Downloaded by guest on September 26, 2021 a Samajdar Rhine a lattice on kagome atoms Rydberg of phases Quantum PNAS simulators quantum Rydberg arrays. atom programmable Rydberg on using based states Our simulators experimental liquid order. quantum regime and and topological theoretical this crystalline with of for that explorations foundation phases argue the more transitions we provide or results phase phases, one solid quantum contain proximate of could the theories of and out model, quan- with lattice dimer associated triangular the parameter tum to mapping order entan- a large From local symmetries. a no lattice has and that identify entropy excitations we Rydberg glement addition, solid dense In complex with symmetries. of regime a variety lattice wide broken a with of phases to presence excitation the cal- laser reveal group culations via renormalization neutral Density-matrix interacting of states. lattice, Rydberg array Paramekanti) atomic kagome an Arun the and of Bhattacharjee on Subhro phases by atoms zero-temperature reviewed 2020; the 29, analyze July review We for (sent 2020 25, November Sachdev, Subir by Contributed and Austria Austria; A-6020, A-6020, Innsbruck Innsbruck Sciences, Innsbruck, of University Physics, Theoretical eie hr h ae rvn nue needn atomic independent lim- induces is “disordered driving gas-like excitations laser the Rydberg to the of contrast where in density regime” interactions, the the star where by the corre- ited by 1E) (identified strongly Fig. space a parameter quantum in of find a regime” via also “liquid states lated We classical mechanism. of manifold order-by-disorder from emerges degenerate actually phases highly ordered a these of one that order. show density-wave We long-range of with formation phases solid the intricate parameters reveal several laser calculations of These function distances. a atomic as and diagram phase rich its perform to establish and computations lattice (DMRG) kagome group the renormalization on density-matrix real- atoms a Rydberg in examine of phases we model Specifically, quantum istic (14–20). found correlated dimensions been interesting two and of have laser one variety that via a (13), interacting investigate display states to arrays, Rydberg we atomic atom here, to neutral excitation consideration, of explo- this states of further- many-body by by avenue induced Motivated promising or ration. origin a geometric couplings—constitute of neighbor either be can quan- which experimental these direct far, evaded thus have states (8–10), (QSL) detection. lattice liquid electronic kagome spin in present tum the are be there may on phase While a systems (3). such code” that “toric indications the some as order topological magnetic time-reversal so-called same a with the of compatible is absence phase the symmetry anyonic In simplest (5). challenging the directly topo- field, are control intrinsic and statistics, their detect anyonic systems, to including experimental properties, of variety logical realized been wide high now, a in by have, found in states states such While Hall (4). quantum fault- fields in fractional magnetic for examples the best-studied are exploited regard The quantum be this (3). many-body principle, computing quantum long-range in tolerant feature can, decades three which systems past entanglement, the Such for 2). matter quantum (1, in research of focus T transitions eateto hsc,HradUiest,Cmrde A02138; MA Cambridge, University, Harvard Physics, of Department ntesac o Ss ytm ihfutain(1 12)— (11, frustration with systems QSLs, for search the In etguefils n noi xiain a enacentral a been has excitations emer- anyonic fractionalization, and with fields, gauge phases gent quantum for search he 01Vl 1 o e2015785118 4 No. 118 Vol. 2021 a,1 e e Ho Wei Wen , | est-aeorders density-wave Z 2 pnlqi 6 ) hc a the has which 7), (6, liquid spin a,b ansPichler Hannes , | unu phase quantum b eateto hsc,Safr nvriy tnod A94305; CA Stanford, University, Stanford Physics, of Department c,d ihi .Lukin D. Mikhail , d nttt o unu pisadQatmIfrain utinAaeyof Academy Austrian Information, Quantum and Optics Quantum for Institute doi:10.1073/pnas.2015785118/-/DCSupplemental at online information supporting contains article This ulse aur 8 2021. 18, January Published 1 the under Published of interest.y competing University no A.P., declare authors and The Sciences; Theoretical for Centre Toronto.y International S.B., W.W.H., Reviewers: R.S., paper.y and the data; wrote analyzed S.S. S.S. S.S. and and and M.D.L., M.D.L., H.P., M.D.L., H.P., H.P., W.W.H., W.W.H., R.S., R.S., research; research; designed performed S.S. and R.S. contributions: Author unit complete of number the denote along us cells Let sites. neighboring constant lattice the where h nenlgon tt n ihyectdRdegsaeof state Rydberg excited highly a and state with ground system internal two-level the a as regarded tv etr fti atc are prim- lattice the and this scaffolding of triangular vectors a itive cell on unit kagome sites Each 1A. three Fig. in comprises sketched arranged as atoms lattice, neutral kagome of a phases on the studying in lies interest Our Model Rydberg Lattice Kagome gauge innate specific and engineering geometries (22). carefully constraints lattice without possi- appropriate even made using interactions, is by this simply Remarkably, ble matter. quantum in studied, entangled accessible investigations of sizes readily experimental be of system direct possibility should the raising the provide regime experiments, this not over that do order demonstrate we results topological topolog- numerical long-range for with our evidence that states quan- While imply hosts for order. then regime phases ical Theories liquid solid this model. these to of of Rydberg part proximate out the transitions phases phase in solid tum regime the liquid We describes the entropy. correctly model dimer entanglement which quantum lattice significant (21), triangular the and to mapping regime order a liquid employ phases the local that solid observe no we strengths regime, has dense interaction a such most in appear for While excitations. owo orsodnemyb drse.Eal [email protected] rrhine or [email protected] Email: addressed. be [email protected]. may correspondence whom To oooia re.Orrslspoiearuet experi- quantum to entangled route highly a probing matter. provide and results realizing Our mentally and order. entanglement quantum for lat- topological long-range candidate with an break promising phase identify a a which hosting we constitutes states correlations, that classical regime the Along to intriguing of lattice. due analysis symmetries kagome tice a extensive theoreti- on an by atoms realized we with be Rydberg work, can such that this phases arranging for quantum In the platforms systems. investigate dynamics cally correlated versatile quantum strongly and as phases of many-body emerged exotic recently atom exploring Rydberg have on based arrays simulators quantum Programmable Significance a a µ n ui Sachdev Subir and , NSlicense.y PNAS by N y µ namnmlmdl ahao a be can atom each model, minimal a In . https://doi.org/10.1073/pnas.2015785118 a stesaigbtentonearest- two between spacing the is a 1 (2a = . a,1 y |g https://www.pnas.org/lookup/suppl/ i i 0) , and and c nttt for Institute |r i a i 2 representing (a = , | √ f9 of 1 3a ),

PHYSICS A BE

C D

√ Fig. 1. Phases of the kagome lattice Rydberg atom array. (A) Geometry of the kagome lattice. The lattice vectors are a1 = (2, 0), a2 = (1, 3). Periodic boundary conditions (PBC) and open boundary conditions (OBC) are imposed along the a2 and a1 directions, resulting in a cylinder. The blue dots are the sites of the original kagome lattice, where the atoms reside, while the red dots outline the medial triangular lattice formed by connecting the centers of the kagome hexagons. (B–D) The various possible symmetry-broken ordered phases. 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-Rb plane. The yellow diamonds and the pink circles are determined from the maxima of the susceptibility at each Rb; the former correspond to the finite-size pseudocritical points delineating the boundaries of the ordered phases. The white bars delimit the extent of the stripe phase. The string phase (Fig. 2) lies at larger detuning, beyond the extent 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 in Figs. 5 and 7, respectively.

the ith atom. The system is driven by a coherent laser field, char- by matrix product state (MPS) ans¨atze of bond dimensions up acterized by a Rabi frequency, Ω, and a detuning, δ. Putting these to d = 3,200. Although (i, j ) runs over all possible pairs of sites terms together, and taking into account the interactions between in Eq. 1, this range is truncated in our computations, where we atoms in Rydberg states (23), we arrive at the Hamiltonian retain interactions between atoms separated by up to 2a (third- nearest neighbors), as shown in Fig. 1A. To mitigate the effects of N the boundaries, we place the system on a cylindrical geometry by X Ω H = |gi hr| + |ri hg|
− δ |ri hr| imposing open (periodic) boundary conditions along the longer Ryd 2 i i i i=1 (shorter) a1 (a2) direction. The resulting cylinders are labeled by 1 X the direction of periodicity and the number of sites along the cir- + V (||x − x ||/a)|ri hr| ⊗ |ri hr| , [1] 2 i j i j cumference; for instance, Fig. 1A depicts a YC6 cylinder. Since (i,j ) the computational cost of the algorithm (for a constant accuracy) scales exponentially with the width of the cylinder (33), here, we where the integers i, j label sites (at positions xi,j ) of the lat- limit the systems considered to a maximum circumference of 12 tice, and the repulsive interaction potential is of the van der lattice spacings. Unless specified otherwise, we always choose the 6 Waals form V (R) = C/R (24). Crucially, the presence of these linear dimensions N1, N2 to yield an aspect ratio of N1/N2 ' 2, interactions modifies the excitation dynamics. A central role in which is known to minimize finite-size corrections and optimize the physics of this setup is played by the phenomenon of the DMRG results in two dimensions (34, 35). Rydberg blockade (25, 26) in which strong nearest-neighbor interactions (V (1)  |Ω|, |δ|) can effectively prevent two neigh- Phase Diagram boring atoms from simultaneously being in Rydberg states. The We first list the various phases of the Rydberg Hamiltonian that excitation of one atom thus inhibits that of another and the can arise on the kagome lattice. Without loss of generality, we associated sites are said to be blockaded. By reducing the lat- set Ω = a = 1 hereafter for notational convenience. At large neg- tice spacing a, sites spaced farther apart can be blockaded as ative detuning, it is energetically favorable for the system to well and it is therefore convenient to parameterize HRyd by have all atoms in the state |gi, corresponding to a trivial “dis- the “blockade radius,” defined by the condition V (Rb /a) ≡ Ω ordered” phase with no broken symmetries (36). As δ/Ω is tuned 6 or equivalently, C ≡ Ω Rb . Finally, we recognize that by identi- toward large positive values, the fraction of atoms in |ri increases fying |gi, |ri with the two states of a S = 1/2 spin, HRyd can but the geometric arrangement of the excitations is subject to also be written as a quantum Ising spin model with C/R6 inter- the constraints stemming from the interactions between nearby actions in the presence of longitudinal (δ) and transverse (Ω) Rydberg atoms. This competition between the detuning and fields (27). the previously identified blockade mechanism results in so- We determine the quantum ground states of HRyd for differ- called “Rydberg crystals” (37), in which Rydberg excitations ent values of δ/Ω and Rb /a using the DMRG (28, 29), which are arranged regularly across the array, engendering symmetry- has been extensively employed on the kagome lattice to iden- broken density-wave ordered phases (19). On the kagome lattice, tify both magnetically ordered and spin liquid ground states of the simplest such crystal that can be formed—while respecting the antiferromagnetic Heisenberg model (30–32). The technical the blockade restrictions—is constructed by having an atom in aspects of our numerics are documented in SI Appendix, sec- the excited state on exactly one out the three sublattices in the tion I. In particular, we work in the variational space spanned kagome unit cell. This is the ordering pattern of the “nematic”

2 of 9 | PNAS Samajdar et al. https://doi.org/10.1073/pnas.2015785118 Quantum phases of Rydberg atoms on a kagome lattice Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 hitnte“tig hs n ics next. we discuss which and lobes, phase “string” another two the is these christen there between detuning, and phase large nematic ordered of the density-wave limit of the boundaries In phase orders. the staggered mark which 1E, simply superposi- in Fig. is their in decline than which drastic rather so- This 40), states tion. a (35, symmetry-broken the to (MES) of states state converges one prefers entropy systematically DMRG minimal and the called because entanglement is low This solid with 5C). the inside Fig. sharply (see drops then phase and (QCP), parti- point critical is tum cylinder one, the ordered an when along subsystem half in each tioned for matrix density unu hsso ybr tm na lattice on kagome atoms Rydberg of phases Quantum al. et Samajdar (EE) entropy state entanglement ground Neumann the von bipartite of the is so of do diagram phase full the bling blockaded. to also up are neighbors stable fourth-nearest remains phase vec- staggered lattice with cell unit 12-site a tors of formed are crystals Rydberg the taking upon fraction” “filling state a limit ordered by classical the characterized be excitations, still Rydberg can of number though is the Even state phase. conserve ground this true within the system, degenerate infinite triply an for so, lattice, kagome (C spontaneously nearest- rotational only order threefold where nematic the regime The breaks a blockaded. in are found sites is neighbor which 1C), (Fig. phase est of density hsstate This 1B. Fig. a encounters in one phases, seen stripe and phase dered classical the any stripe without breaks the solid also quantum wit, a to of sliver analogue, a by one ordered xiain r oiindadsac of the distance so a sites, positioned are third-nearest-neighbor excitations blockade to enough sufficiently are strong when atoms realized phase, Rydberg is neighboring This degeneracy, between interactions ground-state 1D). 12-fold (Fig. density-wave a phase with bears which “staggered” phase the solid it namely another surrounding ordering, yet of hosts array instead point Rydberg dome tricritical nematic a the at it terminates of believe phase throughout. we tip 7), stripe the Fig. the near (see Nev- that data intervening. likely current order is our stripe the on with based one ertheless, two-step phases a disordered always and nematic is signifi- lattice the narrows between phase transition the this increasing of with extent cantly The ordering. nematic the also is in reduction the from simultaneously therefore energy while and density interactions, excitation of the the admixture the maximizing to offsetting coherent due in a assist penalty superposi- atoms with energetic quantum “dressed” state These a state. ground in Rydberg the those the each whereas by are system state, formed sublattices all ground tion two The where the other follows. in configuration the are as a on sublattice window in one on packing narrow atoms geometric a sta- the stripes help in which these optimizes phase (19), of fluctuations the formation quantum The bilize to on cell. attributed density unit be equal distin- the can and is substantial of a both ordering sublattices by QPT stripe two Although nematic the the the order. symmetry, while from first same guished (39), is the model break nematic Potts phases class three-state and universality 1)D stripe the + demarcating in (2 (38) the (QPT) of transition phase quantum qipdwt h nomto bv,w o unt assem- to turn now we above, information the with Equipped uiul,tenmtcpaei eaae rmtetiildis- trivial the from separated is phase nematic the Curiously, rceigt agrbokd ai,w n httekagome the that find we radii, blockade larger to Proceeding 4a 1 hn and i ∼ i hn i 2a δ i hc xlisiseitnea rcro to precursor a as existence its explains which 1/3, δ/ h nun vrg est ntesrp phase stripe the in density average ensuant The . 1 = 1 Ω + C S where /3, ∞, → 3 vN a 2 S ymty codnl,btentedisor- the between accordingly, symmetry; a h soitdcascldniyis density classical associated the ; 1 vN rdal nrae,pasna h quan- the near peaks increases, gradually R ngigfo h iodrdpaeto phase disordered the from going On . b − ≡ R oi sdfcl oacranwhether ascertain to difficult is it so , b /a n r(ρ Tr S i vN 6 hc,i hscs,last a to leads case, this in which, =0, |r ≡ H Ryd rcsottetolbsseen lobes two the out traces 3 r ymtyo h underlying the of symmetry ) i ln i nubae igotcto diagnostic unbiased An . hr ρ r |. R √ ), Z b 7 ρ . 3 r pr.Teresultant The apart. -symmetry-breaking √ en h reduced the being eodwhich beyond 7, H Ryd The 1/6. osnot does ycasclfiln rcin of parame- fractions filling characterized the classical regions three by In yields optimization ( R this interactions. that 2D) interest (Fig. repulsive classical of the the between range competition minimize ter and the simply by detuning can den- solely the one determined fractional is various this, which at see energy, ones To besides staggered (41). phases the ordered sities and additional the nematic in on result the bosons) can hard-core lattice equivalently kagome (or excitations Rydberg l dnie s(h lsia esoso)tefmla nematic familiar the 1 of) (Fig. versions orders classical staggered (the and as identified ily limit. this in set tonian (temporarily) have V we Since interactions. where of limit classical the In Disorder by Order Quantum l ihtesm lsia nry n hsdgnrc grows degeneracy this and states, energy, such classical of lattice. same number the the macroscopic across with a stretch all are bent—that or there straight Interestingly, be strings— may resembles excitations which Rydberg phase, the of of this arrangement to fraction The belonging filling the crystal a of a few with for A phase. patterns string ordering the classical dub possible degenerate we highly what separate forming a state, ground favors system the two, the tigodrdsaeoe nt einbtentenmtcadtestag- the R and nematic (E the possi- phases. between region three (at gered finite energy the a the Comparing over that state (D) find string-ordered arranged lattice. we are the phases, (red) classical span excitations ble that Rydberg (yellow) The “strings” size. in system expo- the fraction—scales filling with same nentially the states—with such of number the here, at states ordered sically i.2. Fig. AB E CD b 3 h hssa lig fatidadasxhcnb read- be can sixth a and third a of fillings at phases The = /δ 5o ie(N wide a on 1.95 steol needn uigprmtrfrteHamil- the for parameter tuning independent only the is V ligo h aoeltie (A–C lattice. kagome the on filling 2/9 at phase Crystalline 3 ersnstesrnt ftethird-nearest-neighbor the of strength the represents ybr rsa omdi h tigpaeat phase string the in formed crystal Rydberg ) f =      1 f = ;1 1/6; 1 2/9; 1/3; = 2 C cylinder. YC8 12) .Wiew kthol he configurations three only sketch we While 2/9. δ/ b f . Ω 2 = V ,i sntdfcl oobserve to difficult not is it 2.25), h eidcarneetof arrangement periodic the ∞, → C /4 /7 3 r rsne nFg 2 Fig. in presented are /9, δ< /δ and < < https://doi.org/10.1073/pnas.2015785118 V V ,rsetvl.I between In respectively. D), 1/7, 3 3 /δ < /δ , = Ω 1/4, )i iiie ythe by minimized is 0) h ratio the 0, = Ω PNAS δ | = Clas- ) A–C. f9 of 3 4.00, [2]

PHYSICS A B Fig. 2E, which illustrates the local magnetizations inside the string phase (at δ = 4.00, Rb = 1.95) on a YC8 cylinder of length N1 = 12 (chosen to be fully compatible with the string order). The ground state found by finite DMRG is patently ordered with the system favoring a configuration of straight strings that wrap around the cylinder, thereby lifting the macroscopic classi- cal degeneracy. This is in contrast to the 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 the kagome lattice. All these ordered states can be identified from either their respective structure factors or the relevant order C D parameters, 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 effective 2 size of Nc = 3N2 . Evidence for ordering or the lack thereof can be gleaned from the static structure factor, which is the Fourier transform of the instantaneous real-space correlation function

1 X iq·(xi −xj ) S(q) = e hni nj i [3] Nc i,j

with the site indexes i, j restricted to the central N2 × N2 region of the cylinder. At a blockade radius of Rb = 1.7, which sta- tions one in the nematic phase (Fig. 3B), the structure factor has Fig. 3. Static structure factors of the various ordered phases. S(q) displays pronounced maxima at the corners of the (hexagonal) extended Brillouin zone, occurring at Q = ±b1, ±b2, ±(b1 + b2), where pronounced and well-defined peaks for the (A) stripe (δ = 2.20, Rb = 1.20), √ √ (B) nematic (δ = 3.30, Rb = 1.70), (C) staggered (δ = 3.30, Rb = 2.10), and b1 = (π, −π/ 3) and b2 = (0, 2π/ 3) are the reciprocal lattice (D) string (δ = 4.00, Rb = 1.95) orders. The dashed white hexagon marks the vectors. A subset of these maxima also persists for the stripe first Brillouin zone of the kagome lattice. The structure factor for the string phase (Fig. 3A)—this is in distinction to the nematic phase phase is computed on the cylindrical geometry shown in Fig. 2E. wherein the peaks at all six reciprocal lattice vectors are of equal strength. In the presence of staggered ordering (Fig. 3C), the peaks are comparatively weaker√ but prominent nonetheless,√ exponentially with the linear dimensions of the system. For appearing at Q =√±b1, ±(π/2, 3π/2), ±(3π/4, π/(4 3)), and example, in Fig. 2A, the positions of all of the atoms in the Ryd- ±(−π/4, 5π/(4 3)). Likewise, in the string phase (Fig. 3D), berg state can be uniformly shifted by ±a2/2 for every other conspicuous maxima are seen to occur at ±2b1/3 for the ground- string without affecting the energy, leading to O(2N1 ) poten- state configuration where straight strings encircle the lattice. tial configurations. Similarly, when the strings are bent, like in While we list here the ordering wavevectors for a finite system, let us briefly note that on an infinite lattice, the structure fac- Fig. 2C, there are O(N2) locations where a kink can be formed N C and, correspondingly, O(2 2 ) states of this type. tors, of course, would additionally include 3-rotated copies of The large classical degeneracy raises the question of the fate the above. of this phase once we reinstate a nonzero transverse field, Ω. One can also directly look at the order parameters that There are two natural outcomes to consider. First, a superpo- diagnose the possible symmetry-broken ordered states. For the sition of the classical ground states can form a quantum liquid nematic phase, an appropriate definition is with topological order, as is commonly seen to occur in quan- ! 3 X X 2 X tum dimer models (42). However, a necessary condition in this Φ = ni + ω ni + ω ni , Nc regard is the existence of a local operator which can connect one i∈A i∈B i∈C classical ground state with another. Since the individual ground states are made up of parallel strings, they are macroscopically where ω ≡ exp(2πi/3) is the cube root of unity, and A, B, far away from each other, and it would take an operator with C denote the three sublattices of the kagome lattice. Simi- support of the size of the system length to move between dif- larly, in the staggered and string phases, one can define the 2 ferent classical configurations, thus violating the requirement of (squared) magnetic order parameter MQ ≡ S(Q), with Q chosen locality. This brings us to the second possibility, namely, that a from among the observed peaks of the structure factor. These quantum “order-by-disorder” phenomenon (43, 44) prevails. In order parameters are more quantitatively addressed (see Fig. 7A, this mechanism, quantum fluctuations lower the energy of partic- which catalogs the ground-state properties calculated at a fixed ular classical states in the degenerate manifold; the system then detuning of δ = 3.3 [dashed line in Fig. 1E]; in particular, we orders in a state around which the cost of excitations is espe- observe that the nematic and staggered order parameters assume cially cheap. In this case, one could anticipate a string-ordered nontrivial values in exactly the regions predicted by the phase solid phase, which should be easily identifiable from the structure diagram.) factor. The DMRG numerics confirm our intuition that such a crys- Mapping to Triangular Lattice Quantum Dimer Models tal should emerge in the phase diagram at sufficiently high At large detuning, we can approximately map the Rydberg sys- detunings. On the YC8 cylinder with N1 = 8, the string phase tem to a model of hard-core bosons at filling f on the kagome appears at detunings beyond the range rendered in Fig. 1E. lattice. The bosonic system (21, 45–49) has an extra U(1) sym- However, it is manifestly observed, for a wider geometry, in metry, which can be spontaneously broken in a superfluid phase;

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(50, , 2 b (0, = 2 .Teciia theory critical The 3B). (Fig. Z π/ 2 pnlqi anthv a have cannot liquid spin √ )= 3) Z N 2 b d 2 pnlqisfound liquids spin 3 ,1) 2, (3, = /2 Z 2) n their and (21), 2 N Z topological d 2 h tri- The . pnliq- spin Z N dimers 2 M d spin 2 = 1 Z = 2 eoddrvtv ftegon-tt energy, ground-state the of derivative second cut. one-dimensional this along properties ground-state the ota n,w eprrl ou naseicbokd radius, blockade specific regime. a this on of focus vicinity R temporarily the we end, in that transitions the To uncover phase to any attempt considerations of first the we existence in section, by previous correlations state Prompted the in regime. more dense described disordered significantly a the in introduces form than blockade excitations earlier, categoriza- the Rydberg defined resists which “liquid,” the but nomenclature that phases The connotes one. solid either two as between 1E— tion Fig. in in interme- lies star an red find which the we by detuning, regime—designated the correlated of diate values large moderately At Regime Liquid The diagram. phase our of regime liquid the leads (δ calculation of This estimate 51). (50, an regime liquid to known) spin QDM (previously the the of on stability of based system Rydberg the for in in further, of range not our do for we below, However, it sizes. system. for system evidence our numerical in clear-cut accessible of find class be universality principle, the in in is QPT Ising This the constraint. dimer the the by condensa- sharp because the a QDMs, by the be bosonic described should the phase of there disordered tion model, the Rydberg to the transition of regime liquid Rydberg of density the (f (∼ that QDM regime note liquid also the in We excitations 7B). Fig. (see regime in fluctuate can dimers of configurations. number dimer total of the model. QDM, superposition Rydberg the the a nematic in as unlike the liquid that, from Note Rydberg depiction separated potential Schematic (B) is a QPT. regime (first-order) of continuous liquid a uniquely by The phase be triangular (21). (staggered) medial now lines) the can of (blue bond points) corresponding (red/green lattice the on lattice dimer kagome a with the model associated of boson site resultant any The of on (A) filling (27). a to state mode at mapped in bosonic is atom be (empty) each can occupied identifying excitations an upon Rydberg as bosons, the hard-core of detuning, system large a of limit the In 4. Fig. b h rtsc bevbei h ucpiiiy enda the as defined susceptibility, the is observable such first The xedn h apn rmteRdegmdlt h QDM the to model Rydberg the from mapping the Extending odd and even the both For (δ 1 = , R .9 orsodnebtenteRdegadqatmdmrmodels. dimer quantum and Rydberg the between Correspondence b ∗ ) 1 = ,adlo tvrain of variations at look and 1E), Fig. in line white (dotted isnFse ofra edter 6–3 n can, and (61–63) theory field conformal Wilson–Fisher pc hr S hs ih eepce oexist to expected be might phase QSL a where space ecmueteparameters the compute we III, section Appendix, SI /6). f = )frtenmtc(tgee)pae boson A phase. (staggered) nematic the for (1/6) 1/3 e 2 = nos uhatasto sntpeetin present not is transition a Such anyons. .981, e xiain aebe rjce out projected been have excitations R b Z 1 = https://doi.org/10.1073/pnas.2015785118 2 pnlqispooe o the for proposed liquids spin scoet hto h odd the of that to close is 0.2) . ,wihpae swithin us places which 997), E 0 ihrsetto respect with , PNAS |r | i f9 of 5 (|gi)

PHYSICS 2 2 the detuning; i.e., χ = −∂ E0/∂ δ . On finite systems, the maxima in a purely bulk observable, its magnitude is diminished: For of the susceptibility can often be used to identify possible QCPs, example, the relative change between the local maximum and which are slightly shifted from their locations in the thermody- the minimum (shoulder) immediately adjacent to it on the right namic limit. In particular, for Rb = 1.9, χ is plotted in Fig. 5A, (left) differs by approximately a factor of 4 (10) between χb and where a single peak in the response is visible at approximately χ for the YC8 cylinder. Hence, the behavior of the susceptibility δ = 2.9. This susceptibility peak—which is recorded by the pink could be indicative of an edge phase transition but whether this circles in Fig 1E—is also reproduced in exact diagonalization is accompanied by, or due to, a change in the bulk wavefunction calculations on a 48-site torus (SI Appendix, section IV). is presently unclear. A similar signature can be discerned in the quantum Next, we investigate the properties of this liquid regime in fidelity |hΨ0(δ)|Ψ0(δ + ε)i| (64, 65), which measures the over- more detail and demonstrate that—as preempted by its name—it lap between two ground-state wavefunctions Ψ0 computed at does not possess any long-range density-wave order. This diag- parameters differing by ε. The fidelity serves as a useful tool nosis of liquidity is best captured by the static structure factor. in studying QPTs because, intuitively, it quantifies the similar- In stark contrast to Fig. 3, S(q) is featureless within the liq- ity between two states, while QPTs are necessarily accompanied uid regime (Fig. 7A) with the spectral weight distributed evenly by an abrupt change in the structure of the ground-state wave- around the extended Brillouin zone. function (66). Zooming in on a narrower window around the This unordered nature is reflected in Fig. 7B, where we plot susceptibility peak, we evaluate the fidelity susceptibility (67), the order parameters characterizing the surrounding symmetry- which, in its differential form, is given by broken states. The order parameters defined earlier are found   to be nonzero in both the nematic and staggered phases but are 1 − |hΨ0(δ)|Ψ0(δ + ε)i| smaller by an order of magnitude in the liquid regime; this is F ≡ 2 2 . [4] 2 2 ε compatible with a vanishing |Φ| and MQ in the thermodynamic limit. In the process, we also find that the transition from the The fidelity susceptibility also displays a local maximum at δ ≈ nematic (staggered) phase to the liquid regime appears to be sec- 2.9, indicating some change in the nature of the ground state as ond order (first order), which is consistent with the expectations we pass into the liquid regime. Unlike the QPTs into the ordered for the QPT into a Z2 QSL in the dimer models, as we have dis- phases, the EE (Fig. 5C) does not drop as we cross this point cussed in the previous section. We do not observe any signatures but rather, continues to increase; however, its first derivative is of a phase transition within the liquid regime. nonmonotonic at δ ≈ 2.9. This suggests that the final liquid state Moreover, one can also define a correlation length from the is likely highly entangled and is not a simple symmetry-breaking structure factor as (73) ground state. Given that we always work on cylinders of finite extent, we cannot exclude the possibility that the peaks in Fig. 5 A and B s 1 S(Q) are due to surface critical phenomena (68, 69) driven by a phase ξ(Q, qmin) = − 1, [5] transition at the edge. Indeed, in Fig. 6A, which shows a profile |qmin| S(Q + qmin) of the liquid regime on a wide cylinder at δ = 3.50, Rb = 1.95, we notice that the edges seek to precipitate the most compatible density-wave order at these fairly large values of the detuning. Nonetheless, the bulk resists any such ordering tendencies and the central region of the system remains visibly uniform, with AB only slight perturbations from the open boundaries. In fact, the bulk fails to order despite being at a detuning for which the system energetically favors a maximal (constrained) packing of Rydberg excitations, as is also evidenced by the nearby staggered and nematic phases above and below the liquid regime, respec- tively. It is perhaps worth noting that in one spatial dimension, the comparable regions lying between the different Zn -ordered states at large detuning are known to belong to a Luttinger liquid phase (70). To eliminate end effects, it is often useful to first evaluate the ground-state energy per site for an infinitely long cylinder by sub- C tracting the energies of finite cylinders of different lengths but with the same circumference (31, 35, 71, 72). Such a subtrac- tion scheme cancels the leading-edge effects, leaving only the bulk energy of the larger system. In particular, this procedure enables us to quantify the influence of the boundaries on thermo- dynamic properties of the system such as the susceptibility. Using two cylinders of fixed width, an estimate of the bulk energy can be found by subtracting the energy of the smaller system from that of the larger. The (negative of the) second derivative of this quantity with respect to the detuning defines the bulk suscep- tibility χb —this gives us the susceptibility in the center of the cylinder with minimal edge effects. Fig. 6B presents the varia- Fig. 5. Signatures of a crossover into the liquid regime. Along a line where the blockade radius is held constant at Rb = 1.90, both (A) the susceptibil- tion of χb with detuning at Rb = 1.90 for the YC6 family: We see ity χ and ( ) the fidelity susceptibility F exhibit a single peak at δ ≈ 2.90. A B that the local maximum of the susceptibility reported in Fig. 5 (C) The behavior of the EE over the same detuning range, however, is dis- is still identifiable, but its precise location is shifted to slightly tinct from the sharp drop observed across the QPTs into any of the ordered higher δ. Analogously, we study the bulk susceptibility for wider phases. On going to higher δ, the system eventually transitions into either YC8 cylinders and find, once again, a distinct peak correspond- the nematic or the string phase, depending on the blockade radius (or ing to the onset of the liquid regime. Although this peak persists potentially, the boundary conditions).

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Fig. r losotrne ntelqi regime. liquid the in ranged short also are where lengths of systems two to method subtraction along here shown susceptibility, lengths bulk with cylinders at YC6 appears two of energies the N between difference In the effects. of edge to sitive BC A A 1 = 2and 12 C Q rpriso h iudrgm.Hr,w ou nteline the on focus we Here, regime. liquid the of Properties ihnteliquid the Within (A) behaviors. boundary and bulk Disentangling ( + i δ ,j and N ≈ ),(k q 1 min ξ 9 eadn h iudrgm.O C yidr (C cylinders YC8 On regime. liquid the heralding 3.09, = C ,l stesm ln ohdrcin ntecylinder the on directions both along same the is ukssetblte:B construction, By susceptibilities: Bulk ) with 5A, Fig. in As 9. ) h orlto egh bandi h liq- the in obtained lengths correlation The Q. stealwdwvvco meitl adjacent immediately wavevector allowed the is 4[ = o 1.9 For 3: h(n B, δ i χ = ≤ · b n 3.50, R j sdtrie rmtescn derivative second the from determined is b )(n ≤ R 2.0, R k b b R · = = b n ξ e o19,acerlclmaximum local clear a 1.90, to set 5 scluae yapyn the applying by calculated is 1.95, l 5tera-pc magnetiza- real-space 1.95—the h − )i ssalrta n atc pcn,s l orltosaesotranged. short are correlations all so spacing, lattice one than smaller is N 1 n = B i 2and 12 · n j χ ihn b hudb insen- be should N k 1 · = n 8. l i] δ = ,the ), [6] h tutr factor structure The (A) radius. blockade the varying while 3. |Φ | 2 and eeso ybr tmcsae opoeavreyo quantum of variety a sub- probe hyperfine to states multiple phases. entangled atomic as using Rydberg by exten- well or e.g., of as envisioned, levels number be arrangements also A atom can regime. model various liquid present the definitively the to and of sions finite- works nature from theoretical the quantify future free establish completely more in nor to uncertainties worthwhile method these be would unbiased it an effects, size neither is DMRG interesting to point investigation. further collectively merit they that physics sizes, system for phase computa- available topological its these a the of and of existence the order none confirm conclusively a topological although tions possible transitions; examined the phase study of associated their numerical signatures of of Our theories number transitions. and of phase models diagram phase dimer quantum global topo- quantum the with lattice in phase placement triangular elusive its an using to order corresponding logical region state a liquid a potentially, the that also, host argued but could We solids, regime. quantum liquid of entangled supports variety highly rich that a phenomena only correlated not strongly plat- studying promising a for constitutes form array the atom that Rydberg showed lattice we analyses, kagome theoretical and numerical on Based Outlook and Discussion quantum thus and different signatures, between false distinguish (77). phases to reliably leading always finite- cylinders, cannot strong on from suffer different effects can a size TEE in solid the bond Additionally, valence (71). a model for a documented for been also evidence has not conclusive does long- this enhanced as S4), an Fig. serve of Appendix, indications (SI find entropy entanglement do range topolog- we is While and origin. in entanglement ical nonlocal the from Importantly, entirely entropy. represent- law arises positive, area TEE and the to universal reduction is constant TEE a ing the of value the phase, nrp TE 7,7)in 76) entanglement topological (75, the (TEE) calculating entropy by QSL a regime of liquid indications the positive in for entanglement search we of super- of Accordingly, degree quantum 74). high absence (12, massive anomalously of an the presence to leading the by position is i.e., QSL essential a the not, that for understood are ingredient been has states it recently, the more fin- ordering, been what long have by QSLs gerprinted Although phase. topological a harbors for expected behavior the with a regime— consistent liquid all the are of properties state these ground the for candidate disordered Z hswr a eetne nsvrldrcin.A the As directions. several in extended be can work This i.S3), Fig. , Appendix (SI gapped a to point numerics our far, So M 2 Q 2 S,s ti aua oakwehrti einpotentially region this whether ask to natural is it so QSL, hrceiigtenmtcadsagrdpae,rsetvl.Both respectively. phases, staggered and nematic the characterizing C eto II section Appendix, SI https://doi.org/10.1073/pnas.2015785118 Z 2 S safinite a as QSL h orlto lengths correlation The ) S ,at (q), o QSL a For . PNAS R b = γ | 5 is 1.95, ∼ f9 of 7 n2 ln

PHYSICS Furthermore, we expect the phase diagram in Fig. 1E to single-particle peaks will be observed in the disordered phase, serve as a valuable guide to the detailed experimental stud- while two-particle continua would appear in a region with Z2 ies of frustrated systems using Rydberg atom arrays. Specif- topological order. Detailed study of such spectra could yield ically, both solid and liquid regimes can be reached starting the pattern of symmetry fractionalization (48, 52, 83). Other from a trivial product ground state by adiabatically changing directions include more direct measurements of the topologi- the laser detuning across the phase transitions, as was demon- cal entanglement entropy (84, 85). Finally, classical and quantum strated previously (15). For experiments with N ∼ O(102) Rb machine-learning techniques (86–89) could be useful for measur- atoms coupled to a 70-S Rydberg state, the typical Rabi fre- ing nonlocal topological order parameters associated with spin quencies involved can be up to (2π) × 10 MHz. With these liquid states. driving parameters, sweeps over the detuning range 0 ≤ δ . 5 Ω, at interatomic spacings such that Rb /a . 3.5, have already Materials and Methods been achieved in one-dimensional atom arrays (78–80). Hence, The DMRG calculations were performed using the ITensor Library (90). coherently preparing all of the different many-body ground Further numerical details are presented in SI Appendix, section I. states and observing their fundamental characteristics should be within experimental reach in two-dimensional systems as well Data Availability. All study data are included in this article and SI Appendix. (81, 82). While the solid phases can be detected directly by evaluat- ACKNOWLEDGMENTS. We acknowledge useful discussions with Subhro ing the corresponding order parameter, the study of any possible Bhattacharjee, Meng Cheng, Yin-Chen He, Roger Melko, Roderich Moessner, William Witczak-Krempa, Ashvin Vishwanath, Norman Yao, and QSL states in the liquid regime (or more generally, on Rydberg Michael Zaletel and especially the team of Dolev Bluvstein, Sepehr Ebadi, platforms) will require different approaches. In particular, mea- Harry Levine, Ahmed Omran, Alexander Keesling, Giulia Semeghini, and suring the statistics of microscopic state occupations (15) or the Tout Wang. We are grateful to Marcus Bintz and Johannes Hauschild for growth of correlations (18) across reversible QPTs could prove pointing out the possibility of an edge transition and sharing their results. We also thank Adrian E. Feiguin for benchmarking the ground-state ener- to be informative. In a Rydberg liquid, one can think of cre- gies observed in our DMRG calculations. R.S. and S.S. were supported by ating and manipulating topologically stable excitations, which the US Department of Energy under Grant DE-SC0019030. W.W.H., H.P., cannot disappear except by pairwise annihilation with a part- and M.D.L. were supported by the US Department of Energy under Grant ner excitation of the same type. The excitation types would DE-SC0021013, the Harvard–MIT Center for Ultracold Atoms, the Office of Naval Research, and the Vannevar Bush Faculty Fellowship. W.W.H. correspond to the three nontrivial anyons of the Z2 spin liq- was additionally supported by the Gordon and Betty Moore Foundation’s uid, and each should manifest as a characteristic local (and Emerging Phenomena in Quantum Systems Initiative, Grant GBMF4306, stable) “lump” in the density of atoms in the excited Rydberg and the National University of Singapore Development Grant AY2019/2020. state; interference experiments between such excitations could The computations in this paper were run on the Faculty of Arts and Sci- ences Research Computing Cannon cluster supported by the Faculty of Arts be used to scrutinize braiding statistics. The dynamic structure and Sciences Division of Science Research Computing Group at Harvard factor can also provide signatures of fractionalization: Dispersive University.

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