Engineering topological phase transition and Aharonov-Bohm caging in a flux-staggered lattice
Amrita Mukherjee, Atanu Nandy, Shreekantha Sil, Arunava Chakrabarti
EEngineering topological phase transition and Aharonov-Bohm caging in aflux-staggered lattice
Amrita Mukherjee, Atanu Nandy, Shreekantha Sil, and Arunava Chakrabarti Department of Physics, University of Kalyani, Kalyani, West Bengal-741 235, India ∗ Department of Physics, Kulti College, Kulti, Paschim Bardhaman, West Bengal-713 343, India † Department of Physics, Visva-Bharati, Santiniketan, West Bengal-731 235, India ‡ Department of Physics, Presidency University, 86/1 College Street, Kolkata, West Bengal - 700 073, India § (Dated: June 11, 2020)A tight binding network of diamond shaped unit cells trapping a staggered magnetic flux distri-bution is shown to exhibit a topological phase transition under a controlled variation of the fluxtrapped in a cell. A simple real space decimation technique maps a binary flux staggered networkinto an equivalent Su-Shrieffer-Heeger (SSH) model. In this way, dealing with a subspace of thefull degrees of freedom, we show that a topological phase transition can be initiated by tuningthe applied magnetic field that eventually simulates an engineering of the numerical values of theoverlap integrals in the paradigmatic SSH model. Thus one can use an external agent, rather thanmonitoring the intrinsic property of a lattice to control the topological properties. This is advanta-geous from an experimental point of view. We also provide an in-depth description and analysis ofthe topologically protected edge states, and discuss how, by tuning the flux from outside one canenhance the spatial extent of the Aharonov-Bohm caging of single particle states for any arbitraryperiod of staggering. This feature can be useful for the study of transport of quantum information.Our results are exact. I. INTRODUCTION
The Su-Schrieffer-Heeger (SSH) model, describingspinless fermions on a one dimensional lattice where theelectron hopping is staggered, stands out as a paradig-matic example of a one-dimensional charge fractional-ized and topologically ordered system.
Symmetry pro-tected topological order, a hallmark of the topologicalinsulators (TI), has enriched and revolutionised conceptsin condensed matter physics by unravelling a novel stateof matter where the energy bands are characterized byquantized topological invariants. Needless to say, the ad-vent of this very special concept of topological order andan exotic quantum phase in condensed matter systemshave spurred immense research activity, both in theoryand in experiments, merging the knowledge and expertiseof research groups from diverse research areas.The SSH model, an emblem of a one dimensional chi-ral symmetric TI, that had initially been proposed toexplain the striking transport characteristics of trans -polyacetylenes, is conveniently dealt with in a tight bind-ing formalism describing non-interacting fermions. Thearrangement of two bonds, described by two different val-ues of the nearest neighbor hopping (overlap) integrals,gives the lattice a bipartite sublattice structure. The sys-tem exhibits nontrivial topological phases enriched withtopologically protected edge states, a signature of theTI’s. The edge states are robust against defects and dis-order, and are fully understood from the single-particleeigenstates.This remarkable feature inspired a series of experi-ments in recent past, where several topological networkshave been realized and explored, bringing to light aplethora of new physics. The experiments cover a widecanvas of physics, encompassing for example, the exotic bulk properties of the topological character of an SSHchain using real space superlattices, demonstration ofrobust edge states in classical systems of coupled mechan-ical oscillators, the dynamics and the topological soli-tonic states in a momentum-space lattice fabricated with Rb atoms, topological Haldane model with K atomstrapped in a honeycomb lattice, both in an ultracold-atomic platform, or the experimentally realized symme-try protected topological phase of interacting bosons us-ing Rydberg atoms - to name a few. Such experimentsalso received good company in recent times in photon-ics, where tailor-made lattice geometries are fabricatedon glassy substrates to explore a wide spectrum of issues,ranging from an extreme localization of photonic excita-tions to the physics of topological systems, revealingthe existence of topological bound states in a Flouquetengineered system resembling the SSH model, or theexperimental realization of a topological insulator usingphotonic Aharonov-Bohm (AB) cages, an important is-sue in the context of the present article.In the present communication we theoretically inves-tigate the topological properties of a flux-staggered ar-ray of diamond shaped cells, axially juxtaposed, as il-lustrated in Fig 1(a). The out-of plane magnetic fieldpiercing the n th cell and the resulting ‘trapped’ flux Φ n can have, in principle, any chosen distribution, offering awide class of staggering patterns. The lattice presentedin Fig. 1(a) is an example of what we shall henceforthrefer to as a topological quantum network (TQN). It isa member of a large group beginning definitely with theparadigmatic SSH chain, and its extensions, and in-clude the photonic diamond TQN’s, the Lieb latticemodels or kagome lattices to name a few. In to-tality, these systems build a formidable (though by nomeans exhaustive) canvas of studies, shedding light on a r X i v : . [ c ond - m a t . m e s - h a ll ] J un the intricacies of the band structure and topological or-der exhibited by the TQN’s. (a)(b)FIG. 1. (Color online) (a) A quasi-one dimensional endlessarray of diamond network trapping a non-uniform distribu-tion of magnetic flux, and (b) the renormalized chain com-prising only the bulk sites B n , and obtained by decimatingthe vertices A n and C n having coordination number equal totwo. Here ˜ (cid:15) and τ n,n ± are the renormalized on-site potentialand the nearest neighbor hopping integrals, in a typical tightbinding scheme. Our choice of the diamond-TQN (DTQN) is motivatedby certain issues outlined next that we find pertinent.Firstly, the flux trapped in a cell can be controlled fromoutside (for example, by controlling the current in a nano-solenoid, that can in principle, be planted in a cell), andthis gives us a complete freedom to continuously ‘de-form’ the Hamiltonian describing the system through thePeierls’ phase associated with the hopping integral, andengineer a topological phase transition (TPT) almost atwill. This, to our mind, is a good choice in many ways aswe do not have to tune the values of the intrinsic system-parameters such as the hopping (overlap) integrals likewhat is done in an SSH case, or control the evanescentcoupling of the waveguides in the photonic lattices or controlling the well-depths in the optical lattice struc-tures, for example. Changing intrinsic parameters es-sentially means that while experimenting, one has to dealwith a different lattice in every occasion. Changing fluxfrom outside gets rid of this ordeal.Secondly, a linear DTQN threaded by a uniform flux,that has been critically examined before and revealed awealth of knowledge, such as, a flux-induced semicon-ducting behavior, spin-filtering effects or a spin-selective Aharonov-Casher caging effect for arbitraryspins, has recently turned into an object of supreme in-terest in state-of-the art photonics, presenting direct ex-perimental evidence of the AB-caging effect. TheAB caging, that is an important issue in the present work,was introduced by Vidal et al.
This a geometry-induced extreme localization of single particle states, seenin periodic lattices. The amplitudes of the wavefunctionare localized over a finite number of lattice sites, beingzero elsewhere. These ‘caged’ states are entirely degener-ate, and are associated with flat (dispersionless) energybands (FB) where the energy turns out to be indepen-dent of the quasimomentum. - Such FB states, form-ing a ‘compacton’ structure, has recently been observed experimentally in an optical waveguide array, and arebeing considered seriously for creation of a whole set of‘diffraction-free’ modes that can transport quantum in-formation up to a long distance. The DTQN systemthus serves as a unique platform to study the prospects ofa tailor-made quantum matter where one can explore thetopological phase transitions, and a geometry-inducedextreme localization of single particle states in view oftransporting quantum information.With this background we present in the following sec-tions the fundamental results obtained and the physicsof topological properties exhibited by the DTQN and itsunique controllable AB caging aspects. Section II dis-cusses the character of the bulk bands in two elementaryvariants of the DTQN. In Section III the AB caging anda possible engineering of its spatial extent is talked abut.The physics of the topologically protected edge modesis presented in Section IV, and conclusions are drawn inSection V. (a)(b)FIG. 2. (Color online) Dispersion relation
E(k) for a dia-mond network with a constant flux trapped in each cell. (a)Φ = 0 . . Here we get one non-dispersive band at E = 0,and two dispersive bands. (b) Φ = 0 . . Here, the entirespectrum collapses into an extreme localization profile, show-ing three flat bands at E = 0, and E = ±
2. where the energy E is expressed in units of t . The on-site potential is set equalto zero for all sites in the DTQN. The Zak-phases in each caseare displayed. II. TOPOLOGY OF THE BULK BANDS
Our model system is depicted in Fig. 1 (a). It is a onedimensional array of diamond shaped cells with infiniteaxial span. The n -th cell traps a finite magnetic flux Φ n .We describe the system by the standard tight-bindingHamiltonian for non-interacting spinless electrons, writ-ten in the Wannier basis, viz., H = (cid:88)
2, when the entirespectrum collapses into just three non-dispersive bandsand the system displays an extreme localization. Thisis the most elementary distribution of an AB-caging, aswill be discussed in the next section.
B. The simplest flux-staggered array
The simplest staggered flux distribution is achieved byperiodically repeating two different magnetic flux values,viz., Φ and Φ in the consecutive cells of an infinite ar-ray. The Hamiltonian matrix in this case has a dimension6 ×
6, and is given by, H ( k ) = t B (1) t F (1) 0 t F (2) e − ika t B (2) e − ika t F (1) 0 0 t B (1) 0 0 t B (1) 0 0 t F (1) 0 00 t F (1) t B (1) 0 t F (2) t B (2) t B (2) e ika t F (2) 0 0 t F (2) e ika t B (2) 0 0 (6)where, t F ( B ) (1) = t exp( ± iπ Φ / ) and t F ( B ) (2) = t exp( ± iπ Φ / ). Straightforward diagonalization of the above matrix provides the entire band diagram. Thedispersion relations are E = 0 (a two-fold degenerateband) and E = ± (cid:113) (4 ± √ (cid:112) (2 + cos(2 π Φ / Φ ) + cos(2 π Φ / Φ ) + F + F + F + F ) (7)where, F = cos[2 ka − ( π/ Φ )(Φ + Φ )], F = cos[2 ka +( π/ Φ )(Φ + Φ )], F = cos[2 ka − ( π/ Φ )(Φ − Φ )] and F = cos[2 ka +( π/ Φ )(Φ − Φ )]. The band at E = 0 doesnot show any response to the applied perturbation butthe four other bands have the obvious flux sensitivity.Fig. 3 displays a subset of the dispersion relations forcertain special values of the magnetic flux, and is going toplay an important role in the subsequent discussion. Wehave calculated the Zak phase in this case of staggeredflux geometry. There is degeneracy in the central band.Following the Wilson loop approach for degenerate bandswe replace Eq. (5) by a product of ‘overlap matrices’ ,viz, Z ( k i ) = − Im log (cid:89) k j S ( k i ,k j ) , ( k i ,k j +1 ) (8)where, S is the overlap matrix .For a binary flux staggered lattice no quantized Zakphase is found. The values are scattered, without show-ing any definite correlation. However, the same val-ues are obtained for symmetrically distributed energyeigenvalues ± E . For example, with Φ = 0 . andΦ = 0 . the energy eigenvalues are E = 0, ± . ± . t (set equalto unity). The corresponding Zak phase values are ob- tained as, Z = 0 . π , 0 . π and − . π respectively.The values keep changing as the flux patterns are madeto vary, retrieving the old value of Z = π at E = 0. Atthis point it becomes a pertinent question to ask our-selves whether there can still be a topological order, ora TPT even if the system doesn’t show up any time re-versal symmetry or any definitive change in the values ofthe Zak phase.We find that, it does really. To address this issue, inthis subsection we take a different approach, based on areal space decimation of a subset of the vertices (theseare A and C -type sites) in the original diamond array tomap it onto an effective SSH chain, where we argue thata topological phase transition (TPT) can be initiated bytuning the values of the magnetic flux. As we mentionedin the introduction, the flux can be tuned from outside and is very much in the hands of an experimentalist.Consider the n -th cell. The difference equation con-necting any j th site with its nearest neighbors l in cellnumber n , is given by,( E − (cid:15) ) ψ j = t (cid:88) l e ± iθ n ψ l (9)where, θ n = 2 π Φ n / ( n = 1, 2). Let us specificallyfocus at the physics when the energy E (cid:54) = 0. This willsuffice. Using Eq. (9) we decimate the top and the down( A n and C n ) vertices (that is, we eliminate the ampli-tudes ψ A n and ψ C n and retain only the amplitudes ψ B n )to map the original DTQN array into an effective one-dimensional chain, described by a difference equation,[( E − (cid:15) ) − t ] ψ B n = 2 t cos 2 θ ψ B n − +2 t cos 2 θ ψ B n +1 (10)One can easily check that this equation yields the exactdensity of states (DOS) profile of the original diamondchain barring the isolated spike at E = 0, the latter beinga contribution from the top vertices A n and C n . But aswe shall see, it is enough to get a flavour of a TPT fromthis equation. Eq. (10) can be written as, E (cid:48) ψ B n = τ ψ B n − + τ ψ B n +1 (11)with E (cid:48) = [( E − (cid:15) ) − t ], and τ n = 2 t cos 2 θ n ( n = 1,2). Eq. (11) describes an SSH chain where the ‘on-site’potential is zero , and the hopping integrals τ and τ alternate periodically.The decimation reduces the number of degrees of free-dom to deal with. The effective SSH chain, now char-acterized by two hopping integrals τ = 2 t cos 2 θ , and τ = 2 t cos 2 θ is thus defined on a reduced (real) sub-space (spanned by a lesser number of the degrees of free-dom). From this angle, it can be taken to describe a‘square root’ topological system. This direct equiv-alence with the basic SSH chain implies that, if we set (cid:15) = 0 and t = 1 in the parent diamond chain, the openingand closing of energy gaps will be observed at E (cid:48) = 0,which amounts to E = ± = Φ ), and its opening (for Φ (cid:54) = Φ ) atthe Brillouin zone boundaries (now at ± π/ H ∗ of a bipartite lattice of length L say, definedon this ‘reduced degrees of freedom’ space as, H ∗ = (cid:18) H ab H † ab (cid:19) (12)Here, a and b refer to the two kinds of vertices, sitting inbetween the pairs of hopping τ − τ and τ − τ respec-tively, in the effective one dimensional bipartite lattice(Fig. 1(b), with τ n − ,n = τ and τ n,n +1 = τ , and re-peating periodically). The mapping obviously relates thetopological properties of the Hamiltonian H ∗ , that nowexhibits a chiral symmetry σ z H ∗ σ z = − H ∗ , with thatshown by the standard SSH model. Since the eigenstatesof H ∗ are also the eigenstates of the original Hamiltonian H , the states with E = ± E = ± effective SSH chain has hopping integrals that areengineered by the applied magnetic field only. It can beeasily checked (and has been depicted in Fig. 3(a) and(b)) that, the gap around E = ± andΦ . This indicates a TPT. We refer the reader to theAppendix for details.Before ending this section, it may be noted that higherorder staggering effects can easily be dealt with now. Ofcourse, a study of the bulk band properties needs han-dling of larger unit cell matrices. We skip such detailshere to save space, though a tertiary flux distributionwill be discussed in the context of AB caging and theedge states in the subsequent sections. III. SELECTIVE AHARONOV-BOHM CAGINGA. The easiest caging with uniform flux
The diamond array considered here exhibits (a) (b)(c) (d)FIG. 4. (Color online) (a) Schematic demonstration of an in-finite array of diamond network threaded with uniform mag-netic flux. The unit cell consists of three atomic sites labelledas A, B and C. (b) U (1) class of compact localized state at E = 0. This is flux-independent. (c) and (d) represent a U (2)class of compact localized state. This is a quincunx profile atthe special value of Φ = 0 . , and at E = ± AB-caging.
This caging effect, in its simplest form,manifests itself through local, clustered distribution ofamplitudes of the wave function. These are the so called compact localized states (CLS).
The consequentialappearance of flat, non-dispersive bands (three in num-ber when the flux is uniform throughout) marks the phe-nomenon. For the simplest caging, the bands are at E = 0, and E = ± E = 0 is independent ofthe choice of the magnetic flux. The only possibility tosatisfy the Schr¨odinger equation is to make the amplitudeof the wave function vanish at the bulk B n sites. Onesuch configuration is shown in Fig. 4(b). This is thesimplest example of a CLS and AB-caging, the strongestof the U (1) class of CLS. The amplitudes are confinedwithin one unit cell of the DTQN. The states at E = 0are robust, and topologically protected, characterized bya quantized Zak phase π .In the case where E (cid:54) = 0, we observe that, as the fluxper plaquette is set at a uniform value Φ = Φ /
2, theeffective hopping integral, connecting a bulk ( B n ) siteto its nearest neighbors becomes zero . This can be di-rectly observed by decimating the top ( A n ) and bottom( C n ) vertices using Eq. (9). The renormalized hoppingon the effective one dimensional chain becomes equal to t eff = [2 t cos( π Φ / Φ )] / ( E − (cid:15) ), and becomes zero atΦ = Φ /
2. This is the basic reason underlying an ex-treme localization in such cases, and the formation of theCLS’s. As for the flat-band energy eigenvalues E = ± /
2, Eq. (9) is easily solved to find a quincunxdistribution of amplitudes, spread over a set of five ver-tices spanning two consecutive unit cells. This is a largerpatch of non-zero amplitudes, and the CLS is in a class U (2). Needless to say, in both the cases, one island ofnon-zero amplitudes is separated from the neighboringislands by sites, where the wave function has a vanishingamplitude. The U (2) class of CLS is depicted in Fig. 4(c) and (d) for E = ± (a)(b)FIG. 5. (Color online) Aharonov-Bohm caging in binary (a),(b), flux staggered lattices. The non-vanishing amplitudes areimprinted by bright cyan dots. The caging, self explanatory,is bordered by dashed boxes. In (b) the distribution shown istypically obtained by setting Φ = Φ /
2, as described in thetext.
B. Extending the quantum prison
The AB-caging at the centre of the band, i.e. for E = 0, is independent of the flux, in general. But, thevanishing of the effective hopping connecting the B n ver-tices on a renormalized linear chain (Fig. 1(b)), when-ever Φ n = Φ /
2, opens up a possibility of engineeringthe ‘physical spread’ (in real space) of the amplitudes ofa CLS when the staggered flux distribution has a higherorder periodicity. One such extended distribution, for abinary ordered staggering with Φ and Φ repeated in analternate fashion, is explicitly worked out and illustratedin Fig. 5, where the sites with non-zero amplitudes arecolored brightly. In the binary flux staggered case the easiest (andstrongest) AB-caging is again observed for E = 0. Thedistribution of amplitudes of the wave function can beworked out analytically. The amplitudes of the wavefunction are pinned at the top ( A n ) and bottom ( C n )sites of any n th diamond cell. The amplitude at every B n site having a coordination number of four is zero. Aspecial situation can be cited as an example. We referto E = 0. We can construct ψ j = − (cos θ / cos θ ) for j ⊂ ( A , C ), and ψ j = 1 for j ⊂ ( A , C ) in Fig. 5(a),and repeating periodically. This is a U (1) class of CLSagain. However, in this special example with E = 0, oneneeds to assure that, Φ (cid:54) = Φ . Other degenerate solu-tions can of course, be worked out. We do not provideall of these, just to save space.The U (2) class of CLS for a Φ -Φ periodic distributioncan be worked out by hand as well. A specific distributionis depicted in Fig 5(b) for Φ = Φ /
2. For such a fluxpattern, we can always work out a particular solution,satisfying Eq. 9. For example, let ψ A = ψ C = xe − iθ , ψ C = ψ A = xe iθ , ψ A = ψ C = y , and ψ B = ψ B = z .If z = 1 is chosen arbitrarily, so that both x and y areevaluated in unit of z , then one can easily work out asolution for E = ±√ θ . Both roots are real,and a Φ - dependent set of amplitudes x = 1 /E , and y = (2 /E ) cos θ , in complete agreement with Eq. (9), canbe obtained. The dependence of the energy eigenvalueand thus, the amplitudes on θ , i.e. on Φ exposes theoptions of tuning the extreme localized states at differentenergies, keeping the pattern of the distribution intact.With a tertiary flux ordering, one has more optionsof tuning the relative magnitudes of the trapped fluxes,and generate both U (1) and U (2) classes of CLS. It canbe easily understood now that, the U (1) class will havea distribution where the amplitudes are pinned at the( A n , C n ) vertices for E = 0. The U (2) class of CLSspreads over a unit cell, that now comprises three con-secutive cells trapping fluxes Φ , Φ and Φ . We haveobtained a variety of distributions, but refrain from over-loading the text with these, as the basic idea, to our mindis conveyed already. As expected, with an increased stag-gering period, the spatial extent of a unit cell increasesand the caging spreads over larger patches of the DTQN. C. Engineered caging in a modulated flux profile
The fact that, setting Φ n = Φ / n -th cellcan make the effective, renormalized coupling betweenthe bulk sites B n vanish, one can, in principle, designthe caging at will. To implement this, we propose anAubry-Andr´e-Harper (AAH) modulation in the fluxdistribution, given by, Φ n / Φ = λ cos( πQn α a ). λ isthe strength of the modulation and the exponent α con-trols the slowness of variation of the profile. In the spe-cial case of Q being an irrational number, the period ofmodulation becomes incommensurate with the underly-ing lattice-period. This particular form of modulations (a) (b)(c) (d)FIG. 6. (Color online) Edge state-amplitudes for a finite diamond array threaded by a constant flux Φ = Φ /
2. (a) At E = √ ψ A = ψ C n = (1 + i ) / ψ L = ψ R = 1 and ψ C = ψ A n = (1 − i ) /
2. Asimilar distribution for the edge state can be obtained for E = −√
2. (b) Plot of probability density for the state at (a). (c) A‘non-half flux’ case, Φ = 0 . . Here, E = ± . has so far been widely explored in relation to the localiza-tion issues in deterministically disordered systems, and recently in the study of topological modes in a onedimensional modulated potential profile. .In the present proposition, the AAH modulation playsa different role. It determines the staggering profile ofthe flux. As can easily be understood, the above dis-tribution of magnetic flux is completely equivalent toan infinite diamond array, immersed in a uniform mag-netic field, and having an axial twist through an n -dependent angle γ n = πQn α a . The effective hoppingbetween the neighboring bulk sites B n is now equal to[2 t / ( E − (cid:15) )] cos[ πλ cos( πQn α a )], and becomes equal tozero for cos( πQn α a ) = 12 λ (13)This in fact, sets a threshold for the flux window λ ,viz, | λ | ≥
1. For a given selection of λ , the choices of arational value of Q and the ‘slowness’ parameter α , whichamounts to a definite set of the angles of axial twist, de-termine the plaquette number n in which the ‘effective’,‘end-to-end’ coupling between the bulk ( B n ) sites disap-pears, leading to an AB caging. It is not unnatural, infact it is definitely possible that, the amplitudes remainnon-zero on all (or at least a majority of) the cells pre-ceding the n -th cell, and after it. This pattern will berepeated as the geometry is periodic as long as Q is arational number. Thus, the effective area of the quan-tum prison is extended beyond just a single plaquette, and that too in an engineered way. Extensive numericalsearch has revealed that cutting off the effective connec-tion in the n th plaquette leads to the formation of 2 n + 1flat bands. These non-dispersive bands crowd towardsthe edges of the energy spectrum, as observed.Emphasizing on the equivalence of the AAH modula-tion with a continuous axial twist, we bring to the noticeof the reader this extremely interesting and non-trivial tilt induced selectivity in the Aharonov-Bohm caging anda lazy extreme localization (dictated by the choice of ra-tional Q -values). The problem seems challenging froman experimental perspective. IV. TOPOLOGICAL EDGE MODESA. A finite array with uniform flux
In Fig. 6 (a) we exhibit a finite diamond array witheach cell pierced by the same magnetic flux Φ. Theextreme sites to the left and to the right of the arrayare marked L and R respectively. Here we observe thatwhen we set Φ = Φ /
2, a pair of topologically non-trivialedge modes at E = ±√ Eachmember of the pair shows amplitudes distributed overjust three vertices (dotted cage with solid blue sites)around each edge. The distribution is robust against aphysical distortion, as has been verified by introducingdisorder in the distribution of the hopping integrals. Theprobability distribution of this sharp pinning of states isshown in Fig. 6(b), and is at per with (a).The paired edge states can also appear for non-halfflux cases, as shown in Fig. 6(c), where Φ = (3 / , and E = ± . FIG. 7. (Color online) Spectral profile as a function of τ (defined as 2 cos( π Φ / Φ ) ) for a finite diamond chain withbinary flux distribution at Φ = 0 that makes τ = 2. (a)A pair of dangling atoms are attached at each of the two ex-treme ends of a finite diamond chain with 20 cells. Here, thetwo in-gap edge states are shown in red color at E = ±
2, andshown in (c). (b) The 20 cell long diamond array without anydangling bonds. There are now four in-gap edge states thatare marked by green and red color respectively, as depictedin panel (d).
B. The binary flux-staggered array
In a conventional, finite SSH chain, comprising twokinds of hopping integrals τ and τ alternating periodi-cally, and beginning with τ (without losing any general-ity), a topologically non-trivial extreme dimerized situa-tion is created by setting τ = 0, and τ (cid:54) = 0. The on-site potential (cid:15) for a purely one dimensional SSH model isconveniently set equal to zero for all sites , leading to theimmediate conclusion that a topologically protected edgestate will appear at an energy E = 0. In our case, we ar-gue that, the localized edge modes in a finite binary-fluxstaggered DTQN can easily be understood by mappingit onto an effective SSH chain. This needs a one steprenormalization process, through the decimation of thetop ( A n ) and the bottom ( C n ) vertices. This is easilyaccomplished by Eq. (10). We discuss two cases to putforward our idea.
1. A finite DTQN with a pair of dangling atoms at eachend
Let us first consider Fig. 7(a) first, where a finiteDTQN has a pair of dangling atoms (or equivalently, bonds) at each end. We have two distinct flux valuesΦ and Φ trapped periodically in this network. With-out losing any generality, we assume the first plaquette tohave flux Φ . As described in Section II, after a square-root mapping, the renormalized DTQN chain resemblesan SSH lattice, described by Eq. (10), or equivalently,by Eq. (11). One identifies E (cid:48) = ( E − (cid:15) ) − t andthe SSH hoppings τ n = 2 cos 2 θ n for n = 1,2, and with θ n = π Φ n / , as already explained in Eq. (11). Thismapping immediately tempts us to believe that, in thisfinite DTQN where all the sites are equivalent (each ‘orig-inating’ from the parent B n sites having a coordinationnumber four), we are going to find localized edge-modesat E (cid:48) = 0. The answer is in the affirmative, but now itis sensitive to and constrained by the relative strengthsof the flux trapped in the consecutive cells. We explainthe condition of existence of such edge modes below.As before, we set (cid:15) = 0 and t = 1. Labelling the left-most B -type vertex (colored blue in Fig 7 (a)) as n = 1,and assuming that, the amplitude of the wave function atthe leftmost ( B n -type) vertex n = 1 (that has a coordi-nation number equal to four), viz, ψ to be finite, we findthat for E (cid:48) = 0 in the square-root mapped SSH chain,that is, for E = ± ψ = 0. This is easily verified using Eq, (10).This immediately relates the amplitudes at all the oddnumbered sites on the mapped SSH chain through therelation, ψ n +1 = ( − n (cid:20) cos 2 θ cos 2 θ (cid:21) n ψ (14)while, by virtue of ψ = 0 the amplitudes ψ n = 0. Fora convergent wave function, we must have (cid:12)(cid:12)(cid:12)(cid:12) cos 2 θ cos θ (cid:12)(cid:12)(cid:12)(cid:12) < onlythen we have a state localized sharply at site number n = 1 on the equivalent one dimensional SSH chain. Interms of the parent diamond lattice we should expectthat at E = ±
2, the amplitudes will spread over a min-imal number of vertices around the extreme left edge.These states should be topologically protected, thanksto the arguments catering a pure, one dimensional SSHmodel.A similar argument leads to the conclusion that, fora flipped convergence condition, viz, | cos θ / cos θ | < all the sites ( B n ) ofthe renormalized one dimensional effective SSH chain ob-tained after decimation, are equivalent (just like a purelyone dimensional SSH lattice where we could set (cid:15) n = 0for all n ). This is precisely what we achieve, when wehave a diamond array with two dangling bonds at eachend. (a) (b)(c) (d)FIG. 8. (Color online) The edge states for a 20-cell long binary flux staggered DTQN with open boundary conditions. (a,b)Φ = Φ /
2, Φ = Φ , and (c,d) Φ = Φ , Φ = Φ /
2. The amplitudes are sharply confined at either edge of the network forenergy E = 2 . ψ A n − = 0 . e − iπ/ , ψ B n = 0 . , ψ C n − = ψ ∗ A n − , ψ A n =0 . e iπ/ , ψ B = − . ψ C n = ψ ∗ A n . For (c) we have found a similar configuration, but with different numerical values,as mentioned in the text. The confinement topography of amplitudes for other edge states with energies E = − . , ± . The topologically protected edge states at E = ± τ is varied,and are seen to lie frozen in the spectral gaps. For thisparticular figure we have set Φ = 0, that makes τ = 2in the effective (mapped) SSH chain, but the scheme andthe result remain similar for any other value of Φ as longas we satisfy the condition Eq. (15). The robustness ofthe state against a varied set of values of the hopping τ implicitly implies a continuous distortion in the lattice,and subsequently, in the Hamiltonian. The edge state isthus protected topologically.
2. Chiral edge states in finite DTQN without danglingatoms
Needless to say that, if we remove the dangling bondsfrom both the ends, we ‘perturb’ the system, and thisperturbation shifts the edge state energy from E = ± τ is varied,keeping τ = 2, are shown in Fig. 7(d).As numerical support to our arguments above, and tounravel chiral edge states, we present some results inFig. 8 where the edge states correspond to FB’s. TheDTQN chosen doesn’t have any dangling edges now.A finite cluster of atomic sites with non zero am-plitudes shows the localization of single particle states at one edge of the system. It is easy to check thatin Fig. 8 (c), the cluster of non-zero amplitudes hosts ψ A = − . . i , ψ L = 0 . . i , ψ C = − ψ ∗ A , ψ A = − . − . i , ψ B = − . − . i , and ψ C = − iψ A . Rest of thelattice points experience zero amplitudes. The FB condi-tion is automatically accomplished by setting either Φ or Φ equal to the half flux quantum. The states re-ported above are chiral in the sense that, they are con-fined to any one edge of the lattice, without having acounterpart sitting at the other end. However, such chi-ral edge states need not necessarily be associated to non-dispersive bands, as can be seen in Fig. 9. In this secondexample, the span of confining cluster becomes larger,encompassing many more lattice points, but localizingeventually. This implies a spatially delayed localization,and an extended quantum prison in the spirit of the ABcaging effect. The importance of the FB states and thetopological edge modes has already been appreciated inliterature .Many more interesting engineering options open up asone heads for increased staggering periods. For example,in case of a tertiary staggered flux distribution with Φ -Φ -Φ repeating periodically (Fig. 10), depending uponwhich cell traps a half flux quantum, the envelope ofthe edge modes can exhibit a diverse span of localiza-tion. For some combination of flux values, it is localized,say, around the left edge and an inversion in the fluxdistribution inverts the scenario, again revealing the chi-ral character of the edge states. Interestingly, if, in aflux period of Φ -Φ -Φ we select Φ = Φ /
2, then edge0 (a) (b) (c)FIG. 9. (Color online) Edge states for a finite diamond chain with binary flux distribution, (a) Φ = Φ , Φ = 3Φ / = 3Φ /
5, Φ = Φ (blue). The amplitudes exhibit chirality, being trapped over a cluster of a few atomic sitesaround one edge only for E = ± . st and 41 st contributions from the finite assemble of sixty one number of single particle states. states are symmetrically pinned at both the edges. Thus,by strategically choosing the distribution distribution ofmagnetic flux, or equivalently, by choosing the angles ofaxial twist while keeping the external magnetic field con-stant, both being controlled externally, one can engineerchiral as well as non-chiral edge modes in this quasi-onedimensional quantum network. A selected set of suchedge modes and their flux-tunability are displayed in theself explanatory Fig. 10. V. CONCLUSION
We have undertaken an in-depth study of a quantumnetwork with diamond shaped cells pierced by a stag-gered magnetic flux distribution. The state-of-the artphotonics has already developed networks trapping syn-thetic gauge fields. A Floquet engineering of the stagger-ing effect in the distribution of magnetic flux or rather, its synthetic equivalent may no longer be a remote possibil-ity. The primary interest is to scrutinize the possibilityof engineering a topological phase transition by tuningthe magnetic flux, an externally controllable parameter,rather than playing around with the system’s intrinsic pa-rameters. The second, and equally important point of in-terest has been an analysis of the Aharonov-Bohm cagingeffect along with the occurrence of flat, non-dispersivebands - hallmarks of an extreme localization, that havefound immense importance in recent literature. In thecase of a binary flux staggered model we use a simplereal space decimation technique to map the system ontoan effective Su-Schreiffer-Heeger chain and show that, itis indeed possible to monitor the magnetic flux to engi-neer a topological phase transition. This can happen evenin the absence of any quantized topological invariant inthe system. The existence of non-chiral and chiral edgemodes when the flux pattern is uniform, or show a non-trivial periodicity, is discussed along with the prospectof achieving a comprehensive control over the spatial lo-cations of such edge modes, using a magnetic flux. TheAharonov-Bohm caging of single particle states, and theconsequential extreme localization of the single particlestates can be induced by special values of the magneticflux trapped in the cells of the network. The compact localized states seen in such a phenomenon are nowa-days drawing huge attention as potential candidates fortransportation of quantum information. We discuss theelementary case of a uniform flux and its variants, andpropose a generalization where the flux pattern followsan Aubry-Andr´e-Harper kind of modulation in its com-mensurate limit. This modulation mimics a continuousaxial twist of the entire quantum network immersed in aconstant magnetic field, and hence opens up, to our mind,new and interesting physics related to what we may call,a twist-induced topological and localization properties inlow dimensional lattice model - an issue that we wish toinvestigate further and report elsewhere.
ACKNOWLEDGMENTS
A. M. acknowledges DST for providing her INSPIREFellowship [ IF (a) (b)(c) (d)(e) (f)FIG. 10. (Color online) Edge states for a diamond array with tertiary flux distribution for a finite size system (a,b,c,d) withΦ = 0 . , Φ = Φ = Φ , and (e,f) Φ = 0 . , Φ = Φ = Φ . (a,b) The amplitudes are strictly pinned at one edge(right) of the system for E = 2 . E = − . , ± . , ± . E = ±√ E = 2 . E = − . , ± . E = 2 . ψ A n − = 0 . e − iπ/ , ψ B n − = 0 . ψ C n − = ψ ∗ A n − , ψ A n − = 0 . e iπ/ , ψ B n = − . ψ C n − = 0 . e − iπ/ , ψ A n = 0 . e − iπ/ , ψ R = 0 . ψ C n = 0 . e iπ/ . (c) is for the energy E = √ ψ A = − (0 . . i ), ψ L = − (0 . . i ), ψ C = − iψ A . (e) is for the energy E = 2 . ψ A = − (0 . . i ), ψ L = − . . i , ψ C = − ψ A , ψ B = 0 . − . i , ψ A = 0 . − . i , ψ C = − iψ A , ψ A n − = 0 . e − iπ/ , ψ C n − = ψ ∗ A n − , ψ B n = 0 . ψ A n = 0 . e iπ/ , ψ R = − . ψ C n = ψ ∗ A n .The number of diamond cells is 21. Appendix A: Topological phase transition in thebinary flux staggered lattice
After decimation of the top and bottom vertices A n and C n in the diamond cells we arrive at a renormal-ized version of the array, that effectively resembles anSSH chain, and is described by the difference equation,Eq (10). This is exactly an SSH chain as we have twoflux values Φ and Φ repeating periodically, and haveeffective nearest neighbor hopping integrals τ and τ ,explained in the main text. Let us suppose that, we aregiven this equation to start an analysis on the existenceof a topological phase transition. We can design a Hamil-tonian describing the unit cell of this effective SSH chainthat leads to such an equation. The Hamiltonian is: H ∗ = (cid:18) τ + τ e − ika τ + τ e ika (cid:19) (A1) FIG. 11. (Color online) Flow of the winding circle in the ζ x − ζ y space. Here, τ = 2 cos( π Φ / Φ ) and τ = 2 cos( π Φ / Φ ). Theeigenvalues are given by, E ± = ± (cid:113) τ + τ + 2 τ τ cos k (A2)2The Bloch eigenvectors corresponding to the eigenvaluescan be written in the form, | u ± ( k ) (cid:105) = 1 √ (cid:18) e ∓ iφ ( k ) (cid:19) (A3)where, the phase factor φ is an explicit function of theflux trapped in the cells, and given by,tan φ ( k ) = cos( π Φ / Φ ) sin ka cos( π Φ / Φ ) + cos( π Φ / Φ ) cos ka (A4)This formulation is well known for an SSH chain. Weexploit this formulation to express our Hamiltonian ina space of reduced degree of freedom (as a result of thedecimation of a subset of vertices) as H ∗ = (cid:126)ζ · (cid:126)σ , where, (cid:126)σ is the vector of the Pauli matrices, and (cid:126)ζ is a threedimensional vector with components, ζ x ( k ) = 2 cos( π Φ / Φ ) + 2 cos( π Φ / Φ ) cos kaζ y ( k ) = 2 cos( π Φ / Φ ) sin kaζ z ( k ) = 0 (A5) The “direction” and the “norm” of the vector (cid:126)ζ containthe information about the eigenstate and its correspond-ing eigenvector. It is now straightforward to observe that,the ‘tip’ of the vector (cid:126)ζ traces out a closed loop, whichis a circle in this case, in the ζ x − ζ y plane. The centerof the circle being given by the point [2 cos( π Φ / Φ ) , π Φ / Φ ), and it is tunable by the flux in anadjacent cell.The above parametrization and picturization immedi-ately make one conclude that a tuning of the flux valuesΦ and Φ is equivalent to deforming the Hamiltonianin a continuous way. Thus, one can drive the systemfrom one insulating phase corresponding to Φ < Φ toΦ > Φ by engineering the flux alone. A topologicalphase transition is definitely on the cards, and happensexactly at Φ = Φ . The circle in the ζ x − ζ y planetouches the origin as flux trapped in one cell equals theother. This can of course, be attained for a continuousdistribution of Φ and Φ . The flow of the trajectory inthe ζ x − ζ y space is shown in Fig. 11. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev.Lett. , 1698 (1979). A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su,Rev. Mod. Phys. , 781 (1988). D. J. Thouless, M. Kohmoto, M. P. Nitingale, and M. denNijs, Phys. Rev. Lett. , 405 (1982). R. Prange and S. M. Girvin,
The Quantum Hall Effect ,2nd ed. Springer-Verlag, New York, (1990). X.-G. Wen, Adv. Phys. , 405 (1995). S. F¨olling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A.Widera, T. M¨uller and I. Bloch, Nature , 1029 (2007). J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V.Porto, Phys. Rev. A , 033605 (2006). C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H.Schomerus, Nat. Comm. , 6710 (2015). C. Liu, W. Gao, B. Yang, and S. Zhang, Phys. Rev. Lett. , 183901 (2017). R. S¨usstrunk and S. D. Huber, Science , 47 (2015). E. J. Meier, F. Alex An, and B. Gadway, Nat. Comm. ,13986 (2016). G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T.Uehlinger, D. Greif, and T. Esslinger, Nature , 237(2014). S. de L´es´eleuc, V. Lienhardt, P. Scholl, D. Barredo, S. We-ber, N. Lang, H. P. B¨uchler, T. Lahange, and A. Browaeys,Science , 775 (2019). V. M. Martinez Alvarez and M. D. Coutinho-Filho, Phys.Rev. A , 013833 (2019). C. Li and A. E. Miroshnichenko, Physics , 2 (2019). S. Mukherjee and R. R. Thomson, Opt. Lett. , 5443(2015). S. Mukherjee, M. Di Liberto, P. ¨Ohberg, R. R. Thomson,and N. Goldman, Phys. Rev. Lett. , 075502 (2018). A. Szameit and S. Nolte, J. Phys. B , 163001 (2010). M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D.Podolsky, F. Dreisow, S. Nolte, M. Segev and A. Szameit,Nature , 196 (2013). M. Kremer, I. Petrides, E. Meyer, M. Heinrich, O. Zilber-berg, and A. Szameit, Nat. Comm. , 907 (2020). W. Jiang, S. Zhang, Z. Wang, F. Liu, and T. Low, NanoLett. , 1959 (2020). R. Chen and B. Zhen, Phys. Lett. , 944 (2017). A. Bhattacharya and B. Pal, Phys. Rev. B , 235145(2019). N. Goldman, D. F. Urban, and D. Bercioux, Phys. Rev. A , 063601 (2011). L. Madail, S. Flannigan, A. M. Marques, A. J. Daley, andR. G. Dias, Phys. Rev. B , 125123 (2019). A. M. Marques and R. G. Dias, Phys. Rev. B ,041104(R) (2019). A. J. Macdonald, P. C. W. Holdsworth, and R. G. Melko,J. Phys. Condens. Matt. , 164208 (2011). L. Fallani, C. Fort, J. E. Lye and M. Inguscio, Opt. Exp. , 4303 (1995). S. Sil, S. K. Maiti, and A. Chakrabarti, Phys. Rev. B ,193309 (2009). A. Aharony, O. Entin-Wohlman, Y. Tokura, and S. Kat-sumoto, Phys. Rev. B , 125328 (2008). A. Aharony, O. Entin-Wohlman, Y. Tokura, and S. Kat-sumoto, Physica E , 629 (2010). A. Mukherjee, R. A. R¨omer, and A. Chakrabarti, Phys.Rev. B , 161108(R) (2019). J. Vidal, R. Mosseri, and B. Do¸ucot, Phys. Rev. Lett. ,5888 (1998). J. Vidal, P. Butaud, B. Do¸uot, and R. Mosseri, Phys. Rev.B , 155306 (2001). D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyat-nikov, Phys. Rev. B , 224203 (2013). D. Leykam, A. Andreanov, and S. Flach, Adv. Phys. X ,1473052 (2018). D. Leykam and S. Flach, APL Photonics , 070901 (2018). W. Maimaiti, A. Andreanov, H. C. Park, O. Gendelman,and S. Flach, Phys. Rev. B , 115135 (2017). A. Ramachandran, A. Andreanov, and S. Flach, Phys.Rev. B , 161104(R) (2017). W. Maimaiti, S. Flach, and A. Andreanov, Phys. Rev. B , 125129 (2019). N. Myoung, H. C. Park, A. Ramachandran, E. Lidorikis,and J-W. Ryu, Sci. Rep. , 2862 (2019). E. Travkin, F. Diebel, and C. Denz, Appl. Phys. Lett. ,011104 (2017). S. Xia, C. Danieli, W. Yan, D. Li, S. Xia, J. Ma, H. Lu, D.Song, L. Tang, S. Flach, and Z. Chen, arXiv.: 1912.09703(2019). G. Gligori´c, P. P. Beli˘cev, D. Leykam, and A. Maluckov,Phys. Rev. A , 013826 (2019). D. L. Bergman, C. Wu, and L. Balents, Phys. Rev. B ,125104 (2008). J.-W. Rhim and B.-J. Yang, Phys. Rev. B , 045107(2019). R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B.Real, C. Mej´ıa-Cort´es, S. Weimann, A. Szameit, and M. I.Molina, Phys. Rev. Lett. , 245503 (2015). M. R¨ontgen, C. V. Morfonios, L. Brouzos, F. K. Diakonos,and P. Schmelcher, Phys. Rev. Lett. , 080504 (2019). J. Zak, Phys. Rev. Lett., , 2747 (1989). M. V. Berry, Proc. R. Soc. Lond. A. , 45 (1984). M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T.Kitagawa, E. Demler, and I. Bloch, Nature Phys. , 795(2013). T. Fukui and Y. Hatsugai, and H. Suzuki, J. Phys. Soc.Jpn. , 1674 (2005). N. Marzari and D. Vanderbilt, Phys. Rev. B , 12847(1997). M. B. de Paz, C. Devescovi, G. Giedke, J. J. Saenz, M.G. Vergniory, B. Bradlyn, D. Bercioux, and A. Garc´ıa-Etxarri, Adv. Quantum Technol. , 1900117 (2020). J. Arkinstall, M. H. Teimourpour, L. Feng, R. El-Ganainy,and H. Schomerus, Phys. Rev. B , 165109 (2017). S. Aubry and G. Andr´e, Ann. Isr. Phys. Soc. , 133 (1980). P. G. Harper, Proc. Phys. Soc. A , 874 (1955). D. J. Thouless, Phys. Rev. Lett. , 2141 (1988). S. Das Sarma, S. He, and X. C. Xie, Phys. Rev. Lett. ,2144 (1988). S. Das Sarma, S. He, and X. C. Xie, Phys. Rev. B , 5544(1990). J. Biddle and S. Das Sarma, Phys. Rev. Lett. , 070601(2010). S. Ganeshan, K. Sun, and S. Das Sarma, Phys. Rev. Lett. , 180403 (2013). J. K. Asb´oth, L. Oroszl´any, and A. P´alya, Springer (2016). G. Pelegr´ı, A. M. Marques, R. G. Dias, A. J. Daley, J.Mompart, and V. Ahufinger, Phys. Rev. A99