Two-body physics in the Su-Schrieffer-Heeger model
TTwo-body physics in the Su-Schrieffer-Heeger model
M. Di Liberto , A. Recati , , I. Carusotto , and C. Menotti INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, 38123 Povo, Italy Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universit¨at M¨unchen, 80333 M¨unchen, Germany (Dated: September 24, 2018)We consider two interacting bosons in a dimerized Su-Schrieffer-Heeger (SSH) lattice. We iden-tify a rich variety of two-body states. In particular, for open boundary conditions and moderateinteractions, edge bound states (EBS) are present even for the dimerization that does not sustainsingle-particle edge states. Moreover, for large values of the interactions, we find a breaking ofthe standard bulk-boundary correspondence. Based on the mapping of two interacting particlesin one dimension onto a single particle in two dimensions, we propose an experimentally realisticcoupled optical fibers setup as quantum simulator of the two-body SSH model. This setup is ableto highlight the localization properties of the states as well as the presence of a resonant scat-tering mechanism provided by a bound state that crosses the scattering continuum, revealing theclosed-channel population in real time and real space.
I. INTRODUCTION
In a perfectly periodic system, states outside the al-lowed bands can appear for both attractive and repulsiveinteractions when composite objects are formed [1, 2].The existence of “exotic” repulsive bound pairs, a.k.a.doublons, has been directly observed for the first timeten years ago by implementing a single-band Hubbardmodel with ultra-cold Bose gases in an optical lattice[3]. The study of doublons has been extended to, e.g.,long range interacting particles [4–6], two-channel models[7, 8], superlattices [9] and spinor gases [10]. Aside frompresenting behaviours and stability properties interestingby themselves [11], doublons deeply affect the dynamicsof the system. For instance very recently it has beenshown that the presence of doublons favours many-bodylocalization in disordered [12] or extended [13] Hubbardmodels.On the other hand, any real crystal is made of a bulkand a surface. The study of how surfaces modify thespectrum of a particle in a finite crystal started with theseminal papers by Tamm [14] and Shockley [15]. Theypointed out the existence of localized states at the sur-face with energy outside the allowed energy bands. Suchsurface states can play an important role in the trans-port properties. Particular attention has been devoted inthe recent years to their characterization in the so-calledtopological insulator materials [16]. The bulk–boundarycorrespondence provides a link between the presence andnumber of in-gap edge states and the topological invari-ants of the bulk crystal. The most famous example isthe chiral state on the edge of a two-dimensional inte-ger quantum Hall system (see e.g. [17]). While most ofthe above mentioned surface states are well explained bysingle-particle band theory, the physics becomes muchmore intriguing in the presence of strong inter-particleinteractions.In this work, we make an important step forward try-ing to combine a topologically non-trivial single-particleband structure with interactions. A prototypical phe- J A B AB B J A B A B A B A BA AB (D1)(D2) (a) (b)
A A AB BB J J J FIG. 1. Sketch of the SSH model considered in this work. ForOBC and even number of sites, one can obtain two dimeriza-tions: (D1) dimerization D J and (D2) dimerization D J . (a) Example of two particles in a dimerized po-tential described by a Su-Schrieffer-Heeger model; (b) Sketchof the mapping onto a 2D single-particle system: strong links J (full lines), weak links J (dashed lines), and local potential U (dark sites). nomenon of this kind is the well celebrated fractionalquantum Hall effect [18, 19]. Here, we focus our atten-tion on the minimal model of two interacting particles ina Su-Schrieffer-Heeger (SSH) lattice. The full two-bodyspectrum can be calculated and very rich physics emergesin spite of the simplicity of the model. In particularwe find: (i) hybridization of different channels leadingto Fano-Feshbach resonances; (ii) existence of out-of-cell(long range) bound pairs; (iii) edge states for the boundpairs. We conclude by proposing an experimentally re-alistic optical fiber setup to quantum simulate the two-body SSH model in the laboratory and experimentallyhighlight our predictions. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n - 0 K E / J ϵ ϵ ϵ d d d NN d d d d d NN (a) (f)(c)(d) y (e) x (b) xy FIG. 2. (a) Two-body spectrum for PBC as a function of the center of mass momentum K for U = 3 J and J = 0 . J ,obtained from scattering theory. The spectrum presents three scattering continua and four bound states d , d NN , d and d . (b) Two-body spectrum as a function of the interaction U for J = 0 . J : The colorscale indicates the sum of onsiteand nearest-neighboring site population and highlights the bound states. (c-f) Bound-state wave functions for U = 3 J and J = 0 . J in a chain of 24 sites at K = 0 for d (c), d (e), d (f) and at K = π for d NN (d). II. MODEL
In this work, we study two interacting bosonic particlesin the dimerized lattice shown in Fig. 1 and governed bythe Hamiltonian H = H + H U , where H = − J (cid:88) i c † A,i c B,i − J (cid:88) i c † A,i +1 c B,i + H . c . (1)is the kinetic part providing the single particle SSHmodel, whereas H U = U (cid:88) i (cid:16) c † A,i c † A,i c A,i c A,i + c † B,i c † B,i c B,i c B,i (cid:17) (2)describes on-site interactions. For later convenience, wedefine a lattice cell by a pair of A and B sites linked bytunneling J > J >
0, and label each lattice cell withindex i . For periodic boundary conditions (PBC) the twopossible dimerizations are obtained via a shift of a singlelattice site, corresponding in practice to the interchangeof strong and weak tunneling. For open boundary con-ditions (OBC) and even number of sites, we define D J and D J (see Fig. 1(D1,D2)).In addition to ultra-cold atom implementations, an in-teresting perspective of our work is to investigate thesame physics with 2D lattices of side-coupled opticalwaveguides, exploiting the mapping of two interactingparticles in 1D (Fig. 1(a)) onto a single particle in 2D[20, 21]. As sketched in Fig. 1(b), the dimerized lat-tice is reproduced by appropriately-tailored spatially-alternating hoppings in the 2D lattice. Two-body on-site interactions in the 1D system are translated into a localpotential U on the diagonal x = y in the single-particle2D model. A straightforward extension of existing exper-iments [22–24], would allow the possibility of observingdistinctive two-body SSH dynamics directly in real spaceand real time. III. BULK SYSTEM
For periodic boundary conditions (PBC), the single-particle SSH model possesses particle-hole symmetry [25]and the spectrum is formed by two Bloch bands with en-ergy E ± ( k ) = ± (cid:112) J + J + 2 J J cos( k ). The two pos-sible dimerizations have the same spectrum, but present aZak phase difference of π [26, 27], corresponding to topo-logically distinct phases identified by different windingnumbers [28]. Apart from the case of hard-core bosonsat half-filling (e.g. [29, 30]), the interacting Hamiltonianbreaks chiral symmetry and a typical two-body spectrumis shown in Fig. 2(a) as a function of the center of massmomentum K .The essential spectrum is not modified by interactionsand consists of three scattering continua (type I), ob-tained by attributing an energy belonging to the single-particle SSH spectrum to each scattering particle. In-stead, the wave functions of scattering states are modi-fied by interactions, showing a depletion at zero relativedistance by increasing U .In one dimension (1D), any interaction introduces adiscrete spectrum, related to the formation of boundpairs. Three bound pairs are readily identified by con-sidering the fully dimerized case J = 0. For each cell i , the strong-link Hamiltonian admits the three differentstates (written in the two-body basis | A i A i (cid:105) , | A i B i (cid:105) and | B i B i (cid:105) ) | d ,i (cid:105) ∝ (cid:18) √ J , U + (cid:113) J + U , √ J (cid:19) , (3) | d ,i (cid:105) ∝ (1 , , − , (4) | d ,i (cid:105) ∝ (cid:18) √ J , U − (cid:113) J + U , √ J (cid:19) , (5)with energies (cid:15) = ( U − (cid:112) J + U ) / (cid:15) = U and (cid:15) = ( U + (cid:112) J + U ) / J finite, the pairs delocalize along the lattice anddevelop narrow bands (see Fig. 2). The three boundstates d α can be well defined for all values of K at energieseither in the band gaps or above the continuum, or crossthe continua in some parameter range (see Fig. 2(b)).Finally, at energies ∼ J , an additional out-of-cell bound state d NN appears, characterized by a predom-inant contribution in neighboring cells ( | A i (cid:105) − | B i (cid:105) ) ⊗ ( | A i +1 (cid:105)−| B i +1 (cid:105) ). Such state arises thanks to an effectivenearest-neighbor interaction due to virtual processes in-volving mainly the d state (see Appendix A 3). The d NN state is present only for momenta around K = π . Thisfact can be understood because the emergent nearest-neighbor interaction, which is responsible for the bind-ing, is very weak compared to the bandwidth 2 J ofthe scattering continuum (see for instance [4]). When (cid:15) = U ∼ J , the d state becomes resonant with d NN and a strong mixing between the two is observedFig. 2(d). A. Scattering theory
The spectra of the bound states can be obtained bysolving the Lippmann-Schwinger equation on the lattice.For two particles, it is useful to describe the externaldegrees of freedom using center-of-mass R = ( x + y ) / r = x − y for the two particlesat lattice positions x and y , and the center-of-mass K = k + k and relative quasi-momentum k = ( k − k ) / K isa good quantum number, allowing to plot the spectrumas E ( K ). For the sake of clarity, in a dimerized latticeof N c cells of lattice spacing D (corresponding to N s =2 N c lattice sites of lattice spacing d = D/
2) the allowed K values in the first Brillouin zone are given by K =2 π(cid:96)/ ( N c D ) for (cid:96) = − N c / , . . . , N c /
2, which, uponBrillouin zone folding, coincide with the allowed K valuesfor a uniform lattice K = 2 π(cid:96)/ ( N s d ) for (cid:96) = − N s / , . . . , N s / p i = 1 √ c A,i + c B,i ) , m i = 1 √ c A,i − c B,i ) , (6) the single-particle Hamiltonian is cast into the form H (cid:48) = − J (cid:88) i (cid:16) p † i p i − m † i m i (cid:17) (7) − J (cid:88) i, ν = ± (cid:16) p † i + ν p i − νp † i + ν m i + νm † i + ν p i − m † i + ν m i (cid:17) . Hamiltonian H (cid:48) describes a particle with pseudo-spindegrees of freedom, labeled as p, m , hopping on a one-dimensional lattice. This transformation is useful to treatthe two-body problem because the center of mass of eachof the two single-particle states p, m is located in the mid-dle of the A − B bond. In a first-quantization description,the two-body wavefunction can be written as | Ψ (cid:105) = (cid:88) x,y,σ ψ σ ( x, y ) | x, y (cid:105) ⊗ | σ (cid:105) , (8)where x, y are, respectively, the unit-cell coordinates ofparticle 1 and 2, and | σ (cid:105) ∈ B σ are the two-body spinstates B σ = | + (cid:105) = | p, p (cid:105) , | (cid:105) = √ ( | p, m (cid:105) + | m, p (cid:105) ) , |−(cid:105) = | m, m (cid:105) , | F (cid:105) = √ ( | p, m (cid:105) − | m, p (cid:105) ) . (9)For the case of indistinguishable bosons discussed here,the amplitude ψ σ ( x, y ) is symmetric when exchanging x ↔ y , except for the σ = F component, which must beantisymmetric in order to provide an overall symmetricwave function. In the pseudo-spin basis, the interactionoperator H U is still local, but not diagonal and repre-sented by the matrix H U = 12 U U
00 2 U U U
00 0 0 0 . (10)In the center-of-mass and relative coordinates, we makethe standard ansatz ψ σ ( x, y ) ≡ e iKR ψ σ ( r ) where K = k + k is the center-of-mass momentum. This choice ofcoordinates and the choice of basis B σ allow to decou-ple the center of mass from the relative motion. Afterstraightforward but tedious calculations, we obtain theSchr¨odinger equation (cid:2) H B + δ r, H U (cid:3) σ,σ (cid:48) ψ σ (cid:48) ( r ) = E ψ σ ( r ) , (11)where the kinetic part of the two-body Hamiltonian reads H = − J − J cos( K/ + r − i J √ sin( K/ + r J √ cos( K/ − r i J √ sin( K/ + r − i J √ sin( K/ + r − iJ sin( K/ − r i J √ sin( K/ + r J + J cos( K/ + r J √ cos( K/ − r − J √ cos( K/ − r − iJ sin( K/ − r − J √ cos( K/ − r . (12)Here, we defined the discrete operators ∆ + r ψ σ ( r ) = ψ σ ( r +1)+ ψ σ ( r −
1) and ∆ − r ψ σ ( r ) = ψ σ ( r +1) − ψ σ ( r − ψ σ ( r ) = (cid:104) rσ | ˆ G ( E ) ˆ H U | ψ (cid:105) = (cid:88) σ (cid:48) , σ (cid:48)(cid:48) (cid:90) dk π e ikr G σσ (cid:48) ( k, E )( H U ) σ (cid:48) σ (cid:48)(cid:48) ψ σ (cid:48)(cid:48) (0)= (cid:88) σ (cid:48) , σ (cid:48)(cid:48) G σσ (cid:48) ( r, E )( H U ) σ (cid:48) σ (cid:48)(cid:48) ψ σ (cid:48)(cid:48) (0) , (13)where we have definedˆ G ( k, E ) = ( E − H B ( K, k )) − . (14)This formalism has been used to calculate the boundstate spectrum shown in Fig. 2(a). B. Resonant scattering
The first noteworthy bulk feature of the two-bodySSH model, persisting for any boundary condition, is thestrong mixing of the d bound-state narrow band andtype I scattering continuum around the resonance condi-tion U = 2 J , where the bound-state energy U matchesthe energy of a scattering state. This mixing leads toa Fano-Feshbach resonance in a lattice [4, 7], and canbe described analytically by using a two-channel scatter-ing theory, as shown below. The occurrence of scatter-ing resonances due to repulsive bound states in multi-band Hamiltonians has been studied also in other con-texts [8, 10].A Feshbach-like resonant scattering process is numeri-cally illustrated in Fig. 3, where we plot the square modu-lus of the two-body wave function ψ ( x, y ) at times before,during and after the collision. At t = 0 we prepare twosingle-particle gaussian wave packets at momenta k = k and k = − k in the upper band of the SSH model, lo-calized at symmetric positions with respect to the latticecenter, sufficiently far from the boundaries and from eachother, as shown in Fig. 3(a). This initial state belongsto the two-body scattering continuum centered aroundenergy 2 J . The time evolution in the presence of inter-actions U is calculated numerically. After collision, weobserve two scattered wavepackets and a sizable popula-tion of a two-body bound wavepacket of type d , highlylocalized along x = y . At the beginning, the population U/J D i a g o n a l d e n s i t y k = 3 /4k = /2 t = 0 t = 32 ħ /J t = 52 ħ /J (a) (b) x x xy
00 0 079 79 7979
FIG. 3. (a) Modulus of the two-body wavefunction | ψ ( x, y ) | for two incident wave packets before, during and after collisionfor J = 0 . J , U = 2 J , k = π/ L = 80 sites; time ismeasured in units of (cid:126) /J ; (b) Diagonal density (cid:80) x | ψ ( x, x ) | after collision as a function of U for two different k and inci-dent energies E k =3 π/ = 1 . J and E k = π/ = 2 . J . of the bound state is localized at the center of the lat-tice, then it expands at a very slow rate along the x = y direction while it decays in scattering states.In Fig. 3(b), we plot the diagonal density (cid:80) x | ψ ( x, x ) | providing a measure of the occupation of the bound stateat a time sufficiently after collision ( t = 68 (cid:126) /J ) for twodifferent values of the incident relative momenta, namely k = 3 π/ k = π/
2, as a function of interaction U . Asexpected, a clear resonance peak is visible at U such thatthe energy of the bound state matches the energy of theincident wave packets. The different heights of the twopeaks can be understood as a consequence of the finitelife-time of the bound state and from the fact that thewave packets are moving with different group velocities.To obtain further understanding of these results, onecan perform a crude approximation and develop a theoryincluding only states | (cid:105) and |−(cid:105) . Indeed, | (cid:105) ≡ ( | p, m (cid:105) + | m, p (cid:105) ) / √ | A, A (cid:105) − |
B, B (cid:105) is the dominant pseudo-spin component for d when J (cid:28) J , U . Analogously, (a) (b) K = = -2 -4 -6 | ψ ( ) | k = Π k = Π U U π /43 π /4 FIG. 4. Population | Ψ (0) | in the | (cid:105) component at r = 0 as afunction of U for: (a) center of mass momentum K = 0 . . k = π/ k = π/ π/ K = 0 .
3. In all cases, the center of the resonancecoincides with the energy E of the scattering particles. the pseudo-spin state |−(cid:105) = ( | A (cid:105) − | B (cid:105) ) ⊗ ( | A (cid:105) − | B (cid:105) ) / ∼ J .The coupling of the other states | + (cid:105) and | F (cid:105) should beintroduced perturbatively in J . However, since Hamil-tonian (12) already contains a coupling between | (cid:105) and |−(cid:105) , the physics is captured in a qualitative manner evenneglecting all other states. The reduced theory thereforereads˜ H eff0 ( k ) = (cid:18) − iJ √ (cid:0) K (cid:1) cos kiJ √ (cid:0) K (cid:1) cos k J + 2 J cos (cid:0) K (cid:1) cos k (cid:19) and ˜ H U = (cid:18) U U/ (cid:19) . (15)The Green’s function can be readily calculated and onefinds G ( r ) = δ r, E , (16) G ( r ) = i e ikr J cos( K/
2) sin k ,G ( r ) = e ikr √ E tan( K/ k + δ r, i √ E tan( K/ , where E = 2 J +2 J cos( K/
2) cos k is the non-interactingspectrum obtained neglecting the off-diagonal terms in˜ H eff0 . The most general solution of the Schr¨odingerequation is given by the two-component spinor Ψ( r ) =(Ψ ( r ) , Ψ − ( r )) T :Ψ( r ) = Φ( r ) + G ( r ) ˜ H U (1 − G (0) ˜ H U ) − Φ(0) , (17)where Φ( r ) is solution of the non-interacting problem.To compare with the numerical results presented above,we take the ansatz Φ( r ) = e ikr (0 , T . According to thisansatz, Φ( r ) populates only the |−(cid:105) component and thetwo particles have relative momentum k , thus modelingtwo incident particles belonging to type I continuum scat-tering off each other.The population of bound state d is described by the | (cid:105) component of Ψ( r ) at r = 0 (on-site pairs), namely Ψ (0). The results are shown in Fig. 4. A sharp reso-nance occurs at U ∼ E , analogous to the one observed inthe numerical simulation of the dynamics of two collid-ing wave packets. When the energy of the incident par-ticles is close to U , which is approximately the energy ofthe bound state, the probability to form the bound statebecomes maximal. Note how the resonance becomessharper when the center of mass momentum K → K = 0 the off-diagonal terms in ˜ H eff0 vanish,thus decoupling the two channels | (cid:105) and |−(cid:105) . IV. EDGE PHYSICS
We now discuss the case of open boundary conditions(OBC) to address the effect of interactions on the finitechain SSH model. As usual, we need to distinguish thetwo possible dimerizations D D
2. Single-particleedge states, typical of dimerization D
2, combined with afreely propagating particle generate two further continuaaround energies ± J (type II). Obviously, such type IIcontinua are absent in D
1, which does not admit single-particle edge states (see Fig. 5(D1,D2)). The two (typeI and type II) continua and the narrow bands of boundstates are independent consequences of the single-particleSSH model and of two-body interactions, respectively.Instead, as a pure consequence of the interplay betweenSSH geometry, interactions and boundary conditions, in-triguing two-body edge bound states (EBS) emerge in thespectrum (see Fig. 5(a-d)). Their presence or absence ishighly non-trivial and essentially driven by a renormal-ization of the edge properties (see also Refs. [31, 32]) .Such EBS can be associated to the different boundstates d i . For PBC the associated bound states - whenwell defined in the whole Brillouin zone - present a two-particle generalized [33] Zak phase difference of π for thetwo dimerizations D D
2. However, as we are goingto show in the following, this does not necessarily corre-spond to the formation of EBS in the finite chain, leadingto a breaking of the standard bulk-boundary correspon-dence. Furthermore, most remarkably, as clearly visiblein Fig. 5(D1,D2), EBS appear not only in dimerization D D
1, which does not admitedge states in the single-particle case.The D /d and D /d , EBS can be interpreted asTamm states of an effective strong-dimerization theory,as it will be detailed below. Localization persists evenwhen the EBS energy enters the scattering continua for U → d NN . Hybridiza-tion between d NN and d around U ∼ J induces avery strong localization at the edges of a wavefunctionwith strong both diagonal and out-of-cell characters (seeFig. 5(b)).
00 000000 232335 35 232323d d NN -d d d (D1) III IIII (c) x(D2)(b)(a) yy (d) xy y
FIG. 5. Spectrum with OBC for J = 0 . J as a function of U for (D1) dimerization D D
2; Thegreen colorscale represents the density in the first and last 2 lattice cells and highlights the localization of EBS states; (a) D /d EBS for U = 3 J and energy E ∼ J , (b) D /d NN − d EBS for U = 2 J and energy E ∼ . J , (c) D /d EBS for U = 2 J and energy E ∼ − . J and (d) D /d EBS for U = J and energy E ∼ . J obtained by exact diagonalization of72 lattice sites. A. Strong-dimerization limit
In order to understand the physics behind bound statesand EBS, it is useful to consider the regime of strongdimerization J (cid:28) J , U . Here, effective models account-ing for the weak tunneling J in second order perturba-tion theory can be developed. The building blocks forthe effective theory are naturally the three strong-linktwo-body eigenstates given in Eqs. (3-5). The effectivelattice is provided by the lattice cells i . More details canbe found in Appendix A.In-cell dimers d α,i can tunnel at second order throughintermediate states given by a particle in link i and aparticle in a neighboring link j . The effective model reads H eff = (cid:88) i,α E α,i d † α,i d α,i + (cid:88) (cid:104) i,j (cid:105) (cid:88) α,β J ijαβ d † α,i d β,j . (18)The parameters that appear in the model above are sec-ond order in the weak tunneling J . The effective modelin Eq. (18) provides an accurate prediction of the boundstate spectrum away from U ∼ J and U ∼ J wherebound state d crosses type II and type I scattering con-tinua, respectively, or U ∼ J where d crosses the lowertype II continuum. Relying on the additional assumptionthat the bound states are well separated in energy andthe coupling among them is weak, effective model (18)can be further simplified through a single band approx-imation, which only keeps J ijαα and E α,i for each state d α,i .Just below the d bound-state narrow band in dimer-ization D
1, one finds EBS D /d (see Fig. 5(a)), whichcan be quantitatively explained as a Tamm state in the framework of effective model (18). The comparison be-tween the results obtained with exact diagonalization andwith the effective model is shown in Fig. 6. The localiza-tion length of EBS D /d is very large. It increases forstrong interactions U (cid:29) J , J , so that, for practical pur-poses, in a finite lattice this state undergoes a crossoverto a not exponentially-localized state.A deeper understanding of the physics underlying thedivergence of the localization length for large interactionscan be obtained via a much simpler strong-interactioneffective model. The states d and d are almost degen-erate for U (cid:29) J , J . In this limit, a more convenientbasis is given by on-site doublons d † A,i | (cid:105) ≡ | A i A i (cid:105) and d † B,i | (cid:105) ≡ | B i B i (cid:105) coupled among each other via second or-der processes. The corresponding effective Hamiltonianis nothing else than an effective single-particle SSH modelwith effective hopping coefficients J eff1 , = − J , /U andeffective on-site energy (cid:15) bulk = U + 2( J + J ) /U . More-over, for dimerization Dσ (with σ = 1 , (cid:15) edge = (cid:15) bulk +∆ E σ ,with ∆ E σ = − J − σ /U . This energy shift at the outer-most sites provides a generalization of the Tamm physicsto the SSH model, which in general, depending on ∆ E ,allows both for Tamm-like states above or below the con-tinua and in-gap states. However, the specific case of oureffective model coincides, in both dimerizations, exactlywith the critical value of ∆ E for which neither Tamm norin-gap states can exist (see discussion in Appendix A 2).This implies that in the strong-interaction limit U (cid:29) J exponentially localized edge states are not to be expectedin finite-size chains, in agreement with the numerical re-sults. i (cell) -5 -4 -3 -2 -1 || Effective evenEffective oddExact evenExact odd
FIG. 6. Exponential localization of the D /d edge statefor U = 3 J and J = 0 . J in a lattice with 36 unit cells(72 sites) in dimerization D
1. We plot the probability tofind two particles in unit cell i calculated with the effectivetheory in the strong dimerization limit (lines) and with exactdiagonalization (markers). Both simulations provide a pair ofeven and odd almost-degenerate eigenstates due to finite size(see legend). For dimerization D
2, a closer inspection of the d and d dimer spectra around their intersection with type IIcontinua shows a peculiar feature: two dimer states,gapped from their continua, appear (see Fig. 5(c-d)).They correspond to pairs of D /d , EBS moved out ofthe corresponding bound-state narrow bands as a conse-quence of the renormalized parameters at the boundaries.These EBS can also be accounted for by effective model(18). In D
2, the effective model is slightly more compli-cated than in D
1, since no full lattice cell is present atthe edges, but rather a single lattice site (see Fig. 1(D2)).In the following, we thus specialize to the case of the D /d state. While in the bulk the bound state pre-serves its standard form | d ,i (cid:105) = ( | A i , A i (cid:105) − | B i , B i (cid:105) ) / √ | d , (cid:105) = −| B , B (cid:105) and | d ,L (cid:105) = | A L , A L (cid:105) . Thistruncated bound-state wave function affects both theeffective hopping J edge,D222 (cid:54) = J bulk22 , the on-site energy E edge,D22 (cid:54) = E bulk2 at the edges, and the on-site energies E , , E ,L − at the outermost complete lattice cells. Asshown in Fig. 7 (blue lines), sufficiently far away fromthe U = J condition, the effective model perfectly repro-duces the numerical spectrum and the presence of gappedstates.The effective theory fails at U ∼ J because the D /d EBS becomes resonant with type II scattering states. Inorder to account for the hybridization mechanism, weconsider a reduced Hilbert space including d -like trun-cated EBS −| B , B (cid:105) and | A L , A L (cid:105) , type II scatteringstates | ψ ji (cid:105) = | ES j (cid:105) ⊗ ( | A i (cid:105) − | B i (cid:105) ) / √
2, where | ES (cid:105) = | B (cid:105) and | ES L (cid:105) = | A L (cid:105) and finally the zero-energy single-particle edge state | B A L (cid:105) (see Appendix A 4). This re-duced theory reproduces very well the avoided crossingaround U = J (see Fig. 7 (red dotted lines)) and pointsout that the D /d EBS is smoothly transformed into a
U/J E / J Exact diagonalizationEffective theoryReduced theory
FIG. 7. Energy spectrum in dimerization D d bound state and the upper type II continuumfor J = 0 . J and 24 lattice sites: Exact diagonalizationresults (black lines), strong dimerization model (blue dashedlines) and reduced model (red dotted lines). type II scattering state when moving away from U ∼ J . V. EXPERIMENTAL OBSERVATION ANDDYNAMICS
In this final section, we present the results of real timesimulations which directly highlight the properties of theSSH model discussed in this work. The most promis-ing idea is that, upon the 1D to 2D mapping introducedin Sec. II, a 2D coupled optical fibers setup can providea quantum simulator of the two-particle 1D SSH model,such that the full two-body dynamics of the system is vis-ibile in real time and real space through the propagatinglight intensity. Beyond the characterization of scattering,bound and edge bound states as discussed in this section,the same setup would allow the visualization of the closedchannel population in a resonant Fano-Feshbach scatter-ing process, already presented in Sec. III B.We study the two-body dynamics, assuming differ-ent initial conditions at t = 0 and different interactionstrengths U . We let the two-body wave function in sec-ond quantization evolve numerically in time via exactdiagonalization. Written in first quantization, the two-body wavefunction can be interpreted as a single particlewave function ψ ( x, y, t ) in 2D ( x and y being equivalentlythe coordinates of the two particles in 1D or the coordi-nates of a single particle in 2D). We address few differentillustrative cases, shown in the following subsections. A. Edge bound state D /d In Fig. 8(a), we plot the wave function of the exactEBS eigenstate for U = 0 . J and J = 0 . J in dimer- (a) (b)xy x FIG. 8. (a) EBS wave function for a lattice with L = 24 sitesin the D2 dimerization for J = 0 . J , U = 0 . J obtainedwith exact diagonalization. (b) Projected wave function on x = y with x ≤
3, used as initial state for the time-evolutionshown in Fig. 9(a) and discussed in Sec. V A. ization D x = y and decay expo-nentially as x grows. Being even and odd states almostdegenerate, one can equally well consider states localizedat either end of the lattice. Therefore, as initial state, wetake the projection of the exact EBS wave function on x = y with x ≤
3, as shown in Fig. 8(b), localized at thebottom left corner of the 2D lattice.We use the observables (cid:104) ˆ x (cid:105) ( t ) ≡ (cid:80) x x | ψ ( x, y, t ) | = (cid:80) y y | ψ ( x, y, t ) | and (cid:112) (cid:104) (ˆ x − ˆ y ) (cid:105) ( t ) to characterize theedge localization properties of the states. In Fig. 9(a) thetime evolution of (cid:104) ˆ x (cid:105) ( t ) is displayed. The plot shows that (cid:104) ˆ x (cid:105) ( t ) (cid:28) L/
2, namely the initially approximate EBS, re-mains localized at one edge of the system. It is remark-able that a very well approximated EBS can be obtainedby initializing the wave function over only three latticesites.However, since the initial state slightly differs from theexact EBS, a small overlap with type II states is presentand observed in a non-vanishing single-particle popula-tion oscillating at x = 0 or y = 0 (see Fig. 9(b)). Thisproduces sizable - but still small when compared to thelattice size - fluctuations (cid:112) (cid:104) ( x − y ) (cid:105) . The visible oscil-lations in both observables arise from the bouncing of thepopulated type II states at the lattice edges. B. Hybridization between d and type IIcontinuum Differently from the previous section, we consider asinitial condition a single doublon localized at the outer-most site of a D U = J . Such initial state has a siz-able overlap with the D /d EBS, the d bound-statecontinuum and type II scattering states.The time evolution shows that the state again remainsmostly localized at one edge. However, (cid:104) ˆ x (cid:105) ( t ) becomeslarger because of the non-negligible population of the d continuum (see Fig. 10(a)). Moreover, oscillations at twodifferent characteristic time-scales are visible. The fast (b)yx t [ /J ] x (x - y) x 10 (a) x FIG. 9. (a) Time evolution of the projected EBS in Fig. 8(b)for dimerization D (cid:104) x (cid:105) (full blue line) and (cid:112) (cid:104) ( x − y ) (cid:105) (dashed red line) as a function of time; (b) Modulus of thetwo-body wavefunction at t = 10 (cid:126) /J . In these simulations U = 0 . J , J = 0 . J and L = 24 sites. (b)y x t [ /J ] x (x - y) x 10 (a) FIG. 10. (a) Time evolution of a doublon initially localized atthe outermost lattice site for dimerization D (cid:104) x (cid:105) (full blueline) and (cid:112) (cid:104) ( x − y ) (cid:105) (dotted red line) as a function of time;(b) Modulus of the two-body wavefunction at t = 10 (cid:126) /J .In these simulations U = J , J = 0 . J and L = 24 sites. (b)y x t [ /J ] x (x - y) (a) x 10 FIG. 11. (a) Time evolution of a doublon initially localizedin the lattice outer most lattice site for dimerization D (cid:104) x (cid:105) (full blue line) and (cid:112) (cid:104) ( x − y ) (cid:105) (dashed red line) as a functionof time; (b) Modulus of the two-body wavefunction at t =10 (cid:126) /J . In these simulations U = 3 J , J = 0 . J and L =24 sites. time scale is present in both observables and it is re-lated to the bouncing of the type II states, as discussedin the previous section. A second slower time-scale isclearly recognizable for the observable (cid:104) ˆ x (cid:105) ( t ) related tothe dynamics of the heavy d bound state and the corre-sponding bouncing off the lattice edges. C. Bound state dynamics ( D ) As initial state, we take again a single doublon local-ized in the outermost site of a D d and type II continuum.At U = 3 J , this initial state has a large overlap withthe d bound states, which are well localized at x = y ,but not necessarily at the edges, and negligible overlapwith the scattering continua. For that reason, the statedelocalizes in the lattice remaining bound at relative dis-tance equal to zero, as shown in Fig. 11. This is re-flected in a negligible value of (cid:112) (cid:104) ( x − y ) (cid:105) during thewhole time evolution and a center-of-mass average posi-tion of the wave packet oscillating significantly in timedue to bounces at the lattice edges. D. Two-body scattering states
As a final example, we show the case where we populateand address scattering states.We take as initial condition a state delocalized in thefirst four lattice sites cells without any double occupation.In our notations, the initial state reads differently in thetwo dimerizations, so that it is convenient to write itexplicitly (symmetrization is assumed): | Ψ D ( t = 0) (cid:105) ∝ ( | A (cid:105) + | B (cid:105) ) ⊗ ( | A (cid:105) + | B (cid:105) ) , (19) | Ψ D ( t = 0) (cid:105) ∝ ( | B (cid:105) + | A (cid:105) ) ⊗ ( | B (cid:105) + | A (cid:105) ) . (20)Due to the vanishing double occupation, the energy of theinitial state is determined by the hopping processes andnot by interactions. It results E ( in ) σ = − J σ dependingon the dimerization Dσ ( σ = 1 , D D
2, the initial statehas non negligible overlap with states in all type I andtype II continua.For that reason, the time evolution, shown in Fig. 12,presents two drastically different behaviours in the twodimerizations: in D D (D1) (D2)xy x max0 FIG. 12. Modulus of the two-body wavefunction after timeevolution of the initial states in Eqs. (19, 20) for (D1) dimer-ization D D t = 75 (cid:126) /J . Inthese simulations U = 1 . J , J = 0 . J and L = 24 sites. VI. CONCLUSIONS
In conclusion, we have studied theoretically the richtwo-particle physics stemming from the interplay of lo-cal interactions with non-trivial single-particle topol-ogy. To this aim, we have considered two particles inthe paradigmatic one-dimensional Su-Schrieffer-Heegerdimerized lattice. We have proposed an experimentallyrealistic system, based on state-of-the-art coupled opticalfiber technology, where the two-body physics in the SSHmodel can be quantum simulated in real time and realspace. Beyond being able of revealing the different scat-tering, bound and edge bound states in finite geometries,experiments along the suggested lines have the potentialof becoming a textbook illustration of the Fano-Feshbachresonance scattering effect.One of our major conceptual results resides in the evi-dence that interactions, in spite of being local, can affectthe boundary conditions over more than one single lat-tice site. Such kind of effects are expected to be evenmore relevant in the presence of non-local interactions.Hence, the most straightforward extension of the presentwork regards the inclusion of nearest-neighbor interac-tions [36].Our work provides a first important progress in theunderstanding of two-body physics in systems with topo-logical properties. In the future, it would be interestingto investigate models in higher dimensions and differentgeometries, where symmetries other than the chiral oneare relevant for the existence of topological states.
VII. ACKNOWLEDGEMENTS
The authors thank M. Burrello, P. ¨Ohberg, C. Ortix,T. Ozawa and H. Price for interesting discussions. A.R.acknowledges support from the Alexander von Humboldtfoundation and W. Zwerger for the kind hospitality atthe TUM. This work was supported by ERC through theQGBE grant, by the EU-FET Proactive grant AQuS,Project No. 640800 and by Provincia Autonoma diTrento.0
Note added . In the final stage of preparation of thiswork, we became aware of a similar and complementaryinvestigation of the two particle SSH model by Gorlachand Poddubny [37].
Appendix A: Effective theories
It this work, we have made extensive use of effectivemodels to describe two bosonic particles in a dimerizedlattice governed by the Hamiltonian H = H J + H J + H U ,where H J and H J are the strong- and weak-tunnelingHamiltonians, and H U accounts for onsite interactions.In this section, we provide the details of their derivation.Consider a Hamiltonian H = H + V . Let us label theset of eigenstates of H as { α } = {| α, m (cid:105)} . Here, the in-dex α indicates a manifold of states (for instance, statesclose in energy to each other that are gapped from therest of the other states), and m labels the states insidethe manifold. Be V a perturbation that weakly couplesthe manifold { α } to the manifold { β } of the remainingeigenstates of H . Including V at second order perturba-tion theory, as shown in [38], one can obtain an effectiveHamiltonian H α eff that describes manifold { α }(cid:104) α, m | H α eff | α, n (cid:105) = E α,m δ m,n + (cid:104) α, m | V | α, n (cid:105) + (A1)+ 12 (cid:88) k,β (cid:54) = α (cid:104) α, m | V | β, k (cid:105)(cid:104) β, k | V | α, n (cid:105)×× (cid:20) E α,m − E β,k + 1 E α,n − E β,k (cid:21) , where E α,m are the eigenvalues of H relative to theeigenstate | α, m (cid:105) .
1. Strong dimerization
In the limit of strong dimerization J (cid:28) J , U , we iden-tify H = H J + H U = (cid:80) i H cell i . Different lattice cellsare decoupled and each cell is described by the strong-link Hamiltonian H cell i , which in the two-particles basis | A i , A i (cid:105) , | A i , B i (cid:105) and | B i , B i (cid:105) takes the form H cell i = U −√ J −√ J −√ J −√ J U . (A2)Its eigenvectors, provided in Eqs. (3-5), have respectivelyenergy (cid:15) = 12 (cid:18) U − (cid:113) J + U (cid:19) , (A3) (cid:15) = U, (A4) (cid:15) = 12 (cid:18) U + (cid:113) J + U (cid:19) . (A5)The states in manifold { α } = {| d ,i (cid:105) , | d ,i (cid:105) , | d ,i (cid:105)} arecoupled through H J in a non-trivial manner via the E E E bulkbulkbulk (a)(b) J J J U/J J J J (c) bulkbulkbulkbulkbulkbulk FIG. 13. Bulk parameters E bulk α and J bulk αβ of the effectivemodel in Eq. (A10) for J = 0 . J , as a function of U . Thedetailed legend can be found in the figure. manifold of virtual states { β } - also eigenstates of H .For PBC, manifold { β } is formed by states of one par-ticle in a cell i and one particle in a cell j , with i (cid:54) = j .There are four possible sets of states | ψ Iij (cid:105) = 1 √ | A i (cid:105) + | B i (cid:105) ) ⊗ √ | A j (cid:105) + | B j (cid:105) ) , (A6) | ψ IIij (cid:105) = 1 √ | A i (cid:105) − | B i (cid:105) ) ⊗ √ | A j (cid:105) − | B j (cid:105) ) , (A7) | ψ IIIij (cid:105) = 1 √ | A i (cid:105) + | B i (cid:105) ) ⊗ √ | A j (cid:105) − | B j (cid:105) ) , (A8) | ψ IVij (cid:105) = 1 √ | A i (cid:105) − | B i (cid:105) ) ⊗ √ | A j (cid:105) + | B j (cid:105) ) , (A9)with energies, respectively, E I = − J , E II = 2 J , E III = E IV = 0. Up to second order in perturba-tion V = H J , one finds the effective Hamiltonian (seeEq. (18)) H eff = (cid:88) i,α E α,i d † α,i d α,i + (cid:88) (cid:104) i,j (cid:105) (cid:88) α,β J ijαβ d † α,i d β,j , (A10)containing renormalized onsite dimer energies and intra-and inter-dimer nearest-neighbor hopping. In general,1coefficients E α,i and J ijαβ have a quite involved analyticalform. The values of the parameters for J = 0 . J asa function of U are shown in Fig. 13 when (cid:104) i, j (cid:105) are inthe bulk of the lattice where no edge effects are involved.The divergencies at U = 2 J are the indication of thecrossing of d with the higher type I continuum.In most regimes, the different bound states are far awayin energy from each other and the coupling among themturns out to be weak. Even if better quantitative predic-tions for the bound states bands can be obtained by in-cluding all the terms, a decoupling of the different boundstates, namely considering for each bound state d α onlythe parameters J ijαα and E α,i , still provides an excellentagreement. In that case, explicit analytical forms can beprovided at least for the simpler case of state d : J bulk22 = − J U J − U J − U , (A11) E bulk2 = U − J bulk22 . (A12)For OBC, bulk and edge parameters differ: in D
1, due tothe missing coupling either on the right-hand or left-handside, one gets a different renormalization of the onsiteenergy at the first and last cells: E edge,D12 = U − J bulk22 , (A13)while the effective hopping parameter is equal at theedges as in the bulk.Different parameters describe dimerization D
2, due tothe presence of half cells (single lattice sites) at the chainedges. The effective tunneling coupling between the first(last) lattice site to the first (last) complete lattice celland the energy offset of the first (last) lattice sites are J edge,D222 = J U √ U J − U , (A14) E edge,D22 = U − √ J edge,D222 . (A15)Finally, the first ( i = 1) and last ( i = L −
1) completecells feel a resulting energy shift given by E , = E ,L − = U + J U J − J U + 2 U J − J U + U . (A16)
2. Strong-interaction limit
In the strong-interaction limit U (cid:29) J , J , the two d , narrow bound-state bands are well separated fromthe rest of the spectrum, as one can deduce fromEqs. (A4,A5) and Figs. 5(D1,D2). We choose linear com-binations of these higher repulsive bound states to con-stitute the manifold { α } for which we write the effectivetheory. This corresponds to consider H = H U and thesubspace of onsite doublons { α } = {| A i , A i (cid:105) , | B i , B i (cid:105)} ≡{ d † A,i | (cid:105) , d † B,i | (cid:105)} , with energy E α = U . The virtual states at energy E β = 0 form manifold { β } = {| A i , A j (cid:105) , | B i , B j (cid:105) | A m , B n (cid:105)} with i (cid:54) = j and ∀ m, n .The coupling is provided by V = H J + H J . Up to sec-ond order in J σ , only | A i , B i (cid:105) and | A i , B i − (cid:105) contribute,corresponding to nearest-neighbor virtual hopping pro-cesses.One obtains an effective single-particle Hamiltonian forthe on-site doublons that reads H eff = 2 J U (cid:88) i d † A,i d B,i + 2 J U (cid:88) i d † A,i +1 d B,i + H.c.+ (cid:20) U + 2 U (cid:0) J + J (cid:1)(cid:21) (cid:88) i (cid:16) d † A,i d A,i + d † B,i d B,i (cid:17) . (A17)The first line clearly shows that the effective model of thebound states is a SSH model with renormalized hoppingcoefficients J eff1 , = − J , U (A18)and effective on-site energies (cid:15) bulk = U + 2 U (cid:0) J + J (cid:1) , (A19)which contains the binding energy U and an on-site en-ergy shift. The latter is generated by similar second-orderprocesses as the ones occurring for the hopping terms: avirtual breaking of the doublon to the left and to theright. However, in the presence of OBC, at the edgesonly one of these two processes will be present, leadingto a different on-site energy at the edge with respect tothe bulk (cid:15) edge = (cid:15) bulk + ∆ E σ , (A20)with ∆ E σ = − J − σ /U , depending on the dimerization Dσ (with σ = 1 , E , the energy of theTamm-like states lies above/below the bands and it de-pends linearly on ∆ E when | ∆ E | is sufficiently large.They appear for | ∆ E | > J eff2 in dimerization D | ∆ E | > J eff1 in dimerization D
2. In-gap states betweenthe bands appear for both dimerizations, but they existin D | ∆ E | > J eff2 and in D | ∆ E | < J eff1 .The in-gap edge states obtained when considering ∆ E as tunable parameter, have a topological origin. The pe-culiar feature of the SSH model is indeed the presence ofzero-energy edge states in dimerization D E , asproven below. As a consequence, topological edge statesin D E ,shift away from zero energy, as discussed in the paragraphabove and shown in Fig. 14(b). Also the in-gap states in2 E / J e ff E/J eff1 (a)(b)
D1D2
FIG. 14. Single-particle SSH spectrum of a finite chain with48 sites for J eff2 = 0 . J eff1 as a function of an arbitrary on-siteenergy shift ∆ E for (a) dimerization D D
2. As discussed in the text, this single-particle modeleffectively describes the bound-states physics in the strong-interaction limit. The red lines are the critical values of ∆ E for the existence of Tamm or in-gap edge states (see text). dimerization D E . In this limit, the system behaves as an ideal D L − | ∆ E | = J eff2 and | ∆ E | = J eff1 , which implies that we are exactly at thecritical values of ∆ E (red lines in Fig. 14) for which nei-ther Tamm nor in-gap states can exist.We now prove the breaking of chiral symmetry inthe effective model. Let us consider a finite chainwith 2 L sites in dimerization D
2. This correspondsto L − d =( d A, , . . . , d A,L , d B, , . . . , d B,L − ), the Hamiltonian takesthe form (up to an overall irrelevant energy shift that wedrop in the discussion below) H eff = d † H eff d = d † (cid:18) H H H † H (cid:19) d . (A21)While H is a L × L matrix that contains the coupling between neighboring sites and has the form H = − J eff1 − J eff2 − J eff1 . . . . . .. . . − J eff1 − J eff2 − J eff1 , (A22) H and H are L × L diagonal matrices, namely H =diag(0 , · · · , , ∆ E ) and H = diag(∆ E , , · · · ,
0) thatdescribe the on-site energy shift, respectively, of the leftand right edge of the chain. If ∆ E were zero, the di-agonal blocks H and H would vanish. Hence, theHamiltonian could be written in the chiral-symmetricform H (0)eff = (cid:18) H H † (cid:19) . In fact, in this case, the op-erator C = σ z ⊗ I provides a chiral symmetry such that {H (0)eff , C} = 0. However, since in our case ∆ E (cid:54) = 0, diag-onal blocks appear in H eff . Chiral symmetry is thereforebroken and zero-energy edge states are not protected [25].
3. Effective theory for the d NN state We develop an effective theory in the strong-dimerization limit J (cid:28) J to qualitative explain theexistence of the d NN bound state. The basis for the ef-fective theory is given by the subspace of states | S ij (cid:105) ∼ ( | A i (cid:105)−| B i (cid:105) ) ⊗ ( | A j (cid:105)−| B j (cid:105) ) with i , j arbitrary cell indices.For i (cid:54) = j , the states | S ij (cid:105) span the upper type I scatter-ing continuum. For i = j , | S ii (cid:105) = ( | d ( U = 0) (cid:105) , whichcan be considered a fairly good approximation for d upto U ≤ J . At first order in J , the energies of states | S ij (cid:105) are E ij = 2 J + δ ij U/
2. The single-particle hoppingamplitude in this subspace is given by J /
2. When U isapproaching 2 J (but sufficiently far to be off-resonantwith the type I scattering states), state d become closerin energy to the S ij manifold. Then, the energy of thestates | S ij (cid:105) with i = j ± J butit is renormalized by second-order processes mediated bythe virtual state d . The energy shift is given by∆ E NN = 2 × J J − U (A23)and provides an effective nearest-neighbor attractive (re-pulsive) interaction when U > J ( U < J ). Therefore,for U > J ( U < J ) we expect a bound state above(below) the continuum. The nearest-neighbor interac-tion is very weak compared to the bandwidth 2 J of thescattering states. This explains the appearance of the d NN state for a limited set of momenta close to K = π [4]. Moreover, for increasing values of U the attractionbecomes weaker and weaker, leading to a progressive dis-appearance of the d NN state. These properties have allbeen observed numerically (see Fig. 2(a-b)).3
4. Reduced theory for the avoided crossing