Quantum anomalous Hall phase in synthetic bilayers via twistless twistronics

Tymoteusz Salamon,1 Ravindra W. Chhajlany,2 Alexandre Dauphin,1 Maciej Lewenstein,1, 3 and Debraj Rakshit1, 4

1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain

2Faculty of Physics, Adam Mickiewicz University, 61614 Poznan, Poland3ICREA, Pg. Lluis Companys 23, Barcelona, Spain

4Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany(Dated: August 10, 2020)

We recently proposed quantum simulators of "twistronic-like" physics based on ultracold atoms and syntheticdimensions [Phys. Rev. Lett. 125, 030504 (2020)]. Conceptually, the scheme is based on the idea that aphysical monolayer optical lattice of desired geometry is upgraded to a synthetic bilayer system by identifyingthe internal states of the trapped atoms with synthetic spatial dimensions. The couplings between the internalstates, i.e. between sites on the two layers, can be exquisitely controlled by laser induced Raman transitions.By spatially modulating the interlayer coupling, Moiré-like patterns can be directly imprinted on the latticewithout the need of a physical twist of the layers. This scheme leads practically to a uniform pattern across thelattice with the added advantage of widely tunable interlayer coupling strengths. The latter feature facilitates theengineering of flat bands at larger "magic" angles, or more directly, for smaller unit cells than in conventionaltwisted materials. In this paper we extend these ideas and demonstrate that our system exhibits topologicalband structures under appropriate conditions. To achieve non-trivial band topology we consider imanaginarynext-to-nearest neighbor tunnelings that drive the system into a quantum anomalous Hall phase. In particular,we focus on three groups of bands, whose their Chern numbers triplet can be associated to a trivial insulator(0,0,0), a standard non-trivial (-1,0,1) and a non-standard non-trivial (-1,1,0). We identify regimes of parameterswhere these three situations occur. We show the presence of an anomalous Hall phase and the appearance oftopological edge states. Our works open the path for experiments on topological effects in twistronics without atwist.

I. INTRODUCTION

Twistronics (from twist and electronics) is a term com-monly used nowadays to describe the physics resulting fromthe twist between layers of two-dimensional materials. Thisterminology was introduced in Ref. [1], which conducted the-oretical studies on how a twist between the layers can changeelectonic properties of bilayer graphene. But the history ofthis new area of research goes back to Ref. [2], whose au-thors suggested that twisted bilayer graphene could providea new material with unprecedented properties. Flat bandsat the magic angle were discovered in 2011 [3], whereasBistritzer and MacDonald showed that for a twisted mate-rial with a “magic angle” the free electron properties radi-cally change [4]. More recently, two seminal experimental pa-pers [5, 6] demonstrated that such twisted bilayer graphene atthe magic angle can host both strongly insulating Mott statesand superconductivity. These impressive results triggered anavalanche of experimental and theoretical works [7–24] (SeeRef. [25] for a recent review article). Many of these recent ac-tivities discuss topological insulators in magic-angle twistedbilayer graphene and the possibility of creating and control-ling topological bands in these systems [24, 26, 27].

Topological order has now become a central research topicin physics, exemplified by the 2016 Nobel Prize for D. J.Thouless, F.D.M. Haldane, and J.M. Kosterlitz [28, 29]. Theintimate relation between topology and condensed mattergoes back to the discovery of the Integer Quantum Hall Ef-fect (IQHE) [30]: a 2D electron gas at low temperature and

under a strong magnetic field presents a quantized transverseconductivity very robust against local perturbations. It wassoon realized [31] that this robustness was coming from anew paradigm: a global topological order which cannot bedescribed by the usual Ginzburg Landau theory of phase tran-sitions. In the particular case of the IQHE, the presence ofa strong magnetic field results in the appearance of flat bands(Landau levels), each of them being characterized by a distincttopological invariant, called Chern number, and the transverseconductivity is equal to the sum of the Chern numbers of theoccupied Landau levels. Soon after, F.D.M. Haldane proposedthe quantum anomalous Hall effect, which presents a quan-tized transverse conductivity but no Landau levels [32]. Sucha toy model turned out to be the crucial ingredient for the orig-inal proposal of topological insulators in graphene [33, 34]and stimulated very rapid progress of the area of topolog-ical/Chern insulators, topological superconductors, topolog-ical flat bands, and even systems with higher-order topol-ogy [35–40]. One of the most challenging and still persist-ing questions are related to the role of interactions, in par-ticular strong interaction and correlations [41]. Interestingly,the interactions do not always destroy the topological phases.Strong interactions in flat band topological materials can leadto the fractional quantum Hall effect [42–44] or to fractionalChern insulators [45–48]. Furthermore, strong interactionscan induce topology through a spontaneous symmetry break-ing mechanism as it is the case in the celebrated topologicalMott insulator [49–56].

Novel insights into the physics of topological order can beprovided by quantum simulators. These highly flexible exper-imental systems are used to mimic complex quantum systemsin a clean and controllable environment. Quantum simulators

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-i! +i!

(a)

Z

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λ = 0 λ = 0.2t

(b)kx ky

ZZ

Z Z

FIG. 1. The supercell structure and effects of complex hopping. (a)The left panel illustrates a sketch of a single plane of the bilayerstructure corresponding to one of the spin states m = ±1/2. Weconsider square lattice potential. The maroon and dark-blue sites areRaman coupled with the second plane under the chosen spatial mod-ulation of the synthetic tunneling. The pink sites (in both the layers)experience a chemical potential, µ. The real-space nearest-neighbortunneling t is shown by the solid lines. The next-nearest-neighborcomplex tunneling depending on directions of hopping and the po-sition of the lattice sites is shown by staggered dotted lines. Thered line shows a top view of the elementary unit cell of the systemcontaining 2 × 8 sites for Raman periodicities lx = ly = 4, wherethe factor of 2 accounts for the two layers. The right top panel de-picts the arrangement of elementary unit cells and the two translationvectors as arrows. This leads to the first Brillouin zone shown in theright bottom panel, indicating also the position of the high-symmetrypoints. (b) The three dimensional view of the six band manifold inthe vicinity of E = −Ω0(1 − α) for Ω0α/h = 40t with α = 0.2,γ = π/2 and µ = 0. The left panel shows the case, when next-nearest-neighbor complex hopping is absent, i.e., λ = 0. It has twoquasi-flat bands at the Dirac points of two dispersive bands in formof Dirac cones. The right panel shows the same set of bands, but inpresence of the complex tunneling, λ = 0.2t. This causes openingof a hard gap between the quasi-flat bands and the nearby dispersivebands.

constitute one of the four major pillars of contemporary quan-tum technologies [57], and can be realized with various plat-forms such as ultracold atoms, trapped ions, superconductingqubits, circuit QED, Nitrogen vacancies in diamond, or nanos-

tructure in condensed matter (for a review see [58–63]). In thiswork, we focus on the simulation of twisted graphene with ul-tracold atoms in optical lattices [64]. Such platforms allowone to simulate diverse geometries leading to, among others,graphene-like physics in synthetic hexagonal and brickwalllattices [65–67] or flat band physics in the Lieb lattice [68, 69].There are indeed other platform for realizing hexagonal lat-tices [70], but atoms combine additional advantages, as ex-plained herafter. Cold atoms provide a unique playground forsynthetic gauge fields [71–73], combined with time dependentlattice modulations/Floquet engineering [74, 75]. In particu-lar, such techniques led to the experimental realization of theHofstadter model [76, 77], the Haldane model [78–80]. Fur-thermore, artificial gauge fields can also be engineered withthe help of synthetic dimensions which allows to engineertopological insulators on ladders with both non trivial Chernnumbers and topological edge states [81–87]. Finally, opticallattices offer possibilities to study bilayer systems and prox-imity effects [88].The rapid development of twistronics in condensed matterphysics of 2D material stimulated extensive quest for quan-tum simulators of twistronics with ultracold atoms [89], andof Moiré patterns in photonic systems [90]. We combinedthese two worlds in a recent work, quantum simulators andsynthetic dimensions, and proposed twistronics without atwist [91]. In this paper we considered a single 2D opti-cal lattice with a desired geometry (honeycomb, brick, or Π-flux square), and created multilayer sytems employing inter-nal state of the atoms inserted in the lattice. These could bein the simplest case fermions with spin 1/2 or 3/2. The moirépatterns were generated by spatial modulations of the Ramantransitions coupling the internal states. Since the strength ofthe Raman coupling can be efficiently controlled in this sys-tem, the appearance of flat bands is expected to occur formuch larger “magic” angles, or better to say for elementarycells of much more modest sizes, like unit cell consisting ofonly 2 × 8 lattice sites. We analyzed the properties of theband structure in such systems, and showed that these expec-tations were indeed correct. In the present paper we developfurther the idea of Ref. [91] and demonstrate that such sys-tem is very flexible and can be tweaked to exhibit topologicalband structure in various situations. In particular, to achievenon-trivial topology we engineer artificial complex next-to-nearest neighbor tunneling analogous to the ones appearing inthe Haldane model [32]. Typically, the energy bands of ourinterest in this system form three groups and the Chern num-ber changes from trivial (0,0,0), to a topological phase with atrivial flat band (-1,0,1) and a topological phase with a non-tivial flat band (-1,1,0). We identify the regimes of parameterswhere these three situations occur, and study properties of thesystem with periodic and open boundary conditions. The pa-per is organized as follows. In Section II we present the de-tails of the model [91], together with modifications requiredfor achievement of topological bands. In Section III we dis-cuss magic configurations and quasi-flat bands. Section IV isdevoted to the investigations of the effects of onset of stag-gered hoping and appearance of topological insulators. Simi-lar effects and topological properties are discussed for dimer-

3

ized lattices in Section V. Section VI contains a short discus-sion of feasibility of experimental realization of the discussedphysical effects. The conclusions and outlook are presentedin Section VII.

II. THE SYSTEM

We consider a system of synthetic spinfull fermions in a bi-layer material. The fermions are subjected to a synthetic mag-netic field which leads to a supercell structure. We proposethe following scheme to realize such Hamiltonian in a quan-tum simulator. We consider ultracold fermionic atoms withfour internal states, m,σ = ±1/2, in a two-dimensional spin-independent optical square lattice with lattice spacing d. Thefour internal states are chosen such that two spin flavours ofelectrons are described by two pairs of the internal states, de-noted by σ. The pair of spin states corresponding to the sameσ are subjected to the Raman coupling. The index m, whichdistinguishes the Raman coupled pairs, is the labelling of thesynthetic dimension. The two possible values ofm effectivelyrealize a synthetic bilayer structure. A moiré-like supercellstructure can be obtained by modulating the Raman couplingstrength, Ω(x, y), in space. The complete Hamiltonian reads

H = Ht +Hλ +HΩ +Hµ, (1)

where

Ht =−∑r,m,σ

t(r)[a†m,σ(r + 1x) + a†m,σ(r + 1y)

]am,σ(r)

+ h.c.

(2)

is the nearest neighbor hopping Hamiltonian with a real andspace dependent tunneling amplitude t(r),

Hλ =∑r,m,σ

λ[exp(iφR(~r))a†m,σ(r + 1x + 1y)+

exp (iφL(~r)) a†m,σ(r− 1x + 1y)]am,σ(r) + h.c.

(3)

is the the next-to-nearest hopping Hamiltonian with a complextunneling amplitude λ and a staggered phase Φ,

HΩ =∑r,m,σ

Ω(r) exp(−iγ · r) a†m+1,σ(r)am,σ(r) + h.c.

(4)denotes the synthetic hopping Hamiltonian with a space de-pendent Raman coupling Ω and a magnetic phase γ = γ(1x+1y), and

Hµ =∑r,m,σ

µ(r) a†m,σ(r)am,σ(r), (5)

is the onsite chemical potential Hamiltonian with an ampli-tude µ(r). Figure 1 provides a schematic depiction of the sys-tem under study.

The spatial modulation of the Raman coupling is chosento be Ω(x, y) = Ω0 [1− α(1 + cos (2πx/lx) cos (2πy/ly))],

where lx (ly) is its periodicity along the x (y) axis. In the fol-lowing, we consider two distinct cases:(i) Staggered complex hopping (SCH). we set t(r) = tand fix the phases associated with the next-nearest neighborcomplex tunneling amplitude by setting φL(r)− φR(r) = π,where φR(r) = (2 r.1y+1)π/2 and φL(r) = (2 r.1y+3)π/2.(ii) Dimerized lattice (DL). We consider dimerized real tun-neling amplitude, such that t(r) takes the form of t1 andt2 in alternative sites in the x and y directions, along withthe next-nearest neighbor complex hopping, where we setφR(r) = φL(r) = π/2.

III. MAGIC CONFIGURATIONS AND QUASI-FLATBANDS

In the following, we study the Hamiltonian (1) under vari-ous boundary conditions: (i) periodic boundary conditions inboth spatial (x and y) direction, (ii) periodic boundary con-dition in one of the spatial direction (here along x), and (iii)open boundary conditions in all spatial directions. We firstconsider the case (i), for which the quasimomentum k =(kx, ky) is a good quantum number. In this case, we can applythe Bloch theorem, along with a gauge transformation, suchthat am,σ(r) =

∑k exp (i(k · r +mγ · r)) am,σ(k). The

Hamiltonian can then be rewritten as H =∑

kHk and canbe diagonalized. The spatial periodicity of the synthetic tun-neling fixes the supercell dimension, and hence the dimensionof Hk. The notation Θ(lx, ly) is introduced to represent cor-responding supercell configuration.

The band structure corresponding to the case with λ = 0and µ = 0 has been studied in detail in the previous work.Remarkably, Θ(4ν, 4ν) configurations, i.e., when lx = ly =4νd, with ν integer, were identified as magic configurations.Θ(4ν, 4ν) configurations host quasi-flat bands surrounded bydispersive Dirac cone spectra with controllable Dirac veloc-ities, and hence share certain characteristics associated withmagic angle twisted bilayer graphene. In this work we focuson the the smallest possible configuration Θ(4, 4) consistingof only 16 lattice sites. The corresponding Brillouin zone ofthe bilayer system showing the high-symmetry points is de-picted in Fig. 1(a). In the following we briefly review fewconsequences for the case with λ = 0 and µ = 0.

In the strong Raman coupling limit (Ω0α) t, isolatedsets of narrow spin degenerate bands appears at the energies±Ω0, ±Ω0(1 − α) and ±Ω0(1 − 2α). We identify the en-ergy spectrum of the system to be symmetric around zeroenergy, i.e. E(~q) = −E(~q) and therefore we restrict ourdiscussion to the negative energy bands. A non-isolated setof six bands around the energy E/t = −Ω0(1 − α) are ofparticular interest in this work. Figure 1(b) shows the en-ergy sptectrum for a representative case with Ω0α/h = 40t,α = 0.2, λ = 0, µ = 0 and the flux γ = π/2 (alsosee Fig. 2(a), which depicts the energy spectrum along thepaths passing through the high-symmetry points). Here theparameters are chosen by considering experimentally acces-sible regime in practice. Within this six band manifold, thetop and bottom three bands are symmetric around the en-ergy −Ω0(1 − 2α). Noticeably, two middle bands, which

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FIG. 2. Magic configuration band structures in presence of stag-gered complex hopping. Band structures corresponding to Θ(4, 4)supercell along the paths passing through the high-symmetry pointsΓ,X,M,Γ,X′,M for Ω0α/h = 40t, α = 0.2, and γ = π/2.Panel (a) shows the the set of six-bands around energy −Ω0(1 − α)with λ = 0.0, while other bands, which are well separated (by atleastthe energy of Ω0α) are not shown. There is no hard gap betweenthe two middle quasi-flat bands and adjacent dispersive Dirac cones.Panels (b-c) reveal evolution of the spectrum with respect to thechemical potential, µ. Panels (a), (b) and (c) corresponds to µ = 0,0.4t and 1.2t, respectively. Finite λ induces a hard gap between thequasi-flat bands and dispersive bands. The chemical potential leadsthe system through a non-trivial gapped-gapless transition, given theFermi energy is adjusted accordingly.

are quasi-flat, are formed closest to −Ω0(1− α) and they aresandwiched between dispersive bands in form of Dirac cones.These well isolated six band manifold with a total dispersionof ∆6 = 4

√2t cos(γ/2) + O(t2/(Ω0α)) are separated from

remaining nearby bands by an energy gap of Ω0α. The flat-ness of the quasi-flat bands (two middle bands) can be tunedprecisely by adjusting the Raman coupling strength. Their ap-proximate width can be derived within the second order per-turbation theory as

∆F =t2 cos2 (γ/2)

Ω0α

(24α3 − 88α2 + 106α− 32

3α3 − 11α2 + 12α− 4

). (6)

The relative flatness of the bands is defined as F = ∆F /∆6.As a result, the relatively flatter middle bands can be obtainedby increasing Ω0α. The parameters here are chosen from ex-perimentally accessible regime. It is worth mentioning thatthe Dirac velocity is proportional to cos(γ/2), and hence thebandwidths of the dispersive Dirac bands can be controlledseparately by tuning γ.

Within these six bands, the system has no gaps. The up-per dispersive Dirac cone touches the middle quasi-flat bandsat the high symmetry point Γ, i.e., kx = ky = 0. A non-zero flux, however, opens a tiny local gap between the lowerdispersive bands and the middle quasi-flat bands at Γ. Finiteflux breaks rotational and time reversal symmetries, c4τ , ofthe system. In the following we discuss strategies for open-ing a gap around the Dirac band touching and the resultingtopological phases of matter.

IV. STAGGERED COMPLEX HOPPING

A. Bulk properties of the system

We investigate the staggered complex hopping case [seeEq. (1) and Fig. 1(a)]. We first set µ = 0 and consider periodicboundary conditions in both x and y directions. A finite valueof λ breaks time-reversal symmetry. We focus on the samesix-band subset of the energy spectrum. A non-zero value ofλ induces a mass term at the Dirac crossing. As a result, threeisolated sets of bands are formed - each of them consists oftwo hybridized bands. The resulting band structure along thehigh-symmetry points is shown in Fig. 2(b) for Ω0/h = 200t,α = 0.2, γ = π/2, and λ = 0.2t. While the quasi-flat bandsremain closest to the energy Ω0(1− α), the top (bottom) dis-persive bands shifts upwards (downwards). The band gapscan be controlled by tuning the value of λ. As we will see,such hopping amplitude λ drives the system into a quantumanomalous Hall phase [32].

We further investigate the influence of a non-zero staggeredchemical potential µ on the system. A Finite value of µ breaksthe inversion symmetry of the system and the energy spectrumis no longer exactly symmetric around the zero energy. Nev-ertheless, the overall qualitative features of the negative andpositive energy bands remain closely similar for a value of µsmaller than the central bandgap. Moreover, this staggeredchemical potential results in a significant asymmetry between

5

C5

ΔE2

0

(a)

(b)

FIG. 3. (a) Chern number of the five lowest bands as a functionof system’ parameters λ/t and µ/t. As seen on (b) the system isgapless for µ = 2λ, which makes C5 undefined in this region. Thegapless region visible on (b) is marked by the red line on (a). Topo-logical transition presented on the Fig.2 can be seen by looking atthe vertical cross-section at λ/t = 0.2. Blue region of chern number-1 indicates the configurations at which system has standard non-trivial topology since C3 = −1 and the order is (−1, 0, 1), whileyellow area depicts parameters’ values for which non-standard topo-logical order (−1, 1, 0) can be obtained. Red and blue points markthe values of the parameters for which edge states have been plottedon Fig.5 (a) and (b),respectively. (b) Second band gap (energy gapbetween the hybridized quasi-flat middle bands and the dispersiveupper band) as a function of λ/t and µ/t. Vanishing energy gapsmark a topological phase transition in the system. Meaning of redand blue points remains the same as on the panel (a).

the two top and two bottom bands within the six band man-ifold under study. As we discuss below, this staggering po-tential has a prominent impact on topological phases of thesystem. Figures 2 (b-d) shows the evolution of the band struc-ture for increasing values of µ: the band gap between the topdispersive bands and the middle quasi-flat bands shrinks, andthe gap closes at µc ' 2λ. For µ > µc, the gap then reopens.Figure 3 shows the energy gap between the set of quasi-flatbands and the upper set of bands. We also observe a gap clos-ing and reopening of the lower gap for increasing staggeringpotential with opposite sign. The latter is reminiscent of theinterplay between the staggering potential and the imaginarynext-to-nearest neighbor hopping in the Lieb lattice [92]. Infact, the system undergoes a topological phase transition. Wecharacterize the topology of the system with the help of theChern number C, a topological invariant for the class of Chern

insulators. The Chern number of the n-th band is defined as

Cn = i

∫BZ

Fxyn (k)dS

= i

NxNy∑l=1

∫Pl

Fxyn,l(k)dS

(7)

the integral of the Berry curvature Fxyn,l(k) = ∇k ×An(k) written in terms of the Berry connection An(k) =〈un(k)| ∂k |un(k)〉. In the second equality, we have rewrit-ten this integral as a sum of integrals over the plaquettes ofthe discretization grid of the Brillouin Zone. For a such adiscretized grid, the Berry curvature can be be computed nu-merically with the help of the FHS algorithm [38]

Cn ≈1

2πi

∑Pl

(〈unk |unm〉 〈unm|uno 〉

⟨uno∣∣unp⟩ ⟨unp ∣∣unk⟩ ), (8)

where Pl denotes a plaquette in the Brillouin Zone with fourvertices (k,m, o, p) labeled in the anti-clockwise order withk being top left vertex and |unk 〉 is a Bloch function corre-sponding to the n-th eigenvalue at point k. The summationis performed over all plaquettes in the Brillouin Zone. Weemphasize that for degenerate bands one has to use the algo-rithm proposed in Ref. [93] to compute the non-abelian Berrycurvature. In this manuscript, we use the total Chern number

Ci =

i∑j=1

Cj (9)

defined as the sum of the Chern numbers of the i first occu-pied bands.

As can be seen in Fig. 3(a), for µ < µc, The hybridizedmiddle set of bands are topologically trivial with zero Chernnumber, while the bottom (top) set of dispersive band is topo-logically nontrivial with C =-1(1). The gap closing leads to atopological phase transition with a transfer of Chern numberfrom the upper set of bands to the middle set of bands [94],and for µ > µc, the middle set of bands becomes non trivialwith C = 1 and the upper set of bands becomes trivial withC = 0. The bottom dispersive bands remain non-trivial withC = −1. In the following subsections, we discuss in detail thetopological edge states appearing in a system with boundaries.

B. Cylindrical geometry and edge states

In order to study the topological edge states of the system,we consider a cylindrical or strip geometry. Calculations areconveniently performed using an enlarged unit cell consist-ing of plaquettes of size 4 × 4 site per layer. More specifi-cally, we consider a bilayer strip of finite length along the xdirection and infinite length in the y direction through periodicboundary conditions. As a result ky remains a good quantumnumber. The enlarged unit cell is repeated nx times in the xdirection.

In the following, we fix the set of parameters as in theprevious section, i.e. Ω0/h = 200t, α = 0.2, γ = π/2,

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E/t

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0 0.4 0.8 1.2 1.6

E/t

qyky0 π /8 π /4 3π /8 π /2

π /4 3π /8 π /2π /80ky

(a)

(b)

FIG. 4. Edge and bulk dispersions for cylindrical geometry. Energyspectrum is shown to demonstrate two distinct cases with (a) µ = 0and (b) µ = 1.2 for the parameters Ω0α/h = 40t, α = 0.2, γ =π/2, λ = 0.2t. The mid-gap states due to open boundaries along thex-axis are shown by red lines. These mid-gap states are edge statesconnecting the energetically separated bulk bands.

and λ = 0.2t, and we concentrate on the same energy win-dow close to the energy −Ω0(1 − α). Figure 4 shows theenergy spectrum as a function of the quasi momentum kyfor nx = 30. Figure 4(a) depicts the energy spectrum forµ = 0 [red dot in Fig. 3(b)]: The system presents three en-ergy bands that were already present in the previous section,which we call bulk energy bands. Interestingly, the bulk en-ergy gaps host topological edge states in accordance to thecelebrated bulk-edge correspondence [93]. These topologicaledge modes are responsible for edge currents and their num-ber at each edge is equal to the sum of Chern numbers of theoccupied bands. Hence, setting the Fermi energy EF at oneof the two energy gaps, one finds one edge state at each edge,which is, as expected. Figure 4 (b) shows the energy spec-trum for µ = 1.2t > µc [blue dot in Fig. 3(b)]. In this casethe first (bottom) energy gap supports topological edge states,while the second energy gap does not. This is again consistentwith the fact that for this case the bottom band has C = −1,and hence there is one edge state at each edge in the first gap,while the sum of Chern numbers of bottom and middle bandsis zero. Hence depending on the choice of EF , we can havea Chern insulator insulator or a trivial insulator. In order to

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200 400 600 800 1000 1200 1400

E/t

n

(a)

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200 400 600 800 1000 1200 1400

E/t

n

(b)

FIG. 5. Energy spectrum of the synthetic bilayer square lattice with2 × 40 × 40 sites and open boundary condition for (a) µ = 0 and(b) µ = 1.2t for the parameters Ω0 = 200, α = 0.2, γ = π/2 andλ = 0.2t. The bulk states are shown by black plus symbols. The newmid-gap states appearing due to open boundary condition is shownby red crosses. These are the edge states. For (a)µ = 0, the edgestates appear in both the energy gaps. However, for µ > µc (shownhere (b)µ = 1.2t, the edge states appear only within the first energygap.

achieve a closer understanding of the edge states and the bulk-boundary correspondence, we further analyze a finite squarelattice in the following discussion.

C. Finite square lattice and edge states

We finally consider a bilayer finite square lattice with 2 ×L × L sites. We do not impose periodic boundary condition,i.e., both the layers are open in both x and y directions. Unlikeprevious cases, the system can not be associated with a goodquantum number due to the absence of any periodicity. Wesolve the Hamiltonian in Eq. (1) by diagonalizing the matrixwith 2L2 × 2L2 entries. We again focus on the energy bandsclose to −Ω0(1 − α) (see Figs. 2 and 4). Figure 5 shows thesorted eigenvalues of the Hamiltonian for a system of lengthL = 40 and for the same set of parameters of the previoussubsections. We again show two different topological phasesfor µ = 0 [Fig. 5(a)] and µ = 1.2t [Fig. 5(b)].

The appearance of the new states due to the absence ofperiodicity are shown by red crosses. These states are de-tached from the bulk states and clearly are a manifestation

7

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DO

S (a

rb. u

nits)

E/t

-162 -161 -160 -159 -158

DO

S (a

rb. u

nits)

E/t

FIG. 6. DOS (in arbitrary units) of the synthetic bilayer square latticewith 2 × 40 × 40 sites and open boundary condition for (a) µ = 0and (b) µ = 1.2t for the parameters Ω0 = 200, α = 0.2, γ = π/2and λ = 0.2t.

of the open boundaries of the layers. These are edge statesand, as we demonstrate below, live on the boundaries of thelayers. For µ < µc, the edge states appear in both energygaps. This is consistent with the discussions for the cylin-drical geometry and the computation of the bulk Chern num-bers. This is illustrated in Fig. 5(a) as an exemplary case withµ = 0. For µ > µc, as expected, the first energy gap hostsnew states, while the second energy gap does not. Figure 5(b)demonstrates this via an example with µ = 1.2t. The cor-responding density of states (DOS) are shown in Fig. 6. Forµ < µc, two regions with low density of states appear ad-jacent to −Ω0(1 − α) under open open boundary condition.These correspond to the energies of the mid-gap states in theenergy gaps of the bulk spectrum. For µ > µc, one of thesetwo regions has vanishing DOS due to absence of the mid-gapstates between the middle band and the upper dispersive band.

In order to characterize the edge states in real space, wedefine probability density of the edge state correspondingto m = ±1 , p±(r) =

∣∣⟨ψE±(r)∣∣ψE±(r)

⟩∣∣2, where ψE±(r)denotes the eigenvector whose energy is the closest to E theand m = ±1 denotes the projection on one of the syntheticdimensions. Interestingly, the edge states have a differentreal space profile depending on the commensurability of thenumber of sites with the supercell: lattices with the lengthcommensurate with the periodicity of the unit cell have two

(a)

(b)

FIG. 7. Probability density plots for an edge mode corresponding toenergy E ≈ −160.28t. (a) The wave function is fully localized onthe edges of the lattice, which in this case has a length of 39 sites.(b) The wave function is localized on two edges of the lattice whichcontain raman coupled sites. In this case has a length of 40 sites,commensurate with the periodicity of the unit cell.

edges with Raman coupled sites and two edges that do notcontain such sites [see Fig. 1(a)], while lattices with thelength mod (LΘ ) = ±1 have all 4 edges of the same kind.Therefore to maintain the symmetry of the edges, wherewe expect the wave function to localize, system depictedon Fig. 7(a) was decreased by one site in each direction toLx,y=39 resulting in symmetric probability distribution overall 4 edges of the lattice. On the other hand, for the latticelengths being a integer multiple of unit cell’s periodicity theedge states are also localized on the borders of the lattice, butthe probability density is not equally distributed, favouringtwo edges, which are not Raman coupled. Such behaviourcan be observed on Fig. 7(b), which shows the spatial densitydistribution, p(r), of a typical edge states in the topologicallyinsulating phases of the system with L=40. We have verifiedthat p−(r) exhibits similar features. The chosen edge stateis a mid-gap state in the first band gap for µ = 0. Thespatial distribution has an asymmetric nature, which comesfrom the finite size synthetic bilayer geometry governed bythe interlayer coupling pattern. The edge states are morelocalized at two adjacent edges of lattice corners, which hostalternative sites with Raman coupled internal states, and

8

are rather weakly localized in rest of the boundaries, whereinternal states in any of the sites are not subjected to suchRaman coupling by construction.

V. DIMERIZED LATTICE

We now focus on the DL case, which is based on the alter-nating NN tunneling both in x and y direction and a complexNNN hopping.

.In particular we consider both dimerized NNN hopping, asin the SCH case as well as unstaggered NNN tunneling withφR = φL that provides a zero net flux per palquette. First,we analyze the possibility of using dimerization as a substi-tute for space dependent chemical potential for obtaining thenon-trivial gap between dispersive and quasi-flat sets of bands.Second, we simplify the NNN hopping leaving the dimeriza-tion untouched, since realizing constant diagonal hopping ex-perimentally is less complicated.

The effect of the lattice dimerization has been primarilystudied in Ref. [95]. The asymmetry of the hopping leads tothe shift of the bands in the energy spectrum together withenergy gap openings. It allows for the isolation of the setof quasi-flat bands, which originally are not separated by theglobal gap from the rest of the spectrum (see Fig. 9(a)). How-ever, this gap opening is trivial and leads to the triplet of Chernnumbers (0, 0, 0).

a. Staggered NNN hopping The interplay of the NNdimerization and the staggered NNN hopping results in atopological phase diagram depicted Fig. 8. Such order wasalso obtained in the SCH case in absence of chemical poten-tial. Hence, one can conclude that dimerization does not affectorder-changing processes but allows one to observe the edgestates of the well separated bands.

b. Uniform hopping We now consider dimerized latticewith uniform NNN hopping within the single layer. As dis-cussed in the previous paragraph, in this approach the open-ing of a global gap is guaranteed by the dimerization of NNhopping. The role of complex NNN tunneling is more com-plicated to explain, since unlike staggered NNN it cannot beassociated with Hall phase nor opens a global gap betweenquasi-flat and dispersive bands, as shown on Fig.9(b). On theother hand, similarly to Ref. [33] in such case the net fluxper plaquette is 0. All above suggests that applying diagonal(NNN) hopping of the complex value does not imply topo-logical non-triviality. Indeed uniform NNN hopping does notopen the global gap under any parameters’ configuration. Onthe other hand, similarly to the effects caused by dimerizationin the absence of NNN hopping, previously quasi-flat bandsbecome dispersive with increasing λ together with the shift inxy-plane. All effects of the uniform complex NNN tunnelingcan be seen on the Fig. 9(c). The effect of dimerized NN hop-ping is similar to band separation observed in staggered NNNhopping except, of course, the topological order of the sys-tem with uniform NNN tunneling varies between the trivialand standard non-trivial with no possibility of reaching non-standard topology, as presented in Fig. 10.

0

(b)

(a)

0

1.4

(a)

(b)

FIG. 8. Chern number of the lower dispersive set of bands (Chernnumber of the set of quasi-flat bands is always 0). In the absenceof the staggered NNN hopping the topology is trivial, regardless thestrength of dimerization( this case is depicted by the first column ofthe plot). Increasing the NNN hopping causes the change of the topo-logical order into the standard nontrivial one after reaching gaplessphase, which is visible on (b) and marked by the red line on (a). Theorder of the signs of the Chern numbers depends on the dimerizationsign. Moreover for t2/t1 = 0 discussed set of bands consists of 3separate subsets of bands with trivial topology( this case is depictedby the lowest row of the plot). Dispersion of each of these subsetsdepends on the value of λ and for λ = 0 all 3 subsets are almostperfectly flat.

VI. EXPERIMENTAL SCHEME

We here discuss a quantum simulation scheme of thesystem. As we will see below, all elements of the proposedscheme have been sucessfully implemented in state-of-the-artexperiments. The challenge consists in combining all thenecessary ingredients. We proceed in this section in steps,with the message directed mostly to the experimentalists.First, we review the basic scheme for twistronics without atwist, proposed already in Ref. [91]. Then, we discuss thenecessary additional ingredients of both proposed schemes:I) Staggered complex hopping, and II) dimerized real tun-nelling. We discuss possible methods that can be used torealize our schemes: i) Laser induced tunneling; ii) Floquetengineering; and iii) Super-lattice and holographic potentialimprinting methods. Finally, we discuss possible detectionschemes of the topological properties of the model.

9

-162

-161

-160

-159

-158E/t

-162

-161

-160

-159

-158

E/t

X M X M-162

-161

-160

-159

-158

E/t

(a)

(b)

(c)

FIG. 9. Magic configuration band structure in presence of uniformcomplex NNN hopping. (a) t2/t1 = 0.7 and λ = 0 shows the effectof pure dimerization of the lattice in the absence of NNN hopping re-sulting in vanishing Dirac cones. In comparison with Fig.2(b), whichrepresents staggered NNN hopping without chemical potential onecan see that dispersion of the quasi-flat bands is much bigger. (b)t2/t1 = 1 and λ = 0.2 represents the spectrum of the system withuniform NNN complex hopping generating the the increased disper-sion of the quasiflat bands and the lack of the global gap and. (c) Inthis case λ = 0.2 t2/t1 = 0.7. Dispersion of the quasiflat bandsis further increased but lattice dimerization provides the global gap.However these two effect compete and the system can be gapless fordifferernt values of λ (see Fig 10(b)).

A. Basic experimental scheme

We proposed in Ref. [91] to use a subset of four states outof the large nuclear spin manifold I = 9/2 of 87Sr, or 173Yb(I = 5/2). The SU(N) invariance inhibits collisional redis-tribution of the atoms among the different states. We selecttwo of them to be σ = ↑, and the other two to be σ = ↓.

0

1

(a)

(b)

FIG. 10. Chern number of the lower dispersive set of bands (Chernnumber of the set of quasi-flat bands is always 0). In the absenceof the uniform NNN hopping the topology is trivial, regardless thestrength of dimerization( this case is depicted by the first columnof the plot). Increasing the NNN hopping causes the change of thetopological order into the standard nontrivial one. However the orderof the signs of the chern numbers depends on the dimerization sign.Moreover for t2/t1 = 0 discussed set of bands consists of 3 separatesubsets of bands with trivial topology( this case is depicted by themiddle row of the plot). Dispersion of each of these subsets dependson the value of λ and for λ = 0 all 3 subsets are almost perfectly flat.

All are subjected to a two-dimensional spin-independent op-tical lattice potential, created by two counter-propagating lat-tice beams. We choose the laser wavelength λL = 813 nm(corresponding to the magic wavelength of the clock transi-tion 1S0 → 3P0). We set a lattice depth to about 8 recoilenergies, 8EL, which yields tunneling of order of 100 Hz.Lattice constant as usual is d = λL/2.

To create the synthetic layer tunneling, we exploit two-photon Raman transitions between spins m = ±1/2. A pairof Raman beams with λR = 689 nm near-resonant to the in-tercombination transition 1S0 → 3P1, would produce a cou-pling of amplitude Ω0 = Ω1Ω2/∆0. Here Ω1 and Ω2 arethe individual coupling amplitudes of the Raman lasers and∆0 the single-photon detuning. The Raman beams propagatein a plane perpendicular to the lattice potential, are alignedalong its diagonal, and form an angle θ with the lattice plane(see Fig. 1 in ref. [91]). This yields an in-plane momen-tum transfer per beam kR = ±2π cos θ/λR, with projec-tions kR/

√2 along the lattice axes. Therefore, the phase

of the synthetic tunneling is γ · r = γ(xx + yy), withγ = ±2π cos θλL/(

√2λR). The sign is determined by the

relative detuning of the Raman lasers. Experimentally, the

10

simplest choice is to use counter-propagating Raman beams(θ = 0), which yields γ = 0.8 (mod 2π). However, othermagnetic fluxes can be easily realized by adjusting the valueof θ.

To implement a periodic modulation of the Raman cou-pling amplitude on the scale of several lattice sites, we pro-pose to exploit a periodic potential created by a laser close-detuned from the excited state to excited state transition3P1 → 3S1 (corresponding to 688 nm [96]). This resultsin a large light shift of the 3P1 excited state of amplitudeδ, leading to a detuning of the Raman beams ∆(x, y) =∆0 + δ(1 + cos (2πx/lx) cos (2πy/ly)). Its effect is to mod-ulate the Raman coupling amplitude Ω(x, y) ' Ω0[(1− α)−α cos (2πx/lx) cos (2πy/ly)], with α = δ/∆0 ∼ 0.2 for real-istic experimental parameters [96, 97]. We therefore named it“modulation laser”.

B. Extensions of the basic scheme

SCH case. Here the NN tunnelling is standard and con-stant, t(r) = t. The staggered chemical potential is given by~µ = µ(x+ y), n = 2d(x+ y), and can be relatively easily re-alized using super-lattice or holographic potential imprintingmethods. The challenging part here is relate to the phases as-sociated with the next-nearest neighbor complex tunnelings,set to φL(~r) − φR(r) = π, where φR(~r) = (2r.1y + 1)π/2.We suggest to use here laser induced tunneling or lattice shak-ing. For laser induced tunneling one possibility would be toto employ the clock transition from 1S0 →3 P0, using appro-priate polarization of the assisting laser to couple to differentexcited states (for instance, coupling +3/2 → +1/2 via σ−polarized light, coupling +3/2→ +3/2 via π-polarized light,and +3/2 → +5/2 via σ+ polarized light). The main prob-lem is that the clock transitions will cross-talk immensely withthe light-shifting scheme of the 3P0 state that we proposed touse to get the moiré pattern. In fact, we should expect that thestaggered complex hoppings will not only be staggered (if wedesign and realize the staggering well), but they will be spa-tially modulated as well. The period of the latter modulationshould follow the period of our “moiré” pattern i.e.

λ(r) = λ0 + ∆λ cos (2πx/lx) cos (2πy/ly). (10)

After a careful study one observes however, that as long as∆λ ∼ t, the effects of the spatial modulation of the NNNcomplex hopping remain marginal. In fact they are limitedto negligible bandwidth corrections, which do not affect topo-logical order nor open/close new gaps in the system.

DL case. In Eq. (1), we considered dimerized real tunnel-ing, such that t(r) takes the form of t1 and t2 in alternativesites in the x and the y directions, along with the next-nearestneighbor complex hopping. Here the situation seems to beeasier from the experimental point of view. The alternatingtunnelling can be achieved using the super-lattice techniques(dimerization). The next-nearest neighbor complex hoppingwith the homogenous phase set to φR(r) = φL(r) = π/2should be accessible via lattice shaking and Floquet engineer-

ing techniques.

C. State-of-the-art experimental techniques

Laser assisted tunneling The idea of employing laser as-sisted tunneling for generation of synthetic gauge fields goesback to the seminal paper of Jaksch and Zoller [71]. It wasgeneralized to non-Abelian fields in Ref. [98]. These ideasall seemed very “baroque” at that time, but finally were real-ized in experiments with amazing effort, but equally amazingresults [73, 76, 77, 99, 100].

Floquet engineering In the context of cold atoms thistechnique goes back to the pioneering theory works of A.Eckardt and M. Holthaus [101], followed by experimentsof E. Arimondo and O. Morsch [102]. In condensed mat-ter the works concerned creation of topological phases ingraphene [103, 104]). The possibility of generating artificialgauge field was first discussed in Ref. [74] and realized in ex-periments of Hamburg group [105]. This culminated with theexperimental realization of arbitrary complex phases [106],and theoretical proposals combining shaking and on-site ex-citations [107]. In the recent years many fascinating resultswere obtain using shaking (cf. [108, 109], for a review see[110]). In a sense, from the perspective of the present papera culmination of these effort consisted in realization of theHaldane model with next nearest neighbor complex tunnel-ings in a brick lattice [66, 78]. Recently, the Hamburg groupcombined the studies of the Haldane model with the use ofmachine learning methods [111–113].

Super-lattice and holographic potential imprintingmethods These are nowadays standard methods, developedalready many years ago and described in textbooks suchas [64]. They have a plethora applications ranging from de-signing traps of special shape, through creation of dimerizedlattices, to imprinting random potentials. All of these meth-ods ban be useful for our purposes, for instance for designinga lattice with dimerized tunneling, etc. .

D. Detection of the topological order

In cold-atom quantum simulators, the standard transportexperiment techniques used to characterize the transverseconductivity in 2D materials are feasible but very demand-ing [114, 115] and there is therefore the need for other de-tection schemes to characterize the topology of the system.In the last decade, many detection schemes have developedfor quantum simulators [72, 116] and we briefly review herethe a non-exhaustive list of techniques that could be appliedto the synthetic twisted bilayer material. The total Chernnumber and the Berry curvature could be measured throughthe anomalous velocity of the center of mass of the atomiccloud [117, 118]. This technique, already applied in recentexperiments [119, 120], would require an additional opticalgradient. Alternatively, the Chern number could measured

11

through the depletion rate of the bands in the presence of heat-ing [121]. This effect, called quantized circular dichroism,has been implemented in state-of-the-art experiment [80] andwould require an additional shaking of the lattice. Finally, thetopology could be characterized through the observation ofthe chiral edge states. The latter could be done by a suitablequench protocol [122, 123].

VII. CONCLUSIONS - OUTLOOK

In the present paper we developed further the idea of“twistronics without a twist” and demonstrated that it can beused to engineer interesting topological band structures un-der various conditions. Focussing on a square lattice systemwith synthetic dimensions, we showed the appearance of ananomalous Hall phase in presence of artificial complex next-to-nearest neighbor interlayer tunneling. Moreover, we dis-cussed the emergence of topological bands via another mech-anism – lattice dimerization. In general, the bands of interestcan be categorized into three groups - trivial, and two cat-egories of topologically non-trivial bands differing by theirChern number combinations: standard nontrivial, and non-standard not-trivial.

Possible directions of this research line in the near fu-ture concern the incorporation of interaction effects. On thetechnical side, in the first stage, incorporation of the interac-tion effects can be carried out via a mean field theory at theHartree-Fock as well as “slave boson/fractionalization” level.Moreover, as our scheme provides the possibility of observingphysics similar to magic angle twisted bilayer graphene withan effectively large rotation angle, implying a much smallersupercell, performing ab-initio calculation could be possible

via advanced tensor network algorithms. The pressing ques-tions in these realm are: (i) the origins of strongly correlatedin strongly correlated phenomena in twisted materials, such asunconventional superconductivity phenomena, (ii) the coexis-tence of superconducting and correlated insulating states inmagic-angle twisted bilayer graphene, and their relationship,(iii) the role of topology in the interacting systems, which canbe probed by altering the quasiflatband topology.

VIII. ACKNOWLEDGEMENTS

We thank Alessio Celi, Christoph Weitenberg, Klaus Seng-stock and Leticia Tarruell for entlighting discussions. M.L.group acknowledges funding from European Union (ERCAdG NOQIA-833801), the Spanish Ministry MINECO andState Research Agency AEI (FIDEUA PID2019-106901GB-I00/10.13039 / 501100011033, SEVERO OCHOA No. SEV-2015-0522, FPI), European Social Fund, Fundació Cellex,Fundació Mir-Puig, Generalitat de Catalunya (AGAUR GrantNo. 2017 SGR 1341, CERCA program, QuantumCAT_U16-011424 , co-funded by ERDF Operational Program of Catalo-nia 2014-2020), MINECO-EU QUANTERA MAQS (fundedby The State Research Agency (AEI) PCI2019-111828-2/ 10.13039/501100011033), and the National Science Cen-tre, Poland-Symfonia Grant No. 2016/20/W/ST4/00314.T.S acknowledges additional support from the Secretariad’Universitats i Recerca de la Generalitat de Catalunya andthe European Social Fund, R.W.C. from the Polish NationalScience Centre (NCN) under Maestro Grant No. DEC-2019/34/A/ST2/00081, A.D. from the Juan de la Ciervaprogram (IJCI-2017-33180) and the financial support froma fellowship granted by la Caixa Foundation (fellowshipcode LCF/BQ/PR20/11770012), and D.R. from the FundacióCellex through a Cellex-ICFO-MPQ postdoctoral fellowship.

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