Abstract
More than 40 years ago, Andreev, Lifshitz and Chester suggested the possible existence of a peculiar solid phase of matter, the microscopic constituents of which can flow superfluidly without resistance due to the formation of zeropoint defects in the ground state of selfassembled crystals. Yet, a physical system where this mechanism is unambiguously established remains to be found, both experimentally and theoretically. Here we investigate the zerotemperature phase diagram of twodimensional bosons with finiterange softcore interactions. For low particle densities, the system is shown to feature a solid phase in which zeropoint vacancies emerge spontaneously and give rise to superfluid flow of particles through the crystal. This provides the first example of defectinduced, continuousspace supersolidity consistent with the Andreev–Lifshitz–Chester scenario.
Introduction
Spontaneous symmetry breaking is a focal principle of condensed matter physics, yet simultaneous breaking of fundamentally different symmetries represents a rare phenomenon. A prime example is the socalled supersolid phase^{1}, which displays both crystalline and superfluid properties, that is, the simultaneous breaking of continuous translational and global gauge symmetry. The first mentioning of such a state goes back to Gross^{2}, who predicted the possibility of a densitymodulated superfluid phase of weakly interacting Bosons described by a classical field. Later, Andreev and Lifshitz^{3}, and Chester (ALC)^{4} conjectured a microscopic mechanism for strongly interacting systems, based on two key assumptions: first, that the ground state of a bosonic crystal contains defects such as vacancies and interstitials and, second, that these defects can delocalize, thereby, giving rise to superfluidity. However, the physical realizability of this scenario has since remained under active debate^{5}.
In 2004, torsional oscillator experiments^{6,7} provided first suggestive evidence for superfluidity in solid ^{4}He through a rapid drop of the resonant oscillation period below a critical temperature, viewed indicative for superfluid decoupling of a fraction of the He crystal. This finding has sparked a host of new experimental activity^{8,9,10,11,12,13,14,15,16,17,18} that, however, challenged the original interpretation and pointed out several artefacts causing a nonsupersolid origin of the observations. Theoretical work has established that crystal incommensurability is a necessary condition for superfluidity^{19} and that zeropoint defects in groundstate solid He are prevented by a large activation energy^{20,21}. In addition, Boninsegni et al.^{21} Rota and Boronat^{22}, Ma et al.^{23}, and Lechner and Dellago^{24} have shown that pointlike defects experience an effective attraction that results in defectclustering and phase separation, ruling out the possibility of defectinduced supersolidity^{21} as in the ALC scenario. Several experiments^{11,14,15,16} have shown that the original observations were caused by shear modulus stiffening of bulk solid He, and later found no signature of superfluidity on avoiding this effect^{25}. As a result, there now seems to be consistent experimental and theoretical evidence for the absence of the longsought supersolid phase in He. The mere existence of continuousspace supersolidity induced by zeropoint defects thus remains an open question.
In this Article, we show that bosonic particles interacting via softcore potentials (see Fig. 1) provide a prototype system for addressing this question. Using exact numerical techniques, we determine the underlying zerotemperature phase diagram, which reveals the emergence of defectinduced supersolidity in the vicinity of commensurate solid phases, as conjectured by ALC^{3,4}.
Results
Supersolidity with softcore bosons
We consider a twodimensional ensemble of N bosons with density ρ, interacting via a pair potential of the type
This interaction approaches a constant value V_{0}/ as the interparticle distance, r, decreases below the softcore distance R_{c} and drops to zero for r>R_{c}. The limiting case γ→∞ yields the softdisc model^{26}, while γ=3 and γ=6 correspond to softcore dipole–dipole^{27} and van der Waals^{28} interactions that can be realized with ultracold atoms^{28,29} or polar molecules^{30,31}. Here we focus on the latter case (γ=6) for which the Hamiltonian reads
where the units of length and energy are R_{c} and ħ^{2}/m, respectively, and m denotes the particle mass. In these units, the zerotemperature physics is controlled by the dimensionless interaction strength U=mV_{0}/(ħ^{2}) and the dimensionless density ρ.
Particles with softcore interactions have been studied previously in the field of soft condensed matter physics^{32,33,34} in the classical hightemperature regime. One of the main findings has been that pair potentials with a negative Fourier component^{32} favour the formation of particle clusters, which in turn can crystallize to form a socalled cluster crystal. In the quantum domain, theoretical work has so far focused on the regime of weak interactions and high particle densities^{27,28,35,36,37,38}, which was shown to be well described by meanfield calculations^{39,40}. In this limit, one finds strongly modulated superfluid states^{2,26,27} with broken translational symmetry in the form of a density wave.
In the following, we investigate the strong coupling domain where correlations and quantum fluctuations are expected to become important. We employ pathintegral Monte Carlo simulations to determine the groundstate properties of the Hamiltonian equation (2) (see Methods section). The obtained phase diagram, shown in Fig. 2, reveals a rich spectrum of phases with varying interaction strength and density.
Small particle densities
At small densities ρ≲0.5, we find two phases: a superfluid and an insulating triangular crystal composed of singly occupied sites, that is, where the number of lattice sites, N_{s}, equals the particle number N. The observed lobe structure of this crystalline region is readily understood by noticing that at very low densities, that is, large interparticle distances >R_{c}, the physics is dominated by the longrange tail of the interaction potential, V~1/r^{6}. For a fixed interaction strength U≳35, we thus find a firstorder liquid–solid quantum phase transition with increasing Aρ, consistent with previous work on bosons with powerlaw interactions^{30,41,42,43}. In particular, the location of the liquid–solid phase transition for very low densities coincides with that for pure van der Waals interactions. With increasing density, however, the average interparticle spacing, , approaches the softcore radius R_{c} and drops to values for which equation (1) strongly deviates from pure ~1/r^{6} interactions and levels off below the turning point R_{o}=(5/7)^{1/6}R_{c}. As a result of the decreasing repulsive interparticle forces, the crystal melts again for increasing densities. As indicated in Fig. 2, we indeed find a reentrant superfluid at particle densities for which <R_{o}.
Intermediate densities
A distinctive consequence of the softcore interaction is that the energy cost for forming close particle pairs is bound by V_{0}. This potentially enables the formation of crystalline phases with N>N_{s} above a critical density where doubly occupied lattice sites become energetically favourable on increasing the lattice constant. As expected for a triangular crystal, the lattice constant decreases as a=(ρ/2)^{−1/2} at small densities. However, around a≈1.4 R_{c} it increases again and settles to a densityindependent value of a_{0}≃1.6 R_{c} on further increase of ρ. The corresponding volume of the unit cell A= provides a measure of the lattice occupancy N/N_{s}=Aρ, which is also shown in Fig. 2. The transition to cluster crystals occurs at Aρ≈1.5. This indeed coincides with the critical density for crystallization of the reentrant superfluid phase.
Around this density, a thin region of phase separation is found to lay in between the cluster crystal and the superfluid phase. Figure 3b shows a typical example for the particle density distribution in this region. Two distinct coexisting phases can be recognized: a crystal phase with exactly two particles per site (upper part of the figure) and a superfluid phase (lower part). We have carefully checked that the occurrence of this phaseseparated state is not an artifact of the simulations by performing accurate annealing and by choosing different initial conditions, such as random and different crystalline configurations.
Above incommensurate lattice occupations N/N_{s}≳1.5, the direct liquid–solid quantum phase transition is replaced by a firstorder transition from a superfluid to a supersolid phase. The supersolid phase is approximately found to occur between the two hyperbola defined by ρU=α, with α≈28 and α≈38, respectively (see dotted lines in Fig. 2). These two lines are derived from the weak interaction limit (U→0 and ρ→∞ with α=const.), where meanfield theory predicts a transition to a densitywave supersolid that is determined only by the value of α (ref. 28). While this meanfield prediction becomes exact in the highdensity limit (see Fig. 2), the situation is dramatically different at moderate densities where the discrete nature of the particles plays a significant role. This gives rise to the emergence of supersolid regions with a lobe structure, which vanish at commensurate lattice occupations N/N_{s}=2 and N/N_{s}=3. There, we find a direct transition between a superfluid and an insulating solid phase.
This behaviour is illustrated in Fig. 3 where the superfluid fraction, f_{s}, is shown as a function of Aρ for α=32, that is, in between the two dotted lines in Fig. 2. As shown in Fig. 3c,e, at Aρ=2 and Aρ=3 one finds a commensurate crystal with exactly N/N_{s}=2 (Fig. 3c) and N/N_{s}=3 (Fig. 3e) particles per lattice site, respectively, and vanishing superfluidity (Fig. 3g). Importantly, the crystal structure in between these two densities is practically unchanged, as seen by comparing the particle density distributions n(r) and density–density correlation functions g_{2}(r) in Fig. 3c–e. However, the incommensurate lattice filling N/N_{s} and the resulting fluctuations of individual site occupations enables particles to tunnel between the sites. This gives rise to a nonvanishing superfluid fraction of the crystal, which can assume sizable values of f_{s}=0.3 for N/N_{s}≈Aρ=2.5.
High densities
For higher densities Aρ>3, the scenario described above changes considerably. As shown in Fig. 3g, the superfluid fraction approaches a constant, densityindependent value f_{s}≈0.24 with increasing ρ. In particular, for an average commensurate filling N/N_{s}=4 there is no direct phase transition between a superfluid and a solid insulating phase, and instead the supersolid phase persists with no significant difference to the case of incommensurate lattice occupancies N/N_{s}≠4. This behaviour signals a crossover to the regime where the supersolid phase can be understood in terms of a densitymodulated superfluid^{2,26,27,28,39}, where the discrete nature of the particles becomes irrelevant. In this limit, the superfluid–supersolid quantum phase transition is well captured by a meanfield description^{26,28,40}. It predicts a transition point at α=ρU=28.2 as well as a superfluid fraction that is solely determined by the value α, and yields f_{s}=0.23 for α=32. As shown in Figs 2 and 3g, both predictions are well confirmed by our Monte Carlo results for Aρ≳3.5, suggesting that the transition to densitywave supersolidity takes place at a surprisingly small number of only N/N_{s}≈3.5 particles per lattice site.
Defect delocalization
The most interesting behaviour takes place around the superfluid–solid quantum phase transition at N/N_{s}=2. Figure 4 provides a more detailed look at the transition between the insulating crystal and the supersolid phase, that is, for U=31 and N/N_{s}≈2. Starting from the insulating solid with doubly occupied lattice sites, we successively remove a small number of particles from randomly chosen sites and monitor the superfluid fraction of the resulting new ground state obtained from our simulations. Removing a small number of particles does not cause structural changes of the ground state but rather creates a small fraction f_{def}=(2N_{s}−N)/N_{s} of zeropoint crystal defects in the form of singly occupied sites. An analysis of the Monte Carlo configurations shows that defects do not cluster and instead delocalize, as illustrated in Fig. 4b. This is also confirmed by the vacancy–vacancy pair correlation function, as shown in Fig. 4c. For r≳a_{0}, it closely resembles the g_{2}(r) of the underlying solid, as expected for a very dilute gas of repulsive bosons. Indeed, we find a finite superfluid fraction even for small defect concentrations, which increases linearly with f_{def}. We have verified that this finding is pertinent to the ground state and not to a metastable configuration by performing simulations with different initial conditions, including clustered defects. The observed behaviour is, thus, consistent with defectinduced supersolidity according to the ALC scenario, and constitutes the central result of this work.
Discussion
Supersolidity in this system is the consequence of two unique features of softcore bosons. First, the energy cost for forming close particle pairs is bound by V_{0}, which facilitates the formation of cluster crystals that naturally entail zeropoint defects. Second, the dynamics and interaction of these defects differs fundamentally from those of conventional solids. In the latter case, vacancies and interstitials induce displacement fields that lead to purely attractive defect interactions^{21,23,24,44} and, therfore, prevent a delocalization of defects^{21}. In cluster solids, on the other hand, defect interactions are purely repulsive, since they interact via the same underlying particle interaction V(r) of equation (1). In the present case, the transition between these two regimes is controlled by the particle density. For Aρ≳1.5 delocalized zeropoint defects allow for the formation of supersolid phases. Below this density particles do not explore the softcore part of the interaction potential, such that defects are attractive and supersolidity is absent consistent with the results of Boninsegni et al.^{21} Around the transition region Aρ≈1.5 neither picture applies, and one observes separation between a superfluid and a doubly occupied, insulating cluster solid. Preliminary calculations based on pathintegral Langevin dynamics^{45} suggest that in this region structural and dynamical heterogeneity can give rise to a quantum glass phase at finite temperature.
Having identified a physical system that facilitates defectinduced supersolidity, we hope that this work will provide useful guidance for future experiments and initiate further theoretical explorations. An important question concerns the general features of the interaction potential that are required to maintain the type of supersolid states described in this work. While the emergence of weakly interacting densitywave supersolids is largely insensitive to the detailed shape of the softcore interaction, the lowdensity physics described in this work may be strongly affected. In fact, it seems reasonable to expect an interesting competition between intra and intersite interactions within the selfassembled crystal that will depend on the longrange tail of the particle interactions. Moreover, the role of the dimensionality and confined geometries of finite systems represent another outstanding issue, and in particular their role for frustration effects with regard to defect delocalization.
While the considered interactions do not straightforwardly occur in natural crystals, they can be designed in ultracold atom experiments. Recent experiments with Bose–Einstein condensates in optical cavities have already demonstrated a densitywave supersolid due to the breaking of a discrete translational symmetry^{46,47} and theoretical work has devised several schemes^{30,31} for the manipulation of longrange interactions between polar molecules by external fields. Moreover, far offresonant excitation of high lying Rydberg states^{28,29,48} in degenerate atomic gases^{49,50,51} was shown to realize interactions of the type of equation (1). Following Maucher et al.^{29}, such a Rydberg dressing of ^{87}Rb condensates to Rb(35p_{3/2}) states^{52} with a laser detuning of ~500 MHz, and an intensity of 100 kW cm^{−2} would produce a sizeable interaction strength of U~35. While we have focussed here on the zerotemperature limit, we have also performed finitetemperature simulations, showing that these parameters will permit the experimental observation of defectinduced supersolid phases for temperatures T≲10 nK, around typical densities ~10^{8} cm^{−2} and with a condensate lifetime of ~30 ms, limited by radiative decay of the weakly admixed Rydberg state. Recent experimental breakthroughs reporting the first observation of Rydberg interaction effects in a laserdriven Bose–Einstein condensate^{53} hold high promise for the prospective realization of the setting described in this work.
Methods
Numerical details
Our numerical results were obtained from pathintegral Monte Carlo simulations^{54} based on the continuousspace worm algorithm^{55} to determine the equilibrium properties of equation (2) in the canonical ensemble, that is, at a fixed temperature T and a fixed particle number, chosen between N=100 and 400. From these simulations we obtain, for example, density profiles, n(r)=‹(r−r_{i}(t))›_{t}, and pair correlation functions, g_{2}(r)=[2πn(N−1)r]^{−1} ‹(r−r_{ij}(t))›_{t}, as well as the superfluid fraction f_{s}, computed from the area estimator, as described in Sindzingre et al.^{56} and Pollock et al.^{57} Here r_{ij}=r_{j}−r_{i}, r_{i} are the positions of the i=1,…,N particles, and ‹..›_{t} denotes an average of the corresponding imaginary time trajectories r_{i}(t). The properties of the system ground state were obtained by extrapolating to the limit of zero temperature, that is, by lowering the temperature until observables, such as the total energy, superfluid fraction and pair correlations did not change on further decrease of T. Moreover, the size of our twodimensional simulation box with periodic boundary conditions was varied to assure insensitivity to system size. The calculated observables were used to construct the phase diagram. The firstorder superfluid–normal solid transition is detected by an abrupt increase of the maximum value (S_{max}) of the static structure factor S(k)=1+ρ∫dre^{ikr}(g_{2}(r)−1) and a simultaneous vanishing of the superfluid fraction. The superfluid–supersolid transition is characterized by a jump of S_{max} and an abrupt decrease of the superfluid fraction from f_{s}≈1 to a finite value f_{s}>0, while the supersolid–normal solid transition is signalled by the vanishing of f_{s}.
Additional information
How to cite this article: Cinti, F. et al. Defectinduced supersolidity with softcore bosons. Nat. Commun. 5:3235 doi: 10.1038/ncomms4235 (2014).
References
 1
Boninsegni, M. & Prokof'ev, N. V. Colloquium: supersolids: what and where are they? Rev. Mod. Phys. 84, 759–776 (2012).
 2
Gross, E. P. Unified theory of interacting bosons. Phys. Rev. 106, 161–162 (1957).
 3
Andreev, A. F. & Lifshitz, I. M. Quantum theory of defects in crystals. JETP 29, 1107–1113 (1969).
 4
Chester, G. V. Speculations on BoseEinstein condensation and quantum crystals. Phys. Rev. A 2, 256–258 (1970).
 5
Balibar, S. The enigma of supersolidity. Nature 464, 176–182 (2010).
 6
Kim, E. & Chan, M. H. W. Probable observation of a supersolid helium phase. Nature 427, 225–227 (2004).
 7
Kim, E. & Chan, M. H. W. Observation of superflow in solid helium. Science 305, 1941–1944 (2004).
 8
Day, J. & Beamish, J. Pressuredriven flow of solid helium. Phys. Rev. Lett. 96, 105304 (2006).
 9
Rittner, A. S. & Reppy, J. Observation of classical rotational inertia and nonclassical supersolid signals in solid He4 below 250 mK. Phys. Rev. Lett. 97, 165301 (2006).
 10
Aoki, Y., Graves, J. C. & Kojima, H. Oscillation frequency dependence of nonclassical rotation inertia of solid ^{4}He. Phys. Rev. Lett. 99, 015301 (2007).
 11
Day, J. & Beamish, J. Lowtemperature shear modulus changes in solid ^{4}He and connection to supersolidity. Nature 450, 853–856 (2007).
 12
Hunt, B. et al. Evidence for a superglass state in solid ^{4}He. Science 324, 632–636 (2009).
 13
West, J. T., Lin, X., Cheng, Z. G. & Chan, M. H. W. Supersolid behavior in confined geometry. Phys. Rev. Lett. 102, 185302 (2009).
 14
Day, J., Syshchenko, O. & Beamish, J. Intrinsic and dislocationinduced elastic behavior of solid helium. Phys. Rev. B 79, 214524 (2009).
 15
Day, J., Syshchenko, O. & Beamish, J. Nonlinear elastic response in solid helium: critical velocity or strain? Phys. Rev. Lett. 104, 075302 (2010).
 16
Reppy, J. D. Nonsuperfluid origin of the nonclassical rotational inertia in a bulk sample of solid ^{4}He. Phys. Rev. Lett. 104, 255301 (2010).
 17
Choi, H., Takahashi, D., Choi, W., Kono, K. & Kim, E. Staircaselike suppression of supersolidity under rotation. Phys. Rev. Lett. 108, 105302 (2012).
 18
Mi, X. & Reppy, J. D. Anomalous behavior of solid ^{4}He in porous vycor glass. Phys. Rev. Lett. 108, 225305 (2012).
 19
Prokof'ev, N. & Svistunov, B. Supersolid state of matter. Phys. Rev. Lett. 94, 155302 (2005).
 20
Ceperley, D. M. & Bernu, B. Ring exchanges and the supersolid phase of ^{4}He. Phys. Rev. Lett. 93, 155303 (2004).
 21
Boninsegni, M. et al. Fate of vacancyinduced supersolidity in ^{4}He. Phys. Rev. Lett. 97, 080401 (2006).
 22
Rota, R. & Boronat, J. Onset temperature of BoseEinstein condensation in incommensurate solid ^{4}He. Phys. Rev. Lett. 108, 045308 (2012).
 23
Ma, P. N., Pollet, L., Troyer, M. & Zhang, F.C. A classical picture of the role of vacancies and interstitials in helium4. J. Low Temp. Phys. 152, 156–163 (2008).
 24
Lechner, W. & Dellago, C. Defect interactions in twodimensional colloidal crystals: vacancy and interstitial strings. Soft Matter 5, 2752–2758 (2009).
 25
Kim, D. Y. & Chan, M. H. W. Absence of supersolidity in solid helium in porous vycor glass. Phys. Rev. Lett. 109, 155301 (2012).
 26
Pomeau, Y. & Rica, S. Dynamics of a model of supersolid. Phys. Rev. Lett. 72, 2426–2430 (1994).
 27
Cinti, F. et al. Supersolid droplet crystal in a dipoleblockaded gas. Phys. Rev. Lett. 105, 135301 (2010).
 28
Henkel, N., Nath, R. & Pohl, T. Threedimensional roton excitations and supersolid formation in Rydbergexcited BoseEinstein condensates. Phys. Rev. Lett. 104, 195302 (2010).
 29
Maucher, F. et al. Rydberginduced solitons: threedimensional selftrapping of matter waves. Phys. Rev. Lett. 106, 170401 (2011).
 30
Büchler, H. P. et al. Strongly correlated 2D quantum phases with cold polar molecules: controlling the shape of the interaction potential. Phys. Rev. Lett. 98, 060404 (2007).
 31
Micheli, A., Pupillo, G., Büchler, H. P. & Zoller, P. Cold polar molecules in twodimensional traps: tailoring interactions with external fields for novel quantum phases. Phys. Rev. A 76, 043604 (2007).
 32
Likos, C. N., Lang, A., Watzlawek, M. & Löwen, H. Criterion for determining clustering versus reentrant melting behavior for bounded interaction potentials. Phys. Rev. E 63, 031206 (2001).
 33
Mladek, B. M. et al. Formation of polymorphic cluster phases for a class of models of purely repulsive soft spheres. Phys. Rev. Lett. 96, 045701 (2006).
 34
Coslovich, D., Strauss, L. & Kahl, G. Hopping and microscopic dynamics of ultrasoft particles in cluster crystals. Soft Matter 7, 2127–2137 (2011).
 35
Sepulveda, N., Josserand, C. & Rica, S. Superfluid density in a twodimensional model of supersolid. Eur. Phys. J. B 78, 439–447 (2010).
 36
Saccani, S., Moroni, S. & Boninsegni, M. Excitation spectrum of a supersolid. Phys. Rev. B 83, 092506 (2011).
 37
Kunimi, M. & Kato, Y. Meanfield and stability analyses of twodimensional flowing softcore bosons modeling a supersolid. Phys. Rev. B 86, 060510 (2012).
 38
Mason, P., Josserand, C. & Rica, S. Activated nucleation of vortices in a dipoleblockaded supersolid condensate. Phys. Rev. Lett. 109, 045301 (2012).
 39
Henkel, N., Cinti, F., Jain, P., Pupillo, G. & Pohl, T. Supersolid vortex crystals in Rydbergdressed BoseEinstein condensates. Phys. Rev. Lett. 108, 265301 (2012).
 40
Macrì, T., Maucher, F., Cinti, F. & Pohl, T. Elementary excitations of ultracold softcore bosons across the superfluidsupersolid phase transition. Phys. Rev. A 87, 061602 (2013).
 41
Astrakharchik, G. E., Boronat, J., Kurbakov, I. L. & Lozovik, Y. E. Quantum phase transition in a twodimensional system of dipoles. Phys. Rev. Lett. 98, 060405 (2007).
 42
Mora, C., Parcollet, O. & Waintal, X. Quantum melting of a crystal of dipolar bosons. Phys. Rev. B 76, 064511 (2007).
 43
Osychenko, O. N., Astrakharchik, G. E., Lutsyshyn, Y., Lozovik, Y. u. E. & Boronat, J. Phase diagram of Rydberg atoms with repulsive van der Waals interaction. Phys. Rev. A 84, 063621 (2011).
 44
Lechner, W. & Dellago, C. Point defects in twodimensional colloidal crystals: simulation vs. elasticity theory. Soft Matter 5, 646–659 (2009).
 45
Markland, T. E. et al. Theory and simulations of quantum glass forming liquids. J. Chem. Phys. 136, 074511 (2012).
 46
Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010).
 47
Mottl, R. et al. Rotontype mode softening in a quantum gas with cavitymediated longrange interactions. Science 336, 1570–1573 (2012).
 48
Pupillo, G., Micheli, A., Boninsegni, M., Lesanovsky, I. & Zoller, P. Strongly correlated gases of Rydbergdressed atoms: quantum and classical dynamics. Phys. Rev. Lett. 104, 223002 (2010).
 49
Heidemann, R. et al. Rydberg excitation of BoseEinstein condensates. Phys. Rev. Lett. 100, 033601 (2008).
 50
Viteau, M. et al. Rydberg excitations in BoseEinstein condensates in quasionedimensional potentials and optical lattices. Phys. Rev. Lett. 107, 060402 (2011).
 51
Schauß, P. et al. Observation of spatially ordered structures in a twodimensional Rydberg gas. Nature 491, 87–91 (2012).
 52
Tong, D. et al. Local blockade of Rydberg excitation in an ultracold gas. Phys. Rev. Lett. 93, 063001 (2004).
 53
Balewski, J. B. et al. Coupling a single electron to a BoseEinstein condensate. Nature 502, 664–667 (2013).
 54
Ceperley, D. M. Path integrals in the theory of condensed helium. Rev. Mod. Phys. 67, 279–355 (1995).
 55
Boninsegni, M., Prokof'ev, N. & Svistunov, B. Worm algorithm for continuousspace path integral Monte Carlo simulations. Phys. Rev. Lett. 96, 070601 (2006).
 56
Sindzingre, P., Klein, M. L. & Ceperley, D. M. Pathintegral Monte Carlo study of lowtemperature ^{4}He clusters. Phys. Rev. Lett. 63, 1601–1605 (1989).
 57
Pollock, E. L. & Ceperley, D. M. Pathintegral computation of superfluid densities. Phys. Rev. B 36, 8343–8352 (1987).
 58
Leggett, A. J. Can a solid be "superfluid"? Phys. Rev. Lett. 25, 1543–1547 (1970).
Acknowledgements
We thank M. Boninsegni, S. Pilati, N.V. Prokof'ev and S.G. Söyler for valuable discussions. This work was supported by the EU through the ITN COHERENCE. G.P. is supported by the ERCSt Grant ‘ColDSIM’ (Grant agreement 307688) and EOARD. W.L. acknowledges support by the Austrian Science Fund through P 25454N27.
Author information
Affiliations
Contributions
F.C. performed the simulations and analysed the data together with T.M.; T.P. drafted the manuscript; and all authors conceived the project, discussed the results and contributed to the final version of the article.
Corresponding author
Correspondence to F. Cinti.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/byncsa/3.0/
About this article
Cite this article
Cinti, F., Macrì, T., Lechner, W. et al. Defectinduced supersolidity with softcore bosons. Nat Commun 5, 3235 (2014). https://doi.org/10.1038/ncomms4235
Received:
Accepted:
Published:
Further reading

Ground state of weakly repulsive softcore bosons on a sphere
Physical Review A (2019)

Quantum hydrodynamics for supersolid crystals and quasicrystals
Physical Review A (2019)

Supersolid phases of Rydbergexcited bosons on a triangular lattice
Physical Review A (2019)

Soliton lattices in the Gross–Pitaevskii equation with nonlocal and repulsive coupling
Physics Letters A (2019)

Slow dynamics coupled with cluster formation in ultrasoftpotential glasses
The Journal of Chemical Physics (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.