Self-organized topological insulator due to cavity-mediatedcorrelated tunnelingTitas Chanda1, Rebecca Kraus2, Giovanna Morigi2, and Jakub Zakrzewski1,3

1Institute of Theoretical Physics, Jagiellonian University in Kraków, Łojasiewicza 11, 30-348 Kraków, Poland2Theoretical Physics, Saarland University, Campus E2.6, D–66123 Saarbrücken, Germany3Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland

Topological materials have potential ap-plications for quantum technologies. Non-interacting topological materials, such as e.g.,topological insulators and superconductors,are classified by means of fundamental symme-try classes. It is instead only partially under-stood how interactions affect topological prop-erties. Here, we discuss a model where topol-ogy emerges from the quantum interferencebetween single-particle dynamics and globalinteractions. The system is composed by soft-core bosons that interact via global correlatedhopping in a one-dimensional lattice. The on-set of quantum interference leads to sponta-neous breaking of the lattice translational sym-metry, the corresponding phase resembles non-trivial states of the celebrated Su-Schriefer-Heeger model. Like the fermionic Peierls in-stability, the emerging quantum phase is atopological insulator and is found at half fill-ings (namely, when the number of sites is twiceas large as the number of bosons). Neverthe-less, here it arises from an interference phe-nomenon that has no known fermionic analog.We argue that these dynamics can be realizedin existing experimental platforms, such ascavity quantum electrodynamics setups, wherethe topological features can be revealed in thelight emitted by the resonator.

IntroductionManifestation of topology in physics [1, 2] created arevolution which is continuing for almost four decades.With the discovery of topological materials, con-densed matter physics has gained a new terrain wherequantum phases of matter are no longer controlled bylocal order parameters as in paradigmatic Landau the-ory of phase transitions but rather by the conservationof certain symmetries [3]. These new phases of mat-ter, so called symmetry-protected topological (SPT)phases, display edge and surface states that can berobust against perturbations, making them genuine

Titas Chanda: titas.chanda@uj.edu.pl

candidates for quantum technologies [4].

To date, a detailed understanding of noninteract-ing topological materials, such as e.g., topological in-sulators and superconductors [5], has been obtainedthrough a successful classification based on funda-mental symmetry classes, the so-called “ten-fold way”[6–8]. On the other hand, interactions are almostunavoidable in many-body systems. It is thereforea central issue to understand whether SPT phasescan survive the inter-particle interactions, or perhapswhether interactions themselves might stabilize SPTphases and even give rise to novel topological prop-erties [9–16]. These questions are at the center ofan active and growing area of research [17, 18]. Re-cent works have argued that the range of interac-tions plays a crucial role on the existence of SPTphases. Specifically, in frustrated antiferromagneticspin-1 chain with power-law decaying 1/rα interac-tions, topological phases are expected to survive atany positive value of α [19]. This behavior sharessimilarities with the topological properties of nonin-teracting Kitaev p-wave superconductors [20] that arerobust against long-range tunneling and pairing cou-plings [21–24]. It was found that for infinite-rangeinteractions the one-dimensional Kitaev chain can ex-hibit edge modes for appropriate boundary conditions[25]. On the other hand, in spin chains and Hub-bard models, the infinite-range interactions suppressthe onset of hidden order at the basis of the Haldanetopological insulator [26, 27].

In this work, we report a novel mechanism wherenontrivial topology emerges from the quantum inter-ference between infinite-range interactions and single-particle dynamics. We consider bosons in a one-dimensional (1D) lattice, where the global interac-tions have the form of correlated tunneling result-ing from the coupling of the bosons with a harmonicoscillator. We identify the conditions under whichthis coupling spontaneously breaks the discrete latticetranslational symmetry and leads to the emergenceof non-trivial edge states at half filling. The result-ing dynamics resembles the one described by the fa-mous Su-Schrieffer-Heeger (SSH) model [28, 29]. Thetopology we report shares analogies with the recentstudies of symmetry breaking topological insulators

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ΩFigure 1: Interference-induced topological phases can berealized in a cavity quantum electrodynamics setup. Thebosons are atoms (red spheres) confined by a one-dimensionaloptical lattice and dispersively interacting with a standing-wave mode of the cavity. The cavity standing-wave field isparallel to the lattice, its wavelength is twice the lattice peri-odicity, and the atoms are trapped at the nodes of the cavitymode. Correlated hopping originates from coherent scatter-ing of laser light into the cavity, the laser Rabi frequencyΩ controls the strength of the interactions. The field at thecavity output is emitted with rate κ and provides informationabout the phase of the bosons.

[14–16, 30, 31]. Nevertheless, differing from previousrealizations, here the interference between quantumfluctuations and global interactions is essential for theonset of the topological phase and cannot be under-stood in terms of a mean-field model.

We argue that these dynamics can be realized, forinstance, in many-body cavity quantum electrody-namics (CQED) setups [32–36], like the one illus-trated in Fig. 1 highlighting the experimental feasi-bility of our proposal. In the following, we first give abrief description of the model in relation to the exper-imental setup. Then, we ascertain the phase diagramof the system at half filling, characterize the topologi-cal phase, and point out how to experimentally verifyits existence.

Bose-Hubbard model with global cor-related tunneling

The model we consider describes the motion of bosonsin a 1D lattice with Ns sites and open boundaries.The specific experimental situation realizing this dy-namics is sketched in Fig. 1. The dynamics is gov-erned by the Hamiltonian H in terms of the field oper-ators bj and b†j , which annihilate and create, respec-tively, a boson at site j in the lowest lattice band.The bosons, tightly confined in the lowest band of thelattice, strongly couple with a cavity standing-wavemode (the harmonic oscillator), which is parallel tothe lattice. When the atoms are transversely drivenby a laser, the Hamiltonian takes the detailed formof a Bose-Hubbard (BH) model with the additional

optomechanical coupling with the oscillator [37]:

H = −tNs−1∑j=1

(b†j bj+1 + H.c.

)+ U

2

Ns∑j=1

nj (nj − 1)

+S(a+ a†)yB + ∆ca†a , (1)

where we have set ~ = 1. Here, the first two terms onthe right-hand side (RHS) of Eq. (1) are the nearest-neighbor hopping with amplitude t and onsite repul-sion with magnitude U . The third term on the RHSis the bosons-cavity coupling, where operators a anda† annihilate and create, respectively, a cavity pho-ton, while the operator B acts on the bosonic Hilbertspace. The last term of (1) gives the cavity oscillatorenergy in the reference frame rotating at the frequencyof the transverse laser with ∆c the detuning betweencavity and laser frequency. Finally, the bosons-cavitycoupling is scaled by the coefficients S and y that canbe independently tuned. The first one, in fact, is pro-portional to the amplitude Ω of the transverse laserfield, c.f. Fig. 1. The coefficient y, instead, dependson the localization of the scattering atoms at the lat-tice sites (see Appendix A).

The specific form of operator B depends on thespatial dependence of the cavity-bosons coupling [38,39]. In the present case, we consider

B =Ns−1∑i=1

(−1)i+1(b†i bi+1 + H.c.

). (2)

The staggered coupling, (−1)i+1(b†i bi+1 + H.c

), orig-

inates from the spatial mode function of the cavitymode along the lattice when the lattice sites are lo-calized at the nodes of the cavity mode, and when theperiodicity of cavity mode standing wave is twice theperiodicity of the optical lattice.

Hamiltonian (1) with (2) is reminiscent of thephonon-electron coupling of the SSH model [40]. Dif-fering from the ionic lattice of the SSH model, thebosons couple to a single oscillator – the cavity mode.Here, the instability is associated with a finite sta-tionary value of the field quadrature x = a + a†: for〈x〉 6= 0 the bonds connecting even-odd and odd-evensites differ by a quantity proportional to 〈x〉. Since xis a dynamical variable, this process is accompaniedby a spontaneous breaking of the lattice translationalsymmetry. In a cavity, for instance, x is the electricfield that is scattered by the atoms and depends onthe atomic mobility. The resulting bosonic dynamicsis thus intrinsically nonlinear.

We analyze the quantum phases of the system inthe limit in which the cavity field (oscillator) can beeliminated from the equations of the bosonic variablesassuming that |∆c| is the largest frequency scale ofthe dynamics. In this limit, the time-averaged field isε(τ) = 1

∆t∫ τ+∆tτ

dta(t) ≈ −SB(τ)/∆c, where τ is thecoarse-grained time in the grid of step ∆t [38, 41]. The

2

Phases Acronyms maxM1(k) kmax ODW OB OS OPSuperfluid SF 6= 0 = 0 = 0 = 0 = 0 = 0

Bond superfluid BSF 6= 0 = ±π/2 = 0 6= 0 = 0 = 0Density-wave DW = 0 – 6= 0 = 0 6= 0 6= 0Bond insulator BI = 0 – = 0 6= 0 6= 0 (= 0) = 0 (6= 0)

Table 1: Different phases, their acronyms, and the corresponding values of order parameters for the Bose-Hubbard model withcavity-mediated interactions.

effective Bose-Hubbard Hamiltonian takes the form

H = −tNs−1∑j=1

(b†j bj+1 + H.c.

)+U

2

Ns∑j=1

nj (nj − 1)+U1

Nsy2B2 ,

(3)where U1 = S2Ns/∆c and the explicit dependenceon Ns warrants the extensivity of the Hamiltonian inthe thermodynamic limit when U1 is fixed by scalingS ∝ 1/

√Ns. The term proportional to B2 emerges

from the back-action of the system through the globalcoupling with the oscillator. It describes a global cor-relation between pair tunnelings.

Before discussing the emerging phases, we notethat the model in Eq. (1) has extensively been em-ployed for describing ultracold atoms in optical lat-tices and optomechanically coupled to a cavity mode[32, 33, 38, 39, 42]. The specific form of the couplingwith operator (2) has been discussed in Ref. [39] andcan be realized by suitably choosing the sign of the de-tuning between laser and atomic transition as in Ref.[34]. The scaling 1/Ns of the nonlinear term corre-sponds to scaling the quantization volume with thelattice size [43]. In the time-scale separation ansatz,the bosons dynamics is coherent provided that |∆c| isalso larger than the cavity decay rate κ: in this regimeshot-noise fluctuations are averaged out. Correspond-ingly, there is no measurement back-action, since theemitted field ε(τ) is the statistical average over thetime grid ∆t [44]. We note that topological phasescan also be realized in plethora of platforms wherethe range of interactions can be tuned and the geom-etry controlled, prominent examples are optomechan-ical arrays [45], photonic systems [46], trapped ions[47], and ultracold atoms in optical lattices [32, 48–53]. Specifically, this model could be also realizedin the experimental setups as discussed in [54–56]. Inthis work, we shall keep on referring to a CQED setup,where the dynamics predicted by Hamiltonian (1) hasbeen extensively studied and can be realized [32–34].

Phase diagram at half fillingIn the rest of this work, we consider the system athalf filling, i.e., density ρ = 1/2. We determine theground state and its properties using the density ma-trix renormalization group (DMRG) method [57, 58]based on matrix product states (MPS) ansatz [59, 60]

– for details see Appendix B. The phase diagram is de-termined as a function of t/U and U1/U . The phasesand the corresponding observables are summarized inTable 1, the observables are detailed below. We char-acterize off-diagonal long-range order by the Fouriertransform of the single particle correlations, i.e., thesingle particle structure factor

M1(k) = 1N2s

∑i,j

eik(i−j)⟨b†i bj

⟩, (4)

which can be experimentally revealed by means oftime-of-flight measurements [61, 62]. The couplingwith the cavity induces off-diagonal long-range order,that is signaled by the bond-wave order parameter

OB = 〈B〉 /(2Ns) . (5)

This quantity is essentially the cavity field in thecoarse graining time scale and is directly measurableby heterodyne detection of the electric field emittedby the cavity [63]. We also consider the density-waveorder parameter, OD = 1

Ns

∣∣∣〈∑j(−1)j nj〉∣∣∣, which sig-

nals the onset of density-wave order and typicallycharacterizes phases when the lattice sites are at theantinodes of the cavity field (see Appendix A). More-over, we analyze the behavior of the string and parityorder parameters:

OS = 〈δnieiπ∑j

k=iδnkδnj〉 , OP = 〈eiπ

∑j

k=iδnk〉 .

(6)

These order parameters depend nonlocally on onsitefluctuations δnj = nj − ρ from the mean density ρ.

We first remark that, for U1 = 0, hence in the ab-sence of global interactions, the phase is superfluid(SF). We also expect that for U1 > 0 the quantumphase of Hamiltonian in Eq. (3) is SF, since the for-mation of a finite cavity field costs energy. Instead,we expect that correlated hopping becomes relevantfor U1 < 0. We have first performed a standardGutzwiller mean-field analysis of the model assumingtwo-site translational invariance. At sufficient largevalues of |U1| mean-field predicts the formation of aSF phase of the even or odd bonds accompanied by afinite value of bond-wave order parameter OB . Thisphase maximizes the cavity field amplitude and wedenote it by Bond Superfluid phase (BSF). We point

3

0.0 0.05 0.1 0.15 0.2 0.25

t/U

−60−50−40−30−20−10

010

U1/U

SF

BSF

BI

maxM1(k)

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.05 0.1 0.15 0.2 0.25

t/U

−60−50−40−30−20−10

010

U1/U

SF

BSF

BI

OB

0.0

0.1

0.2

0.3

0.4

0.0 0.05 0.1 0.15 0.2 0.25

t/U

−60−50−40−30−20−10

010

U1/U

SF

BSF

BI

OS

0.00

0.05

0.10

0.15

0.20

0.25

−1.0 −0.5 0.0 0.5 1.0k/π

0.00

0.05

0.10

M1(k

)

SF, U1/U = 4

BI, U1/U = −12

BI, U1/U = −20

BSF, U1/U = −40

Figure 2: Phase diagram of Hamiltonian (3) in the(t/U, U1/U)-plane determined by means of DMRG. Differentpanels show the behavior of the order parameters indicated inthe plots. We observe the appearance of an insulating phasefor t/U . 0.2 between standard superfluid (SF) and bondsuperfluid (BSF) phases which disappears at larger tunnelingvalues. The blue dashed lines indicate the borders betweenthe regions where the maximum ofM1(k) evaluated over theground state crosses a threshold value which we set at 0.03to compensate finite-size effects. The bottom right panelshows the shape of M1(k) for fixed t = 0.05U . Sharp peaksappear at k = 0 for SF and k = ±π/2 for BSF in contrast tobroad peaks of lower amplitude for BI. Note that the effec-tive strength of the correlated hopping in Eq. (3) scales withy2U1. Here, Ns = 60 and y = −0.0658 (see Appendix A).

out that mean-field predicts that the ground state ex-hibits off-diagonal long-range order for any value ofU1.

We now discuss the quantum phase obtained fromDMRG calculations. The phase diagram is reportedas a function of the ratio t/U , which scales thestrength of the single-particle hopping in units of theonsite repulsion, and of the ratio U1/U , which scalesthe strength of the correlated hopping. We sweep U1from positive to negative values. We note that in acavity the sign of U1 is tuned by means of the signof the detuning ∆c. The effective strength, in par-ticular, shall be here scaled by the parameter y2, de-pending on the particle localization. Here, y is con-stant across the diagram, since we keep in fact theoptical lattice depth constant and tune the ratio t/Uby changing the onsite repulsion U (experimentally,this is realized by means of a Feshbach resonance).Figure 2 displays the DMRG results for (a) the max-imum value of |M1(k)| (4), (b) the bond-wave orderparameter (5), (c) the string order parameter (6), and(d) the dependence of M1(k) on the wave number kfor different values of U1/U . For U1 < 0 the groundstate supports the creation of the cavity field, whichis signaled by the finite value of the bond-wave orderparameter. At sufficiently large values of |U1/U | andt/U the transition is discontinuous, and it separatesthe SF from the BSF phase, where the effective tun-neling amplitudes 〈b†i bi+1 + H.c.〉 attain a staggered

0.0 0.05 0.1 0.15 0.2 0.25

t/U

−60−50−40−30−20−10

010

U1/U

SF

BSF

BI

OMFB

0.0

0.1

0.2

0.3

0.4

0.0 0.05 0.1 0.15 0.2 0.25

t/U

−60−50−40−30−20−10

010

U1/U

SF

BSF

BI

OB −OMFB

0.0

0.1

0.2

0.3

0.4

Figure 3: Left panel: Bond-wave order parameter pre-dicted by mean-field Gutzwiller approach, OMF

B , in the(t/U, U1/U)-plane. The right panel shows the differenceOB − OMF

B , where OB has been determined using DMRG.The blue dashed line indicates the borders between the re-gions same as in Fig. 2. The mean field Gutzwiller approachagrees with DMRG in SF and BSF regimes, but misses theappearance of BI phase.

pattern characterized by a finite value of bond-waveorder parameter OB . The long range coherence of theBSF phase is manifested by narrow peaks of M1(k)centered at k = ±π/2 (see Fig. 2). Remarkably, weobserve a reentrant insulating phase separating the SFand the BSF. The insulator is signaled by vanishingoff-diagonal long-range order and therefore by vanish-ing structure factor M1(k). It is characterized by thenon-zero (zero) values of the string order parameterand by vanishing (non-vanishing) parity order param-eter depending on the boundary sites of these non-local parameters (see the next section for details). Wedenote this phase as a Bond Insulator (BI). This phaseis separated from the SF by a continuous phase tran-sition. The transition BI-BSF seems also continuous.

We note that the bond insulator phase is entirelyabsent in the mean-field approach. Figure 3 displaysthe bond-wave order OMF

B predicted by mean field(left panel) and the deviation of it from the DMRGresult (right panel). While the exact borders betweenvarious phases quantitatively differ, the existence ofboth regular and bond superfluid phases is visible inthe mean field approach. Indeed, the novel BI phaseis entirely due to the interplay between the long-rangecoupling induced by the cavity and the single-particletunneling, and therefore is absent in the mean-fieldconsideration. We further note that studies of theground state of (3), based on exact diagonalization forsmall system sizes, did not report the existence of theBI phase [39]. In the following section, we characterizethe topology associated with the BI phase and arguethat it is stable in the thermodynamic limit.

Emergent topology associated withthe BI phaseBy means of excited-state DMRG, we reveal that theBI phase has triply degenerate ground state (quasi-degenerate for finite Ns) separated by a finite gapfrom the other excited states. The site distributionis visualized in Fig. 4 which shows that the absolute

4

0.00.20.40.60.81.0

〈b† ibi+

1+

H.c.〉

Trivial

0.00

0.25

0.50

0.75

1.00

〈ni〉 Trivial

0.00.20.40.60.81.0

〈b† ibi+

1+

H.c.〉

Topological

0.00

0.25

0.50

0.75

1.00

〈ni〉 Topological

1 10 20 30 39Bonds

0.00.20.40.60.81.0

〈b† ibi+

1+

H.c.〉

Topological

1 10 20 30 40Sites

0.00

0.25

0.50

0.75

1.00

〈ni〉 Topological

Figure 4: Site-dependent properties of the topological andtrivial ground states of the BI phase. Left panels: Effec-tive tunneling amplitudes 〈b†i bi+1 + H.c.〉 as a function ofthe bonds (i, i + 1). Orange (teal) bars denote the even(odd) bond. Right panels: Density 〈ni〉 as a function of thelattice site. The dashed line is a guide for the eye. Observethe characteristic alternate weak/strong pattern in the bondswith weak bonds occurring at the edges for topological statesthat reveal topological particle-hole edge excitations. Here,we set U1/U = −10, t/U = 0.05, and y = −0.0658 (seeAppendix A) to obtain the states using DMRG algorithm.

ground state has a uniform mean half-filling, whilethe other two states possess edge excitations, namely,fractional particle-hole excitations with respect to themean half-filling (bottom two rows of Fig. 4). Suchedge excitations are characterized by the bond-waveorder parameter with opposite sign than the trivialphase. They suggest that the BI phase is a symme-try protected topological (SPT) phase. Similar topo-logical edge states have been reported e.g., for non-interacting system [53] or in superlattice BH model[64], where the superlattice induces a tunneling struc-ture resembling that of the SSH model [28, 29]. Inour case, instead, the effective tunnelings are spon-taneously generated by the creation of a cavity fieldthat breaks discrete Z2 translational symmetry of thesystem. However, bond centered inversion symmetrystill remains intact – it protects this SPT phase. Thissetting is reminiscent of a spontaneous Peierls transi-tion [65] in bosonic systems, like the ones reported in[14–16].

On a further inspection, it is found that the stringorder OS and parity order OP can be non-zero (zero)depending on the location of the two separated sitesthat sit at the boundaries of the non-local operators(sites i and j in Eqs. (6)). We illustrate it in Fig. 5(a).We find that OS 6= 0 and OP = 0, as reported inFig. 2, when the non-local operators start at the sec-ond site of a strong bond (i.e., the bond with largertunneling element) and end at the first site of a weakbond (the bond with smaller tunneling element) fur-ther away. These are unusual properties when com-

1 10 20 30 39Bonds

0.00

0.25

0.50

0.75

1.00

|∞(K

) |/º

Trivial

1 10 20 30 39Bonds

Topological

°50 °40 °30 °20 °10 0 10U1/U

0.000

0.005

0.010

0.015

0.020

Ent

angl

emen

tga

p,

¢"

BSF SF

BI

Ns = 40Ns = 60Ns = 80Ns = 100Ns = 120

Strong bond Degenerate εn

Weak bond Non-degenerate εn(a)

(c)

(d)

(b)

0

2

4

6

8

10

12

14

Ent

angl

emen

tsp

ectr

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," n

Trivial Topological

Figure 5: (a) Illustration of the BI ground-state properties.The ellipsoids with thick black lines indicate the dimerizedstrong bonds with larger effective tunneling amplitudes, whilethin lines indicate the weak alternate bonds with smaller ef-fective tunneling amplitudes. The string order OS becomesnon-zero (with OP = 0) only when it is measured across blueto orange sites in the figure. In all other cases it vanishes,OS = 0 and OP 6= 0. The entanglement spectrum is foundto be degenerate in the bulk of the chain when it is measuredacross the strong bonds, while it is non-degenerate in weakbonds. (b) Entanglement spectrum εn computed at the mid-dle of the chain of Ns = 60 for the trivial and topologicalstates. For trivial state the spectrum is non-degenerate, whileit is doubly degenerate for topological states. (c) Entangle-ment gap (∆ε = ε1 − ε0) as a function of U1 and for fixedt/U = 0.05 across BSF-BI-SF phases. Different curves referto different system sizes. (d) Local Berry phases γ(K) (7),measured across every bonds for trivial (left) and topological(right) states. Here, we consider Ns = 40 and K = 20. In(b)-(d) other system parameters are the same in Fig. 4.

pared to, say, topological Haldane phases of extendedBH models at unit filling [27, 66].

To check whether we are indeed dealing with topo-logical states, we calculate the entanglement spec-trum of the system. For this purpose we partitionthe chain into a right (R) and left (L) subsystem as|ψ〉GS =

∑n λn |ψ〉L ⊗ |ψ〉R where λn are the corre-

sponding Schmidt coefficients for the specific biparti-tion. The entanglement spectrum is then defined asthe set of all the Schmidt coefficients in logarithmicscale εn = − log λn and is degenerate for phases withtopological properties in one dimension [67]. We findthat εn are degenerate near the chain center whenthe bipartition is drawn across a strong bond, whileit is non-degenerate at the weak bonds. In Fig. 5(b),we display the entanglement spectrum for trivial andtopological ground states of the BI phase for Ns = 60,when εn are measured across the bipartition at thechain’s center. The entanglement spectrum, togetherwith the density pattern and the behavior of string

5

and parity order parameters, provide convincing proofof the topological character of the BI phase. Further-more, to show that the BI phase is stable in the ther-modynamic limit, we consider the entanglement gap,∆ε = ε1 − ε0, for different system sizes. Fig. 5(c)presents the variation of ∆ε across BSF-BI-SF phasesfor fixed t/U = 0.05 – confirming the stability of SPTBI phase in the thermodynamic limit.

In order to reveal the bulk-edge correspondence, wedetermine the many-body Berry phase [68] and showthat it is Z2-quantized in the BI phase. We first notethat, because of the strong interactions, the windingnumber or the Zak phase [53, 69, 70] is not a goodtopological indicator in our case. Therefore, we fol-low the original proposition of Hatsugai [71] and de-termine the local many-body Berry phase, which is atopological invariant playing the role of the local “or-der parameter” for an interacting case [71]. For thispurpose, we introduce a local twist t → teiθn in theHamiltonian (3), such that the system still remainsgapped in the BI phase. Then the many-body Berryphase is defined as

γ(K) = ArgK−1∏n=0〈ψθn+1 |ψθn

〉 , (7)

where ψθn’s are the ground states with θ0, θ1, ..., θK =

θ0 on a loop in [0, 2π]. Here, we consider the localBerry phase corresponding to a bond by giving thelocal twist in tunneling strength t → teiθn only onthat particular bond, and take K = 20. The localBerry phases γ(K)’s are displayed in Fig. 5(d) for thesystem size Ns = 40. Similar to the entanglementspectrum, we find γ(K) = π for the strong bonds,while γ(K) = 0 on the weak bonds.

DiscussionThe BI phase of this model is a reentrant phase. Itseparates the SF phase, where correlated hopping issuppressed by quantum fluctuations, from the BSFphase, where correlated tunneling is dominant andsingle-particle tunneling establishes correlations be-tween the bonds. We have provided numerical evi-dence that the emerging topology is essentially char-acterized by the interplay between quantum fluctu-ations and correlated tunneling. Interactions arehere, therefore, essential for the onset of topology.Their global nature is at the basis of the sponta-neous symmetry breaking that accompanies the on-set of this phase and which induces an asymmetrybetween bonds. In this respect, it is reminiscent ofthe Peierls instability of fermions in resonators [31],where the topology is associated with the spontaneousbreaking of Z2 symmetry. Differing from that case,where photon scattering gives rise to a self-organizedsuperlattice trapping the atoms, in our model pho-ton scattering interferes with quantum fluctuations.

Like in [31], gap and edge states can be measuredin the emitted light using pump-probe spectroscopy.The single-particle structure factor may be directlyaccessible by the time-of-flight momenta distributions[61, 62] enabling the detection of insulator-superfluidphase transition. The two combined measurementsof the cavity output and of the structure factor shallprovide a clear distinction between the BI, SF andBSF phases.

We observe that the global long-range interactionof this model inhibits the formation of solitons. Forother choice of periodicity, and thus of the form of op-erator B, one could expect glassiness associated withthe formation of defects [38], whose nature is expectedto be intrinsically different from the one characteriz-ing short-range interacting structures.

To conclude, we have presented a new paradigm oftopological states formation via interference betweensingle particle dynamics and interaction induced hop-ping.

AcknowledgementsWe thank Daniel González-Cuadra and Luca Tagli-acozzo for discussions on tools and methods, andShraddha Sharma for helpful comments. The supportby National Science Centre (Poland) under projectUnisono 2017/25/Z/ST2/03029 (T.C.) within Quan-tERA QTFLAG and OPUS 2019/35/B/ST2/00034(J.Z.) is acknowledged. The continuous support ofPL-Grid Infrastructure made the reported calcula-tions possible. R.K. and G.M. acknowledge sup-port by the German Research Foundation (the pri-ority program No. 1929 GiRyd) and by the Ger-man Ministry of Education and Research (BMBF)via the QuantERA projects NAQUAS and QTFLAG.Projects NAQUAS and QTFLAG have received fund-ing from the QuantERA ERA-NET Cofund in Quan-tum Technologies implemented within the EuropeanUnion’s Horizon 2020 program.

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A Coefficients of the extended Bose-Hubbard modelTo fix the notation, let us consider N atoms of massm confined within an optical cavity in a quasi-one-dimensional configuration (almost) collinear with aone-dimensional optical lattice created by light withwave number kL = 2π/λ, which may be differentfrom k – the wave number of the cavity field. Theoptical lattice is created by the trapping potential,Vtrap = V0 sin2 kLx + V (sin2 kLy + sin2 kLz), whereV V0 making the atomic motion effectively one-dimensional. In our calculations we take V = 50ER.

4 8 12 16 20V0/ER

0.00

0.02

0.04

0.06

0.08

0.10

φ = π/2

−y

0.001 0.01 0.1 0.51/2− φ/π

0.001

0.01

0.1

1

V0 = 4ER

−yz

Figure 6: (Left) The variation of y as a function of the latticedepth V0/ER for φ = π/2. Here, z is identically zero for allvalues of V0/ER. (Right) The variations of y and z as φdeviates from π/2. We consider V0 = 4ER.

The Wannier basis in the lowest band is Wi(x, y, z) =wi(x)Φ0(y, z) with wi(x) being the standard one-dimensional Wannier function centered at site i andΦ0(y, z) - a two-dimensional Wannier function for thetransverse deep lattice. For our purposes, Φ0(y, z)is essentially equivalent to a Gaussian with widthσ = a2

π2

√ER/V , where ER = ~2π2/2ma2 is the recoil

energy and a denotes the periodicity of the opticallattice.

After adiabatically eliminating the cavity field onegets an effective Hamiltonian describing atomic mo-tion in terms of atomic creation/annihilation oper-ators b†i/bi for atoms at site i is decomposed intothe sum H = HBH + HCav, [38, 44] where the mo-tion in the lattice and atom-atom interactions are de-scribed by the standard Bose-Hubbard Hamiltonian(we assume contact interactions and neglect density-dependent tunneling terms known to be small for suchinteractions [72])

HBH = −tNs−1∑j=1

(b†j bj+1 + H.c.

)+ U

2

Ns∑j=1

nj (nj − 1) ,

(8)where Ns denotes the number of lattice-sites. Tun-neling amplitude t and onsite interaction strength Uare given by the integrals

t = −∫dxwi(x)

(~2

2m∂2

∂x2 + V0 sin2(kLx))wi+1(x) ,

(9)

U = g

∫d3r|wi(x)Φ0(y, z)|4 = gπ

2a2

√V

ER

∫dx|wi(x)|4 .

(10)

The rescattering due to the cavity mode leads to along range “all-to-all” interaction terms expressible as

HCav = U1

Ns

∑i,j

b†i bj

∫dx cos(kx+ φ)wi(x)wj(x)

2

.

(11)We rewrite the integrals above as

yij =∫dxwi(x)wj(x) cos(kLxβ + φ). (12)

8

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

USF

BSF

BIDW

max M1(k)

0.00

0.03

0.06

0.09

0.12

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

OB

0.0

0.1

0.2

0.3

0.4

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

ODW

0.0

0.1

0.2

0.3

0.4

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

OS

0.00

0.05

0.10

0.15

0.20

0.25

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

OP

0.0

0.2

0.4

0.6

0.8

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

max M1(k)

0.00

0.03

0.06

0.09

0.12

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10U

1/U

SF

BSF

BIDW

OB

0.0

0.1

0.2

0.3

0.4

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

ODW

0.0

0.1

0.2

0.3

0.4

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

OS

0.00

0.05

0.10

0.15

0.20

0.25

0.001 0.01 0.1 0.31/2° ¡/º

°50

°40

°30

°20

°10

0

10

U1/

U

SF

BSF

BIDW

OP

0.0

0.2

0.4

0.6

0.8

Figure 7: Phase diagrams in the (φ,U1/U)-plane for t/U =0.05 for Ns = 60. The BI phase disappears when φ deviatesfrom π/2 and density wave phase appears. Here, we chooseV0 to be 4ER so that the values of y and z matches that ofFig. 6 (right panel). Blue dashed lines are guide to the eyesto differentiate different phases.

Here, β = k/kL is the ratio between k, the wavenum-ber of the cavity mode, and kL, the wavenumber cor-responding to the optical lattice, and φ is a phase inthe mode function. In our work we consider β = 1,and note that arbitrary values of β would lead to aquasiperiodic Hamiltonian [73, 74] For β = 1 and fori = j the integral becomes independent of i – we de-note is simply as z. For non-diagonal yij , we observethat due to localization of Wannier functions i = j±1terms may be only significant ones. For our choice ofβ = 1, the integral again becomes independent of iand we denote it as y. Then HCav may be put in theform,

HCav = U1

Ns

(z2D2 + yz

(BD + DB

)+ y2B2

)(13)

where

D =Ns∑j=1

(−1)j+1nj , B =Ns−1∑j=1

(−1)j+1(b†j bj+1 + h.c.

).

(14)

Typically, one assumes no phase difference (φ = 0)between the optical cavity and the external opticallattice, i.e., the lattice sites are located at the antin-odes of the cavity mode. Then |z| 6= 0 and |y| ≈ 0, andthe terms proportional to y are dropped off leading tothe standard case considered in the past [75]. Theimportance of additional terms was noticed alreadyin [39] where an identical Hamiltonian is obtained fora slightly different arrangement of the cavity and ex-ternal optical lattice. Here, we have focused on aimmediate vicinity of φ = π/2. This corresponds to

the atoms being trapped at the nodes of the cavitymode, where z vanishes (see left panel of Fig. 6).However, as φ starts to deviate from π/2 the y termrapidly decreases and z term becomes significant (seeright panel of Fig. 6). Note that the quadratic formof HCav is responsible for the long-range characterof the couplings. Squaring D leads then to all-to-alldensity-density interactions, responsible for a sponta-neous formation of density wave phase for sufficientU1 [76]. For the case considered by us, z ≈ 0 andB2 term leads to the all-to-all long-range correlatedtunnelings alternating in sign. Throughout the paper,we have fixed V = 4ER resulting in y = −0.0658 andz = 0 for φ = π/2.

In Fig. 7, we plot the order parameters and thephase diagram in the vicinity of φ = π/2 for t/U =0.05. As φ starts to deviate from π/2, y starts todiminish and z becomes increasingly larger (see rightpanel of Fig. 6). As a result, BI phase is replaced bya more standard density wave (DW) phase [76], whenφ becomes sufficiently different from π/2. In the DWphase ODW as well as OS and OP are both non-zero,while the structure factor vanishes.

B Numerical implementationWe use standard matrix product states (MPS) [59, 60]based density matrix renormalization group (DMRG)[57, 58] method to find the ground state and low-lyingexcited states of the system with open boundary con-dition, where we employ the global U(1) symmetrycorresponding to the conservation of the total numberof particles. For that purpose, we use ITensor C++library (https://itensor.org) where the MPO forthe all connected long-range Hamiltonian can be con-structed exactly [77, 78] using AutoMPO class. Themaximum number bosons (n0) per site has been trun-cated to 6, which is justified as we only consider av-erage density to be ρ = 1/2.

We consider random entangled states, |ψini〉 =1√50

∑49i=0 |ψrand

i 〉, where |ψrandi 〉 are random product

states with density ρ = 1/2, as our initial states forDMRG algorithm. The maximum bond dimensionof MPS during standard two-site DMRG sweeps hasbeen restricted to χmax = 200. We verify the conver-gence of the DMRG algorithm by checking the devi-ations in energy in successive DMRG sweeps. Whenthe energy deviation falls below 10−12, we concludethat the resulting MPS is the ground state of the sys-tem.

To obtain low-lying excited states, we first shift theHamiltonian by a suitable weight factor multipliedwith the projector on the previously found state. Tobe precise, for finding the nth excited state |ψn〉, wesearch for the ground state of the shifted Hamiltonian,

H ′ = H +W

n−1∑m=0|ψm〉 〈ψm| , (15)

9

where W should be guessed to be sufficiently largerthan En − E0.

10