys. Rev. B , 045104 (2012). T.L. Schmidt, S. Rachel, F. von Oppen, and L.I. GlazmanPhys. Rev. Lett. , 156402 (2012). N. Kainaris, I. V. Gornyi, S. T. Carr, and A. D. Mirlin,Phys. Rev. B , 075118 (2014). J. I. V¨ayrynen, M. Goldstein, Y. Gefen, L. I. Glazman,Phys. Rev. B , 115309 (2014). C.W. Huang, Sam T. Carr, D. B. Gutman, E. Shimshoni,and A. D. Mirlin Phys. Rev. B , 125134 (2013). B. B´eri and N. R. Cooper, Phys. Rev. Lett. , 206804(2012). C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. , 1220(1992). T. Giamarchi and H. J. Schulz, Phys. Rev. B , 325(1988). T. Giamarchi,
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