Prediction of Toric Code Topological Order from Rydberg Blockade
PPrediction of Toric Code Topological Order from Rydberg Blockade
Ruben Verresen, Mikhail D. Lukin, and Ashvin Vishwanath
Department of Physics, Harvard University, Cambridge, MA 02138, USA (Dated: November 26, 2020)The physical realization of Z topological order as encountered in the paradigmatic toric code hasproven to be an elusive goal. We show that this phase of matter can be created in a two-dimensionalarray of strongly interacting Rydberg atoms. Our proposal makes use of atoms localized on the sitesof a ruby lattice, coupled via a Rydberg blockade mechanism. First, we show that the blockade modeleffectively realizes a monomer-dimer model on the kagome lattice with a single-site kinetic term,and we obtain its phase diagram using the numerical density matrix renormalization group method.We find a topological quantum liquid (TQL) as evidenced by multiple measures including (i) acontinuous transition between two featureless phases, (ii) a topological entanglement entropy of ln 2as measured in various geometries, (iii) degenerate topological ground states and (iv) the expectedmodular matrix from ground state overlap. Next, we show that the TQL can persist upon includingrealistic, algebraically-decaying van der Waals interactions V ( r ) ∼ /r . Moreover, we can directlyaccess the topological loop operators of this model, which can be measured experimentally using adynamic protocol, providing a “smoking gun” experimental signature of the TQL phase. Finally,we show how to trap an emergent anyon and realize different topological boundary conditions, andwe discuss the implications for exploring fault-tolerant quantum memories. CONTENTS
I. Introduction 1II. Rydberg blockade model 3A. Connection to and differences from dimermodels 3B. Phase diagram 4C. Topological entanglement entropy 5D. String operators and anyon condensation 6E. Topological ground state degeneracy andmodular matrices 7III. Prospects for realization and detection 9A. Quantum liquid for ∼ /r potential and afamily of ruby lattices 9B. Measuring an off-diagonal string bytransforming it into a diagonal string 10IV. Towards fault-tolerant quantum memory 11A. Trapping an e -anyon 11B. Boundary phase diagram 11C. Creating topological degeneracy on the plane 12V. Outlook 13Acknowledgments 14References 14A. Duality between topological string operators 18 I. INTRODUCTION
Nearly five decades ago, Anderson [1] proposed thatquantum fluctuations could lead to a liquid of resonatingvalence bonds, stimulating a vast theoretical effort that continues to this day. Further work related this idea tothe more precise notion of a gapped quantum spin liq-uid , an exotic state potentially realized in frustratedmagnets [2–5]. At the same time, it was understood thatsuch gapped quantum liquids involve topological order[6–8], the simplest example being Z topological order intwo spatial dimensions.Phases of matter with topological order exhibit a num-ber of remarkable properties [9]. First, they imply theemergence of gauge fields, analogous to those describingthe fundamental forces, although the gauge group andother details differ. Thus, Z topological order is associ-ated with a deconfined Z (Ising) gauge group [10, 11].Second, despite being built from bosonic degrees of free-dom, the excitations of such quantum spin liquids arequasiparticles with nontrivial quantum statistics. For ex-ample, the Z spin liquid includes three nontrivial exci-tations, two of which, the electric and magnetic particles, e and m , are bosons, while their combination f = em isa fermion. All three particles acquire a sign change oncircling another anyon, i.e., they have semionic mutualstatistics. These nontrivial statistics immediately lead tothe remarkable property that the ground states of a topo-logically ordered system must be degenerate when real-ized on certain manifolds, such as a torus. Third, thereis a remarkable link between superconductivity and Z quantum spin liquids [12–18]—while the fermion f canbe associated with Bogoliubov quasiparticles, the e, m excitations are related to the superconductor vortices.This led to earlier proposals suggesting that Z topologi-cal order might be key to understanding the phenomenonof high-temperature superconductivity. In this work, we will interchangeably refer to this as a topologicalquantum liquid or spin liquid—even if the bosonic degrees offreedom are not spins but represent, e.g., a two-level atomic state. a r X i v : . [ c ond - m a t . s t r- e l ] N ov (a) (b)(c) FIG. 1.
Rydberg blockade model and relation to dimermodel. (a) Hardcore bosons on the links of the kagome lat-tice (forming the ruby lattice) are strongly-repelling, punish-ing double-occupation within the disk r ≤ r = 2 a . (b) Anexample of a state consistent with the Rydberg blockade atmaximal filling. (c) Since the blockade forbids occupation ofany two touching bonds, we can equivalently draw the config-uration as a dimer covering on the kagome lattice. Finally, a key characteristic of topological order—thelong-ranged nature of its entanglement—was pointed out[19]. On the one hand, this implies that topologically or-dered states of matter realize an entirely new form of en-tangled quantum matter, unlike any other conventionalground states realized to date [20–22]. On the otherhand, this observation also has profound implications inareas such as quantum error correction and fault-tolerantquantum computation. The Z topological order under-lies the ‘toric code’ [19, 23] and ‘surface code’ [24] modelsfor topologically protected quantum memory, which en-code logical quantum bits in degenerate ground states.Since these degenerate ground states cannot be distin-guished by local measurements, quantum information en-coded in them is naturally protected from decoherence.Such intrinsic topological fault tolerance is of great con-sequence in the quest to building robust quantum infor-mation processing devices [25, 26].Due to these considerations, realizing Z topologicalorder has been a major goal of condensed matter re-search. Unfortunately, despite several decades of theo-retical and experimental effort [4, 23, 27–31], no clear-cut realization of Z topological order has been obtainedto date. While topologically ordered states appear inthe context of the fractional quantum Hall effect [32, 33],they are realized under rather special conditions of strongmagnetic fields. In contrast, realizing topological order ina time-reversal invariant system remains a major unful-filled research goal. Such a realization would avoid theneed for applying strong magnetic fields, which is par-ticularly challenging for neutral objects. Furthermore,non-chiral topological orders can be achieved, in which agap can be maintained even at the boundaries. In fact,we note that no realization of topological order in an in-trinsically bosonic or spin system has been conclusivelyidentified to date [34].Recently, a new approach for exploring quantum manybody physics has emerged. It is based on neutral atom arrays trapped in optical tweezer arrays. Tunable atominteractions can be engineered in such systems using theRydberg blockade mechanism [35–38], mediated by laserexcitation of atoms into the Rydberg states [39, 40]. Sig-nificant progress in realizing two dimensional quantumlattice models from the atom arrays was achieved, and arich phase diagram of symmetry breaking orders has beenpredicted [41]. At the same time, the special featuresof the Rydberg atom interactions make them attractiveplatforms for realizing emergent lattice gauge theoriesand quantum dimer models [42–47].Here we introduce a new approach for realizing a Z topologically ordered state as the ground state of a 2DRydberg atom array. We show that this approach doesnot require careful engineering or fine-tuning of the con-straints, enabling the first realization and direct probingof a time reversal and parity invariant topological order,and of emergent deconfined gauge fields in a quantummodel on a near-term quantum device.Our approach for realizing a topological spin liquid isbased on the Rydberg blockade [35–38]: when a neutralatom is excited into a Rydberg state with a high principalquantum number, the resonant excitation of the nearbyatoms is suppressed due to strong atom-atom interac-tions. A minimal effective Hamiltonian for the Rydbergarray—where the possibility of exciting an atom into aRydberg state is described by two-level system—is theso-called P XP model H = (cid:80) i (Ω P σ xi P − δ σ zi ). Here P projects out states that violate the blockade, and Ωis the Rabi frequency between the two levels, which isdriven by a laser with detuning δ . For a fixed blockaderadius, this model, which depends on a single parameter δ/ Ω, has been explored in great detail in one dimension—both theoretically and experimentally—where it led to arich phenomenology including quantum scars [48–56] andlattice gauge theories [57]. In this work, we show thatfor a particular choice of two-dimensional atom arrange-ment, Rydberg blockade radius and laser detuning, a Z spin liquid is stabilized as the ground state of this model.To be specific, we first focus on the P XP model onthe so-called ruby lattice—equivalently, the links of thekagome lattice—with the blockade radius containing sixnearby sites (see Fig. 1(a)). By tuning δ , we find a phasetransition from the trivial phase into another featurelessphase of matter. We determine that the latter is a Z spin liquid using a variety of probes, including the topo-logical entanglement entropy, ground state degeneracies,and modular transformations.These results can be understood by noting that forthe above lattice and blockade radius, the Hamiltonianbecomes equivalent to a dimer-monomer model on thekagome lattice. While it is known that dimer models onnon-bipartite lattices (such as the triangular and kagomelattice) can realize a Z spin liquid, they are notoriouslyhard to implement in experiment. Indeed, even to real-ize the Hilbert space of a dimer model requires specialinteractions. Furthermore, one needs the right Hamilto-nian to drive the model into a spin liquid phase. For in-stance, Ref. [29] discovered a remarkable exactly-soluble Z dimer liquid on the kagome lattice, which however re-quires 32 distinct dimer resonances. If one only includesthe lowest order dimer moves, a valence bond solid is re-alized [58, 59] rather than a spin liquid. The novel insightin the present work is that by including monomers, theeffective Hamiltonian only needs a single-site kinetic term(the creation and destruction of monomers) to perturba-tively generate the multi-site dimer resonances necessaryfor a spin liquid. While dimer-monomer models have arich history [15, 60–64], to the best of our knowledge theyhave not yet been studied with a minimal kinetic termgenerating a rich phenomenology. Dimer-monomer mod-els of this type could provide a new paradigm for thephysical realization of lattice gauge theories, going wellbeyond the example studied in this work.Furthermore, we show that the above findings are notfine-tuned to the P XP model. More precisely, we nu-merically confirm that the spin liquid can also be foundin the full-fledged Hamiltonian with realistic V ( r ) ∼ /r Van der Waals interactions between the Rydberg atomson a particular instance of the ruby lattice.In addition to realizing a Z spin liquid in anexperimentally-relevant model, a very useful property ofthis model is that it also gives a direct handle on thetwo topological string operators. In the language of lat-tice gauge theory, these are the Wilson and ’t Hooftlines. In the context of topological order, these are thestrings whose endpoints host an e - and m -anyon, respec-tively. We explicitly construct these operators on thelattice and confirm the expected behavior of loop opera-tors in the spin liquid, as well as re-interpret the nearbyphases as e - and m -condensates using the Bricmont-Fr¨ohlich-Fredenhagen-Marcu string order parameter [65–70]. These string operators also serve as very usefulprobes to detect the spin liquid in experiments. Whilethe diagonal string operator can be readily measured, it isdesirable to also measure the dual off-diagonal string op-erators. We further show how the string operator for the e -anyon—which a priori involves off-diagonal operationswhich are hard to measure in the lab—can be convertedinto a diagonal string operator by time-evolving with aHamiltonian whose blockade radius has been quenched.Thus, we show that both string operators become mea-surable in the diagonal basis.Finally, we discuss methods to create and manipulatequantum information stored in topologically degenerateground states, paving the way for potential exploration oftopological quantum memories. Two crucial pieces of thepuzzle we identify are the ability to trap an e -anyon andto create distinct topological boundary conditions—bothare straightforwardly achieved by locally changing thelaser detuning. As we will explain, these two ingredientsalready give access to topologically-degenerate qubits inthe plane which can be initialized and read out.The remainder of the paper is structured as follows.Section II concerns the Rydberg blockade model, withsubsection II A comparing it to and distinguishing it from conventional dimer models. Its phase diagram is ob-tained in subsection II B, containing a trivial phase, a Z spin liquid, and a valence bond solid. We confirm thatthe intermediate phase is indeed a spin liquid in termsof its topological entanglement entropy (subsection II C),its topological string operators (subsection II D) and itstopologically-distinct ground states from which we ex-tract part of the modular matrices (subsection II E). Sec-tion III focuses on the experimental feasibility, with sub-section III A showing that the spin liquid persists uponincluding the V ( r ) ∼ /r potential and subsection III Bexplaining how the off-diagonal string operator can be re-duced to a diagonal observable. We end with section IVtaking the first steps towards using this novel realizationfor creating a fault-tolerant quantum memory by show-ing how to trap e -anyons (subsection IV A) and how re-alize distinct boundary conditions (subsection IV B); sec-tion IV C then gives examples of how this can be applied. II. RYDBERG BLOCKADE MODEL
We consider hardcore bosons on the links of the kagomelattice with a two-dimensional version of the Fendley-Sengupta-Sachdev model [71]: H = Ω2 (cid:88) i (cid:16) b i + b † i (cid:17) − δ (cid:88) i n i + 12 (cid:88) i , j V ( | i − j | ) n i n j . (1)We set Ω >
0. For Rydberg atoms, V ( r ) ∼ /r . Wedefer that case to section III. In this and the next twosections, we instead focus on the simpler model where V ( r ) forms a blockade in a particular disk: V ( r ) = (cid:26) + ∞ if r ≤ a r > a. (2)Here the lattice spacing a is the shortest distance betweentwo atoms. As shown in Fig. 1(a), with this interactionrange, a given site is coupled to six other sites, which areordered in pairs at distances r = a , r = √ a ≈ . a and r = 2 a (the next distance would be r = √ a ≈ . a , denoted by the dashed circle in Fig. 1(a)). TheRydberg blockade implies that any two sites within thisdistance cannot both be occupied (Fig. 1(b)), which wecan interpret as a dimer state on the kagome lattice ifthe system is at maximal filling (see Fig. 1(c)). A. Connection to and differences from dimermodels
For a dimer state on the kagome lattice, each vertex istouched by exactly one dimer, such that (cid:104) n (cid:105) = . Our Note that the sign of Ω can be toggled by replacing b i → − b i ,which leaves n i invariant. The only place in this paper wherethe sign of Ω matters is in the definition of the topological stringoperators; see section II D . FIG. 2.
Phase diagram of Rydberg blockade model on the links of the kagome lattice.
The trivial phase at small δ/ Ω is separated from the valence bond solid (VBS) at large δ/ Ω by an intermediate phase which has a large entanglementplateau. We show an exemplary density plot for each of the three phases, which shows that the intermediate phase is featureless.The VBS phase has a 36 site unit cell (72 atoms on the links) highlighted by the gray shaded region—this pattern was studiedbefore in Refs. [58, 59, 72] in the context of the spin-1 / model can have (cid:104) n (cid:105) < , in which case certain verticeshave no dimers—referred to as a monomer. This distin-guishes our system from a usual dimer model. Let usbriefly discuss the implications of this difference. Thereader interested in the numerical results for our modelcan skip ahead to section II B.The constraint of a dimer model—having exactly onedimer per vertex—can be interpreted as a Gauss law.More precisely, the presence or absence of a dimer rep-resents a Z -valued electric field, with the dimer con-straint enforcing the lattice version of the Gauss law ∇ · E = 1 (mod 2). Each vertex thus carries a clas-sical/static electric charge e . For this reason, a dimermodel is also referred to as an odd Z gauge theory[27, 73]. The absence of dynamic matter in a dimer modelimplies that it is a pure Z gauge theory, which has twopossible phases: a deconfined and a confined phase. Theformer is our desired Z spin liquid (or equivalently, aresonating valence bond state), whereas the latter is avalence bond solid . Stabilizing the spin liquid requiresdimer resonances in the Hamiltonian, but due to the lo-cal constraint of a dimer model, these terms typicallyspan many sites. The smallest resonance acts on the sixsites around a hexagon of the kagome lattice. The solv-able dimer model by Misguich, Serban and Pasquier [29]requires 32 distinct types of resonances, the largest span-ning 12 sites. While these conditions can be somewhatrelaxed [74], the direct implementation of dimer models,tuned to a regime of parameter space where a liquid phaseis known to emerge, remains extremely challenging.In contrast, the Rydberg blockade model (1) is a dimer-monomer model. In other words, the Gauss law of the This refers to the freedom of test charges which in this case aremonomers. A more apt name would be a dimer liquid, but here we followthe more common terminology ingrained in the literature. The confined phase is a condensate of the magnetic excitation m .As explained in section II D, this anyon carries a projective rep-resentation under translation such that its condensation impliestranslation symmetry breaking. lattice gauge theory is now ∇ · E = ρ , where ρ is aquantum-mechanical two-level degree of freedom. Thishas two advantages. Firstly, the only explicit dynamicsin our model is a single-site term which creates and de-stroys pairs of monomers/charges (the Rabi oscillation Ωin Eq. (1)). In the limit of large δ Ω , the low-energy theoryis projected into the macroscopically degenerate spaceof (maximally-filled) dimer states. Virtual monomer ex-citations induce dimer resonances between these states.For instance, at leading order in perturbation theory, weobtain H eff = − δ (cid:80) (cid:55) (cid:16)(cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) + h.c. (cid:17) , describ-ing hexagon resonances. Second, since monomers arenow dynamical degrees of freedom, they can be con-densed, driving the system to a translation-symmetrictrivial state . This gives a clear-cut instance of a con-tinuous phase transition between two featureless phasesof matter (as opposed to the valence bond solid, whichhas long-range order), which does not involve any sym-metries.While there are thus clear advantages to not realizinga strict dimer model but rather a dimer-monomer model,it is also advantageous to nevertheless be proximate to adimer model (i.e., have low monomer density). Firstly,it is a good place to hunt for a spin liquid, since—asdiscussed above—a dimer model on the kagome latticecannot realize a trivial phase of matter. Secondly, onehas a direct handle on the topological string operatorsassociated to the Z gauge theory, with anyons living attheir endpoints. We discuss this in detail in section II D. B. Phase diagram
We now study the phase diagram of the model inEq. (1) with the blockade in Eq. (2) using the densitymatrix renormalization group (DMRG) [75–77]. We can In the language of Z gauge theory coupled to matter, this cor-responds to the Higgs phase. . . . . δ/ Ω . . . h n i . . . d h n i / d ( δ / Ω ) . . δ/ Ω . . . n m a x − n m i n (a) (b) FIG. 3.
Detecting phase transitions via filling fraction.
This data is obtained for an infinitely-long cylinder with XC-8geometry. (a) The filling fraction has a singular behavior upontransitioning from the trivial phase into the spin liquid, afterwhich the system enters a regime where (cid:104) n (cid:105) ≈ .
25, consistentwith it being an approximate dimer state. (b) The spin liquidand VBS phase are separated by a first order transition. explicitly enforce V ( r ) = + ∞ by working in the reducedHilbert space where each triangle of the kagome lattice(containing three atoms) only has four states: empty ora dimer on one of the three legs. We cannot similarly set V ( r ) = V ( r ) = + ∞ since the resulting Hilbert space isno longer a tensor product —indeed, this is the magic ofdimer models. Hence, we enforce these constraints ener-getically by choosing a very large V ( r ) = V ( r ) = 50Ω.We have confirmed that our results do not depend onthe details of this choice. We study the model on a cylin-der geometry of fixed circumference (up to XC-12) andinfinite extent [79].When δ/ Ω is low enough, the system is adiabaticallyconnected to the empty state and is thereby completelytrivial. For very large δ/ Ω we enter the regime that is per-turbatively described by a dimer model, as explained insection II A. We find that its ground state spontaneouslybreaks crystalline symmetries and forms a valence bondsolid (VBS). Remarkably, for intermediate δ/ Ω, thesetwo phases are separated by another featureless phase,as shown in Fig. 2 by the diverging correlation length ξ and the entanglement entropy S between two rings of thecylinder. We will argue that this is a Z spin liquid.As a first indication that this intermediate phase is stillwithin the approximate dimer model, we consider thefilling fraction (cid:104) n (cid:105) , shown by the red curve in Fig. 3(a).We see that as δ/ Ω → ∞ , the filling (cid:104) n (cid:105) approaches themaximal 1 / (cid:104) n (cid:105) is still large. It isonly when δ/ Ω is decreased further—entering the trivialphase—that (cid:104) n (cid:105) sharply drops. This is in line with thepossible scenario of exiting the spin liquid by condensingmonomers—as explained in section II A—which would Recently, Chepiga and Mila demonstrated how to performDMRG directly in such constrained Hilbert spaces for a vari-ety of one-dimensional models [78]. It would be very interestingto generalize this method to the current two-dimensional setting. L circ /a − S δ/ Ω = 1 − ln(2)0 1 . . . δ/ Ω γ L circ /a − S δ/ Ω = 1 . − ln(2)0 (a) (b) (c) FIG. 4.
Topological entanglement entropy.
We deter-mine the offset γ in the area law S = αL − γ . (a) For thetrivial phase, this is zero. (b) As we increase δ/ Ω, we enter thespin liquid where γ ≈ ln 2. Here, we plot S L =8 − S L =4 where S L = n is the bipartition entanglement entropy for the XC- n geometry. (c) For an exemplary point in the splin liquid, weextract γ for two distinct geometries (up to XC-12 and YC-8). Note that XC- n (YC- n ) has circumference L circ /a = √ n (2 n ). exhibit itself in a rapid drop of filling density.Moreover, the derivative of (cid:104) n (cid:105) diverges at the transi-tion between the trivial phase and the spin liquid, sig-naling a continuous transition. Indeed, the theoreticalexpectation is that this belongs to the 2 + 1 D Ising uni-versality class (with the trivial phase corresponding tothe ‘ordered’ side), but our available system sizes arenot big enough to accurately extract scaling dimensions.Fig. 3(a) shows no such singularity between the spin liq-uid and VBS phase. However, it turns out that it is afirst order transition which is very hard to diagnose thisway (due to the small energy scales associated to theVBS phase). This is much more easily demonstrated byconsidering the variation of (cid:104) n (cid:105) between different sites:Fig. 3(b) shows that this jumps discontinuously. C. Topological entanglement entropy
One characteristic feature of topological phases of mat-ter can be found in the scaling of the entanglement en-tropy. Gapped phases of matter satisfy an area law: fora region with perimeter L , we have S ( L ) = αL − γ .The constant offset γ is a universal property called thetopological entanglement entropy, encoding informationabout the quantum dimensions of the anyons of the topo-logical order [21]. For a Z spin liquid, γ = ln 2 [20].We take a point in the middle of the presumed spinliquid in Fig. 2, δ/ Ω = 1 .
7, and numerically obtain theentanglement entropy upon bipartitioning the infinitely-long cylinder in two halves. Doing this for different cir-cumferences , we extract γ ≈ ln 2, as shown in Fig. 4(c). As will be explained in section II E, there are degenerate groundstates on the cylinder. To make sure we are comparing applesto apples, we choose the | (cid:105) ground state on each cylinder asdetermined by the string operators discussed in section II D. (a) (b) (c) FIG. 5.
Topological string operators. (a) The two different string operators are defined by their action on a single triangle.We call the diagonal and off-diagonal string operators P and Q , respectively. (b) An example of the action of the stringoperators on a classical dimer state. (c) The definition of the Bricmont-Fr¨ohlich-Fredenhagen-Marcu order parameter [ref]is shown for the diagonal string, (cid:104) P (cid:105) BFFM , which measures the condensation of the m -anyon. The analogous definition for (cid:104) Q (cid:105) BFFM (not shown) measures an e -condensate. Importantly, it has been observed before that one can ob-tain spurious value of γ for very specific cuts in certainlattice models, i.e., one can be deceived into thinkinga trivial phase is in fact topologically ordered [80–82].For this reason, we have extracted γ for two distinct ge-ometries: XC (where the finite periodic direction bisectstriangles of the kagome lattice) and YC (where the cir-cumference runs parallel to one of the axes of the kagomelattice). Both linear fits give a topological entanglemententropy which is remarkably close to ln 2. For compari-son, for a point in the trivial phase ( δ/ Ω = 1) we obtain γ = 0 (Fig. 4(a)).To confirm that the above is not a fine-tuned featureof a particular point in the phase diagram, we extract γ as a function of δ/ Ω. Figure 4(b) indeed shows a plateauwhere γ ≈ ln 2, consistent with a Z spin liquid. Notethat we do not consider γ in the VBS phase since due tothe large unit cell (shown in Fig. 2) the next consistentgeometry is XC-16, which is out of reach with currentmethods. D. String operators and anyon condensation
The advantage of measuring topological entanglemententropy is that it is well-defined for any model even in theabsence of microscopic identification of operators corre-sponding to emergent gauge theory. However, in our Ry-dberg blockade model, a more microscopic understandingof the spin liquid is available. Here, we can identify thetopological string operators associated with this Z lat-tice gauge theory, similar to the toric code model [19].Such an explicit representation of a topological quantumliquid has a variety of uses: in identifying the spin liquid There is also a VBS phase on, e.g., XC-4 and XC-12, but theyhave different patterns and it is thus not meaningful to comparetheir entropies. and its nearby phases (especially in an experimental set-up where, e.g., topological entanglement entropy is notreadily accessible), in creating anyons, in distinguishingtopological ground states and also perhaps for quantuminformation applications, such as the initialization andread-out of topological qubits.A Z lattice gauge theory comes with two string oper-ators determined by the electric field E (defined modulo2) and its conjugate variable, the gauge field A . Thesestrings are the ’t Hooft line e iπ (cid:82) E and the Wilson line e i (cid:82) A , which anticommute at intersection points. As al-ready mentioned in section II A, the binary-valued elec-tric field corresponds to a dimer configuration, with thehardcore dimer constraint acting as a Gauss law. Thestring operator e iπ (cid:82) E thus corresponds to the parity ofdimers along a string. To be precise, we define its actionon a single triangle in Fig. 5(a) (orange dashed line); werefer to this diagonal parity string as P . Due to theGauss law, evaluating it along any closed loop—whichhas to run perpendicular to the bonds of the kagomelattice—measures the charge inside of it. In the ab-sence of monomers—i.e., gauge charge excitations—thisis simply ( − for a contractible loop, asexpected of an odd Z gauge theory. In contrast, non-contractible loops distinguish topologically-distinct sec-tors of the dimer Hilbert space (since this value cannotbe changed by any local operator).In the dimer basis, the dual string e i (cid:82) A has to beoff-diagonal, shuffling the dimers. There is essentiallya unique way of defining such a string that has a well-defined action on single triangles, as shown in Fig. 5(a)(solid blue line); we refer to this string as Q . An exampleis shown in Fig. 5(b). Note that any closed string thatruns parallel to the bonds of the kagome lattice indeedmaps a valid dimer configuration to another valid dimerconfiguration, and it is also easy to see that this string Q anticommutes with P whenever the strings intersect. Tothe best of our knowledge, this definition of the Q stringis novel; for dimer models, one often considers the more FIG. 6.
Diagnosing phases in terms of topologicalstring operators.
Using the definition of the Bricmont-Fr¨ohlich-Fredenhagen-Marcu (BFFM) string order parameterin Fig. 5(c), we find that the trivial phase is an e -condensateand the VBS phase is an m -condensate. These string orderparameters decay to zero in the spin liquid, confirming that itis the deconfined phase of the Z lattice gauge theory. This isfurther backed up by the O (1) results for the P and Q loopsaround the circumference. restrictive strings that have to pass through an alternat-ing series of empty and filled bonds. The advantage ofthis more general definition is twofold: (1) it is also well-defined for states that contain monomers, and (2) withthe definitions for P and Q in Fig. 5(a), there is in facta duality transformation that interchanges them, as dis-cussed in section III B.The electric e and magnetic m excitations of this Z lattice gauge theory live at the endpoints of the Q and P strings, respectively. For instance, Fig. 5(b) shows how an open Q string indeed creates a monomer at eachend. These e and m excitations are topological since theycan only be created in pairs. Moreover, whilst they areindividually bosonic, the anticommuting property of the P and Q string encodes the fact that e and m have non-trivial mutual statistics; equivalently, the endpoint of theproduct string P Q carries an emergent fermion f .The spin liquid is defined by the deconfinement ofthese excitations. The nearby phases correspond to con-densing either the e or the m , which respectively con-fines m or e due to the mutual statistics. Historically,the e -condensate is called the Higgs phase, whereas the m -condensate is called the confined phase (due to thecharged e excitations becoming confined). In an oddgauge theory, with nonzero background gauge charge ateach lattice site, the latter in fact implies spontaneoussymmetry breaking (i.e., a valence bond solid). The rea-son for this is that the m -anyon carries a projective rep-resentation under the Z × Z translation symmetry. Similarly, the open string P in Fig. 5(b) creates m excitations,but this is hard to see since it is acting on a classical dimer state,which is an m -condensate. This is simply a restatement of the Gauss law that the parity
These condensates can be diagnosed by the open P or Q strings attaining long-range order. To properly de-fine what this means, it is important to normalize thesestring operators. Indeed, generically these strings willdecay to zero since the ground state has virtual e and m fluctuations. For this reason, Bricmont and Fr¨ohlich [65]and Fredenhagen and Marcu [66, 67, 69] independentlyintroduced the normalized string operator in Fig. 5(c),which we will refer to as the BFFM string order param-eter. This was also more recently imported into the con-densed matter context—where lattice gauge theories areemergent—by Gregor, Huse, Moessner and Sondhi [70].These two string order parameters are a very useful toolfor diagnosing the different phases of a lattice gauge the-ory, especially since in the presence dynamic matter allloop operators scale with a perimeter law [11].The only remaining technicality to discuss is the phasefactor e iα in the definition of the off-diagonal string Q inFig. 5(a). In general this phase factor cancels out unlessthe Q string changes the total number of dimers, suchas for the open string in Fig. 5(b). Hence, the optimalchoice of e iα depends on the phase difference betweendifferent branches of the ground state wave function withdistinct particle number. In the present model, one canstraightforwardly argue that if the Rabi frequency Ω < > e iα = − Ω | Ω | .We are now in a position to evaluate the open stringand loop operators in the Rydberg blockade model.The results are shown in Fig. 6. As expected, we seethat Q only has long-range order in the trivial phase—corresponding to an e -condensate—whereas P only haslong-range order in the VBS phase—corresponding toan m -condensate. In the intermediate spin liquid, bothBFFM order parameters decay to zero, consistent withthe claim that this is the deconfined phase of the lat-tice gauge theory. Correspondingly, in this regime,the loop operators evaluated around the circumferenceare not suppressed and have an O (1) value (which al-beit decreases with circumference). In fact, the sign ofthis nonzero number labels topologically-distinct groundstates, as we discuss next. E. Topological ground state degeneracy andmodular matrices
Another fingerprint of a topological spin liquid is itstopological ground state degeneracy on manifolds whichare themselves topologically non-trivial [7, 83, 84]. ForAbelian topological order on an infinitely-long cylinder, along a loop surrounding a vertex is −
1: this parity loop can beinterpreted as the anticommutator T x T y T − x T − y for the actionof translations T x,y on the endpoint of a parity string P , i.e., the m -anyon. (a) (b) FIG. 7.
Ground states and modular transformations.
From the ground states on the infinitely-long cylinder, wecan obtain minimally-entangled ground states on the torusgeometry. For the smaller geometry, we show that whilstthe π/ δ/ Ω = 1) orsymmetry-breaking ( δ/ Ω = 2 .
5) phases, it leads to a non-trivial overlap in the spin liquid ( δ/ Ω = 1 . Z spin liquid. The overlaps are shownas a function of the Monte Carlo sweeps, converging towardthe value ≈ . one has a ground state corresponding to each anyon inthe theory. Conceptually, the way these different statesare related is by starting with one of the ground statesand nucleating an anyon pair and separating them in-finitely far along the infinite direction of the cylinder .For the present case, we thus expect four distinct topo-logical ground states, corresponding to 1, e , m and f lines threaded along the infinite axis. Due to the mutualstatistics, these distinct ground states can be diagnosedby measuring the P and Q loops around the circumfer-ence.Numerically, when we repeat DMRG with differentrandom initializations, we find two (quasi-)degenerateground states which are distinguished by the sign of (cid:104) P (cid:105) loop around the circumference . It is tempting to as-sociate these to the trivial anyon and the electric charge,1 and e .To make this concrete, we use the technique ofRefs. [85, 86]: making the resulting matrix product states One could of course instead choose to wrap them around thefinite direction, which would generate a different basis in thisfour-dimensional space of states. However, these states will notbe minimally-entangled on the cylinder, whereas DMRG opti-mizes for that [85, 86]. Equivalently, if one creates an initial state with a given sign ofthe parity loop, DMRG will remain in this sector. This does notwork for the Q loop, presumably because its finite-size effectsalong the circumference are big enough for DMRG to switchsectors. E h P i ≈ +0 . h Q i ≈ +0 . e h P i ≈ − . h Q i ≈ +0 . m h P i ≈ +0 . h Q i ≈ − . f h P i ≈ − . h Q i ≈ − . ≈ . em , f XC-8 ≈ . FIG. 8.
Topological ground state degeneracy.
In thetopological phase, we obtain the 1 and e ground states fromDMRG with random initial states. The m and f states areobtained by starting from fixed-point resonating dimer statesand subsequently applying imaginary time-evolution. The en-ergies shown are for δ/ Ω = 1 .
7. In light gray we also show theeigenvalues of the P and Q loop operators around the circum-ference (for YC-4); the four ground states are characterizedby the signs of these numbers. periodic along the second direction, one obtains wave-functions on a torus geometry as shown in Fig. 7, whichwe denote by | (cid:105) and | e (cid:105) . It can be shown that a π/ (cid:104) | R π/ | (cid:105) using quantum MonteCarlo [87], we see for the smaller torus (of 24 sites) inFig. 7(a) that (i) the ground state of the trivial phaseis completely symmetric, (ii) the ground state of thesymmetry-broken phase gives a vanishing overlap withthe rotated wavefunction, and (iii) the ground state inthe presumed spin liquid gives a finite overlap, suggestingthat it has overlap with a finite number of other states.In fact, its value is universal and can be derived as inRef. [85]. In particular, the relevant 2 × (cid:18) (cid:104) | R π/ | (cid:105) (cid:104) | R π/ | e (cid:105)(cid:104) e | R π/ | (cid:105) (cid:104) e | R π/ | e (cid:105) (cid:19) = 12 (cid:18) (cid:19) . (3)Whereas the value of (cid:104) | R π/ | (cid:105) for the smaller torus isslightly above 1 / e anyon is by constructingthe fixed-point wavefunctions, for which we find a largeoverlap. More precisely, we define | (cid:105) fix as the state onthe cylinder that corresponds to the superposition of alldimer configurations for which (cid:104) P (cid:105) loop = (cid:104) Q (cid:105) loop = 1around the circumference. The other three fixed-pointwavefunctions | e (cid:105) fix , | m (cid:105) fix , | f (cid:105) fix are then obtained by re-spectively applying a Q , P and P Q string along theinfinitely-long axis of the cylinder. We have confirmedthat if we start from the fixed-point wavefunctions for | (cid:105) fix and | e (cid:105) fix and peform imaginary time evolution, weconverge toward the two ground states found by DMRG.This naturally gives us a way of also obtaining the groundstates corresponding to the vison or magnetic particle m ,and the fermion f . We have confirmed that the finite-size(a) (b) (c) FIG. 9.
The ruby lattice. (a) Atoms on the links of thekagome lattice form the vertices of a ruby lattice where therectangle has an aspect ratio ρ = √
3. (b) The ruby latticewith ρ = 1. (c) The ruby lattice with ρ = 3. The radii of thedisks denote the successive interaction distances included inthe numerical calculations for Fig. 10. splitting of these four topological ground states decreaseswith circumference, plotted in Fig. 8.A further characterization beyond topological order in-volves the implementation of symmetry, i.e., symmetryenrichment of topological order [9]. This can be deducedfrom the relation to the kagome lattice dimer model,albeit in the absence of spin rotation symmetry (sincemonomers carry no spin). We expect the relevant pro-jective symmetry group to be that of the bosonic meanfield Q = − Q state of [4], which has been related toother mean field representations in [88–90] . A caveat isthat lattice symmetry enrichment, which implies a back-ground ‘ e ’ particle associated to each kagome site, canmodify ground state overlap matrices for certain systemsizes. III. PROSPECTS FOR REALIZATION ANDDETECTION
In section II we established that the Rydberg blockademodel realizes a Z spin liquid for a range of parameters.The purpose of this section is twofold. First, we we wouldlike to show that this result is not limited to the block-ade model in Eq. (2): the spin liquid persists on adoptingthe realistic Rydberg potential. Second, we would like tohave a way to diagnose the existence of the spin liquidusing probes available in Rydberg experiments. In lightof that, we discuss how the string operators can be mea-sured in the lab. A. Quantum liquid for ∼ /r potential and afamily of ruby lattices We now consider the Rydberg Hamiltonian in Eq. (1)with the algebraically-decaying potential V ( r ) = Ω( r/R b ) ; The Q = − Q state of [4] is equivalent to the [0Hex,0Rhom] of[88], which from Ref. [89] is identified with the Z [0 , π ] β fermionicstate of Ref. [90]. R b is commonly referred to as the (Rydberg) blockaderadius due to sites well within this distance experiencinga large potential, effectively a blockade of the type dis-cussed in section II. Since V ( r ) now explicitly depends onthe distances between the atoms, it is important to dis-cuss the geometry of the lattice. In the blockade model,we specified that the atoms live on the links of the kagomelattice (see Fig. 1(a)). These atoms form the vertices ofthe so-called ruby lattice, demonstrated in Fig. 9(a). Inthis particular case, we see that the rectangles of theruby lattice have an aspect ratio ρ = √
3. However, ρ isa free tuning parameter ; as long as ρ > / √ ≈ . R b to be large enough to enclose thesesix nearest sites (which are enclosed in a disk of radius r /a = (cid:112) ρ ), the resulting model approximates theblockade model. However, due to the 1 /r interaction,we have additional longer-range couplings, and it is non-trivial to know whether or not the spin liquid will bestable to this. For this same reason, we will want totake R b smaller than the next interaction radius, i.e., asa rough guideline for where to search for the spin liquid: (cid:112) ρ < R b a < min (cid:26) √ ρ, (cid:113) √ ρ + ρ (cid:27) . (4)As an example for illustrating the subtle influence ofsuch long-range potential, we focus on ρ = 3, shown inFig. 9(c). The rule of thumb in Eq. (4) suggests that weshould look for spin liquid in the range of 3 . < R b /a < .
9. To numerically simulate the model, we have to trun-cate V ( r ) after some distance. For this large choice of ρ , the triangles are further removed from one another,so a natural choice is to only keep interactions that cou-ple nearest-neighbor triangles. This corresponds to fivedistinct interaction radii, covering the dark blue disk inFig. 9(c). Numerically, we indeed find a Z spin liquidfor a choice of blockade radius R b = 3 . a .We next explore its stability by including yet morefurther-range couplings. We find that upon including oneextra interaction range (the light blue disk in Fig. 9(c)),the spin liquid appears to be destabilized. This result canbe understood intuitively by noting that this couplingpunishes hexagon flipping resonances which are essentialfor a spin liquid. Since this problematic coupling pushesdimers away from hexagons, it stands to reason that in-cluding yet more interactions can perhaps attenuate itsnegative effect. Indeed, upon including all seven distinctinteraction radii shown in Fig. 9(c), we again find a stable Z liquid for R b = 3 . a , as shown in Fig. 10. We observethat as in the blockade model, the spin liquid is char-acterized by a featureless density plots and topologicallydegenerate ground states which are distinguished by the While only ρ = 1 is an Archimedean lattice, the group of crys-talline symmetries is the same for all ρ . FIG. 10.
Spin liquid on ruby lattice ( ρ = 3) with V ( r ) ∼ /r . We consider the lattice in Fig. 9(c) with 7 interactionradii for blockade radius R b = 3 . a . There is a phase transition between two featureless phases, the latter having a largeentanglement plateau. The spin liquid is characterized by the simultaneous vanishing of the off-diagonal string operator (i.e.,the trivial phase is an e -condensate) and emergence of a large signal for the parity loop around the circumference. The latteralso labels two of the degenerate ground states, as annotated on the density plot. The fact that this approximates a dimermodel is evidenced by, e.g., (cid:104) n (cid:105) ≈ .
247 for δ/ Ω = 4 . sign of topological loop operators wrapping around thecircumference.We have also explored some other points in the param-eter space. For example, for ρ = √ R b = 2 . a upon including fourinteraction distances. Here we again see the sensitivedependence on adding further neighbors, but the cur-rent numerical method is not conclusive about the fateof the spin liquid in contrast to the example above. Akey point is that the model has multiple tuning parame-ters that one can use to stabilize this topological phase:the Rydberg blockade radius R b , the detuning δ/ Ω andthe lattice aspect ratio ρ . Having established our maingoal of showing the viability of the Z spin liquid in arealistic set-up, we leave an exhaustive search throughthis three-parameter phase diagram to future numericaland perhaps even experimental work. B. Measuring an off-diagonal string bytransforming it into a diagonal string
In section II D we introduced the two topological stringoperators associated to the Z lattice gauge theory.These can be very useful for identifying the spin liquidand its nearby phases (see Fig. 5). Fortunately, the par-ity string P can be straightforwardly measured in thelab since it is diagonal in the occupation basis and canbe read off from the snapshots of the Rydberg states.The off-diagonal string Q is more challenging to mea-sure directly. We now show that by time-evolving witha quenched Rydberg Hamiltonian, it becomes a diagonalobservable, making it experimentally accessible. Asidefrom its practical significance, this result is also concep-tually valuable since it gives a concrete duality transfor-mation between the two strings. Due to the local con-straint, such a duality is rather non-trivial.To implement this rotation, we consider the RydbergHamiltonian at zero detuning with a complex phase fac- tor in the Rabi oscillation : H (cid:48) = Ω2 (cid:88) i (cid:16) ie iα b † i + h.c. (cid:17) + 12 (cid:88) i , j V ( | i − j | ) n i n j . (5)The essential idea is to consider the evolution under aRydberg blockade localized on individual triangles of theruby lattice, i.e., V ( r ) = + ∞ and V ( r ) = 0 otherwise(see Fig. 1(a) for the definition of r ).Since the blockade now only acts within triangles ofthe ruby lattice, time-evolving with the above Hamilto-nian amounts to an on-site unitary transformation. Itis thus sufficient to consider a single triangle, and bywriting the P and Q operators defined in Fig. 5(a) as4 × t = π √ . (6)Thus, one can effectively measure Q along a string byfirst time-evolving with H (cid:48) and then measuring the P string on the resulting state.If the aspect ratio ρ of the ruby lattice is not too closeto unity, one can approximate this nearest-neighor block-ade Hamiltonian by quenching R b in between the firsttwo radii, i.e., 1 < R b /a < ρ . For instance, we have con-firmed that for ρ = 3, a quench from R b = 3 . a (wherewe found the spin liquid in Fig. 10) to R b = 2 a gives vir-tually indistinguishable results from time-evolving withthe nearest-neighbor blockade (see Fig. 11). In eithercase, we confirm that the value of the diagonal correla-tor at Ω t = τ dual := π √ ≈ .
42 correctly reproduces the This can be engineered by combining the original Hamilto-nian with an appropriately-timed evolution where the detun-ing is dominant, i.e., using e − iασ z / σ x e iασ z / = cos( α ) σ x + i sin( α ) σ y . .
25 0 . √ Ω t/π . . . h ψ ( t ) | P P | ψ ( t ) i (a) (b) FIG. 11.
Measuring the off-diagonal string opera-tor through a quench protocol. (a) The vertical direc-tion is periodic on the cylinder. The off-diagonal string Q (blue wiggly line) can be obtained by measuring the diagonalstring P P (orange dashed lines) after a time-evolution witha nearest-neighbor Rydberg blockade Hamiltonian to timeΩ t = π √ (see Eq. (6)). (b) Starting from the spin liquidground state of the Rydberg Hamiltonian with R b = 3 . a and δ/ Ω = 4 . ρ = 3 (seeFig. 10), we measure (cid:104) P P (cid:105) after time-evolving with eitherthe nearest-neighbor Rydberg blockade model (solid blackline) or the ground state Hamiltonian quenched to R b = 2(red dashed line). The horizontal gray dashed line denotesthe ground state value for (cid:104) Q (cid:105) . ground state expectation value for the off-diagonal stringoperator . IV. TOWARDS FAULT-TOLERANT QUANTUMMEMORY
Part of the reason that topologically ordered phasesof matter are of great interest is that they can serveas a means of potentially creating fault-tolerant quan-tum memories based on degenerate topological groundstates [19]. We have already encountered such degenera-cies associated to a Z spin liquid in section II E. How-ever, this example utilized periodic boundary conditions,which is not natural in an experimental setting. Fortu-nately, topologically-distinct ground states can also arisefor systems with boundaries. This can occur both forsystems with punctures/holes (which one can interpretas a sort of boundary), as well as systems with mixedboundary conditions. Either of these options requires theknowledge of how to realize distinct topological boundaryconditions. Another important ingredient is the trappingof anyons whose braiding implements gates on the quan-tum bits. We first analyse these two ingredients, afterwhich we discuss what one can do with them. Note that the Q -loop in Fig. 11 is dual to two parity strings.This is because it has triangles on both side of the string. Otherloops with triangles on only one side—such as a loop around ahexagon—will be dual to a single parity string. FIG. 12.
A trapping potential for e -anyons. The groundstate (here on a cylinder) for a lattice where four sites arounda vertex have been removed captures an e -anyon. This can beread off from the expectation value of the parity loops (dashedorange lines) around the circumference: if two neighboringloops have opposite sign, then a charge is enclosed. A. Trapping an e -anyon If one wishes to braid with anyons, one has to be ableto localize them to a particular region. Since the e -anyonin this model corresponds to a monomer (e.g., see the dis-cussion in section II A), a natural way of trapping it isby forcing a certain vertex to have no dimer touching it.This can be done by either simply removing the atoms onthese bonds, or by lowering the detuning δ . We numer-ically confirm that this works: Fig. 12 shows the resultof removing two such vertices on XC-8 for the blockademodel at δ/ Ω = 1 .
7. Since parity loops measure thecharge enclosed in a given loop, the nonzero charge lo-calized on these defects can be inferred from comparingthe sign of the parity loops along the cylinder. In fact,we even see that the two e -anyons are connected by agauge string where the parity loops are negative.By adiabatically changing the detuning, this anyon canpotentially be moved around at will, allowing for controlover an e -anyon. Similar approaches can potentially beexplored to trap and control m -anyons as well. Even inthe absence of such an m -anyon, the e -anyon can alreadybe used for non-trivial braiding, as we will discuss insection IV C. B. Boundary phase diagram
There are two topologically-distinct boundary condi-tions for a Z spin liquid. These are characterized bywhether the e or m anyon condenses at the edge. It isno coincidence that the trivial and VBS phase are alsodescribed as condensates (see section II D): if one inter-prets a boundary as a spatial interface from the topolog-ical phase to a non-topological phase, it is natural thatthe characterization of the nearby phases carries over todescribe boundary conditions. Similarly, these e and m condensates along the boundary can be diagnosed us-ing the string operators introduced in section II D. Moreprecisely, m -boundaries ( e -boundaries) have long-rangeorder for the P -string ( Q -string).Simply terminating the lattice—keeping all the Hamil-2(a) (b) (c) (d) FIG. 13.
Boundary phase diagram of the blockade model.
We consider an infinitely long strip of the XC-8 geometry:the bulk is the spin liquid at δ/ Ω = 1 .
7, but we tune δ on the outermost boundary links. (a) The correlation length divergesat two boundary phase transitions; in the intermediate shaded regime, the entanglement is increased. (b) The small and large δ bdy phases have a classical-like dimer filling at the boundary, whereas the intermediate regime has a compressible boundary.(c) By calculating the string operators from boundary-to-boundary, we diagnose the small and large (intermediate) δ bdy phasesas having m -condensed ( e -condensed) boundaries. (d) Density plots (cid:104) n (cid:105) in the three boundary regimes. The strip is infinitelylong (finite) in the horizontal (vertical) direction. tonian terms that fit on the remaining geometry—willtend to stabilize the m -boundary. Indeed, since bound-ary dimers experience less repulsion, they will prefer toarrange in a classical pattern with few fluctuations, giv-ing long-range order to the diagonal string operator P .To stabilize the e -boundary condition, we need to en-hance such boundary fluctuations. One way of doing sois by changing the detuning δ along the boundary sites,searching for the sweet spot where the dimers are sus-pended between the two classical (empty or filled) con-figurations.We numerically determine the resulting boundaryphase diagram for the blockade model on an infinitely-long strip geometry, where we choose the bulk to be deepin the spin liquid at δ/ Ω = 1 .
7. The results are shown inFig. 13. In line with the above expectation, we see thatbefore we change the boundary detuning, i.e., δ bdy = δ ,the strip realizes an m -boundary as evidenced by thelarge response for the end-to-end parity string. As wereach δ bdy ≈ . δ , there is a boundary phase transition(where the correlation length diverges along the infinitedirection) after which the parity string dies out, mak-ing way for a strong signal for the Q string. In thisregime, we stabilize the e -boundary. As we further de-crease δ bdy →
0, we are effectively removing these linksfrom the model, with the remaining geometry again spon-taneously realizing an m -boundary. This picture is alsoconfirmed by the density plots and the (cid:104) n (cid:105) curve: it isonly in the intermediate regime—corresponding to the e -boundary—that the edge dimers are fluctuating. C. Creating topological degeneracy on the plane
With the knowledge of the above boundary phase dia-gram, it is now straightforward to construct a rectangu-lar geometry with a topological ground state degeneracy.A schematic picture is shown in Fig. 14(a): a squareslab where the four boundaries are alternatingly e - and m -condensed. One way of understanding this twofold degeneracy is as follows: one can imagine extracting asingle e -anyon from the top boundary (after all, it is an e -condensate), dragging it through the deconfined bulk,and depositing it at the bottom boundary. Similarly, onecan do the same for an m -anyon from left to right. Dueto the mutual statistics of e and m , these two processesanti-commute, implying a degeneracy.Let us now address how to physically label this two-level system, or equivalently, how to read out a givenstate. If the spin liquid was in a fixed-point limit—similarot the toric code [19]—then the topological string oper-ators P and Q (defined in section II D) would be exactsymmetries of the model. I.e., the logical σ z logic ( σ x logic )operator could then be identified with any P -( Q -)stringconnecting the m -condensed ( e -condensed) boundaries.However, our system is not at a fixed-point limit, suchthat acting with these P and Q string operators need notstay with this subspace; relatedly, we cannot label oursystem in terms of eigenstates of P or Q . Fortunately,using the idea of the BFFM order parameter encounteredin section 5, we can define properly-normalized expecta-tion values: (7)It is worth pointing out that unlike the numerators inEq. (7), the denominators do not depend on the logicalstate of the system and hence they only need to bedetermined once for any particular architecture. To see this, remember that the degeneracy could be interpretedas being a consequence of moving m - or e -anyons between thecorresponding condensed boundaries, but these commute withpairs of topological string operators. FIG. 14.
Topological degeneracy in planar geome-try. (a) Alternating e - and m -condensed boundaries implya twofold degeneracy. One way of understanding this is interms of the Majorana zero modes (red dots) that live at thepoints where the boundary condition changes [91]; due to theglobal emergent fermion parity having to be unity, these fourMajorana modes only give rise to a twofold degeneracy. If welabel states using the P -string connecting the left and rightboundaries (see main text), then pulling an e -anyon out of one e -condensed boundary to another effectively toggles the statesin this two-level system. (b) An annulus geometry with m -condensed boundaries. Moving the e -anyon around the holewill toggle the states. Since e -anyons can only be created inpairs, there will be another e -anyon which we do not move(not shown). To illustrate that this procedure is meaningful andwell-defined, let us consider a simulated example, asshown in Fig. 15. The top and bottom boundaries weretuned to be e -condensed using the boundary phase di-agram in Fig. 13, setting δ bdy = 0 . δ . First, we ob-serve that (cid:104) Q (cid:105) (cid:54) = 0 when it connects the top and bot-tom boundaries; this is consistent with these being e -condensed. Moreover, we see that (cid:104) P (cid:105) ≈ x axis (in the Bloch sphere picture). To confirmthat (cid:104) P (cid:105) ≈ e -condensed boundaries), we con-firm that for two parallel parity strings connecting thetwo m -condensed boundaries, we obtain the nonzero re-sponse |(cid:104) P P (cid:105)| ≈ .
46. As an additional sanity check,we confirmed that this same double-parity-string gives azero response when running from top-to-bottom. Finally,using the BFFM-prescription in Eq. (7), we obtain thatthe logical state indeed lies along the x -axis: (cid:104) σ x logic (cid:105) ≈ {| (cid:105) , | (cid:105)} , defined by (cid:104) n | σ z logic | n (cid:105) = ( − n . We cancreate | (cid:105) by starting with a sample which only has an m -condensed boundary—such that the parity string isa fixed positive value—and then adiabatically create an e -condensed boundary as follows: FIG. 15.
Read-out of a topological ground state.
Weconsider the blockade model for δ/ Ω = 1 . δ bdy =0 . δ . From the boundary phase diagram in Fig. 13, we knowthat this realizes the e -condensed boundary, whereas the leftand right boundaries are m -condensates. For the ground stateof this system, we show the values for the two type of topolog-ical string operators which connect their corresponding con-densates. Upon using the BFFM normalization (see Eq. (7)),the read-out for the logical variables gives a state that liesalong the x -axis of the Bloch sphere. Note that both stringoperators can be experimentally measured using the prescrip-tion in section III B. In the above sequence, we also show the parity stringwhose value will not change throughout this process, suchthat we arrive at | (cid:105) . To initialize into | (cid:105) , we can nowuse the fact that we know how to pin an e -anyon (see sec-tion IV A). We can thus dynamically change the detun-ing to pull an e -anyon out the top e -condensed boundaryand move it into the bottom e -condensed boundary, assketched in Fig. 14(a). This implements the logical σ x logic gate, mapping | (cid:105) → | (cid:105) .One can repeat the above steps for the alternative ar-chitecture of an annulus, shown in Fig. 14(b). In partic-ular, in this case the logical state is toggled by braidingthe e -anyon around the m -condensed hole. More gener-ally, one can create multiple e - and m -condensed holes ina given sample. Braiding these (by dynamically chang-ing the parameters of the Hamiltonian) potentially givesanother handle on topological processing of quantum in-formation [92, 93]. V. OUTLOOK
We have demonstrated that Rydberg blockade on theruby lattice can be utilized to stabilize a Z spin liquid.The underlying mechanism is that of a monomer-dimermodel where single-site monomer fluctuations induce thedimer resonances necessary for a resonating valence bondstate. This same picture also leads to a specific form ofthe two topological string operators. The spin liquid—stable to longer-range V ( r ) ∼ /r interactions—can becharacterized by these string observables in experimentwhere they are measurable by appealing to a dynamic4protocol. Moreover, we showed that this system couldbe used to explore topological quantum memories bylocalizing anyons, realizing conjugate boundary condi-tions which create degeneracy on the plane, and read-ing out quantum states. We note that given the de-tailed differences between our platform and the exacttoric code model, these implementations required newinsights. While the robustness of these techniques in thepresence of realistic imperfections (such as, e.g., spon-taneous emission) will need to be carefully explored, itis important to emphasize that the atom array platformoffers fundamentally new tools for probing and manipu-lating topological quantum matter.Specifically, the theoretical predictions outlined abovecan be probed using programmable quantum simulatorsbased on neutral atom arrays. In particular, the requiredatom arrangements can be realized using demonstratedatom sorting techniques, while relevant effective blockaderange can be readily implemented using laser excitationinto Rydberg states with large principle quantum num-ber 60 < n < P and Q operatorscan be efficiently measured, by either directly analysingthe signal shot images or carrying out this analysis fol-lowing qubit rotation in the dimer basis associated withindividual triangles (as explained in section III B). Thelatter can be realized using resonant atomic driving withappropriately chosen parameters. Moreover, the topo-logical entanglement entropy can be potentially obtainedby measuring the second Renyi entropy [95–99] for differ-ent regions, as described in Refs. [21, 22]. Together withcontrol over boundaries and exploration of samples withnon-trivial topology, these methods constitute a uniqueopportunity for detailed explorations of spin liquid stateswith accuracy and sophistication not accessible with anyother existing approaches. Furthermore, this work opens up a number of very in-triguing avenues that can be explored in the frameworkintroduced here. These range from exploration of non-equilibrium dynamical properties of spin liquid states inresponse to rapid changes of various Hamiltonian param-eters, to experimental realization and detection of anyonswith non-trivial statistics. In particular, anyon braidingcan be explored by using time-varying local potentials(see e.g. section IV A). Moreover, approaches to improvethe stability of TQL and realization of more exotic spinliquid states can potentially be realized by additionalengineering of interaction potentials, using e.g. long-lived hyperfine atomic states [100–102]. In particular,approaches involving optical lattice [100] and Rydbergdressing [103] could be explored to realize a broader vari-ety of spin liquid states. Finally, one can explore whetherrecently developed two-dimensional materials [104] canbe used to realize similar models with desired properties.Potentially, these systems can explored for the realiza-tion of topologically-protected quantum bits, with an eyetowards developing new, robust approaches to manipu-lating quantum information.Note Added: An independent work which will appearin this same posting also studies the quantum phases ofRydberg atoms but in a different arrangement, whereatoms occupy sites of the kagome lattice [105]. ACKNOWLEDGMENTS
We thank Dave Aasen, Marcus Bintz, Soonwon Choi,Sepehr Ebadi, Xun Gao, Marcin Kalinowski, AlexanderKeesling, Harry Levine, Hannes Pichler, Subir Sachdev,Giulia Semeghini, Norm Yao and Mike Zaletel for use-ful discussions. The DMRG simulations were per-formed using the Tensor Network Python (TeNPy) pack-age developed by Johannes Hauschild and Frank Poll-mann [77]. This work was supported by the HarvardQuantum Initiative Postdoctoral Fellowship in Scienceand Engineering (R.V.) and by the Simons Collabora-tion on Ultra-Quantum Matter, which is a grant fromthe Simons Foundation (651440, A.V.). M.D.L. wassupported by the U.S. Department of Energy (GrantDE-SC0021013), the Harvard-MIT Center for UltracoldAtoms (Grant PHY-1734011), the Army Research Of-fice (Grant W911NF2010082), and the National ScienceFoundation (Grant PHY-2012023). 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Here we prove Eq. (6). For this, let us first label the four basis states in a single triangle as follows:Then the P and Q string operators (defined in Fig. 5(a)) can be written as:= q q ∗ and = − − , (A1)where we introduced q = e − iα .The Hamiltonian defined in Eq. (5) does not couple distinct triangles, so it is sufficient to prove the claim for asingle triangle. Then Eq. (5) becomes H (cid:48) = Ω2 (cid:88) i ∈(cid:52) P ( iq ∗ b † i − iqb ) P = i Ω2 − q − q − qq ∗ q ∗ q ∗ = Ω2 × V DV † (A2)where D = √ − and V = 1 √ − iq √ iq √ − −√
31 1 1 √ . (A3)The time-evolution operator is thus t = 2Ω × π × √ ⇒ e − iH (cid:48) t = V e − πi/ e πi/ V † = − q q q − q ∗ − − q ∗ − − q ∗ − . (A4)Then e iHt − − e − iHt = q q ∗ ..