Quantum phases of Rydberg atoms on a kagome lattice
Rhine Samajdar, Wen Wei Ho, Hannes Pichler, Mikhail D. Lukin, Subir Sachdev
QQuantum phases of Rydberg atomson a kagome lattice
Rhine Samajdar a,1 , Wen Wei Ho a,b , Hannes Pichler c,d , Mikhail D. Lukin a , and Subir Sachdev a a Department of Physics, Harvard University, Cambridge, MA 02138, USA; b Department of Physics, Stanford University, Stanford, CA 94305, USA; c Institute for TheoreticalPhysics, University of Innsbruck, Innsbruck A-6020, Austria; d Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, AustriaSubmitted to PNAS on July 28, 2020
We analyze the zero-temperature phases of an array of neutral atomson the kagome lattice, interacting via laser excitation to atomic Ry-dberg states. Density-matrix renormalization group calculations re-veal the presence of a wide variety of complex solid phases withbroken lattice symmetries. In addition, we identify a novel regimewith dense Rydberg excitations that has a large entanglement en-tropy and no local order parameter associated with lattice symme-tries. From a mapping to the triangular lattice quantum dimer model,and theories of quantum phase transitions out of the proximate solidphases, we argue that this regime could contain one or more phaseswith topological order. Our results provide the foundation for theo-retical and experimental explorations of crystalline and liquid statesusing programmable quantum simulators based on Rydberg atomarrays.
Rydberg quantum simulators | Density-wave orders | Quantum phasetransitions T he search for quantum phases with fractionalization, emer-gent gauge fields, and anyonic excitations has been acentral focus of research in quantum matter for the past threedecades (1, 2). Such systems feature long-range many-bodyquantum entanglement, which can, in principle, be exploitedfor fault-tolerant quantum computing (3). The best-studiedexamples in this regard are the fractional quantum Hall statesfound in high magnetic fields (4). While such states have,by now, been realized in a wide variety of experimental sys-tems, their intrinsic topological properties, including anyonicstatistics, are challenging to detect and control directly (5).In the absence of a magnetic field, the simplest anyonic phasecompatible with time-reversal symmetry is the so-called Z spin liquid (6, 7), which has the same topological order as the“toric code” (3). While there are some indications that sucha phase may be present in electronic systems on the kagomelattice (8–10), thus far, these quantum spin liquid (QSL) stateshave evaded direct experimental detection.In the search for QSLs, systems with frustration (11, 12)—which can be either of geometric origin or induced by further-neighbor couplings—constitute a promising avenue of explo-ration. Motivated by this consideration, here, we investigatemany-body states of neutral atom arrays, interacting via laserexcitation to atomic Rydberg states (13), that have been foundto display a variety of interesting correlated quantum phasesin one and two dimensions (14–20). Specifically, we examinea realistic model of Rydberg atoms on the kagome lattice,and perform density-matrix renormalization group (DMRG)computations to establish its rich phase diagram as a functionof laser parameters and atomic distances. These calculationsreveal the formation of several intricate solid phases with long-range density-wave order. We show that one of these orderedphases actually emerges from a highly degenerate manifold of classical states via a quantum order-by-disorder mechanism.We also find a strongly correlated “liquid regime” of parameterspace (identified by the star in Fig. 1) where the density ofRydberg excitations is limited by the interactions, in contrastto the gas-like “disordered regime” where the laser drivinginduces independent atomic excitations. While for most inter-action strengths, solid phases appear in such a dense regime,we observe that the liquid regime has no local order, and sig-nificant entanglement entropy. We employ a mapping to thetriangular lattice quantum dimer model (21), which correctlydescribes the solid phases proximate to the liquid regime in theRydberg model. Theories for quantum phase transitions out ofthese solid phases then suggest that part of this liquid regimecan host states with long-range topological order. While ournumerical results do not provide direct evidence for topologicalorder over the system sizes studied, we demonstrate that thisregime should be readily accessible in experiments, raising thepossibility of experimental investigations of entangled quan-tum matter. Remarkably, this is made possible simply usingappropriate lattice geometries and innate interactions, even without carefully engineering specific gauge constraints (22). Kagome lattice Rydberg model
Our interest lies in studying the phases of neutral atoms ar-ranged on a kagome lattice, as sketched in Fig. 1(a). Eachkagome unit cell comprises three sites on a triangular scaf-folding and the primitive vectors of this lattice are a = (2 a, a = ( a, √ a ), where the lattice constant a is the spacingbetween two nearest-neighboring sites. Let us denote the num- Significance Statement
Programmable quantum simulators based on Rydberg atom ar-rays have recently emerged as versatile platforms for exploringexotic many-body phases and quantum dynamics of stronglycorrelated systems. In this work, we theoretically investigatethe quantum phases that can be realized by arranging suchRydberg atoms on a kagome lattice. Along with an extensiveanalysis of the states which break lattice symmetries due toclassical correlations, we identify an intriguing new regime thatconstitutes a promising candidate for hosting a phase withlong-range quantum entanglement and topological order. Ourresults suggest a novel route to experimentally realizing andprobing highly entangled quantum matter.
R.S. and S.S. conceived the research, R.S. performed the DMRG computations, W.W.H. and H.P.undertook the exact diagonalization studies, and H.P. and M.D.L. explored experimental implica-tions and feasibility. All authors discussed the results and contributed to the manuscript.The authors declare no competing interests. To whom correspondence should be addressed. E-mail: [email protected]
PNAS |
November 26, 2020 | a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov er of complete unit cells along a µ by N µ . In a minimal model,each atom can be regarded as a two-level system with | g i i and | r i i representing the internal ground state and a highlyexcited Rydberg state of the i -th atom. The system is drivenby a coherent laser field, characterized by a Rabi frequency, Ω,and a detuning, δ . Putting these terms together, and takinginto account the interactions between atoms in Rydberg states(23), we arrive at the Hamiltonian H Ryd = N X i =1 Ω2 ( | g i i h r | + | r i i h g | ) − δ | r i i h r | + 12 X ( i,j ) V (cid:0) || x i − x j || /a (cid:1) | r i i h r | ⊗ | r i j h r | , [1]where the integers i, j label sites (at positions x i,j ) of thelattice, and the repulsive interaction potential is of the van derWaals form V ( R ) = C /R (24). Crucially, the presence of theseinteractions modifies the excitation dynamics. A central rolein the physics of this setup is played by the phenomenon of theRydberg blockade (25, 26) in which strong nearest-neighborinteractions ( V (1) (cid:29) | Ω | , | δ | ) can effectively prevent two neigh-boring atoms from simultaneously being in Rydberg states.The excitation of one atom thus inhibits that of another andthe associated sites are said to be blockaded. By reducing thelattice spacing a , sites spaced further apart can be blockadedas well and it is therefore convenient to parametrize H Ryd bythe “blockade radius”, defined by the condition V ( R b /a ) ≡ Ωor equivalently,
C ≡ Ω R b . Finally, we recognize that by iden-tifying | g i , | r i with the two states of a S = 1 / H Ryd can also be written as a quantum Ising spin model with C /R interactions in the presence of longitudinal ( δ ) and transverse(Ω) fields (27). We determine the quantum ground states of H Ryd for differ-ent values of δ/ Ω and R b /a using DMRG (28, 29), which hasbeen extensively employed on the kagome lattice to identifyboth magnetically ordered and spin liquid ground states ofthe antiferromagnetic Heisenberg model (30–32). The techni-cal aspects of our numerics are documented in Sec. I of theSupporting Information (SI). In particular, we work in thevariational space spanned by matrix product state (MPS) an-sätze of bond dimensions up to d = 3200. Although ( i, j ) runsover all possible pair of sites in Eq. (1), this range is truncatedin our computations, where we retain interactions betweenatoms separated by up to 2 a (third-nearest neighbors), asshown in Fig. 1(a). In order to mitigate the effects of theboundaries, we place the system on a cylindrical geometryby imposing open (periodic) boundary conditions along thelonger (shorter) a ( a )-direction. The resulting cylinders arelabeled by the direction of periodicity and the number of sitesalong the circumference; for instance, Fig. 1(a) depicts a YC6cylinder. Since the computational cost of the algorithm (for aconstant accuracy) scales exponentially with the width of thecylinder (33), here, we limit the systems considered to a max-imum circumference of 12 lattice spacings. Unless specifiedotherwise, we always choose the linear dimensions N , N soas to yield an aspect ratio of N /N ’
2, which is known tominimize finite-size corrections and optimize DMRG resultsin two dimensions (34, 35).
Phase diagram
We first list the various phases of the Rydberg Hamiltonianthat can arise on the kagome lattice. Without loss of generality,we set Ω = a = 1 hereafter for notational convenience. At largenegative detuning, it is energetically favorable for the systemto have all atoms in the state | g i , corresponding to a trivial (e)(a) (b) Stripe: δ = 2 . R b = 1 . δ = 3 . R b = 1 . δ = 3 . R b = 2 . Fig. 1. Phases of the kagome lattice Rydberg atom array . (a) Geometry of the kagome lattice; the lattice vectors are a = (2 , , a = (1 , √ . Periodic (open) boundaryconditions, designated by PBC (OBC), are imposed along the a ( a ) direction, resulting in a cylinder. The blue dots are the sites of the original kagome lattice, where the atomsreside, while the red points outline the medial triangular lattice formed by connecting the centers of the kagome hexagons. (b–d) The various possible symmetry-broken orderedphases. Each lattice site is color coded such that green (red) signifies the atom on that site being in the Rydberg (ground) state. (e) Phase diagram of the Hamiltonian (1) in the δ - R b plane. The yellow diamonds and the pink circles are determined from the maxima of the susceptibility at each R b ; the former correspond to the finite-size pseudocriticalpoints delineating the boundaries of the ordered phases. The white bars delimit the extent of the stripe phase. The string phase (see Fig. 2) lies at larger detuning, beyond theextent of this phase diagram, as conveyed by the black arrow. The correlated liquid regime is marked by a red star. The cuts along the dotted and dashed lines are analyzed inFigs. 5 and 7, respectively. | disordered” phase with no broken symmetries (36). As δ/ Ω istuned towards large positive values, the fraction of atoms in | r i increases but the geometric arrangement of the excitationsis subject to the constraints stemming from the interactionsbetween nearby Rydberg atoms. This competition between thedetuning and the previously identified blockade mechanismresults in so-called “Rydberg crystals” (37), in which Rydbergexcitations are arranged regularly across the array, engenderingsymmetry-broken density-wave ordered phases (19). On thekagome lattice, the simplest such crystal that can be formed—while respecting the blockade restrictions—is constructed byhaving an atom in the excited state on exactly one out thethree sublattices in the kagome unit cell. This is the orderingpattern of the “nematic” phase [Fig. 1(c)], which is found in aregime where only nearest-neighbor sites are blockaded. Thenematic order spontaneously breaks the threefold rotational( C ) symmetry of the underlying kagome lattice, so, for aninfinite system, the true ground state is triply degeneratewithin this phase. Even though H Ryd does not conserve thenumber of Rydberg excitations, the ordered state can still becharacterized by a “filling fraction” upon taking the classicallimit δ/ Ω → ∞ , R b /a = 0, which, in this case, leads to adensity of h n i i = 1 /
3, where n i ≡ | r i i h r | .Curiously, the nematic phase is separated from the trivialdisordered one by a sliver of a quantum solid without any clas-sical analogue, to wit, the stripe phase seen in Fig. 1(b). Thisstate also breaks the C symmetry; accordingly, between thedisordered and stripe phases, one encounters a Z -symmetry-breaking quantum phase transition (QPT) (38) in the uni-versality class of the (2+1)D three-state Potts model (39),while the QPT demarcating stripe and nematic is first-order.Although both phases break the same symmetry, the stripeordering is distinguished from the nematic by a substantial andequal density on two sublattices of the unit cell. The formationof these stripes can be attributed to quantum fluctuations (19),which help stabilize the phase in a narrow window as follows.The system optimizes the geometric packing in a configurationwhere all atoms on one sublattice are in the ground state,whereas those on the other two sublattices are each in a quan-tum superposition formed by the ground state with a coherentadmixture of the Rydberg state. These “dressed” atoms as-sist in offsetting the energetic penalty due to the interactions,while simultaneously maximizing the excitation density andtherefore, the reduction in energy from δ . The ensuant averagedensity in the stripe phase is also h n i i ∼ /
3, which explains itsexistence as a precursor to the nematic ordering. The extentof this phase narrows significantly with increasing R b , so itis difficult to ascertain whether the transition between thelattice nematic and disordered phases is always a two-step onewith the stripe order intervening. Nevertheless, based on ourcurrent data (see also Fig. 7), we believe it is likely that thestripe phase terminates at a tricritical point near the tip ofthe nematic dome instead of surrounding it throughout.Proceeding to larger blockade radii, we find that the kagomeRydberg array hosts yet another solid phase with density-waveordering, namely the “staggered” phase [Fig. 1(d)]. This phase,which bears a twelvefold ground-state degeneracy, is realizedwhen interactions between neighboring Rydberg atoms aresufficiently strong enough to blockade third-nearest-neighborsites, so the excitations are positioned a distance of √ - - - - - - - (a) (b)(c) (d)(e) Fig. 2. Crystalline phase at / filling on the kagome lattice . (a–c) Classicallyordered states at f = 2 / ; while we sketch only three configurations here, the numberof such states—with the same filling fraction—scales exponentially with the systemsize. The Rydberg excitations (red) are arranged in “strings” (yellow) that span thelattice. (d) Comparing the three possible classical phases, we find that the energy(at Ω = 0 ) is minimized by the string-ordered state over a finite region between thenematic and the staggered phases. (e) Rydberg crystal formed in the string phase at δ = 4 . , R b = 1 . on a wide ( N = 12 ) YC8 cylinder. cell with lattice vectors 4 a and 2 a + a ; the associatedclassical density is 1 /
6. The staggered phase remains stableup to R b (cid:46) √
7, beyond which fourth-nearest neighbors arealso blockaded.Equipped with the information above, we now turn toassembling the full phase diagram of H Ryd . An unbiased di-agnostic to do so is the bipartite von Neumann entanglemententropy (EE) of the ground state S vN ≡ − Tr ( ρ r ln ρ r ), ρ r be-ing the reduced density matrix for each subsystem when thecylinder is partitioned in half along a . On going from thedisordered phase to an ordered one, S vN gradually increases,peaks near the quantum critical point (QCP), and then dropssharply inside the solid phase [see also Fig. 5(c)]. This isbecause DMRG prefers states with low entanglement and sys-tematically converges to a so-called Minimal Entropy State(MES) (35, 40), which is simply one of the symmetry-brokenstates rather than their superposition. This drastic decline in S vN traces out the two lobes seen in Fig. 1(e), which mark thephase boundaries of the nematic and staggered orders. In thelimit of large detuning, there is another density-wave orderedphase between these two lobes, which we christen the “string”phase and discuss next. Quantum order-by-disorder
In the classical limit of δ/ Ω → ∞ , the periodic arrangementof Rydberg excitations (or equivalently, hard-core bosons) onthe kagome lattice can result in additional ordered phasesbesides the nematic and the staggered at various fractionaldensities (41). To see this, one can simply minimize the PNAS |
November 26, 2020 | lassical energy, which is determined solely by the competitionbetween the detuning and the repulsive interactions. In theparameter range of interest ( R b (cid:46) . f = / V /δ < / , /
9; 1 / < V /δ < / , /
6; 1 / < V /δ, [2]where V represents the strength of the third-nearest-neighborinteractions. Since we have (temporarily) set Ω = 0, the ra-tio V /δ is the only independent tuning parameter for theHamiltonian in this limit.The phases at fillings of a third and a sixth can be readilyidentified as (the classical versions of) the familiar nematic andstaggered orders [Figs. 1(c) and (d)], respectively. In betweenthe two, the system favors a separate highly degenerate classi-cal ground state, forming what we dub the “string” phase. Afew of the possible ordering patterns for a crystal belonging tothis phase, with a filling fraction of f = 2 /
9, are presented inFigs. 2(a–c). The arrangement of the Rydberg excitations re-sembles strings—which may be straight or bent—that stretchacross the lattice. Interestingly, there are a macroscopic num-ber of such states, all with the same classical energy, and thisdegeneracy grows exponentially with the linear dimensions ofthe system. For example, in Fig. 2(a), the positions of all theatoms in the Rydberg state can be uniformly shifted by ± a / O (2 N ) potential configurations. Similarly, when the stringsare bent, like in Fig. 2(c), there are O ( N ) locations wherea kink can be formed, and correspondingly, O (2 N ) states ofthis type.The large classical degeneracy raises the question of thefate of this phase once we reinstate a nonzero transverse field,Ω. There are two natural outcomes to consider. Firstly, a su-perposition of the classical ground states can form a quantumliquid with topological order, as is commonly seen to occurin quantum dimer models (42). However, a necessary condi-tion in this regard is the existence of a local operator whichcan connect one classical ground state with another. Sincethe individual ground states are made up of parallel strings,they are macroscopically far away from each other, and itwould take an operator with support of the size of the systemlength to move between different classical configurations, thusviolating the requirement of locality. This brings us to thesecond possibility, namely, that a quantum “order-by-disorder”phenomenon (43, 44) prevails. In this mechanism, quantumfluctuations lower the energy of particular classical states fromamongst the degenerate manifold; the system then orders in astate around which the cost of excitations is especially cheap.In this case, one could anticipate a string-ordered solid phase,which should be easily identifiable from the structure factor.The DMRG numerics confirm our intuition that such acrystal should emerge in the phase diagram at sufficiently highdetunings. On the YC8 cylinder with N = 8, the string phaseappears at detunings beyond the range rendered in Fig. 1(e).However, it is manifestly observed, for a wider geometry, inFig. 2(e), which illustrates the local magnetizations insidethe string phase (at δ = 4 . R b = 1 .
95) on a YC8 cylinderof length N = 12 (chosen so as to be fully compatible withthe string order). The ground state found by finite DMRG is (a) (b)(c) (d) Fig. 3. Static structure factors of the various ordered phases. S ( q ) displayspronounced and well-defined peaks for the (a) stripe ( δ = 2 . , R b = 1 . ), (b)nematic ( δ = 3 . , R b = 1 . ), (c) staggered ( δ = 3 . , R b = 2 . ), and (d) string( δ = 4 . , R b = 1 . ) orders. The dashed white hexagon marks the first Brillouinzone of the kagome lattice. The structure factor for the string phase is computed onthe cylindrical geometry shown in Fig. 2(e). patently ordered with the system favoring a configuration ofstraight strings that wrap around the cylinder, thereby liftingthe macroscopic classical degeneracy. This is in contrast tothe expectation from naive second-order perturbation theory,which picks out the maximally kinked classical state. Signatures of density-wave orders
In totality, we have thus detected four solid phases on thekagome lattice. All these ordered states can be identified fromeither their respective structure factors, or the relevant orderparameters, as we now show.With a view to extracting bulk properties, in the following,we work with the central half of the system that has an effectivesize of N c = 3 N . Evidence for ordering or the lack thereofcan be gleaned from the static structure factor, which is theFourier transform of the instantaneous real-space correlationfunction S ( q ) = 1 N c X i,j e i q · ( x i − x j ) h n i n j i [3]with the site indices i , j restricted to the central N × N re-gion of the cylinder. At a blockade radius of R b = 1 .
7, whichstations one in the nematic phase [Fig. 3(b)], the structurefactor has pronounced maxima at the corners of the (hexag-onal) extended Brillouin zone, occurring at Q = ± b , ± b , ± ( b + b ), where b = ( π, − π/ √
3) and b = (0 , π/ √
3) arethe reciprocal lattice vectors. A subset of these maxima alsopersists for the stripe phase [Fig. 3(a)]—this is in distinctionto the nematic phase wherein the peaks at all six recipro-cal lattice vectors are of equal strength. In the presence of | taggered ordering [Fig. 3(c)], the peaks are comparativelyweaker but prominent nonetheless, appearing at Q = ± b , ± ( π/ , √ π/ ± (3 π/ , π/ (4 √ ± ( − π/ , π/ (4 √ ± b / C -rotated copies of the above.One can also directly look at the order parameters thatdiagnose the possible symmetry-broken ordered states. Forthe nematic phase, an appropriate definition isΦ = 3 N c X i ∈ A n i + ω X i ∈ B n i + ω X i ∈ C n i ! , where ω ≡ exp(2 πi/
3) is the cube root of unity, and A, B, Cdenote the three sublattices of the kagome lattice. Similarly, inthe staggered and string phases, one can define the (squared)magnetic order parameter M Q ≡ S ( Q ), with Q chosen fromamong the observed peaks of the structure factor. These orderparameters are more quantitatively addressed in Fig. 7(a),which catalogs the ground-state properties calculated at afixed detuning of δ = 3 . Mapping to triangular lattice quantum dimer models
At large detuning, we can approximately map the Rydberg sys-tem to a model of hard-core bosons at filling f on the kagomelattice. The bosonic system (21, 45–49) has an extra U(1)symmetry, which can be spontaneously broken in a superfluidphase; in the Rydberg model without the U(1) symmetry, thedisordered phase is the counterpart of the superfluid. However,any nonsuperfluid topological states of the boson model areinsensitive to the U(1) symmetry, and can also be present inthe Rydberg model.In the limit of strong interactions, hard-core bosons atfilling f = (1 / , / , /
6) on the kagome lattice map (21, 45–47) onto an (odd, even, odd) quantum dimer model (QDM)(50–52) on the medial triangular lattice with N d = (3 , , N d .The triangular lattice of the QDM is formed by joining thecenters of the kagome hexagons, and this correspondence issketched in Fig. 4, which schematically shows the mappingbetween the different Rydberg solids and the phases of theQDM. A key observation here is that both solid phases nextto the liquid regime (marked by the star in Fig. 1) are also phases of the QDM: the nematic phase was found in the QDMwith N d = 2 by Roychowdhury et al. (21), and the staggeredphase is present in the QDM with N d = 1 (50, 51). In bothcases, a Z spin liquid phase with topological order has beenfound adjacent to these solid phases (21, 50, 51) in the QDMs.Making the reasonable assumption that a QDM descriptionfor the Rydberg system holds in the vicinity of the phaseboundaries of these solid states, we expect Z topologicalorder in the liquid regime of the Rydberg model in Fig. 1,proximate to the nematic and staggered solid phases.There is a subtle difference between the Z spin liquidsfound in the N d = 1 , Fig. 4. Correspondence between the Rydberg and quantum dimer models.
Inthe limit of large detuning, the Rydberg excitations can be mapped to a system ofhard-core bosons, upon identifying each atom in state | r i ( | g i ) as an occupied(empty) bosonic mode (27). (a) The resultant boson model is at a filling of f = 1 / ( / ) for the nematic (staggered) phase. A boson on any site of the kagome lattice(red/green points) can now be uniquely associated with a dimer on the corresponding bond of the medial triangular lattice (blue lines) (21). The liquid regime is separatedfrom the nematic (staggered) phase by a continuous (first-order) QPT. (b) Schematicdepiction of a potential Rydberg liquid as a superposition of dimer configurations;note that, unlike in the QDM, the total number of dimers can fluctuate in the Rydbergmodel. picks up a Berry phase of π (2 π ) upon adiabatic transportaround a site of an odd (even) QDM. (2, 50, 51, 53–60). Thisdistinction changes the projective symmetry group of thevisons, and also holds for the Z spin liquids expected in theRydberg model, which must therefore be odd/even as well.Consequently, the spin liquids proposed to be proximate tothe staggered and nematic phases are not identical; one orboth of them could be present in the liquid regime. Moreover,the vison Berry phase places important constraints on the non-topological states obtained by condensing visons: for instance,an odd Z spin liquid cannot have a vison-condensing phasetransition to a trivial “disordered” state with no broken latticesymmetry, which is a manifestation of the Lieb-Schultz-Mattistheorem.Roychowdhury et al. (21) studied the transition from theeven Z spin liquid into the nematic phase. The visons inthis Z spin liquid have an energy dispersion with minima at M = ( π/ , − π/ (2 √ b / M = (0 , π/ √
3) = b / b , b [Fig. 3(b)]. The criticaltheory for this transition is an O(3) Wilson-Fisher theory withcubic anisotropy (21), and this conclusion holds both for theQDM and the Rydberg system. It is interesting to comparethis result to that for the transition from the nematic phaseto the disordered phase of the Rydberg model, which wasmentioned above to be in the universality class of the (2+1)Dthree-state Potts model and hence, weakly first-order (39).Therefore, the nematic phase can melt either by a first-ordertransition to a trivial disordered phase, or via a second-orderone into a topological phase by fractionalizing the nematicorder parameter. So, the observation of a continuous O(3)transition out of the nematic phase to a phase without sym-metry breaking would constitute nontrivial evidence for thepresence of Z topological order in the latter. An apparent PNAS |
November 26, 2020 | econd-order transition in the nematic order parameter canbe seen in Fig. 7(b) below, although our numerical accuracyis not sufficient to determine its universality class.The transition from the staggered phase to the odd Z spinliquid of the N d = 1 QDM is first-order (50, 51), and we expectit to be so for the Rydberg model too. This is compatiblewith the rapid increase of the staggered order parameter outof the liquid regime shown in Fig. 7(b). We also note that thedensity of Rydberg excitations in the liquid regime ( ∼ .
2) isclose to that of the odd QDM ( f = 1 / Z spin liquids proposed forthe liquid regime of the Rydberg model, there should be asharp transition to the disordered phase described by thecondensation of the bosonic e anyons. Such a transition isnot present in the QDMs, because the e excitations have beenprojected out by the dimer constraint. This QPT is in theuniversality class of the Ising ∗ Wilson-Fisher conformal fieldtheory (61–63), and can, in principle, be accessible in oursystem. However, we do not find clear-cut numerical evidencefor it below, for our range of system sizes.Extending the mapping from the Rydberg model to theQDM further, in Sec. III of the SI, we compute the parametersin ( δ, R b )-space where a QSL phase might be expected toexist for the Rydberg system based on the (previously known)regime of stability of the QDM spin liquid (50, 51). Thiscalculation leads to an estimate of ( δ = 2 . R b = 1 . The liquid regime
At moderately large values of the detuning, we find an in-termediate correlated regime—designated by the red star inFig. 1(e)—which lies in between two solid phases but resistscategorization as either. The nomenclature “liquid”, as de-fined earlier, connotes that the Rydberg excitations form adense state in which the blockade introduces significantly morecorrelations than in the disordered regime. Prompted by theconsiderations described in the previous section, we first at-tempt to uncover the existence of any phase transitions inthe vicinity of this regime. To that end, we temporarily focuson a specific blockade radius, R b = 1 . E , with respectto the detuning, i.e., χ = − ∂ E /∂ δ . On finite systems, themaxima of the susceptibility can often be used to identifypossible QCPs, which are slightly shifted from their locationsin the thermodynamic limit. In particular, for R b = 1 . χ is plotted in Fig. 5(a), where a single peak in the responseis visible at approximately δ = 2 .
9. This susceptibility peak—which is recorded by the pink circles in Fig 1(e)—is alsoreproduced in exact diagonalization calculations on a 48-sitetorus (refer to Sec. IV of the SI).A similar signature can be discerned in the quantum fi-delity |h Ψ ( δ ) | Ψ ( δ + ε ) i| (64, 65), which measures the overlapbetween two ground-state wavefunctions Ψ computed at pa-rameters differing by ε . The fidelity serves as a useful tool instudying QPTs because, intuitively, it quantifies the similaritybetween two states, while QPTs are necessarily accompaniedby an abrupt change in the structure of the ground-state wave-function (66). Zooming in on a narrower window around the (a) (b)(c) Fig. 5. Signatures of a crossover into the liquid regime.
Along a line where theblockade radius is held constant at R b = 1 . , both (a) the susceptibility χ and (b)the fidelity susceptibility F exhibit a single peak at δ ≈ . . (c) The behavior of theEE over the same detuning range, however, is distinct from the sharp drop observedacross the QPTs into any of the ordered phases. On going to higher δ , the systemeventually transitions into either the nematic or the string phase depending on theblockade radius (or potentially, the boundary conditions). susceptibility peak, we evaluate the fidelity susceptibility (67),which, in its differential form, is given by F ≡ (cid:20) − |h Ψ ( δ ) | Ψ ( δ + ε ) i| ε (cid:21) . [4]The fidelity susceptibility also displays a local maximum at δ ≈ .
9, indicating some change in the nature of the groundstate as we pass into the liquid regime. Unlike the QPTs intothe ordered phases, the EE [Fig. 5(c)] does not drop as wecross this point but rather, continues to increase; however, itsfirst derivative is nonmontonic at δ ≈ .
9. This suggests thatthe final liquid state is likely highly entangled, and is not asimple symmetry-breaking ground state.Given that we always work on cylinders of finite extent, wecannot exclude the possibility that the peaks in Figs. 5(a,b)are due to surface critical phenomena (68, 69) driven by aphase transition at the edge. Indeed, in Fig. 6(a), whichshows a profile of the liquid regime on a wide cylinder at δ = 3 . R b = 1 .
95, we notice that the edges seek to precipitatethe most compatible density-wave order at these fairly largevalues of the detuning. Nonetheless, the bulk resists anysuch ordering tendencies and the central region of the systemremains visibly uniform, with only slight perturbations fromthe open boundaries. In fact, the bulk fails to order despite being at a detuning for which the system energetically favorsa maximal (constrained) packing of Rydberg excitations, as isalso evidenced by the nearby staggered and nematic phasesabove and below the liquid regime, respectively. It is perhapsworth noting that in one spatial dimension, the comparableregions lying between the different Z n -ordered states at largedetuning are known to belong to a Luttinger liquid phase (70).In order to eliminate end effects, it is often useful to first | (a)(b) (c) Fig. 6. Disentangling bulk and boundary behaviors. (a) Within the liquid regime—depicted here at δ = 3 . , R b = 1 . —the real-space magnetization profile com-municates the absence of density-wave order; note that the edge-induced orderingdoes not permeate into the bulk, which remains uniform. (b,c) Bulk susceptibilities:by construction, χ b should be insensitive to edge effects. In the left panel (b), χ b is determined from the second derivative of the difference between the energies oftwo YC6 cylinders with lengths N = 12 and N = 9 . As in Fig. 5(a), with R b set to . , a clear local maximum appears at δ ≈ . , heralding the liquid regime. OnYC8 cylinders (c), the bulk susceptibility, shown here along R b = 1 . , is calculatedby applying the subtraction method to two systems of lengths N = 12 and N = 8 . evaluate the ground-state energy per site for an infinitelylong cylinder by subtracting the energies of finite cylindersof different lengths but with the same circumference (31, 35,71, 72). Such a subtraction scheme cancels the leading edgeeffects, leaving only the bulk energy of the larger system. Inparticular, this procedure enables us to quantify the influenceof the boundaries on thermodynamic properties of the systemsuch as the susceptibility. Using two cylinders of fixed width,an estimate of the bulk energy can be found by subtractingthe energy of the smaller system from that of the larger. The(negative of the) second derivative of this quantity with respectto the detuning defines the bulk susceptibility χ b —this givesus the susceptibility in the center of the cylinder with minimaledge effects. Figure 6(b) presents the variation of χ b withdetuning at R b = 1 .
90 for the YC6 family: we see that thelocal maximum of the susceptibility reported in Fig. 5(a) isstill identifiable, but its precise location is shifted to slightlyhigher δ . Analogously, we study the bulk susceptibility forwider YC8 cylinders and find, once again, a distinct peakcorresponding to the onset of the liquid regime. Although thispeak persists in a purely bulk observable, its magnitude isdiminished: for example, the relative change between the localmaximum and the minimum (shoulder) immediately adjacentto it on the right (left) differs by approximately a factor offour (ten) between χ b and χ for the YC8 cylinder. Hence, thebehavior of the susceptibility could be indicative of an edgephase transition but whether this is accompanied by, or dueto, a change in the bulk wavefunction is presently unclear.Next, we investigate the properties of this liquid regime inmore detail and demonstrate that—as preempted by its name— it does not possess any long-range density-wave order. Thisdiagnosis of liquidity is best captured by the static structurefactor. In stark contrast to the panels in Fig. 3, S ( q ) isfeatureless within the liquid regime [Fig. 7(a)] with the spectralweight distributed evenly around the extended Brillouin zone.This unordered nature is reflected in [Fig. 7(b)], wherewe plot the order parameters characterizing the surroundingsymmetry-broken states. The order parameters defined earlierare found to be nonzero in both the nematic and staggeredphases but are smaller by an order of magnitude in the liquidregime; this is compatible with a vanishing | Φ | and M Q in the thermodynamic limit. In the process, we also findthat the transition from the nematic (staggered) phase to theliquid regime appears to be second-order (first-order), which isconsistent with the expectations for the QPT into a Z QSL inthe dimer models, as we have discussed in the previous section.We do not observe any signatures of a phase transition within the liquid regime.Moreover, one can also define a correlation length from thestructure factor as (73) ξ ( Q , q min ) = 1 | q min | r S ( Q ) S ( Q + q min ) − , [5]where Q + q min is the allowed wavevector immediately adjacentto the peak at Q . The correlation lengths obtained in theliquid are found to be smaller than the lattice constant, asplotted in Fig. 7(c), thus highlighting the lack of order. Thequalitative behavior of ξ is the same along both directionson the cylinder and mirrors that of the order parameter. Oneither side of the liquid region, the correlation lengths followan increasing trend as long-range order develops deep in thesolid phases. We have further verified that the bond-bondcorrelation functions C ( i,j ) , ( k,l ) = 4[ h ( n i · n j )( n k · n l ) i − h n i · n j ih n k · n l i ] [6]are also short-ranged in the liquid regime.So far, our numerics point to a gapped (see Fig. S3), dis-ordered candidate for the ground state of the liquid regime—these properties are all consistent with the behavior expectedfor a Z QSL, so it is natural to ask whether this region po-tentially harbors a topological phase. Although QSLs havelong been fingerprinted by what the states are not i.e., by the absence of ordering, more recently, it has been understood thatthe essential ingredient for a QSL is the presence of massivequantum superposition leading to an anomalously high degreeof entanglement (12, 74). Accordingly, we search for positiveindications of a QSL in the liquid regime by calculating thetopological entanglement entropy (TEE) (75, 76) in Sec. II ofthe SI. For a QSL phase, the value of the TEE is universaland positive, representing a constant reduction to the area lawentropy. Importantly, the TEE arises entirely from nonlocalentanglement and is topological in origin. While we do findindications of an enhanced long-range entanglement entropy(Fig. S4), this does not serve as conclusive evidence for a Z QSL as a finite γ ∼ ln 2 has also been documented for a valencebond solid in a different model (71). Additionally, the TEEcan suffer from strong finite-size effects on cylinders, leading tofalse signatures, and thus, cannot always reliably distinguishbetween different quantum phases (77). PNAS |
November 26, 2020 | (a) (b) (c) Fig. 7. Properties of the liquid regime.
Here, we focus on the line δ = 3 . while varying the blockade radius. (a) The structure factor S ( q ) , at R b = 1 . , is featureless withno discernible ordering peaks. (b) The order parameters | Φ | and M Q characterizing the nematic and staggered phases, respectively. Both develop a clear trough in the liquidregion between the two phases, indicating the lack of symmetry-breaking order therein. (c) The correlation lengths calculated from Eq. (3) : for . ≤ R b ≤ . , ξ is smallerthan one lattice spacing, so all correlations are short-ranged. Discussion and outlook
Based on numerical and theoretical analyses, we showed thatthe kagome lattice Rydberg atom array constitutes a promisingplatform for studying strongly correlated phenomena thatsupports not only a rich variety of quantum solids, but also,potentially, a highly entangled liquid regime. We arguedthat the liquid region could host a state corresponding toan elusive phase with topological order using its placementin the global phase diagram of triangular lattice quantumdimer models, and theories of their quantum phase transitions.Our numerical study examined a number of signatures of thepossible topological order and its associated phase transitions;although none of these computations conclusively confirm theexistence of a topological phase for the available system sizes,they collectively point to interesting physics that merits furtherinvestigation.This work can be extended in several directions. As DMRGis neither an unbiased method nor free from finite-size effects,it would be worthwhile to more completely quantify theseuncertainties in future theoretical works, and definitively es-tablish the nature of the liquid regime. A number of extensionsto the present model can also be envisioned, e.g., by utilizingvarious atom arrangements as well as multiple hyperfine sub-levels or Rydberg atomic states to probe a variety of quantumentangled phases.Furthermore, we expect the phase diagram in Fig. 1(e) toserve as a valuable guide to the detailed experimental studiesof frustrated systems using Rydberg atom arrays. Specifically,both solid and liquid regimes can be reached starting from atrivial product ground state by adiabatically changing the laserdetuning across the phase transitions, as was demonstratedpreviously (15). For experiments with N ∼ O (10 ) Rb atomscoupled to a 70S Rydberg state, the typical Rabi frequenciesinvolved can be up to (2 π ) ×
10 MHz. With these drivingparameters, sweeps over the detuning range 0 ≤ δ (cid:46) R b /a (cid:46) .
5, have already beenachieved in one-dimensional atom arrays (78–80). Hence,coherently preparing all the different many-body ground statesand observing their fundamental characteristics should bewithin experimental reach in two-dimensional systems as well.While the solid phases can be detected directly by eval-uating the corresponding order parameter, the study of anypossible QSL states in the liquid regime (or more generally, onRydberg platforms) will require new approaches. In particular, measuring the statistics of microscopic state occupations (15)or the growth of correlations (18) across reversible QPTs couldprove to be informative. In a Rydberg liquid, one can thinkof creating and manipulating topologically stable excitations,which cannot disappear except by pairwise annihilation witha partner excitation of the same type. The excitation typeswould correspond to the three nontrivial anyons of the Z spin liquid, and each should manifest as a characteristic local(and stable) “lump” in the density of atoms in the excitedRydberg state; interference experiments between such exci-tations could be used to scrutinize braiding statistics. Thedynamic structure factor can also provide signatures of frac-tionalization: dispersive single-particle peaks will be observedin the disordered phase, while a two-particle continua wouldappear in a region with Z topological order. Detailed studyof such spectra could yield the pattern of symmetry fraction-alization (48, 52, 81). Other directions include more directmeasurements of the topological entanglement entropy (82, 83).Finally, classical and quantum machine learning techniques(84–87) could be useful for measuring nonlocal topologicalorder parameters associated with spin liquid states. Materials and Methods
The DMRG calculations were performed using the ITensor Library(88). Further numerical details are presented in Sec. I of the SI.
ACKNOWLEDGMENTS.
We acknowledge useful discussions withSubhro Bhattacharjee, Meng Cheng, Yin-Chen He, Roger Melko,Roderich Moessner, William Witczak-Krempa, Ashvin Vishwanath,Norman Yao, Michael Zaletel, and especially the team of DolevBluvstein, Sepehr Ebadi, Harry Levine, Ahmed Omran, AlexanderKeesling, and Giulia Semeghini. The authors are grateful to MarcusBintz and Johannes Hauschild for pointing out the possibility ofan edge transition and sharing their results. We also thank AdrianE. Feiguin for benchmarking the ground-state energies observedin our DMRG calculations. R.S. and S.S. were supported by theU.S. Department of Energy under Grant DE-SC0019030. W.W.H.,H.P. and M.D.L. were supported by the U.S. Department of Energyunder Grant DE-SC0021013, the Harvard–MIT Center for UltracoldAtoms, the Office of Naval Research, and the Vannevar Bush FacultyFellowship. W.W.H. was additionally supported by the Gordon andBetty Moore Foundation’s EPiQS Initiative, Grant No. GBMF4306,and the NUS Development Grant AY2019/2020. The computationsin this paper were run on the FASRC Cannon cluster supported bythe FAS Division of Science Research Computing Group at HarvardUniversity.
Note Added:
Another work which will appear in this arXiv postingstudies the quantum phases of Rydberg atoms but in a different | rrangement, where atoms occupy links of the kagome lattice (89).
1. XG Wen, Colloquium: Zoo of quantum-topological phases of matter.
Rev. Mod. Phys. ,041004 (2017).2. S Sachdev, Topological order, emergent gauge fields, and Fermi surface reconstruction. Rep.Prog. Phys. , 014001 (2019).3. AY Kitaev, Fault-tolerant quantum computation by anyons. Ann. Phys. , 2–30 (2003).4. HL Stormer, DC Tsui, AC Gossard, The fractional quantum Hall effect.
Rev. Mod. Phys. ,S298–S305 (1999).5. J Nakamura, S Liang, GC Gardner, MJ Manfra, Direct observation of anyonic braiding statis-tics. Nat. Phys. , 931–936 (2020).6. N Read, S Sachdev, Large- N expansion for frustrated quantum antiferromagnets. Phys. Rev.Lett. , 1773–1776 (1991).7. XG Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders. Phys. Rev. B , 2664–2672 (1991).8. TH Han, et al., Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferro-magnet. Nature , 406–410 (2012).9. M Fu, T Imai, TH Han, YS Lee, Evidence for a gapped spin-liquid ground state in a kagomeHeisenberg antiferromagnet.
Science , 655–658 (2015).10. Z Feng, et al., Gapped Spin-1/2 Spinon Excitations in a New Kagome Quantum Spin LiquidCompound Cu Zn(OH) FBr.
Chin. Phys. Lett. , 077502 (2017).11. L Balents, Spin liquids in frustrated magnets. Nature , 199–208 (2010).12. L Savary, L Balents, Quantum spin liquids: a review.
Rep. Prog. Phys. , 016502 (2016).13. H Weimer, M Müller, I Lesanovsky, P Zoller, HP Büchler, A Rydberg quantum simulator. Nat.Phys. , 382 (2010).14. H Labuhn, et al., Tunable two-dimensional arrays of single Rydberg atoms for realizing quan-tum Ising models. Nature , 667 (2016).15. H Bernien, et al., Probing many-body dynamics on a 51-atom quantum simulator.
Nature , 579 (2017).16. R Samajdar, S Choi, H Pichler, MD Lukin, S Sachdev, Numerical study of the chiral Z quantum phase transition in one spatial dimension. Phys. Rev. A , 023614 (2018).17. S Whitsitt, R Samajdar, S Sachdev, Quantum field theory for the chiral clock transition in onespatial dimension. Phys. Rev. B , 205118 (2018).18. A Keesling, et al., Quantum Kibble–Zurek mechanism and critical dynamics on a pro-grammable Rydberg simulator. Nature , 207 (2019).19. R Samajdar, WW Ho, H Pichler, MD Lukin, S Sachdev, Complex Density Wave Orders andQuantum Phase Transitions in a Model of Square-Lattice Rydberg Atom Arrays.
Phys. Rev.Lett. , 103601 (2020).20. S de Léséleuc, et al., Observation of a symmetry-protected topological phase of interactingbosons with Rydberg atoms.
Science , 775–780 (2019).21. K Roychowdhury, S Bhattacharjee, F Pollmann, Z topological liquid of hard-core bosons ona kagome lattice at / filling. Phys. Rev. B , 075141 (2015).22. MC Bañuls, et al., Simulating lattice gauge theories within quantum technologies. Eur. Phys.J. D , 165 (2020).23. M Saffman, TG Walker, K Mølmer, Quantum information with Rydberg atoms. Rev. Mod.Phys. , 2313–2363 (2010).24. A Browaeys, D Barredo, T Lahaye, Experimental investigations of dipole–dipole interactionsbetween a few Rydberg atoms. J. Phys. B: At. Mol. Opt. Phys. , 152001 (2016).25. D Jaksch, et al., Fast quantum gates for neutral atoms. Phys. Rev. Lett. , 2208 (2000).26. MD Lukin, et al., Dipole Blockade and Quantum Information Processing in MesoscopicAtomic Ensembles. Phys. Rev. Lett. , 037901 (2001).27. S Sachdev, K Sengupta, SM Girvin, Mott insulators in strong electric fields. Phys. Rev. B ,075128 (2002).28. SR White, Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. , 2863 (1992).29. SR White, Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B ,10345 (1993).30. HC Jiang, ZY Weng, DN Sheng, Density Matrix Renormalization Group Numerical Study ofthe Kagome Antiferromagnet. Phys. Rev. Lett. , 117203 (2008).31. S Yan, DA Huse, SR White, Spin-liquid ground state of the S = 1 / kagome Heisenbergantiferromagnet. Science , 1173–1176 (2011).32. S Depenbrock, IP McCulloch, U Schollwöck, Nature of the Spin-Liquid Ground State of the S = 1 / Heisenberg Model on the Kagome Lattice.
Phys. Rev. Lett. , 067201 (2012).33. S Liang, H Pang, Approximate diagonalization using the density matrix renormalization-groupmethod: A two-dimensional-systems perspective.
Phys. Rev. B , 9214 (1994).34. SR White, AL Chernyshev, Neél order in square and triangular lattice Heisenberg models. Phys. Rev. Lett. , 127004 (2007).35. EM Stoudenmire, SR White, Studying two-dimensional systems with the density matrix renor-malization group. Annu. Rev. Condens. Matter Phys. , 111–128 (2012).36. P Nikoli´c, T Senthil, Theory of the kagome lattice Ising antiferromagnet in weak transversefields. Phys. Rev. B , 024401 (2005).37. T Pohl, E Demler, MD Lukin, Dynamical crystallization in the dipole blockade of ultracoldatoms. Phys. Rev. Lett. , 043002 (2010).38. S Sachdev,
Quantum Phase Transitions . (Cambridge University Press, New York), (2011).39. W Janke, R Villanova, Three-dimensional 3-state Potts model revisited with new techniques.
Nucl. Phys. B , 679–696 (1997).40. HC Jiang, Z Wang, L Balents, Identifying topological order by entanglement entropy.
Nat.Phys. , 902–905 (2012).41. D Huerga, S Capponi, J Dukelsky, G Ortiz, Staircase of crystal phases of hard-core bosonson the kagome lattice. Phys. Rev. B , 165124 (2016).42. R Moessner, KS Raman, Quantum Dimer Models , eds. C Lacroix, P Mendels, F Mila.(Springer Berlin Heidelberg, Berlin, Heidelberg), pp. 437–479 (2011).43. J Villain, R Bidaux, JP Carton, R Conte, Order as an effect of disorder.
J. Phys. France ,1263–1272 (1980).44. EF Shender, Antiferromagnetic garnets with fluctuationally interacting sublattices. Sov. Phys. JETP , 178 (1982).45. L Balents, MP Fisher, SM Girvin, Fractionalization in an easy-axis Kagome antiferromagnet. Phys. Rev. B , 224412 (2002).46. SV Isakov, YB Kim, A Paramekanti, Spin-Liquid Phase in a Spin-1/2 Quantum Magnet on theKagome Lattice. Phys. Rev. Lett. , 207204 (2006).47. SV Isakov, MB Hastings, RG Melko, Topological entanglement entropy of a Bose-Hubbardspin liquid. Nat. Phys. , 772–775 (2011).48. GY Sun, et al., Dynamical Signature of Symmetry Fractionalization in Frustrated Magnets. Phys. Rev. Lett. , 077201 (2018).49. YC Wang, M Cheng, W Witczak-Krempa, ZY Meng, Fractionalized conductivity at topologicalphase transitions. arXiv:2005.07337 [cond-mat.str-el] (2020).50. R Moessner, SL Sondhi, Resonating Valence Bond Phase in the Triangular Lattice QuantumDimer Model.
Phys. Rev. Lett. , 1881–1884 (2001).51. R Moessner, SL Sondhi, Ising models of quantum frustration. Phys. Rev. B , 224401(2001).52. Z Yan, YC Wang, N Ma, Y Qi, ZY Meng, Triangular Lattice Quantum Dimer Model Redux:Static and Dynamic Properties. arXiv:2007.11161 [cond-mat.str-el] (2020).53. RA Jalabert, S Sachdev, Spontaneous alignment of frustrated bonds in an anisotropic, three-dimensional Ising model. Phys. Rev. B , 686–690 (1991).54. S Sachdev, Kagome- and triangular-lattice Heisenberg antiferromagnets: Ordering fromquantum fluctuations and quantum-disordered ground states with unconfined bosonicspinons. Phys. Rev. B , 12377–12396 (1992).55. S Sachdev, M Vojta, Translational symmetry breaking in two-dimensional antiferromagnetsand superconductors. J. Phys. Soc. Jpn. Suppl. B (2000).56. T Senthil, MPA Fisher, Z gauge theory of electron fractionalization in strongly correlatedsystems. Phys. Rev. B , 7850–7881 (2000).57. R Moessner, SL Sondhi, E Fradkin, Short-ranged resonating valence bond physics, quantumdimer models, and Ising gauge theories. Phys. Rev. B , 024504 (2002).58. AM Essin, M Hermele, Classifying fractionalization: Symmetry classification of gapped Z spin liquids in two dimensions. Phys. Rev. B , 104406 (2013).59. Y Qi, L Fu, Detecting crystal symmetry fractionalization from the ground state: Application to Z spin liquids on the kagome lattice. Phys. Rev. B , 100401 (2015).60. M Zaletel, YM Lu, A Vishwanath, Measuring space-group symmetry fractionalization in Z spin liquids. Phys. Rev. B , 195164 (2017).61. AV Chubukov, T Senthil, S Sachdev, Universal magnetic properties of frustrated quantumantiferromagnets in two dimensions. Phys. Rev. Lett. , 2089–2092 (1994).62. M Schuler, S Whitsitt, LP Henry, S Sachdev, AM Läuchli, Universal Signatures of QuantumCritical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories. Phys. Rev. Lett. , 210401 (2016).63. S Whitsitt, S Sachdev, Transition from the Z spin liquid to antiferromagnetic order: Spectrumon the torus. Phys. Rev. B , 085134 (2016).64. M Cozzini, R Ionicioiu, P Zanardi, Quantum fidelity and quantum phase transitions in matrixproduct states. Phys. Rev. B , 104420 (2007).65. HQ Zhou, JP Barjaktareviˇc, Fidelity and quantum phase transitions. J. Phys. A: Math. Theor. , 412001 (2008).66. SJ Gu, Fidelity approach to quantum phase transitions. Int. J. Mod. Phys. B , 4371–4458(2010).67. WL You, YW Li, SJ Gu, Fidelity, dynamic structure factor, and susceptibility in critical phenom-ena. Phys. Rev. E , 022101 (2007).68. K Binder, DP Landau, Critical phenomena at surfaces. Phys. A , 17–30 (1990).69. HW Diehl, The Theory of Boundary Critical Phenomena.
Int. J. Mod. Phys. B , 3503–3523(1997).70. P Fendley, K Sengupta, S Sachdev, Competing density-wave orders in a one-dimensionalhard-boson model. Phys. Rev. B , 075106 (2004).71. Z Zhu, DA Huse, SR White, Weak Plaquette Valence Bond Order in the S =1 / Honeycomb J − J Heisenberg Model.
Phys. Rev. Lett. , 127205 (2013).72. Z Zhu, DA Huse, SR White, Unexpected z -Direction Ising Antiferromagnetic Order in a Frus-trated Spin- / J − J XY Model on the Honeycomb Lattice.
Phys. Rev. Lett. ,257201 (2013).73. AW Sandvik, Computational Studies of Quantum Spin Systems.
AIP Conf. Proc. , 135–338 (2010).74. T Grover, Y Zhang, A Vishwanath, Entanglement entropy as a portal to the physics of quan-tum spin liquids.
New J. Phys. , 025002 (2013).75. A Kitaev, J Preskill, Topological Entanglement Entropy. Phys. Rev. Lett. , 110404 (2006).76. M Levin, XG Wen, Detecting Topological Order in a Ground State Wave Function. Phys. Rev.Lett. , 110405 (2006).77. SS Gong, W Zhu, DN Sheng, OI Motrunich, MPA Fisher, Plaquette Ordered Phase andQuantum Phase Diagram in the Spin- J − J Square Heisenberg Model.
Phys. Rev. Lett. , 027201 (2014).78. M Endres, et al., Atom-by-atom assembly of defect-free one-dimensional cold atom arrays.
Science , 1024–1027 (2016).79. H Levine, et al., High-fidelity control and entanglement of Rydberg-atom qubits.
Phys. Rev.Lett. , 123603 (2018).80. A Omran, et al., Generation and manipulation of Schrödinger cat states in Rydberg atomarrays.
Science , 570–574 (2019).81. J Becker, S Wessel, Diagnosing Fractionalization from the Spin Dynamics of Z Spin Liquidson the Kagome Lattice by Quantum Monte Carlo Simulations.
Phys. Rev. Lett. , 077202(2018).82. R Islam, et al., Measuring entanglement entropy in a quantum many-body system.
Nature , 77–83 (2015).83. T Brydges, et al., Probing Rényi entanglement entropy via randomized measurements.
Sci-ence , 260–263 (2019).84. DL Deng, X Li, S Das Sarma, Machine learning topological states.
Phys. Rev. B , 195145(2017). PNAS |
November 26, 2020 |
5. Y Zhang, RG Melko, EA Kim, Machine learning Z quantum spin liquids with quasiparticlestatistics. Phys. Rev. B , 245119 (2017).86. J Carrasquilla, RG Melko, Machine learning phases of matter. Nat. Phys. , 431–434 (2017).87. I Cong, S Choi, MD Lukin, Quantum convolutional neural networks. Nat. Phys. , 1273–1278 (2019).88. M Fishman, SR White, EM Stoudenmire, The ITensor Software Library for Tensor NetworkCalculations. arXiv:2007.14822 [cs.MS] (2020).89. R Verresen, MD Lukin, A Vishwanath. arXiv, to appear (2020). | upporting Information for“Quantum phases of Rydberg atoms on a kagome lattice” Rhine Samajdar, Wen Wei Ho,
1, 2
Hannes Pichler,
3, 4
Mikhail D. Lukin, and Subir Sachdev Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Physics, Stanford University, Stanford, CA 94305, USA Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
The supplementary information presented here contains:I. Technical details of DMRG computations, including a comparison of our results between dif-ferent geometries that may or may not favor the solid phases breaking lattice symmetries.II. Data on the apparent long-range entanglement entropy detected on cylinders of finite length.III. Connection of the Rydberg Hamiltonian to previously studied models of hard-core bosons withring-exchange interactions that are known to harbor a spin liquid phase.IV. Exact diagonalization results on a 48-site torus confirming the existence of all the phases seenwith DMRG and examining the excitation spectrum above the ground state.
I. METHODS: DMRG IN TWO DIMENSIONS
Our numerical results in the main text are obtainedfrom large-scale simulations of the Rydberg Hamiltonianusing the density-matrix renormalization group (DMRG)[1–4], implemented with the ITensor library [5]. The re-markable success of DMRG is today understood to beattributable to an underlying matrix product structure,as the method operates on a particular class of quantumstates [6–8] of the form | Ψ i = X τ ,...,τ n X b ,...,b n − A τ b A τ b b A τ b b · · · A τ n b n − | τ , . . . , τ n i , where A denotes matrices with physical indices τ and linkindices b . The DMRG algorithm finds the optimal matrixproduct state (MPS) representation of the many-bodyground state in this variational space of wavefunctions.While originally formulated as a tool for studyingstrongly correlated one-dimensional quantum systems,DMRG can be extended to two dimensions by mappingthe 2D lattice on to a 1D chain with longer-range inter-actions. Since open boundaries act as effective pinningfields [9, 10] for the Rydberg excitations, the bulk prop-erties of the model are (ideally) best studied on a torus.However, the imposition of fully periodic boundary con-ditions requires squaring the number of states needed fora given accuracy [11], so we instead place the system ona cylinder, with open boundaries along a but periodicones along a . A. Lattice geometry
To avoid spurious effects due to sharp edges, follow-ing Refs. [12–14], we work with a geometry such thateach unit cell at the right boundary of the cylinder con-tains only two sites. The resultant lattice, labeled as YC (2 N ), has a noninteger number of unit cells, with atotal of N × (3 N + 2) sites. Such a geometry has theadded advantage of stabilizing the various ordered states.This implies that within the liquid regime detected inFig. 1(e), the proximate solid orders are unstable despite being explicitly favored, which constitutes further evi-dence against an ordered ground state.In Fig. S1, we demonstrate that our results are qual-itatively the same on lattices with and without an in-teger number of unit cells, and that a featureless liquidstate exists in both cases. The specific density-wave pro-files in the solid phases, however, may be sensitive tothe jagged edges and can differ from those illustrated inthe main text [Figs. 1(b–d)], as typified by the nematicorder in Fig. S1(a): this is because the boundaries seedtwo distinct symmetry-broken configurations from eitherend, which necessarily merge in the center of the system,forming a domain wall. B. Convergence
For the DMRG calculations, our protocol entails firstcarrying out a large number of sweeps at relatively smallbond dimensions ( d ∼ d progres-sively at later stages. At each diagonalization step, weallow for up to six iterations of the Davidson algorithmto facilitate proper convergence. To assist the build-up oflong-range correlations, it is useful to initially add a small“noise” term [15] to the density matrix, which is turnedoff in subsequent sweeps. Recognizing that DMRG is avariational algorithm [16], we specifically ensure that thecalculation is not stuck in a metastable state [17, 18] bychecking for convergence with respect to both bond di-mension and number of sweeps. For instance, in Fig. S2,we show the behavior of ground-state properties as afunction of d for the largest of the cylinders considered D L (a) h n i i (b) (c)(d) FIG. S1.
Ground states on kagome lattices with an in-teger number of unit cells. (a) Nematic ordered state on a8 × δ = 3 . R b = 1 .
7, formed by a superposition oftwo competing C -symmetry-broken states nucleated by theedges. (b) The susceptibility, at R b = 1 .
9, exhibits a distinctmaximum occurring at the same position as for the YC8 cylin-ders in Fig. 5(a). D and L indicate the trivial disordered andcorrelated liquid regimes, respectively. The von Neumann en-tanglement entropy (inset) monotonically increases over thesame range, in exact correspondence with Fig. 5(c). Staticstructure factors at δ = 3 . R b = 1 .
7, and (d) R b = 1 . S ( q ) shows conspicuous ordering peaks for the former, but isfeatureless for the latter. in this work (YC12), at a representative point in theliquid regime ( δ = 3 . R b = 1 . O (10 − ). C. Energy gaps
Along with the ground state | ψ i , a protocol similar tothat described above can be used to also find the excitedstates, although it is computationally more expensive todo so. We can target the first excited state using theHamiltonian H = H Ryd + wP , where P = | ψ ih ψ | is aprojection operator and w is an energetic penalty. Such - - - - FIG. S2.
Convergence of DMRG calculations.
For theYC12 geometry, plotted here are the ground-state energy ( E ,blue) and the entanglement entropy on the central bond ( S ,green) as a function of the bond dimension d at a genericpoint in the liquid regime ( δ = 3 . R b = 1 . < . E upon increasing d from 2400 to 3200. a computation leads to Fig. S3, where we notice that theenergy gap to the first-excited state found with DMRG,∆, is nonzero in the liquid region and its magnitude isconsistent with the values obtained from exact diagonal-ization calculations on a torus with the same width (seeSec. IV).Let us outline here the general expectations for the gapin a Z quantum spin liquid (QSL) phase. As opposedto the fourfold degeneracy on a torus, a Z QSL wouldonly be twofold degenerate on a cylindrical system—thisis because the ground-state energy in two of the sec-tors involves an additional cost due to the presence ofa pair of anyons localized at the ends of the cylinder [19].However, DMRG preferentially converges to one of thequasidegenerate ground states and, in particular, to aminimally entangled state (MES). This well-recognizedentanglement barrier between different MESs [20] ac-counts for the apparent absence of a topological degen-eracy in several numerical studies [10, 21–24]. The split-ting between the two quasidegenerate ground states (on acylinder) belonging to different topological sectors scalesas ∼ L h e − cL v , and is likely to be orders-of-magnitudesmaller than ∆ [23]. For the Rydberg model, it remainsto be seen whether all the four topological sectors of a Z QSL can be constructed in the liquid regime by threadinganyon lines through an infinite cylinder [25].Note that in the ordered phases, we see that the gap isnearly zero, which is a numerical indicator of the groundstate being degenerate [23]. Thus, the observation of afinite gap in the intermediate region rules out any typeof symmetry-breaking order there, including all possibil-ities not considered explicitly via the correlation func-tions [23]. However, a nonzero ∆ does not distinguishbetween a trivial disordered and a potential topologicalliquid phase, and could also arise from edge excitations.
FIG. S3.
Energy gap to the first-excited state detectedwith DMRG.
Along the line δ = 3 .
3, ∆ is nonzero over theintermediate range of R b corresponding to the liquid regimebetween the nematic (left) and staggered (right) orders. Inset:the gap remains nonvanishing on several cylinders of differ-ent circumferences and lengths, so the liquid is demonstrablygapped in the thermodynamic limit. II. EXPLORING TOPOLOGICAL ORDER
For a gapped system in 2D, assuming that the bound-ary, of length L , between two subsystems is smooth (i.e.,devoid of corners), the corresponding von Neumann en-tanglement entropy obeys an “area” law with a potentialconstant subleading correction S vN = αL − γ + O (1 /L ) , (S1)where α is a model-specific nonuniversal coefficient, and γ is the topological entanglement entropy (TEE). In aphase without topological order, γ is trivially zero. For aQSL phase, however, the value of γ is universal and pos-itive, representing a constant reduction to the area lawentropy. The TEE arises entirely from nonlocal entan-glement and is topological in origin: in fact, it is knownthat γ = ln D with D = pP λ d λ , where d λ ≥ λ [26]. Hence, γ probes the anyon content of the topological order, so D > γ >
0) naturally implies that the state supportsfractionalized excitations.Following the prescription proposed by Jiang et al. [22, 23], one can read off γ by placing the system on infi-nite cylinders with varying circumferences L v ≡ N , andextrapolating the EE [27] per the scaling form of (S1).In practice, this procedure is efficient whenever all cor-relation lengths are much shorter than the width of thecylinder [28]—as is the case in Fig. 7(b). As pointed outby Refs. [22, 23], a nonzero γ can potentially arise fromtwo distinct sources: a symmetry-breaking contributionthat enhances the total EE, and a topological piece whichreduces it. However, the former, which arises from globalentanglement of the entire system, can be eliminated byincreasing the length L h of the system at fixed width L v so that the DMRG algorithm converges to an MESamongst the manifold of ground states that are degener-ate in the limit of infinite system size [29]. On the kagomelattice, the infinite-cylinder limit ( L h = ∞ ) is believed tobe well approximated when L h > L v [30]. Using the thusobtained entropies, we compute the TEE for differentpoints along the line R b = 1 .
95. At δ = 0 .
5, where thesystem is in the trivial disordered phase, we find γ ≈ δ = 3 . γ = 0 .
64. Although thisvalue is comparable to the theoretically known TEE ofln 2 for a Z spin liquid [31, 32], it should not be inter-preted as firm evidence for a QSL phase. This is becausestrong finite-size effects on cylinders [33] are known tooften produce spurious contributions to the TEE sincethe “replica” length scale over which γ converges can bearbitrarily larger than the physical two-point correlationlength [28]. - (a)(b) FIG. S4.
Subleading corrections to the area law in theliquid regime. (a) Plotting the variation of the TEE with δ at fixed R b = 1 .
95, we see that γ vanishes in the disorderedphase and starts to deviate from zero in the vicinity of thepreviously determined susceptibility maximum. Inset: linearextrapolation of the EE with the cylinders’ circumferencesaccording to (S1), demonstrating a nonzero and negative in-tercept in the liquid regime. (b) The long-range entanglemententropy grows with increasing δ along R b = 1 .
70, and changessign across the QPT into the nematic phase beyond δ = 2 . Beyond the precise value of γ —which is known to besensitive to numerical details [34] and challenging to es-timate accurately [33, 35–38]—we emphasize the posi-tive sign of the TEE as any discrete-symmetry-breakingstate, should yield a constant correction to the EE of opposite sign [23]. We verify the robustness of this ob-servation by plotting γ as a function of δ , for R b = 1 .
95, inFig. S4(a). The TEE remains zero for an extended rangeof δ and only starts to rise in a crossover region centeredaround the susceptibility maximum at δ = 3 .
2, calculatedin Fig. 5(a)[39]. The sign of γ , and more importantly, thecoincidence of the apparent onset of a nonzero TEE withthe peaks in χ and F determined earlier, are togethersuggestive of a possible transition into a topological QSLstate.Although γ does not seem to saturate to a constantvalue of ln 2 in Fig. S4(a), such variations in the numer-ical TEE, depending on the precise point studied in pa-rameter space, have been previously reported in the lit-erature for other spin-liquid candidates [33, 37] as well.In our case, this effect could stem from the proximity tothe string phase, which sets in at δ ∼ . γ , as defined byEq. (S1), is then a nonzero universal constant, which werefer to as the long-range entanglement entropy (LREE)since it is not of topological origin. The LREE is diffi-cult to determine analytically and the exact value of this geometric constant is not known even for the Ising QCP.To estimate the contribution of the LREE to the TEE inFig. S4(a), we compute γ for variable δ at R b = 1 . O (ln 2), leading us toconjecture that the net long-range entanglement entropyseen earlier in the liquid regime is not merely an artifactof the QCP alone. Admittedly, we cannot unambiguouslyrule out a scenario where there is a single direct QPTfrom the disordered phase to the string-ordered crystal,but we note the absence of any signal of a diverging corre-lation length in our finite-cylinder numerics, which wouldbe expected in the same regime as the enhancement ofthe LREE if that were the case. III. BOSON MODELS WITH RING EXCHANGE
In this section, we demonstrate that the RydbergHamiltonian on the kagome lattice can be related, atleast perturbatively, to certain well-studied models ofhard-core bosons with “ring-exchange” interactions [41–43], which have previously been identified to host a QSLphase.The mapping to hard-core bosons proceeds straightfor-wardly by associating each atom in the Rydberg (ground)state with the presence (absence) of a boson [44] on thecorresponding lattice site. In the bosonic language, H Ryd can be reformulated as H Ryd ≡ H + H ; H = V X h i,j i n i n j + V X hh i,j ii n i n j + V X hhh i,j iii n i n j − δ X i n i , H = Ω2 X i ( b † i + b i ) , (S2)where b † i ( b j ) is the boson creation (annihilation) operator, n i = b † i b i is the number operator, and V i stands for therepulsion strength between i -th nearest-neighbors with V = 27 V = 64 V owing to the van-der-Waals nature of theinteraction. While this Hamiltonian does not conserve the total number of bosons, we first derive an effective Hamil-tonian that recovers the global U(1) symmetry broken by the ( b † i + b i ) terms in Eq. (S2). This is motivated byconsidering the limit of large positive detuning such that boson number is effectively conserved. Using the symbols α, β to label sectors with a fixed number of bosons, and m, n to denote states within each group, the matrix elementsof the effective Hamiltonian are given by [45] h m, α | H eff | n, α i = E m,α δ m,n + h m, α | H | n, α i + X l,β = α h m, α | H | l, β ih l, β | H | n, α i (cid:18) E m,α − E l,β + 1 E n,α − E l,β (cid:19) , (S3)where E is the (purely classical) energy of a given config-uration as determined by H alone. Let us now evaluateEq. (S3) term by term. Consider a second-order hoppingprocess where an existing boson on a given site, say i ,is annihilated first, followed by the creation of a boson on an adjacent site j . Crucially, owing to the Rydbergblockade, the hopping amplitude will be severely reducedif any of the three nearest-neighboring sites of j (besides i ) are occupied. Treating this effect probabilistically,we replace the energy denominators in Eq. (S3) by theirconfigurational averages (cid:28) E m,α − E l,β (cid:29) = (cid:28) E n,α − E l,β (cid:29) = 1 − δ P + 3 V − δ P (1 − P ) , (S4)neglecting terms with p V ( p >
1) in their denominators.Here, P is the probability of finding an unoccupied site;recognizing that the Rydberg liquid appears in proxim-ity to a phase with a filling fraction of one-third, we set P = 2 /
3. With this assumption, the leading-order matrixelements of H eff —from the hopping described above—are given by (Ω / (cid:18) − δ + 1 − δ (cid:19) P . (S5) Next, we consider the reverse process in which a bosonis first created on a given site and then an existing bo-son is annihilated on a neighboring site. Likewise, theapproximate matrix elements of H eff in this case are(Ω / (cid:18) δ − V + 1 δ − V (cid:19) P . (S6)Note that Eq. (S6) is already of order 1 /V and can beneglected in comparison to Eq. (S5) since V (cid:29) δ .Naively, the analysis above suggests that there are pro-cesses by which a particle can hop arbitrary distances,but these cancel between the contributions of the N − N +1-boson subspaces [45] at this order, and the onlysurviving hopping terms connects adjacent sites. Thisleads us to the effective Rydberg Hamiltonian H eff = − X h i,j i t (cid:16) b † i b j + H.c. (cid:17) + V X h i,j i n i n j + V X hh i,j ii n i n j + V X hhh i,j iii n i n j − δ X i n i , (S7)with t = Ω P / (4 δ ). The reason behind this formal manipulation is that it allows us to rewrite H eff = − t X h i,j i (cid:16) b † i b j + H.c. (cid:17) + 2 V X h i,j i n i n j + X hh i,j ii n i n j + X hhh i,j iii n i n j − δ X i n i + H def (S8)= − t X h i,j i (cid:16) b † i b j + H.c. (cid:17) + V X (cid:20)(cid:16) n − µ V (cid:17) − µ V (cid:21) + H def ≡ H b + H def , (S9)where H b only includes homogeneous interactions while H def can be viewed as a deformation thereof that en-compasses all the distance-dependent nonuniformity in H eff . In Eq. (S9), n is the number of particles in eachof the hexagons of the kagome lattice, µ = δ + 2 V isthe effective chemical potential, and V is a single short-range repulsion strength that we will specify shortly. Itis easy to see that for µ = (4 , , V , the second termof Eq. (S9) is minimized by having (1 , ,
3) bosons perhexagon respectively or equivalently, a filling fraction of f = (1 / , / , / H b , at half-filling, is known to exhibit a superfluid-insulator transi-tion at ( V /t ) c ≈ .
8, and the insulating phase is a topo-logically ordered Z Mott insulator [42]. However, atboth 1 / / Z spinliquid regime as shown by Ref. [43] following a mappingonto the triangular-lattice quantum dimer model. As theratio V /t has to exceed a certain critical value to ob-tain the QSL phase, one should compare V in H eff tothe smallest interaction scale in Eq. (S7); accordingly,we identify 2 V = V . Supplementing this equation withthe relation δ = 6 V and the derived expression for t , onecan easily solve for { δ/ Ω , V } . Roychowdhury et al. [43] showed that the parameter ranges realizing the spin liq-uids at 1 / / V /t ) c byIsakov et al. [42] in our calculation. Taking, for instance, V /t = 20 & ( V /t ) c , we find R b /a = 1 .
997 and δ/ Ω = 2 . IV. EXACT DIAGONALIZATION STUDIES
In this section, we supplement the DMRG simulationsof the main text with exact diagonalization (ED) stud-ies of the kagome lattice Rydberg Hamiltonian. WhileED techniques are restricted to system sizes smaller thanthose accessible with DMRG, they are completely unbi-ased and offer a complementary viewpoint as one is ableto probe features that are harder to extract using DMRG,such as spectral gaps to higher excited states as wellas the full distribution of the ground-state wavefunctionover the computational basis states. Furthermore, withED, one has the ability to impose arbitrary boundaryconditions such as toroidal ones, which help circumventedge effects but are challenging to handle with tensornetwork methods. We find that the ED numerics con-firm the existence of at least four phases (see Fig. S6), aswell as the natures of the ordered phases. However, wedo not exactly observe the fourfold near-degeneracy ofthe ground state that one would expect for a topological Z liquid—this is not surprising given the large finite-sizeeffects known to affect ED studies of spin systems on thekagome lattice [46, 47].We consider here a 48-site cluster of linear dimensions N = N = 4 with fully periodic boundary conditions suchthat an atom located at position r is identified with thoseat positions r + N µ a µ ( µ = 1 , ≈ . × , so we instead operate in the so-called “Rydberg-blockaded” space, where no two neighboring atoms onthe lattice are allowed to be simultaneously excited. Thisleads to an effective Hamiltonian H eff = X i Ω Y h i,j i (cid:0) − n j (cid:1) S xi − δ X i n i + X a< || x i − x j ||≤ a V ij n i n j , (S10)where S xi = ( | r i i h g | +H.c.) /
2, and V ij ≡ V (cid:0) || x i − x j || /a (cid:1) ;hereafter, we will set Ω = a = 1 as before. The first termdescribes a spin-flip in the blockaded space, and the rela-tion h i, j i in the projector specifies that sites i and j arenearest neighbors (NNs). In the last term, we sum overpairwise interactions of Rydberg atoms with mutual dis-tances corresponding to second- and third-NNs; the first-NN repulsion strength is formally infinite due to the hardblockade constraint. The distance between any two sitesis taken to be the shortest one on the torus. For R b & R b ∼ .
9, theeffective model should presumably bring out its existenceas well as its universal properties.It is useful to note that the Hamiltonian H eff is also in-variant upon translations in the a and a directions, andthe spectrum can therefore be decomposed into momen-tum sectors. For concreteness, let T and T represent theoperators implementing such shifts in the respective di-rections. Given that T N µ µ = 1 and N µ = 4, the eigenvaluesof T µ = e ik µ range across k µ = 2 πn µ / n µ = 1 , , , k , k ) = ( π/ , π/ π/ , π ), ( π, π ), (2 π, π ),and have Hilbert space dimensions of 7587799, 7587792,7590567, and 7590689, respectively. In total, six sectorsare equivalent to ( π/ , π/ π/ , π ), threeto ( π, π ), and one to (2 π, π ). This momentum resolutionis crucial for our ability to numerically treat a system of48 spins with ED. FIG. S5.
Spectral gaps of low-lying eigenstates.
UsingED, we trace the evolution of the gaps E i − E (in units ofΩ) for the first few states, with energies E i , of the effectiveHamiltonian (S10) as a function of R b at fixed δ = 3 . E . Note that thelowest blue curve is twofold degenerate. The region R b < . R b > . In Fig. S5, we display the low-lying gaps E i − E be-tween eigenstates with energy E i and the ground state,which has energy E , working at a fixed detuning, δ = 3 . R b < .
85, there are two eigenstates with minus-cule energy differences ( ∼ − in units of Ω) above theground state. This implies that we should consider thefirst three states of the system as belonging to a “ground-state manifold”. Indeed, it is easy to check that the de-composition of the ground-state wavefunction in termsof the computational basis states in the z -direction at R b = 1 . R b > . ∼ − – 10 − Ω). Inthis case, we predominantly find the classical configu-rations corresponding to staggered order in the wave-function decomposition, in alignment with the predic-tions from DMRG. However, for intermediate values ofthe blockade radius, where 1 . . R b . .
0, we notice amarkedly different spectra, clearly showing the presenceof an intervening phase.Prompted by the observations above, we now investi-gate the various phases realized on the 48-site kagomelattice in further detail. In parallel with the main text,we first plot, for three different values of the blockade ra-dius, the susceptibility χ = − ∂ E /∂ δ , which can serveas a convenient probe to hunt for quantum phase tran-sitions (QPTs). In the thermodynamic limit, the sus-ceptibility diverges at a quantum critical point but ona finite lattice, this divergence is inevitably rounded offand manifests itself as a local maximum of χ . Focusingon this diagnostic, in Fig. S6(b), we see a peak occurring ( a ) ( b ) ( c ) D L N D S L FIG. S6.
Susceptibilities observed in ED simulations .The variation of χ = − ∂ E /∂ δ with the detuning δ at aconstant R b = (a) 1 .
95, (b) 1 . .
00, brings outseveral noticeable local maxima, which could be indicativeof quantum phase transitions. The shorthand labels N , S ,and 2 / around δ ∼ . R b = 1 . δ is expected to be trivially disordered, we infer, albeitindirectly, that the region on the other side of the peakis possibly the correlated liquid regime—we will verifythis hypothesis shortly. We emphasize, however, that ascaling analysis of the height of the peak with systemsize is necessary to confirm whether this signal is due toa genuine QPT. At even larger δ , the system undergoestwo successive transitions into first the string phase andthereafter, the nematic.Figure S7 collates the momentum-resolved energyspectrum, the ground-state decomposition, and thestructure factor in each of the four regions identified fromthe susceptibility. Inspecting the ground-state wave-function at δ = 3 . δ = 5 .
25) and nematic ( δ = 6 .
00) phases seen inFigs. S7(c) and (d), respectively. One can also gen-erate representative classical configurations constitutingthe ground-state wavefunctions by sampling these dis-tributions: from such snapshots, we find that the mi-crostates in the liquid regime do not bespeak any par-ticular order, whereas for the string and nematic phases,the ideal density-wave ordering is readily visible in lo-cal patches. This liquidity is also reflected in the staticstructure factor, which is mostly uniform in Fig. S7(b)but develops prominent features for higher values of δ .Taken together, these three pieces of information indi-cate that the liquid state is not ordered. Finally, let usmention that the dimer-dimer correlator [Eq. (6)] is zeroby construction (since we work in the blockaded space),thereby ruling out any valence bond crystal phases.The energy spectra arrayed above present another in- dependent method to potentially distinguish between thedisordered and liquid regimes: for instance, in the zero-momentum sector, the first-excited state is always doublydegenerate in the former but unique in the latter. For atrivial paramagnet, the lowest-energy translationally in-variant excitation should be a superposition of spin-flips,which is a property that holds throughout the phase.Therefore, the change in the character of the excitationsbetween Figs. S7(a) and (b), conveyed by the differing k = 0 spectra, suggests that the two regimes could beof different natures. Another nontrivial distinction per-tains to which six states in the ( k , k ) sectors constitutethe lowest-lying ones (above the ground state at the Γpoint): in the disordered phase, these are the states inthe ( k , k ) = (0 , π ), and equivalent, sectors but, for theliquid, they belong to the ( k , k ) = (0 , π/
2) and associ-ated sectors.Similar considerations apply for the R b = 2 . E i − E , is very small at ∼ .
05 Ω.In the thermodynamic limit, three of these excited statescould, in principle, have energies that approach that ofthe ground state with the rest remaining gapped, therebyforming the expected ground-state manifold of a Z spinliquid wherein each of the four states corresponds to ananyon type in the theory. While we are unable to conclu-sively detect whether this scenario occurs based on ourED simulations, we note that such a drawback is alsopresent for several other ED studies [46–51] of kagomesystems such as the spin-1 / ( a )( b )( c )( d ) FIG. S7.
Excitation spectra and ground-state properties at R b = 1 . . Three of the chosen points in parameter spacebelong to the (a) disordered, (c) string, and (d) nematic phase, whereas (b) lies within the liquid regime. The leftmost columnshows the low-lying spectrum. If two states are degenerate within machine precision, they are plotted as concentric circles; forthe sake of visual resolution, if two levels are spaced less than 10 − apart, they are depicted as being split horizontally. Thecentral column illustrates the ground-state decomposition (GSD) in the zero-momentum sector of the state circled in red in theleft panels—the blue bars represent the probability for each classical configuration, indexed along the horizontal axis. In thesolid phases (c, d), the classical density-wave ordered configurations have the maximum weights ∼ O (10 − ); the magnitudes ofthese peaks are much larger than any in the disordered or liquid regimes. The presence or absence of ordering is also registeredin the associated static structure factors (right). ( a )( b )( c )( d ) FIG. S8.
Excitation spectra and ground-state properties at R b = 2 . . Here, we plot the same quantities as in Fig. S7,but for a higher blockade radius. On increasing detuning, the sequence of phases now encountered in going from the disorderedto the ordered side is slightly different: from the staggered phase (b), the system enters the liquid regime (c), followed bya transition into the string phase(d). This suggests a scenario where the tip of the staggered lobe bends downwards in thephase diagram and is briefly intersected by the R b = 2 . C -rotated copies of the latter. [1] S. R. White, “Density matrix formulation for quan-tum renormalization groups,” Phys. Rev. Lett. , 2863(1992).[2] S. R. White, “Density-matrix algorithms for quantumrenormalization groups,” Phys. Rev. B , 10345 (1993).[3] U. Schollw¨ock, “The density-matrix renormalizationgroup,” Rev. Mod. Phys. , 259 (2005).[4] U. Schollw¨ock, “The density-matrix renormalizationgroup: a short introduction,” Phil. Trans. R. Soc. A ,2643 (2011).[5] M. Fishman, S. R. White, and E. M. Stoudenmire, “TheITensor Software Library for Tensor Network Calcula-tions,” arXiv:2007.14822 [cs.MS] (2020).[6] I. P. McCulloch, “From density-matrix renormalizationgroup to matrix product states,” J. Stat. Mech. ,P10014 (2007).[7] F. Verstraete, V. Murg, and J. I. Cirac, “Matrix prod-uct states, projected entangled pair states, and varia-tional renormalization group methods for quantum spinsystems,” Adv. Phys. , 143 (2008).[8] U. Schollw¨ock, “The density-matrix renormalizationgroup in the age of matrix product states,” Ann. Phys. , 96 (2011).[9] S. R. White and D. J. Scalapino, “Checkerboard patternsin the t − J model,” Phys. Rev. B , 220506 (2004).[10] S. Yan, D. A. Huse, and S. R. White, “Spin-liquid groundstate of the S = 1 / , 1173 (2011).[11] E. M. Stoudenmire and S. R. White, “Studying two-dimensional systems with the density matrix renormal-ization group,” Annu. Rev. Condens. Matter Phys. , 111(2012).[12] Y.-C. He and Y. Chen, “Distinct Spin Liquids and TheirTransitions in Spin-1 / XXZ
Kagome Antiferromag-nets,” Phys. Rev. Lett. , 037201 (2015).[13] F. Kolley, S. Depenbrock, I. P. McCulloch,U. Schollw¨ock, and V. Alba, “Phase diagram ofthe J − J Heisenberg model on the kagome lattice,”Phys. Rev. B , 104418 (2015).[14] H.-C. Jiang, T. Devereaux, and S. A. Kivelson, “HolonWigner Crystal in a Lightly Doped Kagome QuantumSpin Liquid,” Phys. Rev. Lett. , 067002 (2017).[15] S. R. White, “Density matrix renormalization group al-gorithms with a single center site,” Phys. Rev. B ,180403(R) (2005).[16] J. Dukelsky, M. A. Martin-Delgado, T. Nishino, andG. Sierra, “Equivalence of the variational matrix productmethod and the density matrix renormalization groupapplied to spin chains,” EPL , 457 (1998).[17] S. R. White and D. J. Scalapino, “Density Matrix Renor-malization Group Study of the Striped Phase in the 2D t − J Model,” Phys. Rev. Lett. , 1272 (1998).[18] H. Takasaki, T. Hikihara, and T. Nishino, “Fixed pointof the finite system DMRG,” J. Phys. Soc. Jpn. , 1537(1999).[19] D. Poilblanc and N. Schuch, “Simplex Z spin liquidson the kagome lattice with projected entangled pairstates: Spinon and vison coherence lengths, topologi- cal entropy, and gapless edge modes,” Phys. Rev. B ,140407 (2013).[20] A. Vishwanath, “Identifying a spin liquid on the Kagomelattice using quantum entanglement,” Journal Club forCondensed Matter Physics (2012).[21] H. C. Jiang, Z. Y. Weng, and D. N. Sheng, “DensityMatrix Renormalization Group Numerical Study of theKagome Antiferromagnet,” Phys. Rev. Lett. , 117203(2008).[22] H.-C. Jiang, Z. Wang, and L. Balents, “Identifying topo-logical order by entanglement entropy,” Nature Phys. ,902 (2012).[23] H.-C. Jiang, H. Yao, and L. Balents, “Spin liquid groundstate of the spin- square J - J Heisenberg model,” Phys.Rev. B , 024424 (2012).[24] S. Depenbrock, I. P. McCulloch, and U. Schollw¨ock, “Na-ture of the Spin-Liquid Ground State of the S = 1 / , 067201 (2012).[25] Y.-C. He, D. N. Sheng, and Y. Chen, “Obtaining topo-logical degenerate ground states by the density matrixrenormalization group,” Phys. Rev. B , 075110 (2014).[26] S. Dong, E. Fradkin, R. G. Leigh, and S. Nowling,“Topological entanglement entropy in Chern-Simons the-ories and quantum Hall fluids,” JHEP , 016 (2008).[27] L. Cincio and G. Vidal, “Characterizing Topological Or-der by Studying the Ground States on an Infinite Cylin-der,” Phys. Rev. Lett. , 067208 (2013).[28] L. Zou and J. Haah, “Spurious long-range entanglementand replica correlation length,” Phys. Rev. B , 075151(2016).[29] Y. Zhang, T. Grover, A. Turner, M. Oshikawa, andA. Vishwanath, “Quasiparticle statistics and braidingfrom ground-state entanglement,” Phys. Rev. B ,235151 (2012).[30] D. Poilblanc, N. Schuch, D. P´erez-Garcia, and J. I. Cirac,“Topological and entanglement properties of resonatingvalence bond wave functions,” Phys. Rev. B , 014404(2012).[31] N. Read and S. Sachdev, “Large- N expansion for frus-trated quantum antiferromagnets,” Phys. Rev. Lett. ,1773 (1991).[32] X. G. Wen, “Mean-field theory of spin-liquid states withfinite energy gap and topological orders,” Phys. Rev. B , 2664 (1991).[33] S.-S. Gong, W. Zhu, D. N. Sheng, O. I. Motrunich, andM. P. A. Fisher, “Plaquette Ordered Phase and QuantumPhase Diagram in the Spin- J − J Square HeisenbergModel,” Phys. Rev. Lett. , 027201 (2014).[34] L. Tagliacozzo, A. Celi, and M. Lewenstein, “TensorNetworks for Lattice Gauge Theories with ContinuousGroups,” Phys. Rev. X , 041024 (2014).[35] Z. Zhu, D. A. Huse, and S. R. White, “Weak PlaquetteValence Bond Order in the S =1 / J − J Heisenberg Model,” Phys. Rev. Lett. , 127205 (2013).[36] Y. Zhang, T. Grover, and A. Vishwanath, “Topologi-cal entanglement entropy of Z spin liquids and latticeLaughlin states,” Phys. Rev. B , 075128 (2011). [37] S.-S. Gong, D. N. Sheng, O. I. Motrunich, and M. P. A.Fisher, “Phase diagram of the spin- J - J Heisenbergmodel on a honeycomb lattice,” Phys. Rev. B , 165138(2013).[38] W. Zhu, S. S. Gong, and D. N. Sheng, “Interaction-driven fractional quantum Hall state of hard-core bosonson kagome lattice at one-third filling,” Phys. Rev. B ,035129 (2016).[39] Beyond δ ∼ .
60 (at R b = 1 . γ should drop sharply. However, the compatibility ofthe string order itself depends on the system size, leadingto variations in the exact location of the QCP on finitecylinders, and consequently, an inability to reliably ex-trapolate the EEs within this region per the scaling formof Eq. (S1).[40] M. A. Metlitski, C. A. Fuertes, and S. Sachdev, “Entan-glement entropy in the O ( N ) model,” Phys. Rev. B ,115122 (2009).[41] L. Balents, M. P. Fisher, and S. M. Girvin, “Fractional-ization in an easy-axis Kagome antiferromagnet,” Phys.Rev. B , 224412 (2002).[42] S. V. Isakov, Y. B. Kim, and A. Paramekanti, “Spin-Liquid Phase in a Spin-1/2 Quantum Magnet on theKagome Lattice,” Phys. Rev. Lett. , 207204 (2006).[43] K. Roychowdhury, S. Bhattacharjee, and F. Pollmann,“ Z topological liquid of hard-core bosons on a kagomelattice at 1 / , 075141 (2015). [44] S. Sachdev, K. Sengupta, and S. M. Girvin, “Mott insu-lators in strong electric fields,” Phys. Rev. B , 075128(2002).[45] S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, New York, 2011).[46] A. M. L¨auchli, J. Sudan, and E. S. Sørensen, “Ground-state energy and spin gap of spin- Kagom´e-Heisenbergantiferromagnetic clusters: Large-scale exact diagonal-ization results,” Phys. Rev. B , 212401 (2011).[47] A. M. L¨auchli, J. Sudan, and R. Moessner, “The S = 1 / , 155142 (2019).[48] P. W. Leung and V. Elser, “Numerical studies of a 36-sitekagom´e antiferromagnet,” Phys. Rev. B , 5459 (1993).[49] P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, andP. Sindzingre, “Order versus disorder in the quantumHeisenberg antiferromagnet on the kagom´e lattice usingexact spectra analysis,” Phys. Rev. B , 2521 (1997).[50] C. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier,P. Sindzingre, P. Lecheminant, and L. Pierre, “First ex-citations of the spin 1/2 Heisenberg antiferromagnet onthe kagom´e lattice,” Eur. Phys. J. B , 501 (1998).[51] H. Nakano and T. Sakai, “Numerical-diagonalizationstudy of spin gap issue of the Kagome lattice Heisenbergantiferromagnet,” J. Phys. Soc. Jpn. , 053704 (2011).[52] P. Mendels and F. Bert, “Quantum kagome frustratedantiferromagnets: One route to quantum spin liquids,”C. R. Phys.17