Creating, probing, and manipulating fractionally charged excitations of fractional Chern insulators in optical lattices
Mantas Račiūnas, F. Nur Ünal, Egidijus Anisimovas, André Eckardt
CCreating, probing, and manipulating fractionally charged excitations of fractionalChern insulators in optical lattices
Mantas Rači¯unas, ∗ F. Nur Ünal, † Egidijus Anisimovas, ‡ and André Eckardt § Institute of Theoretical Physics and Astronomy,Vilnius University, Saul˙etekio 3, LT-10257 Vilnius, Lithuania Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany (Dated: January 3, 2019)We propose a set of schemes to create and probe fractionally charged excitations of a fractionalChern insulator state in an optical lattice. This includes the creation of localized quasiparticlesand quasiholes using both static local defects and the dynamical local insertion of synthetic fluxquanta. Simulations of repulsively interacting bosons on a finite square lattice with experimentallyrelevant open boundary conditions show that already a four-particle system exhibits signatures ofcharge fractionalization in the quantum-Hall-like state at the filling fraction of / particle perflux quantum. This result is favorable for the prospects of adiabatic preparation of fractional Cherninsulators. Our work is inspired by recent experimental breakthroughs in atomic quantum gases: therealization of strong artificial magnetic fields in optical lattices, the ability of single-site addressingin quantum gas microscopes, and the preparation of low-entropy insulating states by engineering anentropy-absorbing metallic reservoir. I. INTRODUCTION
Topologically ordered states of matter that supportanyonic excitations are a fascinating example for theemergence of intriguing properties from the interplay ofmany interacting degrees of freedom [1]. Moreover, ithas been shown that such states can form a platformfor robust (topologically protected) quantum informationprocessing [2], provided one is able to create and ma-nipulate the anyonic excitations in a coherent fashion.Well known examples of topologically ordered states arefractional quantum Hall (FQH) states [3] and the closelyrelated fractional Chern insulators (FCIs) in lattice sys-tems [4–6]. However, in solid-state devices even the di-rect observation of individual anyonic quasiparticles isextremely difficult (see Ref. 7 for a recent proposal), notto mention their coherent creation and manipulation.Atomic quantum gases have been considered as analternative environment for studying FQH physics al-ready for more than a decade [8–11]. However, for along time the achievable effective magnetic fields werenot strong enough to observe quantum Hall physics andalso the entropies (or temperatures) were rather high.This situation has now changed with several experimen-tal breakthroughs. On the one hand, strong artificialmagnetic fields and two-dimensional spin-orbit couplingwere achieved in optical lattice systems [12–27] allowingfor the observation of topologically nontrivial band struc-tures [24, 26], a (quantized) bulk Hall response [19, 23],and chiral edge transport [20–22]. On the other hand,quantum gas microscopes were established as tools for ∗ [email protected] † [email protected] ‡ egidijus.anisimovas@ff.vu.lt § [email protected] manipulating and imaging atoms on single lattice sites[28–39]. Using digital micromirror devices they allow fortailoring light-shift potentials with high spatio-temporalresolution. This technique was very recently employedto prepare low-entropy insulating states by engineering apotential that gives rise to an entropy-absorbing metallicshell at the boundary of a small system [37].Inspired by these developments, in this paper we ad-dress two crucial questions regarding the possibility torealize and observe FQH physics in a quantum gas mi-croscope. The first question is: Provided a FCI state hasbeen realized, what are experimentally feasible probesthat can reveal its characteristic signatures? Here, ourapproach is to design schemes for creating and manip-ulating the elementary excitations of the system andto probe their fractional “charge” (i.e. particle number).This also paves the way towards studying their fractional(anyonic) statistics in the future. The second question is:Are characteristic features of a FCI state still accessiblein small systems of just a few particles? This questionis of eminent practical importance because, in contrastto solid-state systems, quantum gases are well isolatedrather than kept at a given temperature by their environ-ment. This implies that the state of the system has to beprepared adiabatically, starting from a topologically triv-ial ground state and passing through a continuous phasetransition into the desired FCI state [40, 41] while relyingon a finite-size gap in the spectrum.In order to address these questions, we propose threedifferent probes that can be implemented in quantum gasmicroscopes, and simulate them exactly for small systemsof four particles on about fifty sites. We consider the ex-perimentally relevant scenario of repulsively interactingbosons on a square lattice subjected to a homogeneousflux and focus on regimes where one expects a Laughlin-like FCI state at the filling of ν = 1 / particles per fluxquantum. Note that it is not our aim to study fundamen-tal properties of FCI states as such, as they have been a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n investigated already in a number of previous studies (see,e.g., Refs. 40–46 and also 11, 47–50 for other types of op-tical lattices). We rather address how and whether it ispossible to observe signatures of FCIs in small realisticsystems. The first probe is the interaction energy (whichcan be obtained from measuring site occupations) as afunction of the filling factor ν . As expected for a FCI,we find a pronounced minimum in the interaction energyaround ν = 1 / . The second probe is the accumulationof quantized fractional charge ν near engineered local de-fects as a unique fingerprint revealing the localization ofindividual fractionally charged excitations. Finally, as athird probe, we study the adiabatic creation and pump-ing of fractionally charged excitations via the insertionof a flux quantum through a thin solenoid localized onsingle plaquettes (as they can be implemented using Flo-quet engineering [51]). Simulations of these probes indi-cate FQH physics already in the small systems studiedhere. II. MODEL
We consider interacting bosons moving in a two-dimensional square lattice subjected to an effective mag-netic field. The system is described by a Bose-HubbardHamiltonian, written in terms of annihilation and densityoperators, a (cid:96) and n (cid:96) = a † (cid:96) a (cid:96) for bosons on site (cid:96) , H = − t (cid:88) (cid:104) (cid:96) (cid:48) (cid:96) (cid:105) e iθ (cid:96) (cid:48) (cid:96) a † (cid:96) (cid:48) a (cid:96) + U (cid:88) (cid:96) ˆ n (cid:96) (ˆ n (cid:96) − (cid:88) (cid:96) V (cid:96) ˆ n (cid:96) . (1)Here, t describes nearest-neighbor tunneling. The Peierlsphases θ (cid:96) (cid:48) (cid:96) , which encode a uniform flux φ = 2 πα andpossibly also local fluxes on certain plaquettes, can beimplemented using Floquet engineering [14]. Whereas ahomogeneous flux has already been achieved experimen-tally [18, 19, 25], the creation of solenoid-type single-plaquette fluxes is proposed in Refs. 51 and 52. TheHubbard parameter U describes on-site interactions. Wealso consider energy offsets V (cid:96) on some sites, to be usedfor creating and/or trapping quasiparticle and quasiholeexcitations.Motivated by a recent experiment [19], we focus onthe regime α = 1 / . In this situation, the band struc-ture consists of four bands. The lowermost (which isadiabatically connected to the lowest Landau level in thelow-flux regime) is protected by a gap much wider thanits band width. The formation of a topological bandstructure was observed by measuring the Chern number C = 1 with convincing precision and fueled the hope for apossible many-body experiment. For strong interactionsthis system is expected to stabilize an incompressible FCIstate at certain fractional fillings ν , such as a ν = 1 / -Laughlin-type state [40–46, 53]. V/t
V/t x y x y QuasiparticleQuasiholeBackground
V/t (b)(c) (d)(e) (f)(a) +V - V FIG. 1. Lattice geometry and charge redistribution due toadditional potential offsets. Typical geometry of × sites.Potential offsets ± V are each distributed uniformly over foursites. The narrow ‘control’ region in the center (green shade)separates the left (blue) and right (red) regions where densitychange is calculated. Charge accumulation/depletion withquantized steps of magnitude one half is seen in lattices (b) × and (c) × operating in the topological regime;here U/t = 7 . and ν = 1 / . (d) A trivial band insulatorleads to continuous charge flow. Density change induced bythe impurity potentials: for (e) fractional quantum Hall stateat ν = 1 / , V = 3 , U = 7 . and (f) integer quantum Hallstate at ν = 1 , V = 9 , U = 0 . III. FINITE-SIZE SYSTEM
In our numerical simulations, we consider a squarelattice of N x × N y sites with open boundary conditions,which contains ( N x − N y − plaquettes. Such asystem can be realized in current quantum gas micro-scope setups. To reach the strongly interacting regime, U (cid:29) t , we set U/t = 7 . . However, varying the ratio U/t we find little qualitative dependence on its actualvalue and most of the presented results are also repro-duced in the hard-core limit. The samples are piercedby a uniform background flux, and further perturbed byeither: (i) introducing localized attractive and repulsivepotential offsets, each distributed over four lattice sitessurrounding a single plaquette, or (ii) inserting two addi-tional localized solenoid-type fluxes of equal magnitudeand opposite signs, each penetrating a single plaquette[51]. Figure 1(a) shows a typical example with dimen-sions N x × N y = 9 × . Note that the two locations ofthe potential offsets or the solenoid fluxes (the affectedplaquettes are marked by dots) are well separated fromeach other and avoid the boundary sites. In Fig. 1(a) wealso mark three regions in which charge redistributionwill be monitored. The broad region on the left (markedwith blue shade) accommodates the quasihole, and itscounterpart on the right (red shade) accommodates thequasiparticle. The two regions are separated by the nar-row ‘neutral’ stripe (green shade) that helps verify theirlocalization.In an infinite square lattice (or for periodic bound-ary conditions) the number of sites N sites (i.e. of single-particle states) matches the number of plaquettes N plaq .Thus, the filling factor ν is uniquely defined as the ratiobetween the number of particles per plaquette N/N plaq and the number of flux quanta per plaquette α , ν = N/N plaq α . (2)Any deviation from the nominal rational value of the fill-ing factor that corresponds to a given FQH state resultsin creation of quasiparticle excitations. In a finite sys-tem, an ambiguity arises because of extra sites lying atthe system boundary. Counting sites rather than plaque-ttes in Eq. (2) would produce an alternative filling factor: ˜ ν = ν ( N x − N y − /N x N y . This issue — being of em-inent importance for the preparation of FQH states inquantum gas microscopes — poses a natural question:What is the optimal filling for the stabilization of a FQHdroplet in a small system? FIG. 2. Scaled interaction energy, as it can be measured ina quantum gas microscope, versus filling factor, for configura-tions × (a) and × (b) obtained for U/t = 7 . . Thefilling is tuned via plaquette flux φ . Broad minima appearclose to the expected filling factor ν = 1 / . A. Interaction energy
In order to address the posed question, we consider N bosons on a square lattice of N x × N y sites (denoted N @ N x × N y hereafter). We focus on a homogeneous sys-tem with uniform plaquette flux φ = 2 πα and in theabsence of on-site potentials, i.e. V (cid:96) = 0 , ∀ (cid:96) . In an exper-iment, a continuous control of the flux can be achieved,e.g., by implementing the moving superlattice used forthe Floquet engineering of the plaquette fluxes by a dig-ital micromirror device [25]. The contribution to theground-state energy that comes from the interaction of particles should attain a minimum at fractional filling fac-tors that correspond to the formation of the FCI state.This approach was introduced for interacting fermions[54] on finite lattices with periodic boundary conditions,and conspicuous minima at odd-denominator fractionalfillings were demonstrated. We define the average inter-action energy as E int = (cid:68) Ψ gs (cid:12)(cid:12)(cid:12) U (cid:88) (cid:96) ˆ n (cid:96) (ˆ n (cid:96) − (cid:12)(cid:12)(cid:12) Ψ gs (cid:69) , (3)where | Ψ gs (cid:105) denotes the ground state obtained from exactdiagonalization.We begin with our principal × configuration,which gives ν = 1 / for α = 1 / in Eq. (2), and varythe filling ν by tuning α . Figure 2(a) shows that thebroad minimum of E int has its deepest point at ν ≈ . which is just slightly off the nominal value ν = 1 / . Thealternative definition ˜ ν ≈ . ν would lead to an obviousdeviation. In Fig. 2(b) we repeat the numerical exper-iment for another lattice geometry × , which is ofcomparable area. The minimum of the interaction energyis again centered around ν = 1 / but would shift whenplotted against ˜ ν ≈ . ν . Note that for the parametersand protocol of Fig. 2 we do not expect a FCI state atquarter filling. In this scenario, the filling is adjusted bytuning the plaquette flux and ν = 1 / corresponds to φ = π that neither breaks time-reversal symmetry norgives rise to separate Bloch bands with nonzero Chernnumbers.In both Fig. 2(a) and (b), we observe rather broadminima of width ∼ . with respect to the filling. Below(see Fig. 3), we will see that the corresponding filling fac-tors coincide with those intervals of ν for which we alsofind the quantized fractional charge transport expectedfor a FCI state. This finding suggests a robustness of theFCI state with respect to filling, which should greatly fa-cilitate its experimental preparation. We attribute thiseffect to the fact that in our small systems, a finite frac-tion of the total particle number can be accommodatedin gapless edge modes without significantly changing thebulk properties of the system (which are protected bya gap). This effect is another advantage of consideringsmall systems that complements their advantage for adi-abatic state preparation.We stress that besides serving as a numerical bench-mark, the interaction energy can also be used as an exper-imental indicator for the formation of FQH states. Thesingle-site resolution of quantum gas microscopes allowsfor extracting both the mean and the fluctuations of theon-site occupations via measuring their statistics in re-peated experiments. This turns E int into an experimen-tally accessible quantity. B. Incompressibility and charge fractionalization
While the minimum of the interaction energy aroundhalf filling is consistent with a FCI ground state, it doesnot reveal specific signatures, such as charge fraction-alization, of this state and could also indicate, e.g., agapped density wave at half filling. Therefore, we proposeanother experimentally feasible method to probe the frac-tional excitations of this system. The method is based onthe idea to introduce localized impurity potentials [46] topin quasiparticles or quasiholes. For that purpose we as-sume that the system is prepared in the presence of twospatially-separated local defects in the bulk: a potentialdip and a potential bump. Monitoring the integratedparticle density (“charge”) in the vicinity of each defect,we expect two signatures for a FCI state: First, the in-compressibility associated with the bulk gap should makethe system stiff against weak impurities. Second, ramp-ing up the impurities further, we expect the dip and thebump to attract a quasiparticle and a quasihole, respec-tively, corresponding to a quantized charge accumulationof + ν and − ν .We observe that in our situation potentials situated ona single site (cf. Ref. 46) do not constitute the optimalchoice. In fact, due to the finite extent of the quasiparti-cles and quasiholes, it is preferable to work with poten-tials | V (cid:96) | = V / distributed over four neighboring sites,as depicted in Fig. 1(a) for the × lattice. We ver-ified that results do not deteriorate when the impuritypotentials are smeared out over more than four sites asmodeled by Gaussians with a width of a lattice constant.We calculate the ground state as a function of V andplot the charge accumulated in the three control regions[indicated by the background colors in Fig. 1 (a)] inFig. 1 (b). We see that as the potential strength V growsfrom to t , the number of particles in the left (right) endof the elongated lattice gradually grows (decays) fromthe initial value of around . to a stable value of . ( . ) (depicted with dashed lines) and remains pinned tothis value in a broad region of potential strengths up to V ≈ t . We note, that the initial region of the curves alsodisplays insensitivity against weak potentials. Thus, wemake two observations that point to the successful cre-ation of fractionally charged excitations in our small sys-tem: not only the transferred charge equals ± / with agood accuracy but also the initial and the final states dis-play rigidity typical for incompressible FCI states. Thesmall deviation from the expected quasihole charge of − / can be attributed to a finite-size effect.To further corroborate the observations, we show theresults of analogous simulations in panels (c) and (d) ofFig. 1, respectively, for a sample of different shape ( × )and for a topologically trivial system (with four Blochbands created by a superlattice potential rather than bymagnetic flux). In the former case the results reproducethe features seen for lattices × while in the latter casewe see a featureless uniform growth of the transferredcharge.In Fig. 1(e), we plot the density change induced bylocal potentials of strength V = 3 [considering the con-figuration of Fig. 1(a) and (b)]. For completeness, wehave also plotted the density change found for an integer quantum Hall state on the same lattice but with ν = 1 and U = 0 . Here a larger impurity V = 9 is requiredto create integer charged quasiparticles/holes, since thesingle-particle band gap is much larger that the many-body gap protecting the FCI. C. Creating excitations and fractional chargepumping
A further hallmark of FCI states is fractionally quan-tized adiabatic charge pumping in response to the inser-tion of a flux quantum [40, 51, 55]. In order to probe thiseffect, we consider the configuration shown in the insetof Fig. 3, where additional Peierls phases of δφ are addedon the bonds marked by yellow arrows. This leads tolocal solenoid-type fluxes ± δφ piercing the marked pla-quettes A and B on top of the homogeneous backgroundflux φ . When δφ is ramped up slowly from to π , weexpect that a quasihole of charge − ν is created at pla-quette A and a quasiparticle of charge ν at plaquetteB. This corresponds to the adiabatic pumping of frac-tional charge ν from A to B. In order to prevent theso-created excitations from dispersing, we also add smallstatic local energy offsets of the same form as the onesconsidered previously [Fig. 1(a)]. We choose a very smallvalue V = ± . t , which hardly alters the ground-statecharge distribution [as can be seen from Fig. 1(b)], butis still large enough to pin the dynamically created localquasiparticles and quasiholes [46].For this protocol, we simulate the time evolution.Starting from the ground state of a small system( × ), δφ is switched on linearly from to π moder-ately slowly, with the ramp time τ = 5 (cid:126) /t . We note thatthe ramp time τ should not be too large as the pumpedcharge would have time to disperse during the slow ramp,whereas the opposite limit of too fast ramping would heatthe system up. In Fig. 3 we depict the so-induced changein the particle numbers in the three regions indicatedwith different background colors in the inset. Here thefilling factor ν is again controlled by changing the back-ground flux φ . The dashed lines correspond to the frac-tional charge transfer predicted for the FCI state at halffilling. We find that the pumped charge saturates approx-imately to the expected value only in the region markedby the rectangular box, where the particle density in the(green) control region also remains fixed. In additionto being centered around ν = 1 / , this region matchesthe broad minimum of the interaction energy shown inFig. 2(a) for the same system. This provides anotherindication that a system as small as the one consideredhere can support a FCI-type ground state. IV. CONCLUSIONS AND OUTLOOK
In summary, we have proposed and investigated twocomplementary schemes for probing charge fractionaliza- ν x A B
Quasiparticle
Quasihole
Background
FIG. 3. The net charge transferred in a × lattice in re-sponse to the insertion of a flux quantum plotted as a functionof the filling factor. The rectangle drawn in the black dashedline marks the vicinity of ν = 1 / . The background flux is setto α = 1 / , interaction U/t = 7 . , and additional potential V = ± . is distributed over the A/B plaquettes. tion in optical lattice systems as a signature of FCI states.They rely on different strategies for creating and trans-porting quasiholes and quasiparticles: either by staticimpurities or by the local insertion of a flux quantum.As alternative probes are not easy to implement, the pro-posed density-based protocols will be crucial for measur-ing unique signatures of a FCI state at a particular filling ν . For example, observing the fractionally quantized bulk Hall response to a homogeneous force is difficult becauseof the external confinement, and so far, there is also nostraightforward method proposed for measuring the ex-change phase πν characteristic to the anyonic quasipar-ticles. Also probing chiral edge modes via the dynamicsof defects [45] or the (experimentally challenging) mea-surement of single-particle coherence [40] do not allow fora clear distinction between integer and fractional Cherninsulator states. Additionally, we have pointed out thatalso the interaction energy is a measurable quantity thatprovides further support regarding the realization of in-sulating states of matter at fractional filling. Simulatingthese probes for a bosonic system on a square lattice,we found that already small systems of just four parti-cles feature signatures of FQH state at half filling. Thisresult is of immediate practical relevance, since the adia-batic preparation of such a state relies on the finite-sizegap of the system.In future work, it will be interesting to look for signa-tures of charge fractionalization for FCI states at fillingfactors different from ν = 1 / , either Laughlin states atlower filling or hierarchy states. Another fascinating per-spective is to study whether the techniques introducedhere can be extended also to measure signatures of any-onic statistics. ACKNOWLEDGMENTS
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