Topological invariant in three-dimensional band insulators with disorder
TTopological invariant in three-dimensional band insulators with disorder
H.-M. Guo
Department of Physics, Capital Normal University, Beijing, 100048, China andDepartment of Physics and Astronomy, University of British Columbia,Vancouver, BC, Canada V6T 1Z1
Topological insulators in three dimensions are characterized by a Z -valued topological invariant,which consists of a strong index and three weak indices. In the presence of disorder, only the strongindex survives. This paper studies the topological invariant in disordered three-dimensional systemby viewing it as a super-cell of an infinite periodic system. As an application of this method weshow that the strong index becomes non-trivial when strong enough disorder is introduced into atrivial insulator with spin-orbit coupling, realizing a strong topological Anderson insulator. Wealso numerically extract the gap range and determine the phase boundaries of this topologicalphase, which fits well with those obtained from self-consistent Born approximation (SCBA) and thetransport calculations. PACS numbers: 73.43.-f, 72.25.Hg, 73.20.-r, 85.75.-d
Time reversal invariant band insulators of non-interacting electrons are basically divided into twoclasses: the ordinary insulator and the topological insula-tor [1–4]. The latter is a novel phase of quantum matter.It has an insulating bulk gap and gapless edge or surfacestates. These gapless states are topologically protectedand are immune to non-magnetic disorder. Recently an-other kind of non-trivial quantum phase termed topo-logical Anderson insulator (TAI) has been predicted toexist in two dimensions (2D)[5–7] and three dimensions(3D) [8], which makes the situation more interesting. Inthe TAI phase, remarkably, the topologically protectedgapless states emerge due to disorder.For systems without disorder, the topological phasescan be characterized by studying the gapless states asobtained e.g. from diagonalizing the Hamiltonian in ageometry with edges or surfaces. They can also be char-acterized by the topological invariants calculated fromthe bulk Hamiltonian, which have been well studied inrecent literature [9, 10]. However in the disordered sys-tems, gapless modes alone cannot unambiguously iden-tify the topological phases because they may be local-ized in space. Instead, the transport properties are usu-ally used to find these extended topologically protectedmodes. Similarly we may also use the topological in-variant to characterize the topological phases induced bydisorder. A question naturally arises: how to calculatethe topological invariant in the presence of disorder.At first glance, it is not obvious how to generalize thepresent methods from translation invariant band insula-tors to the disordered systems. Let us first recall theinteger quantum Hall effect (IQHE) where the general-ization to the case with disorder is well understood. Thetopological quantum number in IQHE, which character-izes the quantized Hall conductivity, is known as the firstChern number [also refered to as the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) integer] and is closely re-lated to Berry’s phase. In the presence of disorder, theTKNN integers defined for a clean system can be general- ized. By introducing generalized periodic boundary con-ditions and averaging over different boundary conditionphase, an invariant expression can be constructed appli-cable to the situation where many-body interaction andsubstrate disorder are present [12, 13]. Since the bound-ary condition phases can be transferred to the Hamil-tonian by a unitary transformation, such considerationsare actually equivalent to thinking of the system as asuper-cell of an infinite system. Since the infinite systemwhich is periodic in the super-cell has translation sym-metry, the wave vectors can be well defined, which infact correspond to the boundary condition phases. Theadvantage of such consideration is that since the bandstructure is recovered, one can use the known methodsto calculate the topological invariant for the infinite sys-tem. The finite system under consideration, which isnow a super-cell of the infinite system, shares the sametopological properties as the infinite system. This ideahas been used to study the phase transition in the pres-ence of disorder in 2D quantum spin Hall system (QSHE)[14], where the authors found that a metallic region al-ways appears between ordinary and topological insulatorin spin-orbit coupled systems with disorder when thereis no extra conservation law (Recently the same result isalso obtained via C*-Algebras [15]).In this paper, we study the topological invariant of thedisordered 3D system by viewing it as a super-cell of aninfinite system. We start from a trivial insulator withspin-orbit coupling and find that when strong enoughdisorder is introduced into the system, a gap appears athalf filling and the corresponding topological invariantis non-trivial, confirming the existence of the disorder-induced non-trivial topological phase.To be concrete, we consider a model describing itiner-ant electrons with spin-orbit coupling on a cubic latticewith the Hamiltonian in the momentum space [16, 17], a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p e i φ e -i φ FIG. 1: (Color online)The cubic lattice on which the Hamil-tonian is defined. As an example, the super-cell has a size 2 (red closed circles). The sites which have connections withnext (previous) super-cell obtain phases e iφ i ( e − iφ i ), where φ i = 0 or π and i = x, y, z depends on the TRIM and thedirection of the bonds. H ( k ) = d ( k )I × + d ( k ) d ( k ) 0 d − ( k ) d z ( k ) − d ( k ) d − ( k ) 00 d + ( k ) d ( k ) − d z ( k ) d + ( k ) 0 − d z ( k ) − d ( k ) (1)where d ± ( k ) = d ( k ) ± id ( k ), d ( k ) = (cid:15) − t (cid:80) i cos k i , d i ( k ) = − λ sin( k i ) and d ( k ) = 2 γ (3 − (cid:80) i cos k i )( i = 1 , , Bi Se family [18–21]. At half filling,depending on the parameters, the system can be a triv-ial insulator or a topological insulator. To simulate theeffects of disorder we consider a random on-site potentialΣ j U j Ψ + j Ψ j , with U j uniformly distributed in the range( − U / , U / L x × L y × L z (thelattice constant a = 1). There is no translation symmetryin the system when disorder is present. However takingthe system as a super-cell of an infinite system, transla-tion symmetry is recovered. The lattice vector becomes a (cid:48) i = L i (the size of the super-cell in i = x, y, z direc-tion) and the corresponding reciprocal lattice vectors are b i = πL i .In 3D, the topological invariant consists of four Z numbers forming an index ( ν ; ν ν ν ), which distinguish16 topological classes [10, 22]. Usually it is a difficultproblem to evaluate them for a given band structure.However in the presence of inversion symmetry, the prob-lem can be greatly simplified. It has been shown thatthey can be determined from the knowledge of the par-ity ξ m ( Γ i ) of the 2 m -th occupied energy band at the 8time reversal invariant momenta (TRIM) Γ i that satisfyΓ i = Γ i + G . The 8 TRIM can be expressed in termsof primitive reciprocal lattice vectors as Γ i =( n n n ) = ( n b + n b + n b ) /
2, with n j = 0 ,
1. Then ν α is de-termined by the product ( − ν = (cid:81) n j =0 , δ n n n , and( − ν i =1 , , = (cid:81) n j (cid:54) = i =0 , n i =1 δ n n n , where the parityproduct for the occupied bands δ i = (cid:81) Nm =1 ξ m (Γ i ). Totake advantage of the simplification, we only consider dis-order configurations with inversion symmetry. For largeenough super-cell such consideration will not change theunderlying physics. So to calculate the topological in-variant, we only need to consider the Hamiltonian at theeight TRIM and they are equivalent to those of a finitesystem with boundary conditions which are periodic upto phases φ x , φ y , φ z =0 or π for boundary sites, as shownin Fig. 1.To understand what happens when introducing thesuper-cell, we first calculate the Z topological invari-ant of a clean (1; 000) strong topological insulator. Theresults are shown in Fig. 2 (a). At half filling, we findthat δ = − δ = 1 at other TRIM.This result is consistent with the ordinary band structurecalculations and the reason is explained below. Takinga super-cell means enlarging the original unit cell in the a , a , a directions ( a , a , a are the lattice vectors ofthe original lattice). The resulting new Brillouin zone(BZ) is folded in the corresponding directions and shrunkin size. The occupied states at the eight TRIM for thenew BZ contain those at the eight TRIM for the origi-nal BZ. Though more states which are at other momentaof the original BZ will reside on the TRIM of the newBZ, they do not change the parity product for the oc-cupied states. Thus the product of all eight δ s from thesuper-cell calculation still yield the same ’strong’ indexof the Z topological invariant. However the ’weak’ in-dex cannot be obtained from this method if the super-cell contains an even number of unit cell in the corre-sponding direction. As mentioned above, the weak in-dices ν , ν , ν are the product of four δ s which are in theplanes k = π , k = π , k = π respectively. Suppose thatthe super-cell contains an even number of unit cell in the a direction, then the k = 0 and k = π planes will col-lapse onto the k (cid:48) = 0 plane in the new BZ. Thus the weakindex determined by the four TRIM on the k (cid:48) = π planein the new BZ must be 0. From another point of view,this is understandable because the ’weak’ indices are re-lated to layered 2D quantum spin-Hall states and the 3Dsuper-cell naturally fails in calculating the quantities re-flecting the 2D physics. When considering systems withdisorder, the weak indices are eliminated and only thestrong index remains robust. The topological invariantfor the system with disorder is the ’strong’ index of the Z topological invariant for clean 3D band insulators. Inthe following, we simply call it the topological invariantfor the 3D systems with disorder.Now we take into account disorder as described aboveand start from a trivial insulator. Fig. 2 (b) shows theresult of such a calculation on a 8 lattice with U = 150 Energy Level (meV)-100 0 100 200 300 400 500 k= (0,0,0) k= (0,0,π) k= (0,π,0) k= (0,π,π) k= (π,0,0) k= (π,0,π) k= (π,π,0) k= (π,π,π) Energy Level (meV)-20 0 20 40 60 (a) (b) P a r i t y P r odu c t FIG. 2: (Color online) The parity product for the occupied states at the eight TRIM for a 8 super-cell without disorder (a)and with disorder (b). The crosses mark the eigenvalue of the system and the corresponding parity product for the filling upto this energy level. Here we use parameters: t = 24meV, λ = 20meV, γ = 16 meV and (a) (cid:15) = 134meV, corresponding to m = −
10 meV, where the system is a (1; 000) strong topological insulator; (b) (cid:15) = 145 meV and U = 150 meV, correspondingto m = 1 meV, where the clean system is a trivial insulator. The red lines in both figures show the gap range which appearsat half filling. L=4
40 60 80 100
L=6
100 120 140 160 180 200
L=8
Number
FIG. 3: (Color online)Distributions of the number of Z -oddband pairs from 500 disorder realizations. The bar heights isthe fraction of disorder realizations that have a given numberof band pairs with Z = 1 (The numbers are all odd here).The parameters are the same as those in Fig. 2 (b). meV. Generally the disorder will eliminate all degenera-cies except those protected by time reversal symmetry.So at the TRIM, each eigenvalue of the Hamiltonianis doubly degenerate. Though the disorder makes theenergy spectrum more continuous, a small gap remainsclear at half filling. Here the gap location is shifted to thehigh energy, which is different from the case in the cleansystem where the gap appears symmetrically at E = 0.The parity eigenvalue products δ = − δ = 1 at other TRIM for half filling show that thetopological invariant of the system is 1, so the systemwill exhibit non-trivial topological properties. We haveextended the above calculation to 500 different disorderrealizations. In all 500 disorder realizations, the topolog-ical invariant is 1 for half filling, which further confirmsthe topological phase in the system. Thus through calcu-lating the topological invariant of the system, we obtainfurther confirmation of the topological phase induced bydisorder. This new phase has been termed as ’strong topological Anderson insulator’ (STAI) [8]. Due to thenon-trivial topological properties in STAI, topologicallyprotected surface states will appear at the surfaces asthe case in the ’strong topological insulator’ phase of theclean system, which has already been verified by trans-port calculations.It is also interesting to look at the number of Z -oddband pairs. In 2D QSHE, the number of Z -odd Karmerspairs increases linearly with the system size in the metal-lic region and grows slowly for the topological insulator[14]. Here in 3D, we find a similar result. In Fig. 3,we show the distribution of the number of Z -odd bandpairs at three lattice sizes 4 , 6 and 8 . We find thatthe location of the mean roughly scales with L ( L is thelattice size). A detailed analysis on the limited data alsoshows that the number growth is somewhat slower thanlinear. With enough data and larger sizes, finite-size scal-ing could be carried out. However this is not accessibleat present due to the limited computer resources.For different disorder realizations, though there stillexits a gap at half filling, its position on the energy axischanges randomly. We extract the energy levels for themost energetic electron at half filling and the one justabove it and show their distributions in Fig. 4. Thecurves in Fig. 4 show Gaussian-like distributions. Thedistributions for half filling and half plus one at someTRIM have no overlaps while on the other TRIM haveoverlaps. This means there is no ’true’ gap range evenfor the eight TRIM. However the overlaps are alreadyvery small for the lattice size of 8 . We also performedthe same calculations on lattice sizes of 4 and 6 . Withthe increasing lattice size the widths of the distributionsdecrease. We therefore attribute the overlaps for thepresent lattice size to finite size effect and expect themto diminish for larger lattice sizes.For the parameters used the physical gap size is deter-mined by gap at k = (0 , ,
0) TRIM. We can obtain thegap value from the peaks of the distributions and thisvalue will approximate the one for the larger lattice size.We have determined the phase diagram in the U − E F plane from calculations on a 6 system and the result isshown in Fig. 5. The range of STAI in the U − E F planehas also been obtained from conductivity calculationsand the SCBA, where the weak-disorder boundary fitswell with each other while the strong-disorder boundarydoes not [8]. It was found that the weak-disorder bound-ary marks the crossing of a band edge and the strong-disorder boundary marks the crossing of a mobility edge,and The SCBA does not work for the strong-disorderboundary [7].The result in Fig. 5 shows that the phase boundariesobtained from the present calculation fits well for boththe weak-disorder and strong-disorder phase boundariesdetermined from the conductivity calculation [8]. Therange of STAI phase should be determined by the gapbetween the two edges containing extended states. Webelieve that we have extracted such a gap. The reason isthat the localized states are closely related to the disor-der realizations and can be removed by an average overdisorder realizations. In the phase diagram, increasingthe strength of disorder, the system first undergoes asharp phase transition, which happens at a critical dis-order strength where the gap closes. Then the systementers a stable topological phase for a wide range of dis-order strength. Finally the system experiences anotherphase transition to a diffusive metal. Close to the lattertransition, the finite system under consideration is in amixed phase, where some fraction of disorder realizationsyield a non-trivial topological invariant.In conclusion, we have generalized the method of cal-culating the topological invariant in disordered 2D QSHEto 3D disordered systems. In this method, the finite 3Dsystem is viewed as a super-cell of a large lattice with welldefined wave vector, which allows us to directly use thedefinition of topological invariants for clean band insula-tors. The obtained topological invariant can be thoughtof as describing the finite system with disorder. Using theinversion symmetry preserving disorder configurations,we carried out explicit calculations on a model Hamilto-nian (describing the physics of insulators in Bi Se fam-ily) with on-site disorder. We found that the topologicalinvariant for a system that is a trivial insulator in theabsence of disorder can become nontrivial when strongenough disorder is introduced. This result confirms theexistence of the strong topological Anderson insulator, atopological phase in three space dimensions whose exis-tence is fundamentally dependent on disorder [8].As additional application of this method we countedthe number of Z -odd band pairs. Though this number p )(0, p ,0) (0, p , p )( p ,0,0) ( p ,0, p )( p , p ,0) ( p , p , p ) Energy (meV) P r obab ili t y FIG. 4: (Color online) Distributions of energy levels at halffilling and half plus one filling from 500 disorder realizations.The system sizes are 4 (blue),6 (green) and 8 (red). The pa-rameters are the same as those in Fig. 2 (b). U (meV)0 100 200 300 E F ( m e V ) -200204060 Super cellSCBA FIG. 5: (Color online) Phase boundaries for the STAI phasein the U − E F plane obtained from the super-cell calcuations.The red crosses represent the peaks in energy level distribu-tions for half and half plus one fillings and are obtained fromcalculations of 500 disorder realizations on 6 system. Thephase boundaries from the SCBA are also shown for compar-ison. varies with the disorder realization, the total number isalways odd when the system is in the STAI phase andit scales roughly linearly with the number of the latticesites. Finally, we obtained the phase boundaries for theSTAI phase by extracting the gap range, which fits wellwith the known results.In the presence of on-site disorder, the diagonal el-ements of the Hamiltonian matrix are random, drawnfrom a statistical distribution. The Hamiltonian can beregarded as a random matrix which can be studied inthe framework of the random matrix theory (RMT). TheRMT has already been applied to IQHE and disorderedsuperconductors [23–25]. We expect some insights intothe present problem from RMT, which we leave to futurework. Acknowledgment .— Authors are indebted to I. Garate,M. Franz, G. Refael, G. Rosenberg, C. Weeks and M.Vazifeh for stimulating discussions. Support for this workcame from NSERC, CIfAR and The China ScholarshipCouncil. [1] B.A. Bernevig, T.L. Hughes, and S.-C. Zhang, Science , 194 (2010).[3] M.Z. Hasan, C.L. Kane, arXiv: 1002.3895.[4] X.-L. Qi, T.L. Hughes, and S.-C. Zhang, Phys. Rev. B , 045302 (2007).[11] D.J. Thouless, M. Kohmoto, M.P. Nightingale, and M.den Nijs, Phys. Rev. Lett.
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