Interaction induced doublons and embedded topological subspace in a complete flat-band system
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Interaction induced doublons and embedded topological subspace in a completeflat-band system
Yoshihito Kuno, Tomonari Mizoguchi, and Yasuhiro Hatsugai
Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (Dated: September 18, 2020)In this work, we investigate effects of weak interactions on a bosonic complete flat-band sys-tem. By employing a band projection method, the flat-band Hamiltonian with weak interactionsis mapped to an effective Hamiltonian. The effective Hamiltonian indicates that doublons behaveas well-defined quasi-particles, which acquire itinerancy through the hopping induced by interac-tions. When we focus on a two-particle system, from the effective Hamiltonian, an effective subspacespanned only by doublon bases emerges. The effective subspace induces spreading of a single dou-blon and we find an interesting property: The dynamics of a single doublon keeps short-rangedensity-density correlation in sharp contrast to a conventional two-particle spreading. Furthermore,when introducing a modulated weak interaction, we find an interaction induced topological subspaceembedded in the full Hilbert space. We elucidate the embedded topological subspace by observingthe dynamics of a single doublon, and show that the embedded topological subspace possesses abulk topological invariant. We further expect that for the system with open boundary the embeddedtopological subspace has an interaction induced topological edge mode described by the doublon.The bulk–edge–correspondence holds even for the embedded topological subspace.
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I. INTRODUCTION
Flat-band systems so far have been receiving a lot ofattentions in condensed matter community. Physics ofthe flat-band is rich. Interesting subjects exist such asflat-band ferromagnetism [1–3], flat-band superconduc-tors [4–8], and topological flat bands [9–14], etc. Fur-thermore, localization properties of flat-band systems areexotic. Flat-band systems possesses an interesting sin-gle particle mode, “compact localized state” [15], whichhas a spatial spread of a few sites. Then, the flat-bandsystem exhibits an exotic dynamical aspect [16, 17]: Aspecific initial state of one particle tends to be localizeddynamically. Furthermore, for complete flat-band sys-tems where all bands are flat, a complete dynamical lo-calization called Aharanov-Bohm (AB) caging is knownto exist. Historically, a complete flat-band model wasfirst proposed in Ref. 18 and the AB caging has beenexpected there. The AB caging is also a key factor of adisorder free localization, which is a hot topic [15, 19]. Onthe experimental side, two decades ago, the AB cagingwas first realized in GaAs/GaAlAs system [20]. Recently,photonic experiments also realized a complete flat-bandsystem [21, 22] and succeeded in the observation of theAB caging. All of these are noninteracting systems.Here, it is very natural to ask whether or not theAB cages survive for interacting systems or what kindof changes the interaction gives to the AB caging. Sofar, two decades ago, the first research was done for theabove asking by Vidal, et.al. [23]: They found that, oncea weak Coulomb repulsive force is switched on, ballis-tic spreading of two-particles has been expected undera flat-band diamond chain. In other words, interactionsinduce delocalization and spreading. This seems to besomewhat counter-intuitive. Even in recent theoretical studies, the fate of the ABcaging under interactions has been investigated in [24–27], and the effects of interaction have been also inves-tigated from the viewpoint of the disorder–free many–body–localization [28–31]. However, we have not fullyelucidated the effects of the interaction for complete flat-band systems.In this work, we investigate the fundamental effectsof the interaction for a complete flat-band system byconsidering an interacting two-particle system on a sim-ple one–dimensional flat-band lattice, namely flat-bandCreutz ladder [32, 33]. The Creutz ladder has been re-cently implemented in coldatoms [34], and also can beimplemented for photonic crystals with some weak non-linearities of the medium like Kerr nonlinearity. In ourtheoretical analysis, we make use of a band projectionmethod [35–37]. The Creutz ladder with weak inter-actions can be mapped into an effective Hamiltonian,which indicates the presence of a paired particle, namely,doublon. We focus on two-particle systems, and inves-tigate characteristic properties of the single doublon, es-pecially for its dynamical aspects and topological prop-erties. Beyond the previous study [23], we will explicitlyshow the presence of the subspace constituted only bydoublon bases and clearly show the interaction-induceditinerancy of single doublon in details. We find an in-teresting property on the complete flat-band with weakinteractions, that is, the doublon spreading keeps a short-range density-density correlation which does not appearin a conventional two-(free)particle spreading.Much recently, Pelegr´ı et al. [38] studied a two-particlesystem in a diamond chain with complete flat-bands,which can be feasible in coldatoms [39]. There, theyshowed from the numerical diagonalization for the opensystem that a suitable interaction tuning induces dou-blon edge modes. In this regard, we also introduce amodulated interaction for our Creutz ladder, study thedynamical aspect, and in particular focus on the study ofthe bulk topology in the two-particle doublon subspace,which is a different perspective from the reference [38].Detection of a band topology through single particle dy-namics in topological systems attracts great interest inboth theory [40–43] and experiment [44, 45]. Such a de-tection can be also applicable to our target system asshown in this work. We show that for the two-particlesystem a modulated interaction induces a topological sub-space of the doublon embedded into the full Hilbert space ,and that the topology of that subspace is also clearly vis-ible from spreading of a single doublon, and the topolog-ical invariant can be extracted from the dynamics. Wenumerically demonstrate the dynamics of a single dou-blon, where the dynamics is governed only by the topo-logical subspace, of which the Hamiltonian correspondsto the low energy effective Hamiltonian obtained by theband projection method. Then we clearly show the pres-ence of the bulk–edge–correspondence (BEC) [46] for thetopological subspace embedded in the full Hilbert space.To our knowledge, the BEC for the subspace is also anew finding.This paper is organized as follows. In Sec. II, we in-troduce the Creutz-ladder model with weak on-site inter-actions. Section III shows the band projection methodand the effective Hamiltonian. The properties of energyspectrum and eigenstates for two-particle system are dis-cussed in Sec. IV. Base on the properties, in Sec. V, weclarify the dynamics of the two-particle system inducedby weak uniform on-site interactions. In Sec. VI, we dis-cuss the effects of a modulated interaction. The proper-ties of energy spectrum and eigenstates for two-particlesystem are discussed in detail. In particular, we showthat a part of the subspace is topological. From the sin-gle doublon dynamics the bulk topological invariant isextracted. And we clarify the presence of the BEC. Sec-tion VII is devoted to the conclusion.
II. CREUTZ LADDER
In this work, we consider the Creutz ladder (itsschematic picture is shown in Fig. 1). The model wasoriginally introduced by Creutz [32, 33], whose Hamilto-nian is given by H C = X j (cid:20) − it ( a † j +1 a j − b † j +1 b j ) − t ( a † j +1 b j + b † j +1 a j ) (cid:21) , (1)where a ( † ) j and b ( † ) j are the boson annihilation (creation)operators on the upper and lower chains, respectively. j refers to a unit cell. t and t are the intra-chain andinter-chain hopping amplitudes, respectively. When t = t two complete flat-bands with E = ± t appear due FIG. 1: Creutz ladder model: the yellow circle represents aunit cell, the blue and magenta shade objects are compactlocalized states, W + ,j and W − ,j . to the interference of the hoppings. In what follows, weset the flat-band condition, t = t . Under the flat-bandcondition, compact localized eigenstates of the upper andlower flat-bands are given by [35, 36] W †− ,j = 12 (cid:20) ia † j +1 + b † j +1 + a † j + ib † j (cid:21) , (2) W † + ,j = 12 (cid:20) ia † j +1 + b † j +1 − a † j − ib † j (cid:21) . (3)The W ± ,j operator can be regarded as bosonic particles,which satisfy[ W α,j , W † β,j ′ ] = δ αβ δ jj ′ , [ W ( † ) α,j , W ( † ) β,j ′ ] = 0 , where [ · ] implies a (bosonic) commutation relation. Ac-cordingly, we call the particle represented by W ± ,j oper-ator a “W-particle”.When one substitutes Eqs. (2) and (3) into H C , theCreutz ladder is written by H C = L − X j =0 (cid:20) − t W †− ,j W − ,j + 2 t W † + ,j W + ,j (cid:21) . (4)Here, the W-particle picture includes no hoppings andonly on-site potential terms. The number operators ofthe W-particle W †± ,j W ± ,j is a conserved quantity. Thepresence of the conserved quantities leads to a disorderfree localization [25–31]. As discussed in [35], the modelalso has topological properties. For a finite system withopen boundary, the edge mode γ ℓ ( ℓ = L ( R )) appears.Then, the Creutz ladder with open boundary is writtenby H C = L − X j =0 (cid:20) − t W †− ,j W − ,j + 2 t W † + ,j W + ,j (cid:21) + ǫ edge ( γ † L γ L + γ † R γ R ) , (5)where γ ( † ) L ( R ) is a annihilation (creation) operator of theleft (right) edge mode, and ǫ edge is an energy for the edgemodes. For noninteractiong case, ǫ edge = 0. It shouldbe noted that in W -particle sector, one lattice site isvanished due to the presence of the edge mode γ R ( L ) . Inwhat follows, we set t = 1. III. PERTURBATION: WEAK ON–SITEINTERACTION
Let us introduce a weak interaction in the flat–bandCreutz ladder. We consider the following weak on-siteinteraction: δV = L − X j =0 (cid:20) U aj a † j a † j a j a j + U bj b † j b † j b j b j (cid:21) . (6)In this work we assume that the interaction strength ismuch smaller than the band gap, U aj , U bj ≪ | t | (= 4).Let us express the interaction term with W-particlesby using a j = 12 (cid:20) W − ,j − W + ,j + iW − ,j − + iW + ,j − (cid:21) , (7) b j = 12 (cid:20) W − ,j − + W + ,j − + iW − ,j − iW + ,j (cid:21) . (8)Here, we employ the band projection method [35–37].Since the interaction is weak, we assume that the W -particle picture is valid and that when the interactionterm is expanded by W-particle operators one can ignoreterms containing W + or W † + . Under this situation, wecan construct a lower flat-band projected effective Hamil-tonian under periodic boundary condition, which is givenas H p e = L − X j =0 U j (cid:20)(cid:18) − B † j B j − + h.c. + B † j B j + B † j − B j − (cid:19) +4 W †− ,j W − ,j W †− ,j − W − ,j − (cid:21) + E self . (9)Here B † j ≡ W † j W † j , describing a doublon, E self is a self-energy coming from the on-site interactions, and the dou-blon hopping terms emerge. Note that the higher flat-band projected effective Hamiltonian can also be con-structed as well. In this work, based on the effectiveHamiltonian H p e , we will discuss the physics of two-particle. IV. TWO-PARTICLE SPECTRUM FORUNIFORM INTERACTION
Let us consider a uniform interaction U aj = U bj = U in a two-particle system. This situation elucidates theessence of the interaction effects in the complete flat-band system. To clarify the properties of the system wecalculate all energy spectrum and eigenstates for a finitesystem with periodic boundary condition.For noninteracting situation, the two-particle eigen-states are constituted just by occupying two of W -particle states. This picture mostly survives when theweak interaction is switched on. Figure 2 (a) is the fullspectrum of the two-particle system with finite U [47]. Here, there are three bands: approximately, the lowerband is constructed by states where two of W †− ,j are oc-cupied, the middle band by one W †− ,j and one W † + ,j areoccupied, and the higher band is by two of W † + ,j are oc-cupied.In particular, we show the closeup of the lowest bandspectrum in Fig. 2 (b). For the lowest band the effec-tive Hamiltonian H p e is valid. From the form of H p e wecan further separate the lowest band space by using thefollowing three kinds of bases: | W apk,ℓ i = W †− ,k W †− ,ℓ | i , | k − ℓ | ≥ , (10) | W NNj i = W †− ,j W †− ,j +1 | i , (11) | B j i = W †− ,j W †− ,j | i , (12)where | W apk,ℓ i is a two particle state that two W-particlesdo not overlap, | W NNj i is a two particle state that the twoW-particles share two sites, and | B j i is a single doublonstate of W-particles; | i is a vacuum. One can easilyconfirm that | W apk,ℓ i and | W NNj i become eigenstates for H p e ; H p e | W apk,ℓ i = E self | W apk,ℓ i , (13) H p e | W NNj i = (cid:18) U E self (cid:19) | W NNj i . (14)On the other hand, the doublon base | B j i is not an eigen-state for H p e due to the presence of the doublon hoppingterm. In the lowest band, the doublon hopping term of H p e acts as an off-diagonal operator for | B j i . Accordingly, H p e leads to a set of the eigenstates, which is generallygiven by the linear combination of | B j i basis: | φ Bℓ i = X j c ℓj (cid:18) √ | B j i (cid:19) , (15) X j | c ℓj | = 1 . (16)The eigenstate | φ Bℓ i are regarded as an extended state ofthe doublon if c ℓj has a broad distributions with a finitevalue.Actually, such eigenstates | φ Bℓ i can be detected even ifthey are buried in the full set of the eigenstates. Let usintroduce the following doublon subspace projector: P B = 12 L − X j =0 | B j ih B j | . (17)Note that the eigenstate | φ Bℓ i are normalized such that h φ Bℓ | P B | φ Bℓ i = 1. This means that the eigenstate | φ Bℓ i is spanned only by | B j i basis. Numerically, we calcu-lated P B for all of the lowest band eigenstates. The re-sult is shown in Appendix. We can numerically find L eigenstates spanned only by | B j i basis. The set of theeigenstates | φ Bℓ i is embedded in the full Hilbert space. In state (cid:1) (cid:0) E ne r g y state (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) E ne r g y FIG. 2: (a) The entire energy spectrum for two-particle sys-tem. We see three bands. Here we set L = 20, U = 0 .
1. (b)The closeup of the lowest band spectrum. Between 170-thand 209-th states, a mini dispersive band structure appears,where some extended doublon eigenstates | φ Bℓ i are lurking. what follows we call the set of the eigenstates “doublonsubspace”. The dimension of the doublon subspace is L .We further characterize the doublon subspace. Sincethe effective Hamiltonian H p e possesses the doublon hop-ping term, we expect that the hopping term induces ex-tended eigenstates of the doublon. Let us verify whetheror not such an extended state exists as the subspace. Tothis end, we observe the response to a flux insertion [48–50], which can be introduced by a phase twist for the hop-ping amplitude: t → t exp( iθ/L ), t → t exp( iθ/L )[51]. When one varies θ from − π , the single magnetic fluxis injected into the periodic system. As for the generalresponse of the flux insertion, the followings are known[48–50]: While an eigenstate composed of single parti-cles exhibits 2 π –periodicity, an eigenstate constituted bya paired object bases exhibits π –periodicity since thecharge of the paired object is twice as large as that ofa single particle. We observe the flux response for thelowest band spectrum of the two-particle system. Thenumerical results are shown in Fig.3 (a) and (b). Here,for comparison, we also investigate the case of break-ing the flat-band condition (i.e., t = t ). For nonflat-band and noninteracting case, all spectrum in the low-est band exhibit 2 π –periodicity. On the other hand, forflat-band and weak interaction case (Fig.3 (b)), interest-ingly enough some spectrum exhibit π -periodicity whilethe others do not response to the flux insertion. Actu-ally, the orange spectrum lines in Fig.3 (b) correspondto eigenstates constituted by the doublon base, that is,they belong to the doublon subspace. On the other hand,the blue lines in Fig.3 (b) are much degenerate, and thecorresponding eigenstates are somewhat localized. Suchstates do not respond to the flux insertion [52]. Thesestates are constituted by | W apk,ℓ i and | W NNj i bases. Fromthe response to the flux insertion, we confirm that the ex-tended eigenstates constituted by the doublon base existwithin the full Hilbert space. So far such a response hasbeen reported as for the many-body groundstate [49, 50],but in our data, we observed that excited states also ex-hibit such π -periodicity. (a) (b) FIG. 3: Flux insertion response: (a) nonflat-band case, t =1 . U = 0 .
1, (b) Flat-band case, t = 1, U = 0 .
1. Theorange lines represent the eigenstates constituted only by thedoublon bases | B j i , that is | φ Bℓ i . The number of the orangespectrum lines is L . V. UNITARY DYNAMICS FOR UNIFORMINTERACTION
So far, we have clarified the spectrum and eigenstatesfor the two-particle system with the uniform interaction.In this section, we show that the two-particle system gov-erned by H p e exhibits interesting dynamics. From H p e ,the doublon hopping terms make the doublon state ex-tended. Then, if one puts a single doublon at j , whatdynamics does the initial single doublon show? With-out interaction, the doublon initial state is just two non-interacting W -particles. On the flat-band system, eachparticle does not spread out, that is, exhibit just the ABcaging. On the other hand, for finite U , where the flat-band is partially distorted, some eigenstates turn intoextended states while the rest of the eigenstates are lo-calized states. From such extended states, if one puts asingle doublon as initial state, the dynamics is expectedto exhibit spreading. Interestingly enough, the dynamicskeeps a correlation. In what follows, we discuss such asingle doublon dynamics and show the explicit analyti-cal description for it, and numerically demonstrate thedynamics of the single doublon.We investigate the dynamics of the initial state | B j i ,where j = ( L − /
2. The dynamics is given by a unitarydynamics. Here, we recognize that there are the followingrelations for the two W-particle bases: h B j | B j ′ i = 2 δ jj ′ , h B j | W NNj ′ i = 0 , h B j | W apk,ℓ i = 0 . (18)Furthermore, since W + and W − are orthogonal to eachother at single-particle level, all eigenstates in the middleand higher bands described by | φ mh ℓ i is orthogonal to | B j i , h φ mh ℓ | B j i = 0. Then, the unitary dynamics can begiven by the following form (we set ¯ h = 1) U ( t ) = U L ( t ) + U mh ( t ) , (19) time (a) (b)(c) r = 1 r = 2 r = 3 r = 4 site ti m e time site(d) (e)(f) ti m e r = 1 r = 2 r = 3 r = 4 timetime FIG. 4: (a) Time evolution of doublon density for flat-band case ( U = 0 . t = 1). The initial state is B j | i . (b) Thedynamical behavior of the eigenvalue of the doublon projector, h Ψ( t ) | P B | Ψ( t ) i for flat-band case. (c) The dynamical behaviorof the density-density correlation C ( r, t ) for flat-band case. (d) Time evolution of doublon density for nonflat-band case( U = 0 . t = 1 . B j | i . (e) The dynamical behavior of the eigenvalue of the doublon projector, h Ψ( t ) | P B | Ψ( t ) i for nonflat-band case. (f) The dynamical behavior of the density-density correlation C ( r, t ) for nonflat-bandcase. where U L ( t ) = X ℓ e − iǫ Bℓ t | φ Bℓ ih φ Bℓ | + X ℓ e − iǫ NNℓ t | W NNℓ ih W NNℓ | + X ℓ e − iǫ apℓ t | W apℓ ih W apℓ | , (20) U mh ( t ) = X ℓ e − iǫ mh ℓ t | φ mh ℓ ih φ mh ℓ | . (21)Here, ǫ bℓ , ǫ NNj and ǫ apℓ are eigenvalues of the eigenstatesfor the subspace of the lowest band, and ǫ mh ℓ is an eigen-value of the eigenstates in the middle and higher band.Due to the orthogonal relations in Eq. (18) and h φ mh ℓ | B j i = 0, the unitary dynamics for the initial state | B j i is governed only by U L , the dynamics is given as asimple form: | Ψ( t ) i = U ( t ) | B j i = U L ( t ) | B j i = X ℓ e − iǫ bℓ t | φ Bℓ ih φ Bℓ | B j i , (22)that is, the dynamics of the single doublon is governedonly by the doublon subspace, i.e., the set of the eigen-states | φ Bℓ i . The extended eigenstate | φ Bℓ i makes theinitial localized single doublon spread along the timeevolution. Accordingly, the weak interaction U inducesspreading phenomena on the flat-band system, which issomewhat counter-intuitive.We further show that the dynamics of the single dou-blon possesses an interesting correlation property. Let usintroduce a density-density correlation: C ( r, t ) = 1 L X i h Ψ( t ) | n αi n αi + r | Ψ( t ) i (23) where n αi = a † i a i ( α = a ), ( b † i b i ) ( α = b ), i is a latticesite for each chain in the Creutz ladder. In what follows,we set α = a . One can show that C ( r, t ) has a specificproperty by the following calculation.To begin with, we remark that the following conditionshold: h B j | n ai | B j i = ( i ∈ ˜ j )0 ( otherwise ) (24)where ˜ j is the set of sites resided in the doublon at j , | B j i (there are four sites in a set ˜ j ). From this relationof Eq. (24), the following conditions are obtained: h B j | n ai n ai | B j i = ( δ j j × (const.) ( i , i ∈ ˜ j )0 (otherwise) . (25)Let us consider i + 1 = i ( r = 1) case. Then thecondition of Eq. (25) becomes h B j | n ai n ai +1 | B j ′ i = 14 δ jj ′ , ( i , i + 1 ∈ ˜ j ) , (26)where i , i + 1 ∈ ˜ j . By using the above condition ofEq. (26) and the eigenstates | φ Bℓ i of Eq. (15), we canexpress the short-range correlation, C ( r = 1 , t ) as C (1 , t ) = 1 L X i h Ψ( t ) | n ai n ai +1 | Ψ( t ) i = 1 L X i ,ℓ,ℓ ′ ,j,j ′ K ℓ ′ ∗ j ′ ( t ) K ℓj ( t ) h B j | n ai n ai +1 | B j i = 14 L X j ,ℓ,ℓ ′ K ℓ ∗ j ( t ) K ℓ ′ j ( t ) , (27) state − − E ne r g y state − − − − − − E ne r g y FIG. 5: (a) The entire energy spectrum, we see three bands.Here we set L = 20, U e = 0 . U e = 0 .
05. (b) The closeupof the lowest band spectrum. The band splits to small sub-bands due to the modulated on-site interaction. The smallband gap originates from the doublon subspace. where K ℓj ( t ) = e − iǫ bℓ t c ℓj c ℓ ∗ j . (28)From this form, C (1 , t ) has possibility to take a finitevalue with oscillation coming from the exponential factorin Eq. (28). On the other hand, for r ≥
2, we expect thatthe correlation suddenly vanishes, C ( r, t ) = 0, due tothe condition of Eq. (25). Therefore, the density-densitycorrelation behaves as C ( r, t ) = ( (finite) r ≤
10 ( r ≥ . (29)This implies that the unitary dynamics of the single dou-blon possesses the short-range density-density correla-tion, but no long-range one during the time evolution.Note that for spreading of free two particles, C ( r, t ) haspossibility to have a finite value even for r ≥ • If the doublon spreading occurs, the eigenvalue of P B keeps almost unity during time evolution. • Even under the doublon spreading, the density-density correlation function C ( r, t ) keeps short-range correlations.Figure 4 is the numerical results for the flat-band case.For the calculation of the dynamics, we set the unit oftime as ¯ h/t = 1. We first calculated the density ofthe doublon defined by h Ψ( t ) | B † j B j | Ψ( t ) i for each site j during time evolution. The result is shown in Fig. 4(a). Here, we remark that h Ψ( t ) | B † j B j | Ψ( t ) i = 4 meansthe single doublon localizes at the site j . For the weakinteraction the initial localized single doublon exhibitsan isotropic ballistic expansion, and also the expectationvalue of the doublon projector P B stays unity during timeevolution as shown in Fig. 4 (b). This indicates thatspreading of the single doublon occurs with the doublonpairing retained. As shown in Fig. 4 (c) the numericalresults for the density-density correlation C ( r, t ) is con-sistent with the analytical prediction of Eq. (29), that is, site site time ti m e (a) (b) (c) (d) ti m e time (flat)(nonflat)(uniform, flat) (flat)(nonflat)(uniform, flat) FIG. 6: Time evolution of doublon density for flat-band (a)and nonflat-band ( t = 1 .
1) (b). (c) The behavior of thedoublon projector, h Ψ( t ) | P B | Ψ( t ) i . (d) Time evolution of thedoublon MCD for various parameter cases. the short–range correlation C (1 , t ) is only finite duringthe time evolution and the more longer correlations arezero.On the other hand, for nonflat-band case, the initialsingle doublon spreads and decays as shown in Fig. 4 (d),and the expectation value of P B decreases during timeevolution, as shown in Fig. 4 (e). The initial doublondisappears with spreading, and also the density-densitycorrelation C ( r, t ) behaves differently from the flat-bandcase, the long-range correlations develop during time evo-lution [Fig. 4 (f)]. VI. EMERGENT TOPOLOGICAL SUBSPACE
In this section, we consider spatially–modulated on-site interaction. The modulation makes the band struc-ture of the doublon subspace gapped. The gapped dou-blon can subspace exhibit non-trivial topology. Fromnow on, we discuss the origin of the topological characterand numerically-demonstrate the topological properties.Let us set U aj ∈ odd ( even ) = U bj ∈ odd ( even ) = U o ( e ) and con-sider two-particle system. In this case, the band projec-tion method is valid and the lower band effective Hamil-tonian only for the doublon base | B j i is given by H pB e = L − X j =0 U j (cid:18) − B † j B j − + h.c. + B † j B j + B † j − B j − (cid:19) , (30)If we focus only on the doublon subspace, the above ef-fective Hamiltonian can be regarded as the Su-Schrieffer-Heeger (SSH) model [53]. The spectrum described by H pB e can be regarded as a single particle spectrum fordoublon. Figure 5 (a) is the full spectrum for the two-particle system. Globally, three bands appear again.But, if one looks at the lowest band in more detail,the lowest band exhibits a small gap as shown in Fig. 5(b). Here, the doublon subspace is embedded in the low-est band spectrum; its numerical verification is shownin Appendix. The small band gap in the lowest bandcomes from the gapped doublon subspace. The gap sizeis | U e − U o | × H pB e since all other eigenstates in the Hilbert space are or-thogonal to the doublon states. The bulk topology ofthe doublon subspace is expected to be characterized bythe winding number. To extract the bulk topology forthe doublon subspace, we introduce the doublon meanchiral displacement (MCD) [40, 44, 45], which can ex-tract the winding number from the dynamics of a singledoublon. The doublon MCD for the doublon subspacecan be given byMCD( t ) = h Ψ( t ) | Γ | Ψ( t ) i , (31)Γ = 14 L/ − X m =0 m (cid:20) | B m ih B m | − | B m +1 ih B m +1 | (cid:21) , (32)where Γ is a multiple operator created by the position andchiral operator for the doublon. If the doublon subspacehas a non-trivial topology, MCD( t ) oscillates around 1 / /
2, it is the signal that the winding number of the dou-blon subspace is one [40–42, 44, 45]. We note that thewinding number also corresponds to the Zak phase [55].For trivial topology, the MCD( t ) oscillates around zero.This means that the winding number is zero.Let us show numerical results, we investigate the uni-tary dynamics of the initial state | B j i , where j =( L − /
2. We set U o = 0 . U e = 0 .
05. The density of thedoublon is shown in Fig. 6 (a) and (b). For the flat-bandcase anisotropic spreading of the doublon appears whilefor the nonflat-band case, the density of the doublon de-cays with spreading during time evolution. The expec-tation values of P B for various cases are shown in Fig. 6(c). For the flat-band cases, the values of P B stay unity,the doublon paring is preserved, but for the nonflat-bandcase, the value of P B exhibits sudden decrease, meaningthat the doublon suddenly decays during time evolution.The result of MCD( t ) is shown in Fig. 6 (d), under theflat-band and modulated interaction case, the value ofMCD( t ) exhibits oscillation around 1 /
2, this is the sig-nal that the doublon subspace possesses the non-trivial (a) (b) t(cid:7)(cid:8)(cid:9) (cid:10)(cid:11)(cid:12)(cid:13)
FIG. 7: (a) Disorder-averaged expectation value of P B duringtime evolution. (b) Disorder-averaged doubon MCD. we set L = 20, and used 300 quench disorder samples. topology, that is, the winding number γ w = 1. For othercases, such a doublon MCD behavior does not appear,that is, the system is in trivial phase. From these, thedoublon subspace under the modulated interactions hasa non-trivial topology characterized by the winding num-ber. A. Robustness for weak disorder
Let us investigate the robustness for the topologicalproperties for the doublon subspace. We introduce auniform disorder for the interaction as U aj ∈ odd ( even ) = U bj ∈ odd ( even ) = U o ( e ) → U o ( e ) + δw j , where δw j ∈ [ − w , w ]. How do the doublon projector and thedoublon MCD behave? Figure 7 shows the effects of(quenched) disorder for P B and MCD( t ). For P B , forweak disorder w = 0 .
01, the value of P B is a little sup-pressed however, it does not decay so much during timeevolution. On the other hand, for more large disorder w = 0 .
03 and 0 .
05 larger than the bulk gap of the dou-blon subspace, the value of P B decays to some extent, thesaturation value seems to be ∼ .
5. Accordingly, the ro-bustness of P B for a disorder is qualitatively determinedby the subspace bulk gap. This is consistent with thebehavior of MCD. As shown in Fig. 7 (b), for a weakdisorder, MCD( t ) is not suppressed and almost oscillatearound 1 /
2. It implies that the doublon subspace stillhas non–trivial topology. On the other hand, for largerdisorder w = 0 .
03 and 0 .
05, MCD( t ) are highly sup-pressed, it implies that the doublon subspace no longerhas non-trivial topology. B. Topological doublon edge mode
We observed that the doublon subspace has the bulktopological invariant, characterized by the dynamics of adoublon. According to the spirit of the BEC, the bulktopology in the periodic system induces some edge modesunder open boundary condition [46]. Here, we numeri-cally verify the presence of edge states of the doublonand BEC for the doublon subspace.When we consider the system with open boundaries,as shown in Eq. (5), the original Creutz ladder possesses state − − E ne r g y state − − − − − − E ne r g y (a) (b) site FIG. 8: (a) The entire energy spectrum, there are five bands.(b) The lowest band spectrum. The band splits to the minisub-band due to the modulated on-site interaction. Withinthe band gap, there appears a single edge mode (the red cir-cle) described by the doublon. The inset shows the doublondensity for the edge mode. Here,we set L = 20, U o = 0 . U e = 0 . the single–particle edge modes. The W-particle W ± ,j residing at edges turn into left and right edge modes.Let us look for an edge mode different from the single-particle ones. The effective Hamiltonian by using theband projection method is given by H o e = L − X j =0 U j (cid:20)(cid:18) − B † j B j − + h.c. + B † j B j + B † j − B j − (cid:19) +4 W † j W j W † j − W j − (cid:21) + E self . (33)Here, if one focuses on the two-particle case and thedoublon subspace, the model corresponds to the SSHmodel described by B j with open boundaries. If L iseven, the SSH chain has an odd number of the totalsite. This leads to a single edge mode localized at ei-ther edge. Let us numerically treat the two-particle sys-tem with open boundaries. The entire spectrum of thetwo-particle system is shown in Fig. 8 (a). Comparedto the periodic case in Fig. 5 (a), the open boundarycase exhibits five bands. This is due to the presenceof the original (single particle) edge states γ † R ( L ) withzero energy. And for the lowest band as shown in Fig. 8(b), we find that a single edge mode resides within thesmall gap induced by the doublon SSH model in H o e . Seethe inset in Fig. 8 (b), the density distribution of dou-blon certainly reflects an edge state, given by | edge i ∝ [ B † L − + ( U e /U o ) B † L − + ( U e /U o ) B † L − + · · · ] | i , whichcan be regarded as a doublon edge mode. We empha-size that the doublon edge mode is different from thesignle-particle edge mode since the doublon edge modeis induced by the modulated interactions and is also atwo-particle object.From these facts, we conclude that the BEC holds forthe doublon subspace embedded in the full Hilbert space. VII. CONCLUSION
In this work, we considered two-particle physics ina complete flat-band model with weak on-site interac-tions. The weak interactions induce a subspace spannedby doublon bases, which is embedded in the full Hilbertspace. The eigenstates of the doublon subspace are ex-tended. This fact leads to spreading for a single dou-blon. The spreading has an interesting property: thedensity-density correlation exhibits short-range correla-tions. Furthermore, we clarified that a spatially mod-ulated weak interaction induces a topological subspacedescribed by the doublon bases. Numerically, we showedthat the bulk topological invariant for the doublon sub-space can be extracted by observing the unitary dynam-ics of the single doublon and then, for open boundarycase, we have found a doublon edge mode which is dif-ferent from the original topological edge modes in thenoninteracting Creutz ladder. Accordingly, we clearlyshowed that the topological doublon subspace embeddedin the full-Hilbert space exhibits the BEC. We hope thesefindings will be verified in future experiments. In thiswork, we considered repulsive interactions for the Creutzladder, but we expect that interaction-induced itinerantdoublons and their topology will appear in broader classof models, including attractively interacting systems.
Acknowledgments
The work is supported in part by JSPS KAK-ENHI Grant Numbers JP17H06138 (Y.K, Y.H.) andJP20K14371 (T.M.).
Appendix: Eigenvalues of doublon projector
We plot the eigenvalues of the doublon projector P B foreigenstates for two-particle system. Figure. 9 is the re-sult of the eigenvalues of the doublon projector P B for alleigenstates in the lowest band. The result for flat-bandand uniform interaction indicates that L eigenstates havethe nonzero eigenvalue 1, the others have zero eigenvalue[Fig. 9 (a)]. The former eigenstates are composed onlyby the doublon bases | B j i . For the modulated interac-tion case shown in Fig. 9 (b), the results also exhibit L eigenstates having the nonzero eigenvalue almost 1. Onthe other hand, for nonflat-band and uniform interactioncase shown in Fig. 9 (c), there is no eigenstate having thenozero eigenvalue 1. [1] A. Mielke and H. Tasaki, Comm. Math. Phys. , 341(1993). [2] H. Tasaki, Prog. Theor. Phys. , 489 (1998). state (a) state (b) state ((cid:14)(cid:15) FIG. 9: Distribution of eigenvalues of the doublon projector P B . (a) Flat-band and uniform interaction, t = 1, U = 0 . t = 1, U o = 0 . U = .
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