Phase diagram of soft-core bosons in two dimensions S. Saccani1, S. Moroni1 and M. Boninsegni2 1SISSA Scuola Internazionale Superiore di Studi Avanzati and DEMOCRITOS National Simulation Center, Istituto Officina dei Materiali del CNR Via Bonomea 265, I-34136, Trieste, Italy and 2Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 (Dated: January 14, 2011) The low temperature phase diagram of Bose soft disks in two dimensions is studied by numer- ical simulations. It is shown that a supersolid cluster phase exists, within a range of the model parameters, analogous to that recently observed for a system of aligned dipoles interacting via a softenedpotentialatshortdistance. Thesefindingsindicatethatalong-rangetailoftheinteraction 1 isunneededtoobtainsuchaphase,andthatthesoft-corerepulsiveinteractionistheminimalmodel 1 for supersolidity. 0 2 PACSnumbers: 67.80.K-,67.85.Hj,67.85.Jk,67.85.-d,02.70.Ss n a J Introduction. The supersolid phase of matter, display- tial, have yielded evidence of the same instability of a 3 ing simultaneously crystalline order and dissipation-less gas of point defects (vacancies) observed in the quantum 1 flow, is a subject of long standing interest in condensed system.7 This suggests that the origin of such instability matterandquantummany-bodyphysics. Inrecentyears, may lie in the strong interaction (specifically the hard ] the attention of theorists and experimenters alike has core repulsion) among particles, which quantum delocal- s a focused on solid 4He, following the observation of non- ization cannot overcome. One is then led to pose the g classical rotational inertia by Kim and Chan.1 At the theoretical question of which type of inter-particle inter- - t present time, agreement is still lacking, as to whether action (or, class thereof) might underlie supersolid be- n experimental findings indeed mark the first observation haviour. In particular, can an interaction featuring a a of supersolid behaviour.2 The most reliable theoretical “softer” core, saturating at short distance to a value of u q studies, based on first-principle numerical simulations, the order of the characteristic zero-point kinetic energy . show that superfluidity if it occurs at all in solid helium, of the particles, result in the appearance of a supersolid t a is not underlain by the mechanism originally envisioned phase ? m in the seminal works by Andreev, Lifshitz and Chester, This question might have seemed little more than “aca- - i.e., through Bose Condensation of a dilute gas of va- demic”untilnotsolongago,forhowwouldonegoabout d cancies or interstitials,3,4 but involves instead extended creating artificially such an interaction, which does not n defects, such as dislocations.5 In particular, a dilute gas occur in any known naturally occurring quantum many- o of point defects in solid helium has been predicted to be body system ? However, impressive advances in cold c [ thermodynamically unstable.6 atomphysicsappeartoallowonetodojustthat,namely Regardless of how the current controversy over the in- to“fashion”artificialinter-particlepotentials,notarising 1 v terpretation of the present 4He phenomenology is even- in any known condensed matter system. It makes there- 0 tually resolved, it seems fair to state that solid helium fore sense to search theoretically for supersolid, or other 0 does not afford a direct, simple, and clear observation exotic phase of matters, based on more general types 6 of the supersolid phenomenon. Still, among all simple of interactions among elementary constituents than the 2 atomic or molecular condensed matter systems, helium onesconsideredsofar,withtherealisticexpectationthat 1. shouldbethebestcandidatebyfar,duetothefavourable such interactions might be realizable in the laboratory. 0 combination of large quantum delocalization of its con- Inarecentpaper,8 atwo-dimensionalsystemofbosons 1 stituent(Bose)particles,andweaknessoftheinteratomic was studied, interacting via a purely repulsive pair-wise 1 potential. Butwhatexactly,inthephysicsofthissimple potential, decaying as 1/r3 at long distance but saturat- : v crystal,contributestosuppress(ifnoteliminateentirely) ing to a finite value as particles approach one another. i its superfluid response ? Thisparticularformofpotentialmightbefeasibleincold X The thermodynamics of solid 4He, as it emerges from atomicsystems,throughamechanismknownasRydberg r a first-principlequantumsimulations,islargelydetermined blockade.9 Such a system displays at low temperature a by the strong repulsive core of the pair-wise interatomic crystalline phase in which unit cells feature more than potential at short distance. For example, a very simple one particle. In turn, such a crystal turns superfluid at model of Bose hard spheres reproduces surprisingly ac- sufficiently low temperature, phase coherence being es- curately the phase diagram of condensed helium. Such a tablished by quantum-mechanical tunnelling of particles repulsivecoreisaubiquitousfeatureofordinaryinterac- across adjacent lattice sites. tionsbetweenatomsormolecules, arisingfrom thePauli Multiple occupancy crystals (or cluster crystals) are exclusion principle, acting between electronic clouds of a well known subject in classical physics, as a model for different atoms. Computer simulation studies of clas- polymersinteraction.10Inparticular,acriterionisknown sical crystals, making use of the Lennard-Jones poten- forthequantitativepredictionofclustering,basedonthe 2 form of the molecular interatomic potential.11 It is rea- sonable to expect that the basic physics should remain relevant for a quantum-mechanical system as well. How- ever, it is not clear what role, if any, the long-range re- pulsive tail of the interaction plays, in the occurrence of superfluidity of the cluster crystal. In Ref. 8, supersolid cluster crystal phases were ob- served in numerical simulations with long-range tails other than 1/r3 (e.g., 1/r6), and indeed it is simple to convince oneself that a long-range tail is not required, in order for a cluster crystal phase to exist. For example, the classical ground state of a system of particles inter- acting via the following, soft core potential (cid:26) V if r ≤a v(r)= (1) 0 if r >a FIG.1. Qualitativelowtemperaturephasediagramforhigh andlowDasafunctionofµ. Thepanelsshowtypicalspatial will be a cluster crystal at densities for which the mean configuration of the world lines resulting from simulations at inter-particle distance d is less than the soft-core diame- low T, referring to the various phases. Results shown in the tera. TheaveragenumberofparticlesK perclustercan upper part of the figure correspond to simulations with D = also be easily established to be equal to γa2/d2, where γ 60, whereas the lower part to D=3. is a number (slightly greater than one) that depends on dimensionality. In other words, K is independent of V. Inordertoinvestigateingreaterdetailtheimportance chemical potential µ). Because this methodology is by played by the tail of the inter-particle interaction in the now well-established, and is thoroughly described else- stabilization of a supersolid cluster solid, as well as to where,weomitheretechnicaldetails,andrefertheinter- contribute to the search for the “minimal model” of su- estedreadertotheoriginalreferences.12,13 Thesystemis persolidity,wehaveinvestigatedinthisworkthelowtem- enclosedinacellwithperiodicboundaryconditions. We perature properties of a two-dimensional system of Bose denote by N the average number of particle and express soft disks, i.e., particles interacting via the simple po- the density ρ in terms of the dimensionless parameter (cid:112) tential given by Eq. (1). In spite of its simplicity, to our r =1/ ρa2. s knowledge(andsurprise)thishasnotbeenthesubjectof Results. We present here results obtained varying µ any prior theoretical study. Our calculations are numer- and D, in the T → 0 limit; that is, in all cases shown ical, based on the Continuous-space Worm Algorithm. explicitly, thevalueofthetemperatureissufficientlylow Our main result is that the same phase(s) observed in thatestimatescanberegardedasessentiallygroundstate Ref. 8 are present in the system considered here, which ones (typically T ∼(cid:15) , for most quantities). ◦ isthereforearguablythesimplestmodelsystemunderly- Figure 1 summarizes qualitatively the ground state ing a supersolid phase. Because of the substantial irrele- phase diagram in the D −µ plane, with the aid of in- vanceofthelong-rangeformoftherepulsivetail,wemay stantaneous (in the Monte Carlo sense) many-particle conclude that the supersolid cluster crystal phase should configurations (i.e., world lines). For D >> 1 (D = 60 be observable in a relatively broad class of interactions. in the figure), the physics of the system is essentially Model and Methodology. WeconsiderasystemofBose that of the hard-sphere fluid. At low density, the system particles of spin zero in two dimensions. The Hamilto- is a superfluid gas, which undergoes solidification into a nian of the system in reduced units is triangular crystal on increasing µ. For sufficiently large valuesofD,thenumberofparticlespercluster(unitcell) N 1(cid:88) (cid:88) is K = 1. However, on further increasing the chemical H=− (cid:53)2+D Θ(|r −1|), (2) 2 i ij potential, particles bunch into clusters (also referred to i=1 i>j as “droplets”) which organize in a solid preserving the wherer isthedistancebetweenparticlesiandj,alldis- triangular structure. ij tances are expressed in units of the soft-core diameter a, Theappearanceathighdensityofsuchaclusterphase while all energies are expressed in units of (cid:15) =h¯2/ma2. is a classical effect, directly related to the finite energy ◦ TheparameterD ≡V/(cid:15) canalsobeexpressedas(a/ξ)2, cost associated to particles being at a distance less than ◦ whereξ isthequantum-mechanicalpenetrationlengthof the soft core diameter. Indeed, it is relatively simple to a potential barrier of height V. In the limit ξ → 0 the estimate the number of particles per cluster at a given model (2) reduces to the hard-sphere gas. nominal density r , by minimizing the potential energy s Quantum Monte Carlo simulations of the system de- per particle. Cluster formation becomes favourable for scribed by (2) have been performed by means of the r < 1, and one finds K = α/r2, where α is a number s s Continuous-space Worm Algorithm, in the grand canon- < 2, independent of D. ∼ ical ensemble (i.e., at fixed temperature T, area A and For lower values of D, the main qualitative change 3 0 r = 0.8 T = 0.25 9 s rs = 0.5 -1 T = 1 7 rs = 0.25 -2 T = 4 ) ) P -3 g(r 5 g (10 -4 o 3 L -5 1 -6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -7 0 20 40 60 80 100 r Length FIG. 2. Ground state pair correlation function for D = 5, at three different values of r . The solid line (r = 0.8) s s FIG. 4. Frequency of occurrence P of permutation of dif- corresponds to a superfluid gas, the dashed one (r = 0.5) s ferent length (i.e., involving different numbers of particles), to a supersolid, and the dotted one (r = 0.25) to a non- s at temperature T= 4 (dark boxes), T=1 and T=0.25 (light superfluid cluster crystal. boxes). Temperatures are in units of (cid:15) . Here, D = 5 and ◦ r =0.5. Thesystemisintheclustercrystalphaseatallthe s temperatures. is that the system crystallizes at higher densities (i.e., smaller values of r ), directly into a cluster crystal with s K > 1. The formation of the cluster crystal is signalled In the low T limit, this system may develop phase co- in the pair correlation function g(r), as shown in Figure herence and display dissipation-less flow, besides diago- 2) for the case D = 5. The pair correlation function is nal order. We refer to this phase as “cluster supersolid”, essentially featureless in the case of a gas, i.e., g(r) ∼ 1 completely analogous to the one described in Ref. 8. In with only a slight depression for r ≈ 1. For a cluster theclustersupersolidphase,particlestunnelbetweenad- crystal,g(r)developsapeakatr =0,signallingmultiple jacentclusters,andthepaircorrelationfunctiontakeson occupation of a single unit cell, as well as robust oscil- afinitevaluebetweensuccessivepeaks. Intheinsulating lations, with period consistent with the lattice constant clustercrystal,ontheotherhand,g(r)isessentiallyzero of the cluster crystal. As the density increases, clusters between peaks (see Figure 2). The presence of a global comprisealargernumberofparticles, asreflectedbythe superfluid response, extending to the whole system, is increasingly greater strength of the peak at r =0. assessed numerically through the direct computation of the winding number.14 A typical result is shown in Fig- ure 3, for D = 5 and r = 0.5. As in any simulation s 0.14 study, the superfluid transition is smeared by finite-size N = 72 0.12 N = 144 effects, and accurate finite-size scaling analysis of the re- sult obtained on systems comprising significantly differ- N = 288 0.10 entnumbersofparticlesisrequired,inordertodetermine accuratelythetransitiontemperature. Theresultsshown 0.08 in Figure 3 are consistent with a superfluid transition in (cid:108)s 0.06 the Kosterlitz-Thouless universality class,15 as expected foratwo-dimensionalsystem. Itisworthnotingthatthe 0.04 superfluid fraction does not saturate to a value of 100% 0.02 as T →0. This is in line with the spontaneous breaking oftranslationalinvarianceassociatedtocrystallineorder, 0.00 as first pointed out by Leggett.16 0.0 0.5 1.0 1.5 2.0 2.5 Theonsetofsuperfluidityiswellknowntobeunderlain T by long cycles of exchanges of identical particles (per- mutations). Fig. 4 shows histograms of frequency of FIG. 3. Superfluid fraction of as a function of T (in units occurrence of exchange cycles of different “length”, i.e., of (cid:15) ) for a system of soft disks with D=5. Here, µ is set involvingdifferentnumbersofparticles,atthreedifferent ◦ so that r = 0.5. Data shown are for three system sizes, temperaturesforasystemintheclustercrystalphase. At s comprising N=72 (circles), N=144 (diamonds) and N=288 the highest T, P drops sharply for cycles involving more (squares) particles. When not shown, statistical errors are than the number of particles per cluster (approximately smaller than symbol sizes. 7). At the lowest T, exchanges involving all particles in 4 the system set in. However, even at the highest temper- D <10,onincreasingµ,asthenumberK ofparticlesper ature clusters are individually superfluid, even though cluster becomes large, in this case the potential energy global phase coherence does not exist. barrier for tunnelling growing linearly with K. Our data at finite T are consistent with an exponential decrease with D or µ of the superfluid transition temperature of the cluster crystal, but we cannot rule out continuous quantum phase transitions between a supersolid and a normal cluster crystal, driven by either µ or D. The re- sulting schematic finite temperature phase diagram, for a value of D for which a supersolid phase is observed, is shown in Figure 5. Conclusions. In summary, we have studied by Monte Carlo simulations the phase diagram of a two- dimensionalsystemofbosonsinteractingviaarepulsive, short-range soft-core potential. This system displays a low-temperaturesupersolidphase,whereinparticlestun- nel across nearest-neighbouring, multiply occupied unit cells. The same qualitative behaviour was previously found in similar work,8 where, in addition to the soft core, a long-range repulsive interaction between parti- FIG. 5. Qualitative low D finite T phase diagram of the cles was included. The results reported here show that system. Thick lines show first-order, dashed lines continuous phase transitions. nolong-rangetailisnecessaryfortheonsetofsuperfluid- ity. Themostimportantconclusion,besidesthefactthat soft-corebosonsseemstobethe“minimal”modelforsu- The supersolid phase described above has been ob- persolidity, is that the physics illustrated here should be served in this work at low T, for D < 10 and r ≈ 0.5, ∼ s observable under relatively broad conditions, if soft-core which corresponds to a unit cell occupation K ∼ 5. For pair-wise interaction potentials can be fashioned. large values of D (i.e., D ≥10), clusters forming a crys- tal are increasingly compact, i.e., particles pile up on a This work was supported by the Natural Science and small spatial region, and an exponential decrease of the Engineering Research Council of Canada under research tunnelling rate for particles across clusters is expected, grant G121210893, and by the Italian MIUR under asthepotentialenergybarrierassociatedwithtunnelling COFIN07. Oneofus(MB)gratefullyacknowledgeshos- grows linearly with D. The same effect takes place for pitality of IOM-CNR at SISSA. 1 E. Kim and M. H. W. Chan, Nature, 427, 225 (2004); 9 M. Lukin et al., Phys. Rev. Lett. 87, 037901 (2001). Science 305, 1941 (2004). 10 B. M. Mladek, P. Charbonneau, C. N. Likos, D. Frenkel 2 See, for instance, N. V. Prokof’ev, Adv. Phys. 56, 381 and G. Kahl, J. Phys.:CM 20, 494245 (2008). (2007). 11 C.N.Likos,A.Lang,M.WatzlawekandH.Lo¨wen,Phys. 3 A. F. Andreev and I. M. 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