Dimensional evolution between one- and two-dimensional topological phases
DDimensional evolution between one- and two-dimensional topological phases
Huaiming Guo ∗ , Yang Lin Department of Physics, Beihang University, Beijing, 100191, China
Shun-Qing Shen
Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong
Dimensional evolution between one- (1 D ) and two-dimensional (2 D ) topological phases is in-vestigated systematically. The crossover from a 2 D topological insulator to its 1 D limit showsoscillating behavior between a 1 D ordinary insulator and a 1 D topological insulator. By construct-ing a 2 D topological system from a 1 D topological insulator, it is shown that there exist possiblyweak topological phases in 2 D time-reversal invariant band insulators, one of which can be realizedin anisotropic systems. The topological invariant of the phase is Z = 0. However the edge statesmay appear along specific boundaries. It can be interpreted as arranged 1 D topological phases, andhave symmetry-protecting nature as the corresponding 1 D topological phase. Robust edge statescan exist under specific conditions. These results provide further understanding on 2 D time-reversalinvariant insulators, and can be realized experimentally. I. INTRODUCTION
Topological insulator (TI) is a novel quantum state ofmatter, which is determined by the topological proper-ties of its band structure. It has generated great interestsin the field of condensed matter physics and materialscience due to its many exotic electromagnetic proper-ties and possible potential applications . Its discoveryalso deepens understanding on the time-reversal invari-ant band insulators. In two dimensions, ordinary insu-lator and quantum spin Hall insulator are characterizedby a Z invariant ν : ν = 0 for a conventional insulatorand ν = 1 for a quantum spin Hall insulator . In threedimensions (3 D ) time-reversal invariant band insulatorscan be classified into 16 topological classes distinguishedby four Z topological invariants, and thus the ordinaryinsulator is distinguished from ’weak’ and ’strong’ TIs .The 3 D TIs have been proposed and verified in manymaterials . However the 2 D TIs have only been real-ized in HgTe/CdTe and InAs/GaSb/AlSb quantum wellsystems . Theoretical studies have suggested that the2 D TI may be achieved in a thin film of 3 D TI. In thethin film, the quantum tunneling between the two sur-faces generates a hybridized gap at the Dirac point. De-pending on the thickness of the quantum wells, the sys-tem oscillates between an ordinary insulator and quan-tum spin Hall insulator . Typical TI materials, suchas Bi T e and Bi Se have a layered structure consist-ing of weakly coupled quintuple layers, which makes itrelatively easy to grow high quality crystalline thin filmsusing molecular beam epitaxy. Until now the thin filmsof 3 D TIs have been successfully fabricated experimen-tally and the gap-opening has been observed . Thispaves the way to realize the quantum spin Hall insula-tor from the present various 3 D TIs, which will greatlyenlarge the family of 2 D TIs. Furthermore by introduc-ing ferromagnetism in thin film, the quantum anomalousHall effect can be realized, which has been experimentallyconfirmed in magnetic TIs of Cr-doped (
Bi, Sb ) T e . Also it has been shown that the Chern number of quan-tum anomalous Hall effect can be higher than one bytuning exchange field or sample thickness . Com-pared to the 3 D strong TIs, the weak TIs are relatedto the topological property of the lower dimensions in amore direct way, since it can be interpreted as layered 2 D quantum spin Hall insulator. Though no weak TIs havebeen reported experimentally, they are expected to haveinteresting physical properties . Besides the studieson the 2 D limit of 3 D TIs, recently a theoretical formal-ism has been developed to show that a 3 D TI can bedesigned artificially via stacking 2 D layers . It providescontrollable approach to engineer ’homemade’ TIs andovercomes the limitation imposed by bulk crystal geom-etry.The above studies show the connection between the 2 D and 3 D TIs. It is notable that recently there are increas-ing interests in 1 D topological phases . Speciallythey have been studied experimentally using ultra-coldfermions trapped in the optical superlattice and photonsin photonic quasicrystals and metamaterials . Withthe developments of these techniques, various 1 D modelswith topological properties may be realized . Also thesetechniques can be easily extended to 2 D or 3 D cases. Itis also desirable to study the 1 D topological phase in realmaterials. A natural thought is to narrow a 2 D TI andthe narrow strip may be a 1 D TI. Also a formalism onhow to construct 2 D TIs from 1 D ones is needed. Theunderlying question is the connections between the 1 D and 2 D TIs.In the paper, the question is studied systematically. Itis found that the crossover from the 2 D quantum spinHall insulator to its 1 D limit shows oscillatory behaviorbetween a 1 D ordinary insulator and 1 D TI . Gener-ally the 2 D TI is a ’genuine’ 2 D phase and cannot beunderstood simply from the corresponding 1 D phases.However by arranging 1 D TIs, it is found that there ex-ists a 2 D weak topological phase in anisotropic systems.In contrast to the quantum spin Hall insulator: the 2 D weak TI has topological invariant Z = 0; the edge states a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l are mid-gap ones and only appear along specific bound-aries. These results provide further understanding on 2 D time-reversal invariant insulators, and can be realized ex-perimentally. The paper is organized as the following: InSec.II, the model Hamiltonian is introduced to describethe 1 D and 2 D TIs; In Sec.III, an oscillatory crossoverfrom 2 D to 1 D topological phases is observed; In Sec.IV, a 2 D weak TI is identified by arranging 1 D topolog-ical models; In Sec. V, the physical properties of the 2 D weak TI are studied; Finally Sec. VI is the conclusion ofthe present paper. II. THE D AND D TI MODELS G ap M/B -12 -8 -4 0 402 sp x y Eigenstate (b) (c)(a)
FIG. 1: (a) The phase diagram of the Hamiltonian Eq.(1) andEq.(2). (b) The schematic diagram of the 2 D model Eq.(1),in which the A -kind (blue solid lines) and B -kind (dashed andsolid black lines) hoppings along y -direction are shown. (c)The real hopping between the NN chains are converted to thehopping between the eigenstates of the NN chains. The starting point of the present work is the 2D tight-binding model for a quantum spin Hall insulator , H D = (cid:88) i ( M + 4 B )Ψ † i I ⊗ σ z Ψ i (1) − (cid:88) i, ˆ x B Ψ † i I ⊗ σ z Ψ i +ˆ x − (cid:88) i, ˆ y B Ψ † i I ⊗ σ z Ψ i +ˆ y − (cid:88) i, ˆ x sgn (ˆ x ) iA Ψ † i s z ⊗ σ x Ψ i +ˆ x − (cid:88) i, ˆ y sgn (ˆ y ) iA Ψ † i I ⊗ σ y Ψ i +ˆ y where I is the identity matrix and σ j , s j ( j = x, y, z )are the Pauli matrices representing the orbit and spin, respectively; Ψ i = ( s i ↑ , p i ↑ , s i ↓ , p i, ↓ ) T with s i ↑ ( ↓ ) ( p i ↑ ( ↓ ) )electron annihilating operator at site r i . The first termis the on-site potential, which has different signs for the s − orbit and p − orbit. The second and third terms arethe hopping amplitudes among the s − orbits or p − or-bits, which are differed by a sign. The third and fourthterms are the hopping amplitudes between the s − orbitand p − orbit electrons, which is due to the spin-orbitcoupling. A and B are the hopping amplitudes and inthe following of the paper we take B positive and set A = 1 as a unit of the energy scale. The Hamiltonian isinvariant under time-reversal T = is y ⊗ I K . It belongsto the AII class and its topological property is describedby a Z index . For M > , M < − B it is a trivialinsulator. For − B < M < Z = 1. In theHamiltonian, the subsystems of spin-up and -down aredecoupled.Based on the spin-up subsystem and reducing one di-mension (such as the y − dimension), a 1D spinless topo-logical model is obtained , H D = (cid:88) i ( M + 4 B )Ψ † i ↑ σ z Ψ i ↑ − (cid:88) i, ˆ x B Ψ † i ↑ σ z Ψ i +ˆ x ↑ (2) − (cid:88) i, ˆ x sgn (ˆ x ) iA Ψ † i ↑ σ x Ψ i +ˆ x ↑ . At half-filling, the system is a non-trivial insulator for − B < M < − B and a trivial insulator for M > − B or M < − B . In the momentum space itbecomes: H D ( k ) = [ M +4 B − Bcos ( k )] σ z +2 Asin ( k ) σ x .The Hamiltonian possesses a particle-hole symmetry σ x H ∗ D ( k ) σ x = − H D ( − k ), a pseudo-time-reversal sym-metry σ z H ∗ D ( k ) σ z = H D ( − k ) and a chiral symmetry σ y H D ( k ) σ y = − H D ( k ). It belongs to the BDI class andits topological invariant is a winding number ν , which isan integer . The winding number of the HamiltonianEq.(2) is ν = 1. In the case, it is equivalent to the Berryphase, which describes the electric polarization. TheBerry phase in the k space is defined as: γ = ¸ A ( k ) dk with the Berry connection A ( k ) = i (cid:104) u k | ddk | u k (cid:105) and | u k (cid:105) the occupied Bloch states . Due to the protectionof the symmetries in BDI class the Berry phase γ mod2 π can have two values: π for a topologically nontrivialphase and 0 for a topologically trivial phase. The topo-logical property is manifested by the boundary states ofzero energy on an open chain. The spin-down subsystem,which is the time-reversal counterpart of Eq.(2), has awinding number ν = −
1. Then the combined systemis time-reversal invariant with the time-reversal operator(the time-reversal operator T is the same as the one forEq.(1) ). Then the combined system belongs to DIII classand its topological invariant is a Z . The Z topologicalinvariant for s z conserved system can be calculated usingthe berry phase of either spin subsystem. III. THE OSCILLATORY CROSSOVER FROM D TO D TOPOLOGICAL PHASES
We consider the 2D TI model Eq.(1) in a narrowstrip configuration. Its finite-size effect has been studiedpreviously . It is found that on a narrow strip the edgestates on the two sides can couple together to producea gap in the spectrum. The finite-size gap ∆ ∝ e − λL ,decays in an exponential law with the width L . Thedecaying length scale is determined by the bulk gap ofEq.(1). As shown in Fig.2, the bulk gap vanishes at M/B = 0 , − , −
8, near which the finite-size gap is max-imum. Interestingly the narrow strip as a quasi- 1 D sys-tem shows 1 D topological phase in an oscillatory way.It is a 1 D topological phase with boundary states whenthe width L is odd, while trivial when the width L iseven. Since spin s z is conserved in Eq.(1), the oscillatorycrossover happens also for each spin subsystem. If thespins are coupled (such as by the Rashba spin-orbit cou-pling described in Sec. V), the above results still persist. -10 -8 -6 -4 -2 0 2 E n -202 PBCOBC M/B -10 -8 -6 -4 -2 0 2 E n -202 L=3L=4(a)(b) FIG. 2: (Color online) The eigenenergies at half filling andthe one above of Eq.(1) vs. M on a thin strip with the width:(a) L = 3; (b) L = 4. The length of the strip is N = 50with periodic boundary condition (PBC) or open boundarycondition (OBC). The oscillatory behavior happens near
M/B = − M/B = 0 , −
8, which sepa-rate a TI from a trivial insulator. It is noted in Fig.1that
M/B = − D topological phase ofEq.(2), which is reduced from Eq.(1). So the oscillatorybehavior is closely related to the corresponding 1 D topo-logical phase. In the following section, we construct the2 D TI model Eq.(1) from the point of view of coupled 1 D models Eq.(2) to understand the oscillatory behavior. IV. CONSTRUCTING A D MODEL FROM D TOPOLOGICAL MODEL
The 2 D model in Eq.(1) can be viewed as a set of thecoupled 1 D models in Eq.(2). In the limit of zero cou-pling, the boundary states of the 1 D topological modelform two flat bands at the two edges of the 2 D system,which are topological protected by the symmetry of theisolated 1 D model. Next we study the evolution of theboundary states as the 1 D chains are coupled by the hop-ping terms. Suppose the Hamiltonian of a 1 D open chainalong x − direction is H open,n D [the same as the one inEq.(2)] with n denoting the n − th chain (it is identical fordifferent n ). Since the Hamiltonian H open,n D has the chi-ral symmetry, its eigenenergies are symmetric to 0 and welabel them: { E } n = − E ( n )0 , E ( n )0 , − E ( n )1 , E ( n )1 , ... ( E ( n )0 WTI 1WTI 2WTI 2 TI 1TI 2TI 1 TI 2trivial trivial FIG. 4: The phase diagram of the modified HamiltonianEq.(1) with anisotropic B -kind hoppings in the ( M, B (cid:48) ) plane.In TI 1 (TI 2) the crossing of the edge states is at k y = 0( π )with y − directed boundary. The color represents the gap ofthe bulk system. In the previous section a weak 2 D TI is identified withthe mid-gap edge states along specific edges and the topo-logical invariant Z = 0. More generally the Hamiltonianin Eq.(1) can be modified by changing the amplitudeof the B -kind hopping along y − direction to B (cid:48) whichcan be tuned. With the points on which the gap closes,the phase diagram of the modified Hamiltonian in the( M, B (cid:48) ) plane can be obtained. As shown in Fig.4, be-sides the quantum spin Hall insulators and trivial insula-tors, the 2 D weak TI exists in three regions of the phase M M M M WTI 2WTI 2 W T I W T I trivialtrivialtrivial trivialWTI 1WTI 1 FIG. 5: The phase diagram of the modified HamiltonianEq.(1) with anisotropic mass in the ( M , M ) plane. Theinset schematically shows the pattern of the mass on the lat-tice. The regions without notation represent the quantumspin Hall insulator. The color represents the gap of the bulksystem. diagram. We distinguishes the 2 D weak TIs with themid-gap edge states appearing on x − edge (WTI 2) or y − edge (WTI 1), and the quantum spin Hall insulatorswith the gapless crossing appearing at k y = 0 (TI 1) or k y = π (TI 2).It is noticed that the 2 D weak TI exists in the regionswith anisotropic B -kind hoppings ( B (cid:48) (cid:54) = B ). Indeed theanisotropy is key to realize the phase . The anisotropycan also be induced in the parameter M of Eq.(1). Con-sider the case shown in Fig.5: the mass M is uniformalong y − direction, but has alternating values M , M along x − direction. The phase diagram in the ( M , M )plane is shown in Fig.5, in which the 2 D weak TI is iden-tified in six regions.Till now the 2 D weak TI is identified in the anisotropicsystems. Its topological invariant Z is zero, but thephase is different from the Z = 0 trivial insulators.It is desirable to characterize its topological property.It has been known that the Z topological invariantof a 2D time-reversal invariant insulator is defined as:( − ν = (cid:81) n j =0 , δ n n , with δ n n the time-reversal po-larization at the four time-reversal invariant momentaΓ i =( n n ) = ( n π ˆ x + n π ˆ y ), with n j = 0 , 1. The aboveconstructed 2 D Hamiltonian has the inversion symmetry H ( − k ) = ˆ P H ( k ) ˆ P with ˆ P the inversion operator. Inthe presence of the inversion symmetry, δ i can be deter-mined by the parity of the occupied band eigenstates: δ i = (cid:81) Nm =1 ξ m (Γ i ), where ξ m is the parity eigenvaluesof the 2 m -th occupied states. The topological propertyof 2 D time-reversal invariant insulators is determined byfour time-reversal polarizations δ i . In general δ i is notgauge invariant, while in the presence of inversion sym-metry δ i is gauge invariant. So for the Z = 0 time-reversal invariant insulators with inversion symmetry, thedetails of the four δ i should distinguish the 2 D weak TIsand the trivial insulators. - ++ - ky kx - + + - ky kx +++ + ky kx -- ky kx - ++ - ky kx - ++ - ky kx (a) (b) (c)(d) (e) (f) -- WTI 1 - WTI 2 trivial FIG. 6: Depicts δ i at the time-reversal invariant momentaof 2 D weak TIs and trivial insulators. The upper ones arefor the case of anisotropic B -kind hoppings: (a) WTI 1;(b)WTI 2;(c) trivial insulator. The lower ones are for the case ofanisotropic mass: (d) WTI 1;(e) WTI 2;(f) trivial insulator. For the case of anisotropic B -kind hoppings, the inver-sion operator is ˆ P = I ⊗ σ Z and δ i = − sgn ( M (Γ i )) with˜ M ( k ) = M + 4 B − B cos( k x ) − B (cid:48) cos( k y ). In WTI 1, ifthe boundary is along y − direction, the two δ i projectedon k y = 0( π ) have different signs, which is in contrastto the trivial insulator. Define the Z invariant π k µ theproduct of two δ i on the line k µ ( k µ = n i π, µ = x, y ),then the additional Z indices π k µ distinguish the 2 D weak TI and trivial insulators. π k µ is directly related tothe existence of the edge states on µ -directed boundaryand π k µ = − k µ .For example in the WTI 1 phase shown in Fig.6(a), if theboundary is along y − direction, k y remains good quan-tum number and π k y =0 = π k y = π = − 1. So there appearmid-gap edge states with two crossings at k y = 0 , π onthe boundaries. Also in quantum spin Hall insulators, π k µ determines the position of the crossing of the edgestates, which happens at k µ with π k µ = − P = I ⊗ diag ( σ z , e − ik y / σ z ) (the inversion centeris chosen on a site with the mass M ). The δ i of the 2 D weak TI phases are calculated, which is shown in Fig.6.It seems that the above discussion is inapplicable. Ac-tually to characterize the topological property properly,the δ i should be defined compared to the corresponding δ triviali of the trivial insulator in the same model, i.e.,˜ δ i = δ i δ triviali . With ˜ δ i , Fig.6(d) [(e)] is the same asFig.6(a) [(b)], and the 2 D weak TI is characterized cor-rectly.So to correctly characterize the 2 D weak TI with in-version symmetry, the δ i should be defined compared tothat of the trivial insulator in the same model. It can beunderstood from the view of band inverting. We take thecase of anisotropic mass as an example. Its Hamiltonianin the momentum space writes as, H M ( k ) = (cid:18) h ( k y ) h ( k x ) h † ( k x ) h ( k y ) (cid:19) , (3) M -20 -10 0 10 20 M -20-1001020 M -20 -10 0 10 20-20-1001020-20 -10 0 10 20 M -20-1001020 -20 -10 0 10 20-20-1001020 M -20 -10 0 10 20 M -20-1001020 +++- -- ++++ ++---- --+-+- +- WTI 2WTI 1 (a) (b) FIG. 7: (Color online) (a) The zero-gap line in ( M , M ) planefor the case of anisotropic mass at the four time-reversal in-variant momenta. + / − is the parity of the occupied band.(b) The phase diagram obtained by combining the zero-gaplines in (a). The arrow dashed line 1(2) is the path to WTI1(2) (the ones corresponding to (d) and (e) in Fig.6) from atrivial insulator. where h , ( k y ) = [ M , + 4 B − B cos( k y )] σ z +2 A sin( k y ) σ x and h ( k x ) = − B (1 + e − ik x ) σ z − iA (1 − e − ik x ) σ y . The eigenenergies and eigenvectors at the time-reversal invariant momenta can be obtained analytically.In Fig.7 (a) the zero-gap line in the ( M , M ) plane ateach time-reversal invariant momentum and the paritiesof the occupied bands are shown. When the zero-gap lineis crossed, there occurs a band inverting. By combiningall the zero-gap lines, the phase diagram in Fig.5 is re-covered. Any phase in the phase diagram can be reachedby band inverting starting from a trivial insulator. Thus˜ δ i = δ i δ triviali records the number of band inverting froma trivial insulator. ˜ δ i = − i -th time-reversal invariant momentum. Thenthe topological property can be analyzed correctly with˜ δ i .For the general case without inversion symmetry, thetopological property of the 2 D weak TI can be under-stood from the Berry phase of one spin subsystem, sincethe 2 D weak TI is closely related to 1 D topological phase.The Berry phase defined with k y ( k x ) at fixed k x ( k y ) canbe calculated. The symmetries which protects the 1 D topological phase is broken except at specific k x ( k y ). Ifthere are two k x ( k y ) at which the Berry phase is π , whichmeans the edge states exist and have two crossings in thepresence of x ( y )-directed boundary, the system is a 2 D weak TI.The above analysis is based on the system with spin s z conservation. However the result is applicable to anytime-reversal invariant insulators. Next we study the caseof the spin-up and -down subsystem coupled by Rashbaspin-orbit coupling, which preserve the time-reversal in-variant symmetry, H R = − (cid:88) i, ˆ x sgn (ˆ x ) iλ R Ψ † i s y ⊗ I Ψ i +ˆ x (4)+ (cid:88) i, ˆ y sgn (ˆ y ) iλ R Ψ † i s x ⊗ I Ψ i +ˆ y . Adding the above term to the modified version of theHamiltonian Eq.(1). For the case of anisotropic B -kindhoppings, the total Hamiltonian in the momentum spaceis, H T ( k ) = ˜ M ( k ) I ⊗ σ z + ˜ A ( k x ) s z ⊗ σ x + ˜ A ( k y ) I ⊗ σ y + ˜ λ R ( k x ) s y ⊗ I − ˜ λ R ( k y ) s x ⊗ I. (5)Its energy spectrum is:( E T ( k )) = [ ˜ A ( k x )] + {± (cid:113) [ ˜ λ R ( k x )] + [ ˜ λ R ( k y )] (6)+ (cid:113) [ ˜ M ( k )] + [ ˜ A ( k y )] } . with ˜ A ( k x ) = 2 Asin ( k x ), ˜ A ( k y ) = 2 Asin ( k y ), ˜ λ R ( k x ) =2 λ R sin( k x ), ˜ λ R ( k y ) = 2 λ R sin( k y ). As has been known,the Rashba spin-orbit coupling does not break the quan-tum spin Hall effect when it is small. In the following weshow that the weak TI is also robust to it. From Eq.(6),it is noticed that the gap closing is independent of λ R for λ R < A . Since the topological quantum phase transitionoccurs when the gap closes, the weak TI is not affected bythe Rashba spin-orbit coupling in this case. For λ R > A ,the gap of the bulk system vanishes at a critical λ cR , whenthe weak TI is broken. The calculated energy spectrumand time-reversal polarization δ i are consistent with theabove analysis. For the case of anisotropic mass, the re-sult is similar. (k y =0) a b M/B -6 -4 -2 B ' / B -2-1012 a b WTI 1TI 1 y k x / π E k x / π E (c)(a)(b) FIG. 8: (Color online) (a) The bulk gap at k y = 0 withthe parameters along the boundary of WTI 1 and TI 1 in thephase diagram (the bulk gap at k y = π is all zero). The energyspectrum on a y -directed strip with the width: (b) L = 50 and(c) L = 10. The parameters are: M/B = − , B (cid:48) /B = 0 . As has been stated, since the mid-gap edge states in the2 D weak TI are related to the corresponding 1 D bound-ary modes, it can be destroyed by the disorder. Howeverunder specific conditions, they can be robust as those inthe Z = 1 quantum spin Hall insulators. It can happenin the region near the boundary between WTI and TIphases. For example, for the case with anisotropic B -kindhoppings, in the middle of the boundary between WTIand TI phases, the gaps at the two valleys k x ( y ) = 0 , π isdifferent. For the case shown in Fig.8, the gap at k y = π is nearly zero while is still large at k y = 0. Consideringa narrow strip with its edges along y − direction, due tothe finite-size effect, the edge states at k y = π is gapped,while the ones at k y = 0 persist. Thus a single robustgapless crossing at k y = 0 is realized in the finite-sizegap. VI. CONCLUSIONS AND DISCUSSIONS Dimensional evolution between 1 D and 2 D topologicalphases is investigated systematically. The crossover froma 2 D TI to its 1 D limit shows oscillatory behavior be-tween a 1 D ordinary insulator and 1 D TI. By construct-ing a 2 D topological system from 1 D TI, it is shownthat there exist the weak topological phase in 2 D time-reversal invariant band insulators. The phase can be re-alized in anisotropic systems. In the weak phase, thetopological invariant Z = 0 and the edge states only ap-pear along specific boundaries. Since the edge states areclosely related to the boundary states of the correspond-ing 1 D topological phase, they may be destroyed by dis-order and have symmetry-protecting nature as the corre-sponding 1 D topological phase. The effect of the Rashbaspin-orbit coupling, which preserves time-reversal invari- ant symmetry, but couples the spins, is also studied.These results provide further understanding on 2 D time-reversal invariant insulators.Finally we discuss the relevance of the results to ex-perimental measurements. It is unclear whether the2 D weak TI materials exist in nature. However sincethe anisotropy is important, it should be searched inanisotropic materials. Besides real materials, recentlythe double-well potential formed by laser light has beendeveloped , in which s and p orbital cold-atoms can beloaded. 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