Tuneable defect interactions and supersolidity in dipolar quantum gases on a lattice potential
Point defects in self-assembled crystals, such as vacancies and interstitials, attract each other and form stable clusters. This leads to a phase separation between perfect crystalline structures and defect conglomerates at low temperatures. We propose a method that allows one to tune the effective interactions between point defects from attractive to repulsive by means of external periodic fields. In the quantum regime, this allows one to engineer strongly-correlated many-body phases. We exemplify the microscopic mechanism by considering dipolar quantum gases of ground state polar molecules and weakly bound molecules of strongly magnetic atoms trapped in a weak optical lattice in a two-dimensional configuration. By tuning the lattice depth, defect interactions turn repulsive, which allows us to deterministically design a novel supersolid phase in the continuum limit.
pacs:61.72.jd, 67.85.-d, 64.70.Tg, 05.30.Rt
Defects are crucial for the determination of macroscopic mechanical, optical, and electronic properties of solids TAYLOR (); DISLOCATIONS (); HIRTH_LOTHE (). One key aspect is the mutual effective interactions between point defects such as vacancies or interstitials, corresponding to the lack or excess of crystal particles, respectively. In self-assembled classical crystals the effective interactions between point defects is attractive for all combinations of defects (vacancy-vacancy, interstitial-interstitial, interstitial-vacancy) in a wide range CLASSICALDEFECT2 (); DEFECTS_CLASSICAL (); L_DISPLACEMENT (); L_ELASTCITY (). This interaction leads to the formation of string-like defect clusters COLLOIDEXP (). The mechanism behind the attraction of defects is a result of non-linear effects in the displacement fields of multiple defects and cannot be described as a simple pair interaction within elasticity theory. In the quantum regime, the interaction between vacancies is even less well understood, and it is an open question whether the classical results may be directly used to infer many-body properties in the quantum regime. There, the interaction between vacancies is a crucial part of the theory for the supersolid phase ALC (); ALC2 (); GROSS (); ENIGMA (); ANDERSON (); BoninsegniRevModPhys (). The supersolid is conjectured to be a result of delocalized vacancies. However, the precise role of the defect dynamics to establish both superfluid and crystalline orders has been the object of intense investigations in the last decades SUSOEXP (); DEFECTS_FATE (); Boninsegni2005 (); KIMCHAN (); CINTI (); SOFTCORE2 (); SOFTCORE1 (); REATTOREV (); RICA1 (); RICA2 (); MACRI1 (); MACRI2 (); KUNIMI (); ANCILOTTO ().
Here, we show that effective interactions between point defects in a self-assembled crystal can be systematically tuned from attractive to repulsive by means of external periodic superlattices. The lattice spacing of the optical lattice is chosen to be identical or double the lattice spacing of the self assembled crystal. In this setup, the interaction between point defects is the result of an interplay between displacement induced attraction and energy induced repulsion. By properly choosing the parameters of the superlattice, the relative strength of the displacement-induced energy and entropy parts can be tuned, thus making the interaction an experimentally accessible variable.
Based on this microscopic picture of tunable defects, we investigate the many-body quantum regime. As an example, using exact quantum Monte-Carlo methods, we study the phases of a dipolar bosonic gas in the presence of a weak triangular optical lattice. Unlike previous works that focused on the tight-binding limit valid for deep lattice potentials Danshita (); Pollet2010 (); Boninsegni2005 (); BoninsegniRevModPhys (), here we focus on the continuous space limit where the band structure of the lattice is not formed or barely formed. We investigate the phase diagram by varying the lattice depth and the strength of the dipole-dipole interaction both for a commensurate and an incommensurate filling of the lattice potential, around a small density . In the commensurate case, we observe a superfluid to insulating quantum phase transition. Most notably, introducing defects in the presence of the weak periodic potential allows for the realisation of defect-induced supersolidity as originally proposed by Andreev and Lifshitz ALC2 (). In our case, the triangular crystalline structure for the dipoles is either due to direct strong dipolar interactions or to ”pinning” of the weakly interacting superfluid by the lattice potential (without opening of a spectral gap), depending on the system parameters. We present a zero-temperature phase diagram which furnishes a complete description of the superfluid-supersolid and crystal-supersolid phase transition. We find that superfluity and -condensate fraction are comparatively robust against finite temperature, and estimate possible experimental parameters for experiments with ground state polar molecules recentJILA (); JILA (); MOL1 (); MOL2 (); MOL3 (); LICS (); NAK (); HEXTRANS () and weakly bound molecules of strongly dipolar magnetic atoms Ferlaino (); Laburthe-Tolra (); Pfau (); Lev (). These results differ qualitatively from their deep-lattice counterparts, where a superfluid state was found for these low densities Danshita (); Pollet2010 ().
We note that investigations of the quantum mechanical phases of lattice Hamiltonians in the continuum limit have so far focused on contact interactions Buechler (); Astra2007 (); Filinov (). This has led to the prediction of generalized superfluid-Mott insulator transitions in continuous space Pilati (), as well as of solid and superfluid phases of, e.g, He, He and D adsorbed on solid-state surfaces such as, e.g., graphite Pollet (), graphane Nava (); Carb (); Boronat2009 (); Boronat2010 (), Alkali substrates ALKALISUBSTRATE (); Caz2013 (). The present work takes a step in a similar direction for the case of finite range interactions.
The methodology presented in this paper introduces a new toolbox for the manipulation of complex matter, in analogy to the techniques developed to tune and shape the direct interactions between particles in systems as diverse as classical colloids Likos2001 (); Int0 (); Int1 (); Int2 (); Int3 (); Int4 (); Int5 (); BECHINGER (), as well as atomic and molecular systems in the quantum regime RYD1 (); Chin2010 (); Jones2006 (); Buechler (); Gorshkov2013 (); RYD3 (), which is the basis for the success in the realization of many-body phases in these systems BlochRMP (); Baranov (); Tura (). The proposal is based on self-assembled 2D crystals of polar molecules. For details on the required parameters for trapping and interaction strengths see Ref. Buechler ().
The paper is organised as follow: in Section II we introduce the model Hamiltonian. In Section III we show and discuss in detail results concerning defect interactions in the classical regime. The study of the quantum phases is proposed in Section IV. Section V discusses the influence of temperature on supersolidity as well as parameters for possible physical realisations of self-assembled crystals in cold dipolar quantum gases Buechler (). Finally, in Section VI we draw some conclusions and future outlooks.
We consider ultracold polar molecules which are trapped in two dimensions with an additional triangular optical lattice [Fig. 1(a)] described by the Hamiltonian
Here is the mass, while represents the single particle momentum. In the presence of a perpendicular electric field, the dipole-dipole interaction is purely repulsive Buechler () with strength . The last term of Eq. (1) represents the external potential of a triangular optical lattice with depth
which can be implemented with two standing laser beams SUPERLATTICE (). In the following energies and distances are given in units of , being the optical lattice’s constant. The particle density is given in units of , where is the size of the system and the particle number. As a consequence, the filling fraction is for particles with average distance and for , respectively. We consider a crystal commensurate with respect to the lattice if the ratio between density and filling fraction is an integer value and the crystal has triangular symmetry. Therefore, both and are commensurate configurations.
Hamiltonian Eq. (1) can be realized with cold polar molecules (e.g. KRb JILA (), RbCs Innsbruck (), NaK NAK () or LiCs LICS ()), where particle densities are usually Ye2013 (). Alternatively, Rydberg-dressed atoms Biedermann2014 (); Pfau2014 (); RYD1 (); RYD2 (); RYD3 (); Pohl2010 (); Pohl2014 () or ground state magnetic atoms Ferlaino (); Laburthe-Tolra (); Pfau (); Lev () may be used, with . In the latter case, recent experiments with weakly bound Er molecules composed of two highly-magnetic ground state atoms open the way to the realisation of cold quantum gases with comparatively large dipole moments and high densities Dalmonte (); Ferlaino2015 ().
Iii Defect Interactions
Interstitials (vacancies) are point defects which result from adding (removing) a single particle to (from) a self-assembled crystal. Due to entropy, a finite number of defects exists even in thermal equilibrium HIRTH_LOTHE (). These point defects induce a long-range displacement field L_DISPLACEMENT (); L_ELASTCITY () with a non-linear short range part that is responsible for the complex dynamics of isolated defects REICHHARDT () and the interaction between them DEFECTS_CLASSICAL (); L_INTERACTION (). On a quantitative level, effective interactions can be understood as the change in free energy, , as a function of the defect distance for various and fixed parameters (, , ) with
Here, denotes ensemble averages and is the probability to find two defects at distance . The first equality corresponds to the so-called reversible work theorem CHANDLER ().
The parameter in Eq. (3) describes the distance between the two defects and is determined using the following protocol, already introduced e.g. in Refs. L_INTERACTION (); COLLOIDEXP (). In each time-step, a virtual triangular lattice with lattice spacing equivalent to the average particle distance is considered. Then, each particle is associated with the closest virtual lattice site. This implies that in the presence of an interstitial there will be exactly one doubly-occupied virtual lattice site. The position of this lattice site is defined as the position of the interstitial. For vacancies, the unoccupied lattice site is the position of the vacancy. The free energy profile of two defects as a function of the distance corresponds to the effective interaction between the two defects. In the following, Eq. (3) is evaluated using Monte Carlo sampling in combination with the self-consistent histogram method (see Ref. FRENKEL ()). The effective force between the defects is then the negative slope of the free energy.
In free space () vacancies and interstitials attract each other in all combinations (vacancy-vacancy, interstitial-interstitial) and form string-like defect clusters, as shown in Ref. COLLOIDEXP (). Examples of this behaviour are given in Figs. 1 (b1) and (c1) for two interstitials and two vacancies, respectively. The resulting effective interaction potential is purely attractive and increases monotonically with for all cases, as shown in Fig. 2(a-d) for (black continuous line).
We find that the presence of an additional optical lattice [Fig. 1(a)] changes dramatically the energetics, dynamics and interaction of defects. In particular, by increasing the lattice depth can become repulsive. Example results for between interstitials and vacancies for various lattice depths are shown in the upper and lower panels of Fig. 2, respectively, for two choices of particle densities [panels (a) and (c)] and [panels (b) and (d)]. In all cases, the figure shows that for interstitials the turning point where turns first from attractive to repulsive is , while for vacancies a larger depth of is required. For comparatively large lattice depths (e.g., ) the interaction starts to approach the black dashed lines, which correspond to analytical results from a discrete lattice model introduced below. The sign and strength of the effective interactions is however density dependent for intermediate lattice strengths. For example, in the case of interstitials displays a non-monotonic dependence on for densities close to [panel(b)]. For the dynamics of defects, this implies phase separation and cluster formation for . An example of this behaviour is given in Fig. 1 (b3). Finally, we find that for the dynamics of defects is effectively frozen, as the energy necessary to hop from one site to the next becomes increasingly prohibitive.
The limit of a large lattice depth in the commensurate crystal (with any filling) can be understood from the following simple lattice model. In this model, particles are fixed to lattice positions and can hop between sites with given (temperature-dependent) rates. Each lattice site can be un-occupied, occupied or doubly-occupied. As above, we assume that the direct interaction between individual particles is . We remove the divergence at by fixing the energy of a doubly-occupied site to .
Let us first consider the interaction between two interstitials in this model. In this case, all sites will be occupied and two sites are doubly-occupied, for the case of unit filling plus two additional particles. The energy of the system is then . The first term corresponds to the sum of all interactions in the system with particles and the second term is the offset from the two interstitials. The first term is a function of the distance between the additional particles, say and . The effective potential between the defects reads . The term and all contributions in the first term up to last term cancel. This is identical to the interaction between two particles and therefore, the effective potential is .
For vacancies, the situation is less obvious. In this case, the system contains occupied lattice sites and two un-occupied lattice sites. Here, the distance between the two vacancies is defined as the distance between the two un-occupied lattice sites. Considering again , we find that when summing up all energy contributions in the system the energy behaves exactly as . Therefore, also two vacancies in the lattice are repulsive with . These exact results are depicted in Fig. 2 as dashed lines.
Based on this microscopic classical model of tunable defect interactions, in the next Section we present results on defect-induced quantum phases using exact quantum Monte Carlo simulations for bosonic dipoles.
Iv Quantum Phases
The present section examines the applicability of classical predictions to the quantum regime,
considering commensurate and incommensurate filling values around .
We chose this value as the classical groundstate configuration at
is the commensurate triangular crystal that best minimises lattice-induced frustration: the free-space triangular lattice is very little distorted by the external lattice potential. In addition, previous studies using tight-binding models for bosonic particles valid in the limit of very large lattice depths have mainly focused on experimentally challenging (for molecules) higher densities as Pollet2010 (); Boninsegni2005 (); Wessel2005 (); Heid2005 (); Melko2005 (). This regime of lower densities is thus essentially unexplored from the point of view of the investigation of quantum many-body phases.
In the following we will be specifically concerned with the emergence of supersolid behaviour for densities close to the commensurate filling .
In our analysis, we use an exact numerical quantum Monte-Carlo algorithm in the continuous-space path integral (PIMC) representation WORM (); Ceperley (); Buechler (). Our PIMC code is based on the so-called worm algorithm, which is known to efficiently provide numerically exact estimates of thermodynamic quantities such as the superfluid density and the structure factor, which can be used as order parameters to determine the nature of superfluid and solid phase, respectively. The superfluid density reads
with the winding number estimator along the and directions SF (). The static structure factor is instead defined as
with a crystal vector, and characterizes diagonal order. We use up to =188 particles and about 750 sites to minimize finite-size effects and defect concentrations of up to four percent.
Quantum phases of Eq. (1) with have been investigated for the case of bosonic polar molecules in Refs. Buechler (); Astra2007 (); Filinov (); Moroni2014 (). The phase diagram is characterised by a melting quantum phase transition from a triangular crystal phase [with and ] to a homogeneous superfluid [with and ] by decreasing the interaction strength, defined as
below a critical value . Here we are interested in investigating the phase diagram as a function of
and of the depth of an additional optical lattice.
Figure 3 (upper panel) displays the ground state limit of vs. for an interaction strength , corresponding to a superfluid for . In the figure, we consider both the case of a lattice commensurate with the dipole density at filling (black circles) and the case of non-commensurate filling with a small density of interstitial defects (equaling 0.02 and 0.04 for the red empty and full squares, respectively).
For (black circles), the figure displays a sudden drop of , suggesting a quantum phase transition from a homogeneous superfluid () to an insulating phase () induced by the increase of the strength of the lattice potential, occurring at the critical value . This picture is corroborated by a sudden increase of the static structure factor at the same value [Fig. 3 (lower panel)], showing that the insulating phase is in fact a crystal. This transition is thus driven by the suppression of quantum kinetic energy with increasing Bloch2008 (). As we discuss below, for physical realizations with, e.g., polar molecules KRb or Er, the value of corresponds to a very small value of the lattice depth, where the two-dimensional band structure is not formed. Thus, this quantum phase transition can be regarded to happen in the continuum, as a two-dimensional analog of the so-called pinning quantum phase transition that has been predicted in one dimension Dalmonte (); Buchler2003 (); Giamarchi (). For short-range interactions, the one-dimensional analog has been observed experimentally in 1D ExpNaegerl (); Buchler2003 () while the theree-dimensional case has been studied via quantum Monte-Carlo simulations in Ref. Pilati ().
The scenario of the superfluid-crystal quantum phase transition described above changes drastically when defects are introduced: in this case remains finite but not unitary, as expected for a non-homogeneous superfluid leggett (). This is shown in Fig. 3 (upper panel)] in the parameter regime , where the superfluid density increases with the concentration of interstitial defects (see, in particular, the cases and 12).
Interestingly, we find that in the whole parameter range the order parameter is finite and maximal at the crystal wave-vector corresponding to the commensurate dipolar crystal with (as in the case described above), implying a diagonal crystalline order with a periodicity that is different from that one of the underlying lattice potential Note1 (). This is in contrast, e.g., to the case with short-range interactions Pilati (), where the periodicity of the (Mott) insulating phase is trivial, in that it coincides with the one of the underlying lattice.
Here, the coexistence of a finite and in the parameter regime with demonstrates the realisation of a supersolid state of matter with coexisting diagonal order and superfluidity.
Figure 4 shows the complete zero-temperature phase diagram as a function of and , again keeping the defect density fixed at 0.02. As described above, we find regions of superfluid, crystal and supersolid behaviour. In particular, for we re-obtain the phase diagram for a bosonic dipolar gas discussed above Buechler (), characterised by the quantum melting transition of the triangular crystal into a homogeneous superfluid at . We find that increasing from initially has the simple effect to shift the quantum melting transition to smaller values of , consistent with the example of Fig. 3 (black empty dots). However, the nature of the transition changes dramatically above : a defect-induced supersolid phase intervenes between the superfluid and the crystal. For all phases can be observed by simply tuning .
An interesting example of this latter situation is shown in Fig. 5, which presents results for , corresponding to a cut in the phase diagram of Fig. 4 along the indicated dashed line. Figure 5 shows that, while grows monotonically with increasing , the superfluid density displays an interesting re-entrant behavior: it is for , then drops to zero for , and then becomes again finite with - for ; this latter behavior corresponds to a supersolid. We notice that in this parameter region with the crystalline structure is purely imposed by the presence of the lattice potential, which pins the strongly interacting superfluid, however, crucially without opening a gap. This is similar to the lattice supersolid discussed in Refs. Pollet2010 (); Boninsegni2005 (), however it occurs for shallow lattice depths, where the band structure is not formed (see below).
For the triangular crystalline structure is present also for . As the crystal is essentially classical (however, see HEXTRANS ()), we expect that the results on the tunability of defect interactions derived in the Sec.III above should provide directly insights into defect dynamics in this parameter regime. Indeed, in the PIMC quantum calculations we find that the effect of a sufficiently deep lattice in this crystalline case (e.g., ) is to originate a finite superfluid density coexisting with crystalline order, when a finite density of defects is present. This defect-induced supersolidity is only possible for effective repulsive interactions between the defects, as would be predicted by the classical results given above.
Finally, for sufficiently large the superfluid fraction vanishes altogether, and the groundstate evolves into an insulating lattice-type crystal. This is similar to the observed frozen dynamics in the classical regime.
In order to visualise the difference between the solid and supersolid phases, Fig. 6 shows snapshots of the projection of world-lines onto the xy-plane taken from the PICM simulations, obtained by tracing over imaginary time, at in the absence of defects [Fig. 6(a)] and with two interstitials [Fig. 6(b)] for and . As explained in literature Ceperley (), these projections (that are for illustration purposes only) are the closest representation of the square of the wave-function for the many-body system that can be obtained in a simulation, where overlapping paths imply exchanges among the bosonic particles and superfluidity. The figure shows that in the absence of defects, paths remain localised around the local minima of the lattice potential . However, in the supersolid phase localised paths coexist with paths that are delocalized throughout the system, representing cyclic exchanges (permutations) among bosons [see Fig. 6(b), red thick line]. This residual exchange mechanism is consistent with defect-induced supersolidity as originally proposed by Andreev and Lifshitz ALC2 () and only recently demonstrated via exact theoretical techniques for bosons with cluster-forming interactions CINTI (). Our analysis shows that defect-induced supersolidity can be originated in the continuum also for non-cluster forming liquids, using periodic external potentials. Apart of the qualitative difference, as explained below, this should provide quantitative advantages in the experimental realisation of the supersolid phase, as it could results in, e.g, higher temperatures than possible in the tight-binding regime.
Particle delocalization is also reflected in the (quasi-) condensate fraction, which is easily accessible in experiments Bloch2008 (), defined as the asymptotic (i.e., ) behaviour of the angle averaged one-body density matrix
and  the particle annihilation [creation] operators at position Pethick (). In the presence of long-range off-diagonal order associated to a finite condensate fraction, factorizes at large separation as
with the condensate wave-function. Employing the same set of parameters of Fig. 6(b),
Fig. 6(c) shows that a constant value for the supersolid phase with finite defect
concentration is here realised (full squares), corresponding to a finite (quasi-)condensate fraction at . The latter disappears in the case of commensurate filling [parameters as in Fig. 6(a)], where decays exponentially with distance [empty square in panel (c)], as expected for an insulating crystalline phase.
V Finite temperature and possible physical realizations
The supersolid phase described here is considerably resilient towards finite temperature effects. An example of this is shown in Fig. 7, where we present results for both and as a function of the rescaled temperature , expressed in units of the quantum kinetic energy at the mean inter particle distance . The figure shows that supersolid behaviour survives up to temperatures of the order of 1. The latter is consistent with a BKT transition for with a comparatively large transition temperature (see below).
v.1 Polar molecules:
As an example of experimental realisation of the phase above, here we consider first a gas of bosonic KRb molecules with dipole moment Debye trapped on a lattice with spacing nm. The lattice recoil energy is kHz (in frequency units), while the unit of energy reads kHz. By re-expressing the value in convenient units, the numbers above suggest that the supersolid phase described above would be realised for a weak lattice potential with a depth of just . This corresponds to a situation where the band structure of the two-dimensional lattice has not formed, which is consistent with our claim that this quantum phase transition occurs in the continuum. The quantum kinetic energy at the average particle density in the absence of the optical lattice reads Hz. . Thus, from the results of Fig. 7 we obtain that the supersolid phase would survive up to temperatures of the order of nK or larger.
v.2 Magnetic quantum gases:
Magnetic quantum gases of have been recently trapped in lattices with a small spacing nm Ferlaino2015-2 (). For molecules with magnetic dipole moment of (with the Bohr magneton) this implies kHz, which is comparable to the case of polar molecules. Extrapolation from our numerical results (see Fig. 4) implies that a lattice depth of order of is necessary to induce supersolidity.
Here, the recoil energy is kHz, and thus the condition on the lattice depth for inducing a supersolid behaviour reads , which is well in the continuum limit. The quantum kinetic energy reads Hz, for a low density . This corresponds to a critical temperature for observing supersolidity of the order of nK, which is within experimental reach. We note that for the case of strongly dipolar magnetic gases, reaching densities of order unity in the lattice is possible. This in principle would allow for the observation of some of the effects described above, such as the superfluid-insulator quantum phase transition for commensurate lattice fillings, at considerably higher temperatures.
Vi Conclusions and Outlook
The results presented here may help to settle the long-standing question of the role of defect interactions for the formation of the supersolid phase DEFECTS_FATE (); ANDERSON () and enable the experimental realization of a supersolid with quantum dipolar gases in combination with tunable optical lattices. We note that while here we have focused on specific examples with fixed defect densities, defect-induced supersolidity is expected to appear for a large range of densities for a careful choice of interaction parameters, e.g., similar to the case of Ref. CINTI (). This should make the observation of defect-induced supersolidity possible in experiments where particle density is tunable within a few percent only. The tunability of defect interactions opens also interesting prospects for the observation of other phases, such as solitons and breathers in the classical and quantum regimes BlatterPRB2014 (); Dalmonte ().
Acknowledgements - We thank M. Troyer, P. Zoller, T. Macrì and F. Mezzacapo for fruitful discussions. WL acknowledges support by the Austrian Science Fund (FWF): P 25454-N27. GP is supported by the ERC-St Grant ColdSIM (No. 307688), EOARD, RySQ, UdS via IdEX and ANR via BLUESHIELD.
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