One-dimensional topological metal
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r One-dimensional topological metal
Masoud Bahari and Mir Vahid Hosseini ∗ Department of Physics, Faculty of Science, University of Zanjan, Zanjan 45371-38791, Iran (Dated: April 23, 2019)We propose a new type of topological states of matter exhibiting topologically nontrivial edge states (ESs)within gapless bulk states (GBSs) protected by both spin rotational and reflection symmetries. A model pre-senting such states is simply comprised of a one-dimensional reflection symmetric superlattice in the presenceof spin-orbit (SO) coupling containing odd number of sublattices per unit cell. We show that the system has arich phase diagram including a topological metal (TM) phase where nontrivial ESs coexist with nontrivial GBSsat Fermi level. Topologically distinct phases can be reached through subband gap closing-reopening transitiondepending on the relative strength of inter and intra unit cell SO couplings. Moreover, topological class of thesystem is AI with an integer topological invariant called Z index. The stability of TM states is also analyzedagainst Zeeman magnetic fields and on-site potentials resulting in that the spin rotational symmetry around thelattice direction is a key requirement for the appearance of such states. Also, possible experimental realizationsare discussed. Introduction. —The search for exotic quantum states ofmatter has attracted a great deal of attention since discov-ery of topological insulators (TIs) [1] and topological super-conductors (TSs) [2] in condensed matter physics. Furtherinvestigations have also revealed a novel nontrivial topologi-cal states in the so-called Weyl semimetals possessing gaplessbulk and Fermi arc surface states [3]. In contrast, TIs and TSshave symmetry protected edge states (ESs) inside gapped bulkstates. Apart from condensed matter systems, some schemeshave been proposed to realize topological phases using coldatoms in optical lattices [4] and employing light in photoniccrystals [5]. Also, the exploration of topological states hasbeen extended even to classical systems [6].In most of the TIs, the symmetry protected ESs, making rel-evant requirement for topological quantum computations [7],play dominant role only in a limited certain range of energies.Moreover, due to smallness of energy gap, ESs may also befaded by excitations of bulk states at finite temperature. Onthe other hand, the coexistence of ESs and bulk states occursin a narrow energy window in three-dimensional TI candidatematerials such as Bi Se and Bi Te [8]. However, it is in-triguing to have a situation in which dominant symmetry pro-tected ESs exist not only in the bandgap but also within thegapless bulk states (GBSs). Therefore, such systems would bein turn served as topological metals (TMs) [9] even by shiftingFermi level toward conduction or valence bands.Several one-dimensional (1D) models have been studied torealize new classes of topological phases [10] concerning bothTIs [11] and TSs [12]. These studies stimulate to look for newpossibilities for nontrivial topological states mimicking nei-ther TIs nor TSs. So far, however, metallic phase being quasi-degenerate with topologically protected ESs has not been re-ported to be nontrivial in topology. Hence, it is interestingto develop a minimal and feasible model by which TMs canbe emerged easily. In the present Letter, we consider a 1Dspin-orbit (SO)-coupled superlattice with odd number of sub-lattices per unit cell, as shown in Fig. 1(a), featuring variousnontrivial phases [Figs. 2(a) and 2(b)]. Surprisingly, we findthat topological ESs while keeping their non-triviality can ex- FIG. 1. (Color online) (a) Schematic illustration of reflection sym-metric superlattice containing three sublattices per unit cell (blue,red and green balls). The mirror point is located at the second sub-lattice of middle unit cell. The energy spectrum dependence of ˆ H with its IPR on (b) [(e)] λ ′ with δt = − (+) t/ and λ = − λ ′ ( λ = λ ′ + 2 . t ) for 60 unit cells. Solid lines and hexagrams rep-resent the spectra of ˆ h U = − and ˆ h U =+ , respectively. Panels (d) and(g) [(c) and (f)] are the corresponding invariant Z ( Z U = ± ). tend into GBSs with increasing SO interaction as shown inFigs. 1(b) and 1(e). This results in nontrivial TM phase dueto settling SO coupling on odd number of sublattices providedthat spin rotational and reflection symmetries are not broken.It should be noted that our findings are valid in general physi-cal grounds, independent of our model parameters, and there-fore may be realized in a variety of platforms such as con-densed matter systems and quantum Fermi gases. Model. —We consider a 1D multipartite SO-coupled super-lattice along x-axis with period T ≥ described by the totaltight-binding Hamiltonian as ˆ H = ˆ H t + ˆ H so , (1)where kinetic ( ˆ H t ) and SO ( ˆ H so ) terms are given by FIG. 2. (Color online) Topological phase diagram in the plane ( λ ′ , λ ) with T = 3 and 160 unit cells for (a) δt = − t/ , (b) δt = + t/ . The system is a TM at E = 0 indicated by dashedlines. ˆ H t = P n,σ P Tα [ t α ˆ c † α,n,σ ˆ c α +1 ,n,σ + h.c. ] and ˆ H so = P n,σ P Tα [ λ α ˆ c † α,n,σ ˆ c α +1 ,n, − σ + h.c. ] , respectively. The op-erator ˆ c † α,n,σ (ˆ c α,n,σ ) is fermion creation (annihilation) oper-ator of electrons with spin σ = ( ↑ , ↓ ) on α sublattice of n thunit cell. t α ( λ α ) denotes the hopping (SO coupling) ampli-tude. Considering periodic boundary conditions and perform-ing fourier transformation, the Hamiltonian (1) can be writtenin the basis of ˆ ψ = (ˆ c ,k,σ , ˆ c ,k,σ , ..., ˆ c T,k,σ ) T with ˆ c α,k,σ =(ˆ c α,k, ↑ , ˆ c α,k, ↓ ) yielding a compact form ˆ H = P k ˆ ψ † ˆ H ( k ) ˆ ψ with ˆ H ( k ) = h ˆ h T e − ik ˆ h . . . . . .. . . . . . ˆ h T − ˆ h T e ik ˆ h T − T × T , (2)where ˆ h α = t α I + λ α τ x with I and τ x being identity ma-trix and the x-component of Pauli matrix acting on spin sub-space, respectively, and α = 1 , ..., T . When λ α = λ T − α and t α = t T − α , our model preserves unitary reflection symmetryabout a 1D mirror point (located at the middle of unit cell) as R ˆ H ( k ) R − = ˆ H ( − k ) with R = δ i,T +1 − j ⊗ τ x where δ i,j is Kronecker delta. Also, because the x-component of spin isa good quantum number, the lattice has a U(1) spin rotationalsymmetry. So ˆ H ( k ) is invariant under spin rotation operator U = I T ⊗ τ x around the x-axis where I T is an identity matrixof size T . Although the usual time-reversal symmetry is bro-ken due to the presence of SO coupling, however, an effectivetime-reversal symmetry can be determined as T ˆ H ( k ) T − =ˆ H ( − k ) with T = I T ⊗ τ x K where K is the complex con-jugate. Note the reflection, spin rotation, and time-reversaloperators show the properties R = U = T = 1 . Sincethe symmetries of Hamiltonian are based on the conventionalsymmetries, thus the topology of ESs can be classified follow-ing the general classification [13, 14] of topological systems.Due to [ R , T ] = 0 , the topological classification of systembelongs to AI class with topological index Z .Since the spin rotation operator commutes with ˆ H ( k ) , theHamiltonian can be block-diagonalized into two T × T Hamil-tonians as ˆ H ( k ) = ˆ h U = − ( k ) ⊕ ˆ h U =+ ( k ) whose decoupled subspaces are spanned by eigenstates of U with eigenval-ues ± . This can be done through a unitary transforma-tion ˆ H ( k ) = U ˆ H ( k ) U † where U is constructed from theeigenspace of U and will be characterized below. Each blockof ˆ H ( k ) takes the form ˆ h U = ± ( k ) = ± Γ ± T e ik Γ ± . . . . . .. . . . . . Γ ± T − Γ ± T e − ik Γ ± T − T × T , (3)where Γ ± α = t α ± λ α . Note that ˆ h U = ± has both reflectionand time-reversal symmetries because of [ U , R ] = 0 and [ U , T ] = 0 . Now, we define the topological invariant Z asfollows [15]. Each of ˆ h U = ± commutes with reflection opera-tor at reflection symmetric momenta k ref = (0 , π ) , thus eigen-states of ˆ h U = ± have a well-defined parity ζ U = − (+) ( k ref ) = ± at those points. This, subsequently, allows for specifying aninteger invariant N i, U = ± = | n ,i, U = ± − n ,i, U = ± | to clas-sify ˆ h U = ± . Here, we have defined n ,i, U = ± and n ,i, U = ± asthe number of negative parities related to the energy bandsof ˆ h U = ± ( k ) in the i th bandgap at k ref = 0 and k ref = π , re-spectively. So, topological number Z for multi-subspace andmulti-band structure of the system can be defined as Z := X j = ± Z U = j = X j = ± T − X i =1 N i, U = j , (4)giving the number of localized ESs under open boundary con-ditions. Here, Z U = ± denotes the topological invariant of sub-spaces.Interestingly, each subsystem, described by ˆ h U = ± , is simi-lar to a 1D spinless system consisting of T ”super-sublattices”per unit cell. Each super-sublattice is comprised of a sublatticewith opposite spin species so that the new hopping amplitudebetween two adjacent super-sublattices in the subsystem la-belled by U = ± is Γ ± α . This can be illuminated by a transfor-mation from the old basis to the new one through the unitarymatrix U T × T as ˆΨ i = P Tj =1 U i,j ˆ ψ j where i = 1 , ..., T .The non-zero matrix elements of U are U α, T − α +1) = 1 √ , U T − α +1 , α − = 1 √ ,U α, T − α )+1 = − √ , U T − α +1 , α = 1 √ . Therefore, the new basis is ˆΨ = ( ˆΨ − , ˆΨ + ) T where ˆΨ − (+) corresponds to the basis of eigenspace of U with eigenvalues − whose entries are super-sublattices given by ˆΨ ± α =1 / √ c T − α +1 ,k, ↓ ± ˆ c T − α +1 ,k, ↑ ) .On the other hand, it is also easy to obtain real-spaceoperator for the spin rotational symmetry as U = I T N ⊗ τ x where I T N is an identity matrix of size
T N with N being thenumber of unit cells. Therefore, the real-space Hamiltonian(1) can be brought into two block-diagonal matrices in theeigenspace of U = ± as ˆ H = ˆ h U = − ⊕ ˆ h U =+ . In orderto study the localization of states, we calculate normalizedlogarithm of the inverse participation ratio (IPR) of aneigenvector | ψ E i associated with eigenenergy E as definedby I E = Ln ( P T Ni =1 |h i | ψ E i| ) / Ln (2 T N ) where | i i is basiselements [16]. Here, I E = − denotes delocalized states,whereas for much more localized ones I E = 0 . Results and discussion. —Without loss of generality, we fo-cus on the case of three sublattices per unit cell, T = 3 , asthe model illustrated in Fig. 1(a). Reflection symmetry re-quires λ = λ ≡ λ ′ and t = t ≡ t ′ . Here, the intraand inter unit cell hoppings, respectively, are t ′ = t − δt and t = t + δt where t ( δt ) stands for hopping energy (hop-ping modulation strength). Energy spectrum of ˆ H and its IPRas a function of intra unit cell SO coupling strength λ ′ underopen boundary conditions is shown in Fig. 1(b) with inter unitcell SO coupling strength λ = − λ ′ and δt < . The solidlines and hexagrams represent the eigenvalues of ˆ h U = − and ˆ h U =+ , respectively. As λ ′ increasing, two gap closings occursimultaneously in the subbands of ˆ h U = − away from Fermienergy at λ ′ = 0 . t and then two degenerate localized ESsemerge in the bandgaps. Interestingly, with further increasein λ ′ , the pairs of ESs enter to the GBSs of ˆ h U =+ at λ ′ = t .These topological ESs meet each other at Fermi energy with λ ′ = t ′ = 1 . t leading to the appearance of strongly local-ized fourfold degenerate states. Finally, they stay in the GBSswhile preserving their localization with an extra increase of λ ′ . The corresponding topological Z integer is plotted in Fig.1(d). In the parameter region where the ESs appear, Z integertakes value 2 demonstrating the existence of two pairs of lo-calized ESs steaming from Z U = − = 2 (shown in Fig. 1(c))whereas Z U =+ = 0 . As a result, topologically protected ESsof an eigenspace could penetrate into trivial GBSs of the otherone.In particular, both of the eigenspaces may host nontrivialtopological phases by choosing appropriate SO coupling val-ues in a way that topological phase transitions occur in bothsubspaces. This will result in appearance of four pairs of ESs.In Figs. 1(e) and 1(g), respectively, the dependence of en-ergy spectrum ˆ H and topological invariant Z on λ ′ is pre-sented for δt > and λ = λ ′ + 2 . t . As shown in Figs.1(e) and 1(f), the first topological phase transition happensin the ˆ h U =+ spectrum leading to appearance of highly lo-calized ESs in the bandgaps (GBSs) for the parameter space λ ′ ∈ ( − . t, − . t )[( − . t, − . t )] . After taking placeof the second topological phase transition in the ˆ h U = − spec-trum at λ ′ = − . t , two new ESs are emerged in addition tothe former ones with Z U = − = 2 . Therefore, the system hostsfour topological ESs Z = 4 , as shown in Fig. 1(g). When λ ′ further increases, surprisingly, topological ESs of the ˆ h U = − spectrum reside inside the nontrivial GBSs of ˆ h U =+ and sys-tem re-enters to the TM phase. This is in contrast to the case oftopological bound states embedded in non-topological contin-uous spectrum [17]. Moreover, the electron-like and hole-likeESs intersect each other at λ ′ = ± t ′ = ± . t [Fig. 1(e)]. Remarkably, from both Figs. 1(b) and 1(e), one can see thatESs of an eigenspace at Fermi energy are quasi-degeneratenot only with their own highly degenerate GBSs but also withGBSs of the other eigenspace establishing TM phase. Notealso that the Z values change at which the subband gap clos-ing/reopening occurs [Figs. 1(d) and 1(g)].In order to shed light on the mechanisms underlying theabove-mentioned behaviors, we focus on understanding theeffect of spin and sublattice degrees of freedom on band struc-ture. In fact, the odd number of sublattices provides bulkmetallic ground states resulting in breaking particle-hole andchiral symmetries. Therefore, possible bandgaps can only oc-cur away from Fermi surface. Now, exploiting SO couplingbreaks spin degeneracy and subsequently each band splits intotwo subbands corresponding to two different helical compo-nents. The resulting spin helical subbands retain the nontriv-ial ESs and at the same time push them into the metallic bulkstates as a consequence of U(1) symmetry. As such, if the sys-tem is in a topologically nontrivial phase then the Fermi levelcrosses at some of the topological ESs embedded in GBSs.The topological phase diagram of system in the plane( λ ′ , λ ) is depicted in Figs 2(a) and 2(b) for δt = − t/ and δt = + t/ , respectively. The black solid lines denote the bor-der between topologically different phases. Also, the dashedlines correspond to TM states at Fermi level. The topologicalphase diagrams are categorized into three distinct phases ac-cording to Z = 0 , , and implying none, two, and four pairsof ESs, respectively. The regions Z = 2 and are dividedinto subcategories depending on the appearance of ESs eitherwithin GBSs or in the bandgaps. In both diagrams, there arethree possibilities for the region Z = 4 : i) four pairs of ESswithin GBSs, ii) four pairs of ESs in the bandgaps, and iii)two pairs of ESs in the bandgaps and the other two pairs ofESs within GBSs. In addition, the regions Z = 2 have twopossibilities that are two pairs of ESs within GBSs and withinbandgaps. Also, the regions of Z = 2 and Z = 4 with ESs inthe bandgaps for δt = + t/ are more dominant than those of δt = − t/ . Stability of ESs. —The existence of topological ESs that arequasi-degenerate with GBSs is ensured by the presence ofspin rotational symmetry. To illustrate this feature, let us in-vestigate ESs stability against perturbations originated fromon-site potential and Zeeman magnetic field. We add theterm ˆ H ′ = ˆ H V + ˆ H B to Hamiltonian (1) including on-siteHamiltonian ˆ H V = P n,σ P Tα V α,n ˆ c † α,n,σ ˆ c α,n,σ and ZeemanHamiltonian ˆ H B = P n,σ,σ ′ P Tα ˆ c † α,n,σ ( M n,α · τ )ˆ c α,n,σ ′ .Here, V α,n defines the amplitude of on-site potential, τ isthe Pauli spin vector and the Zeeman field vector is M n,α =( M n,x,α , M n,y,α , M n,z,α ) . For concreteness, we will inspectthe effects of these perturbations on the topological propertiesseparately.We first investigate only the effect of Zeeman field. Sincethe lattice is invariant under rotations about the x-axis, we ap-ply Zeeman field along the y-axis. This Zeeman field violateslattice U(1) symmetry and, in consequence, ˆ H ( k ) can not be FIG. 3. (Color online) Band structure with its IPR as a function of λ ′ for (a) δt = − t/ , (b) δt = + t/ in the presence of y-componentof staggered Zeeman field chosen as M y, = M y, and M y, = − M y, = t/ . Other parameters of panels (a) and (b) are the sameas Figs. 1(b) and 1(e), respectively. block diagonalized. Accordingly, the topological invariant (4)is no longer valid. To discriminate the role of U(1) symme-try from that of reflection symmetry, we break U(1) symmetrysuch that the reflection symmetry remains untouched. To doso, we need to set M y, = M y, . Under such situation, theenergy spectrum of Figs. 1(b) and 1(e) is re-calculated andplotted in Figs. 3(a) and 3(b), respectively, in the presenceof y-component of staggered Zeeman field. Interestingly, onecan observe that ESs within GBS can not survive resultingin termination of ESs from at least one end with no gap clo-sure. While in-gap states preserve their degeneracies owingto the reflection symmetry. Although this feature is obtainedfor staggered Zeeman field but similar results hold for uni-form Zeeman field. Otherwise, preserving the spin rotationalsymmetry and violating reflection one lead to destroying ESsand making the system topologically trivial (not shown). Asa result, the TM phase manifests itself whenever both reflec-tion and U(1) symmetries are present simultaneously in thesystem.In addition, the x-component of Zeeman field or on-site po-tential conserve the U(1) symmetry. One readily finds thatthese terms have diagonal entries in each block of transformedHamiltonian ˆ H ( k ) = (ˆ h U = − ( k ) + ˆ V U = − ) ⊕ (ˆ h U =+ ( k ) +ˆ V U =+ ) with ˆ V U = ± = µ ± i δ i,j where µ ± i = V T − i +1 ± M x,T − i +1 . Here, reflection symmetry requires V i = V T − i +1 and M x,i = M x,T − i +1 , for which we set V = V and M x, = M x, in the case of T = 3 . Obviously, the uni-form on-site potential (x-component of Zeeman field) shiftsthe energy levels of each block of ˆ H ( k ) to the same (oppo-site) direction. These enable us to shift the energies of ESsappearing away from Fermi level toward E = 0 while theirquasi-degeneracy with GBSs remain intact. Therefore, theTM phase will be accessible for much larger range of SOcouplings. Moreover, interestingly, either alternating on-sitepotential or x-component of Zeeman splitting can impose gapclosing in blocks of ˆ H ( k ) independently. So, the number ofESs would be asymmetric about Fermi level resulting in in-ducing odd number of ESs. The energy spectra of a finitechain as functions of λ ′ in the presence of a uniform Zeemanfield along x-axis and of V are depicted in Figs. 4(a) and 4(d), FIG. 4. (Color online) Energy spectrum of ˆ H with its IPR versus(a)[(d)] λ ′ ( V ) with δt = − t/ : λ = λ ′ and M x, = M x, = M x, = t/ ( λ ′ = t, λ = 2 . t, V = V and V = − t ) for 60 unitcells. Panels (c) and (f) [(b) and (e)] are the corresponding invariant Z ( Z U = ± ) . respectively. One can see from Fig. 4(a) that the x-componentof Zeeman field splits the energy spectra of ˆ h U = − and ˆ h U =+ relative to each other, as already mentioned. Also, three pairsof ESs can be seen in the energy spectrum of Fig. 4(d). Thecorresponding invariants Z ( Z U = ± ) are shown in panels (c)and (f) [(b) and (e)] of Fig. 4. Experimental proposal. —Recent experimental achieve-ments make it possible to realize 1D dimerized lattice model,known as Su-Schrieffer-Heeger (SSH) model [11], by fabri-cating heterostructures of alternating thin films of band in-sulators and TIs [18] relying on condensed matter physics.Also, modulated SO coupling can be implemented either byapplying local electric field [19] or by using cluster of heavyatoms [20]. In the latter case, due to proximity effect, SO cou-pling can be transferred from the bands of heavy atoms to thebands of system such that other properties of the structure it-self remain unaffected. On the other hand, using cold atomsin optical lattices provide an excellent playground with theeasy tunability of control parameters to simulate topologicalbands [21] in artificial quantum systems like SSH chain [22].For cold-atom experiments, we suggest to employ superpo-sition of retroreflected laser beams or to imprint superlatticewith a spatial light modulator producing extended SSH modelwith three number of sublattices [23]. Furthermore, it is pos-sible to engineer SO interaction in tripartite lattices [24] evenfor neutral cold-atomic gases [25]. Using spatially resolvedradio-frequency spectroscopy [26], the ESs within GBSs canbe recognized from the local density of states.
Conclusions. —We revealed a new kind of exotic topolog-ical states characterized by the coexistence of topologicallynontrivial ESs and nontrivial GBSs either at or far from theFermi level. The main ingredient of these exotic metallicstates arises from the coupling of odd number of sublatticesto spin degree of freedom in a 1D periodic arrays. We foundthat such systems undergo a topological phase transition un-der subband gap closure conditions. The effects of Zeemanfields and on-site potentials on the topological phases indicatethat TM states are protected by both reflection and spin rota-tional symmetries. The concept of nontrivial TM phase maybe generalized to higher invariants and non-hermitian case, aswell as including interaction effects.
ACKNOWLEDGMENT
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