We analyze the energy spectrum and eigenstates of cold atoms in a tilted
brick-wall optical lattice. When the tilt is applied, the system exhibits a
sequence of topological phase transitions reflected in an abrupt change of the
eigenstates. It is demonstrated that these topological phase transitions can be
easily detected in a laboratory experiment by observing Bloch oscillations of
cold atoms.Comment: 3 pages, 4 figure

Kolovsky, Andrey R.

English

arXiv

We analyze the energy spectrum and eigenstates of cold atoms in a tilted brick-wall optical lattice. When the tilt is applied, the system exhibits a sequence of topological phase transitions reflected in an abrupt change of the eigenstates. It is demonstrated that these topological phase transitions can be easily detected in a laboratory experiment by observing Bloch oscillations of cold atoms

Andrey R. Kolovsky

Siberian Federal University Digital Repository

PHYSICAL REVIEW A 98, 013603 (2018)Topological phase transitions in tilted optical latticesAndrey R. KolovskyL. V. Kirensky Institute of Physics SB RAS, 660036 Krasnoyarsk, Russiaand School of Engineering Physics and Radio Electronics, Siberian Federal University, 660041 Krasnoyarsk, Russia(Received 7 March 2018; published 5 July 2018)We analyze the energy spectrum and eigenstates of cold atoms in a tilted brick-wall optical lattice. When thetilt is applied, the system exhibits a sequence of topological phase transitions reflected in an abrupt change ofthe eigenstates. It is demonstrated that these topological phase transitions can be easily detected in a laboratoryexperiment by observing Bloch oscillations of cold atoms.DOI: 10.1103/PhysRevA.98.013603I. INTRODUCTIONThe Bloch band theory is a cornerstone of solid statephysics. It gave a simple explanation for the fact that somecrystalline materials are electric insulators while the othersare conductors. However, after discovery of the quantum Halleffect in 1980 it was gradually recognized that this divisionof solid crystals into band insulators and metals is incompleteand there are materials which are insulating in the bulk butconducting at the edges. These materials with an unusual Blochband structure were named topological insulators and theirstudy became a blossoming branch of solid state physics andother fields [1–4].An excellent playground for studying different models oftopological insulators is provided by cold atoms in opticallattices. For example, in Ref. [5] the authors realized theone-dimensional Su-Schrieffer-Heeger (SSH) model [6] andmeasured the Zak phases for two different dimerization ofthe SSH lattice. In Ref. [7], the two-dimensional lattice witheffective magnetic field was created and the Chern numberof the ground magnetic band was measured by observing theanomalous velocity of atoms. The cold atom realization of theHaldane lattice [8] was reported in Ref. [9].In this work, we discuss a class of topologically nontrivialsystems which should be treated as quasi-one-dimensional.These are the tilted bipartite lattices. Topological phase tran-sitions in tilted bipartite lattices were mentioned in Ref. [10],which studied the biased dice lattice. The dice lattice has arather particular geometry that leads to the Bloch spectrumconsisting of three bands, with one band being completelyflat. While the emphasis of Ref. [10] was on flat bands, inthe present work we focus on the topological phase transitionsand their physical manifestations that can be detected in thepresent-day laboratory experiments with cold atoms. Puttinglaboratory accessibility on the first place, we consider thebrick-wall lattice which was used earlier in the experiment [11]to study the effect of Dirac cones. The network connectivityof the brick-wall lattice is shown in Fig. 1 where the solidand dashed bonds correspond to the strong J1 and weak J2coupling, respectively. Here we analyze the simplest casewhere the brick-wall lattice is tilted in the y direction. In otherwords, the atoms are subject to a potential field F alignedwith the y axis. The field breaks translational invariance of thesystem in the y direction and causes the atoms to be localizedwithin the Stark localization length ∼ 1/F . However, in thex direction translational symmetry is preserved and, hence,atomic eigenstates are extended Bloch-like states. Localizationin one direction together with translational invariance in theother relates the system to the SSH model. However, due tothe presence of an extra parameter F , the system is essentiallyricher than the celebrated SSH lattice. In particular, the tiltedbrick-wall lattice preserves its topological properties even ifthe hopping matrix element J2 is strictly zero.II. WANNIER-STARK ENERGY BANDSWe begin with the energy spectrum of a quantum particlein the tilted brick-wall lattice. As for any bipartite lattice, itconsists of pairs of one-dimensional energy bands arrangedinto the Wannier-Stark ladder [12]; see Fig. 2. Numericallythis spectrum can be found by using at least two differentmethods—by diagonalizing the truncated Hamiltonian matrixor by calculating and diagonalizing the Floquet operator, whichis the system evolution operator over the Bloch period. Thelatter approach has certain advantages over the former becauseit (i) does not require truncation of the Hilbert space and (ii)opens prospects for analytical studies of the spectrum [12,13].In the framework of the Floquet operator approach, theeigenstates of a quantum particle in a tilted bipartite latticeare conveniently characterized by the column vector func-tion Y(θ ; κ) = [YA(θ ; κ),Y B(θ ; κ)]T whose Fourier transformgives occupation amplitudes of the lattice sites,ψA,Bl,p ∼ eiκdl∫ 2π0YA,B(θ ; κ)eipθdθ. (1)In Eq. (1), d denotes the lattice period, l = x/d and p = y/dlabel the lattice sites, and κ is the transverse quasimomentum(i.e., the quasimomentum of the Bloch wave in the x direction).According to Ref. [10], it can be shown that the functionY(θ ; κ) obeys the ordinary differential equationidFdYdθ= G(θ ; κ)Y, (2)2469-9926/2018/98(1)/013603(4) 013603-1 ©2018 American Physical SocietyANDREY R. KOLOVSKY PHYSICAL REVIEW A 98, 013603 (2018)−3 −2 −1 0 1 2 3−3−2−10123xyFIG. 1. The brick-wall lattice. The bonds marked by the solid anddashed lines corresponds to the different hoping matrix elements J1and J2. Notice that for J2 = 0 the brick-wall lattice transforms into thedeformed honeycomb lattice with A and B sites marked by asterisksand open circles.where the matrix G(θ ; κ) is of the formG(θ ; κ) =(0 ff ∗ 0), (3)and f (θ ; κ) = J2 + J1(ei2dκ + 2eidκ cos θ ). Using Eq. (2), oneconstructs the 2 × 2 unitary (Floquet) matrixU (κ) = êxp[idF∫ 2π0G(θ ; κ)dθ], (4)FIG. 2. Energy spectrum of the tilted brick-wall lattice. Param-eters are J1 = 0.5, J2 = 0, and dF = 1/1.9 (left panel), dF = 1/2(middle panel), and dF = 1/2.1 (right panel). Through the paper, weuse arbitrary energy units since only the ratio between the hoppingand Stark energies matters.1 2 3 4 5-4-2024W1 2 3 4 51/dF00.51d/2FIG. 3. Upper panel: Winding number of the phase χ vs theinverse field magnitude. Lower panel: The energy band E(1)n=0 as acolor map. The limits of the energy axis are Emin = 0 (dark blue)and Emax = dF/2 (bright yellow). The values of the hopping matrixelements are the same as in Fig. 2.whose eigenvalues λ1 and λ2 = λ∗1 determine the Wannier-Stark spectrum through the relationE(1,2)n (κ) = dFn + idF2πln λ1,2(κ). (5)In what follows, we take the convention that index 1 alwaysrefers to the upper band (solid blue lines) and index 2 refers tothe lower band (dashed red lines).We mention in passing the analogy between the Wannier-Stark spectrum of the brick-wall lattice and the Bloch spectrumof the SSH lattice. The latter is known to be given byeigenvalues of the 2 × 2 Hermitian matrixH (κ) =(0 ff ∗ 0), (6)where f (κ) = J2 + J1eidκ [1]. If ratio J2/J1 is varied, theBloch bands develop a conical intersection when J1 = J2 andthe SSH model shows a topological phase transition reflectedin a change of the Zak phase. Notice that the existence ofthis topological invariant is ensured by the particular structureof H (κ), which is the same as for the matrix G(θ,κ) in thedefinition the Floquet operator (4). Thus we may expect asimilar phenomenon in the tilted brick-wall lattice where theWannier-Stark energy bands are sensitive not only to the ratioJ1/J2 but also to variation of F . For future reference, thelower panel in Fig. 3 shows the upper band with the ladderindex n = 0 as a function of the quasimomentum and inversefield magnitude. Our particular interest in this figure are pointsof conical intersections between energy bands of differentsymmetry, seen as the dark and bright spots.III. TOPOLOGICAL INVARIANTWe proceed with topological phase transitions. The objectto study are eigenvectors Y1,2(κ) = [YA1,2(κ),Y B1,2(κ)]T of theunitary matrix (4). Because of the particular algebraic structure013603-2TOPOLOGICAL PHASE TRANSITIONS IN TILTED … PHYSICAL REVIEW A 98, 013603 (2018)of the matrix (3), the eigenvectors can be shown to have thefollowing simple form:Y1,2 =(1√2eiχ± 1√2e−iχ), (7)where χ is a function of the quasimomentum and without anyloss of generality we can require χ (κ = 0) = 0. FollowingRef. [10] we introduce a topological invariant—the windingnumber W [14],W = 12πi∫ 2π0angle(YAjYBj)dκ = 1πi∫ 2π0e−iχddκeiχdκ,(8)which is an integer number because Y1,2(κ) are periodicfunctions of the quasimomentum. Notice that W is not affectedby gauge transformations when the eigenvectors are multipliedby a phase factor exp[i(κ)] with (κ) being an arbitraryperiodic function of κ . As F is varied, the winding number(8) may change its value only at some critical values of thefield, where energy bands of different symmetry touch eachother; see upper panel in Fig. 3. The observed jumps of thewinding number is a formal manifestation of topological phasetransitions.Let us focus on the single topological transition, say atFcr =1/2, where the bands shown in Fig. 2 touch each other at κ = 0and κ = π . In vicinity of this critical value, the matrix (4) isclose to the identity matrix,U (κ = 0) =( √1 − 	2 i	i	√1 − 	2), (9)where 	 is proportional to F = F − Fcr . It is easy to see thatfor 	  1 eigenvectors of the matrix (9) are given by Eq. (7)with χ = 0 and eigenvalues by λ1,2 ≈ exp(±i	). The crucialpoint is that off-diagonal elements of the matrix (9) changetheir sign when F crosses Fcr . Then the symmetric eigenvectorY1 = [1/√2,1/√2]T , which has been associated with theupper band, skips to the lower band [16]. This seemingly simpleresult has important consequences, which can be detectedexperimentally by inducing Bloch oscillations of atoms in thex direction. We discuss the general scheme of the proposedexperiment in the next section.IV. DETECTING PHASE TRANSITIONSWe consider a BEC of noninteracting cold atoms (i.e., acoherent wave packet with vanishing phase difference betweendifferent sites) and apply an external field F which is slightlymismatched with the y axis, Fx  Fy ≈ F . In this setup,the strong component Fy creates the ladder of Wannier-Starkenergy bands and the weak component Fx induces Blochoscillations of atoms in these bands. These oscillations can beconveniently described in terms of the Wannier-Stark states|(1,2)n (κ)〉 as soon as we specify the expansion coefficienta(1,2)n (κ) = 〈(1,2)n (κ)|ψ(t = 0)〉. Since the occupation ampli-tudes of A and B sites in the initial state |ψ(t = 0)〉 haveequal phases, we predominantly populate κ = 0 vicinity ofupper bands if Fy < Fcr , and κ = 0 vicinity of lower bands0 100 200 300 400-100 100 200 300 400-100102030405060<x><x>t/T-50-40-30-20-10010FIG. 4. Bloch oscillations of a localized wave packet for F =1/1.9 (upper panel) and F = 1/2.1 (lower panel). Time is mea-sured in units of the tunneling period T = h/J1. Shown are themean value x(t) = 〈ψ(t)|xˆ|ψ(t)〉, blue solid lines, and y(t) =〈ψ(t)|yˆ|ψ(t)〉, red dotted lines. The ratio of the field componentsFx/Fy = 0.005.if Fy > Fcr . Thus, the weak component Fx induces Blochoscillations either in upper or lower bands depending oninequality relation between F and Fcr . This is illustrated inFig. 4, which shows the results of straightforward numericalsimulations of the wave-packet dynamics for Fx/Fy = 0.005and F = 1/1.9, upper panel, and F = 1/2.1, lower panel. Itis seen that the wave-packet displacement in the x directionreproduces the dispersion relation of upper bands (blue solidlines) in Fig. 2(a) and lower bands (red dashed lines) inFig. 2(c), respectively. (Since dFy > J1, the wave-packetdisplacement in the y direction can be neglected.) In oursimulations, we intentionally use very small ratio Fx/Fy sincefor a larger ratio the Bloch oscillations become complicatedby interband Landau-Zenner transitions. Yet, qualitatively theeffect remains the same—the wave packet moves in oppositedirections for F smaller and larger than Fcr .V. CONCLUSIONIn summary, we analyzed topological phase transitionsin the tilted brick-wall optical lattice which are induced byvariation of the field magnitude. These transitions correspondto a qualitative change of the two-dimensional Wannier-Stark states and can be detected by observing atomic Blochoscillations, which nowadays is a routine procedure in thecold-atom physics. We notice, in passing, that one gets similarresults for the more familiar honeycomb lattice. Moreover,since in our numerical illustrations we used J2 = 0, theresults depicted in Fig. 3 can be directly applied to thehoneycomb lattice after rescaling the abscissa axis by the factor√3/2.To conclude the work, we mention that the discussed phasetransitions can be also observed in the curved two-dimensional013603-3ANDREY R. KOLOVSKY PHYSICAL REVIEW A 98, 013603 (2018)photonic crystals, where the inverse curvature radius playsthe role of the field magnitude [17–19]. The latter systemalso admits to study the other manifestation of topologicalphase transitions, namely, a rearrangement of the edge states.The detailed analysis of this rearrangement will be given in aseparate paper.ACKNOWLEDGMENTSThe authors acknowledge fruitful discussions with S. Flachand D. N. Maksimov and the hospitality of Max-PlanckInstitute for Complex Systems in Dresden where this workwas completed.[1] J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Courseon Topological Insulators: Band Structure and Edge States inOne and Two Dimensions, Lecture Notes in Physics Vol. 919(Springer, Berlin, 2016).[2] B. A. Bernevig, Topological Insulators and Topological Su-perconductors (Princeton University Press, Princeton, NJ,2013).[3] M. Franz and L. Molenkamp, Topological Insulators, Contem-porary Concepts of Condensed Matter Sciences Vol. 6 (Elsevier,Amsterdam, 2013).[4] S.-Q. Shen, Topological Insulators: Dirac Equation in Con-densed Matter, Springer Series in Solid-State Sciences Vol. 187(Springer, Singapore, 2017).[5] M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T.Kitagawa, E. Demler, and I. Bloch, Direct measurement ofthe Zak phase in topological bloch bands, Nat. Phys. 9, 795(2013).[6] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons inPolyacetylene, Phys. Rev. Lett. 42, 1698 (1979).[7] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Bar-reiro, S. Nascimbène, N. R. Cooper, I. Bloch, and N. Goldman,Measuring the Chern number of Hofstadter bands with ultracoldbosonic atoms, Nat. Phys. 11, 162 (2015).[8] F. D. M. Haldane, Model for a Quantum Hall Effect WithoutLandau Levels: Condensed-Matter Realization of the “ParityAnomaly,” Phys. Rev. Lett. 61, 2015 (1988).[9] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger,D. Greif, and T. Esslinger, Experimental realization of thetopological Haldane model with ultra cold fermions, Nature(London) 515, 237 (2014).[10] A. R. Kolovsky, A. Ramachandran, and S. Flach, Topo-logical flat Wannier-Stark bands, Phys. Rev. B 97, 045120(2018).[11] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger,Creating, moving, and merging Dirac points with a Fermi gas ina tunable honeycomb lattice, Nature (London) 483, 302 (2012).[12] D. N. Maksimov, E. N. Bulgakov, and A. R. Kolovsky, Wannier-Stark states in double-periodic lattices. II. Two-dimensionallattices, Phys. Rev. A 91, 053632 (2015).[13] D. N. Maksimov, E. N. Bulgakov, and A. R. Kolovsky, Wannier-Stark states in double-periodic lattices. I. One-dimensionallattices, Phys. Rev. A 91, 053631 (2015).[14] Introduced winding number should not be mixed with thatdiscussed in Ref. [15]. The former is a characteristic of the eigen-states of tilted bipartite lattices while the latter is a characteristicof the spectrum of tilted topological lattices (for example, theHaldane lattice) in the limit F → 0.[15] W.-R. Lee and K. Park, Direct manifestation of topological orderin the winding number of the Wannier-Stark ladder, Phys. Rev.B 92, 195144 (2015).[16] We mention that this result, when taken together with the factthat eigenvectors are periodic functions of the quasimomentum,necessarily implies a change of the winding number. The latterhappens through developing of a nonanalyticity of χ = χ (κ) atκ = 0 and κ = π .[17] H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov,A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, andF. Lederer, Bloch oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices, Phys. Rev. Lett. 96, 053903(2006).[18] A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, U. Peschel,and F. Lederer, Optical Bloch oscillations in general waveguidelattices, J. Opt. Soc. Am. B 24, 2632 (2007).[19] G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R.Osellame, Fractional Bloch oscillations in photonic lattices, Nat.Commun. 4, 1555 (2013).013603-4

Topological phase transitions in tilted optical lattices

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Topological phase transitions in tilted optical lattices

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