Self-bound crystals of antiparallel dipolar mixtures

Recent experiments have created supersolids of dipolar quantum droplets. The resulting crystals lack, however, a genuine cohesive energy and are maintained by the presence of an external conﬁnement, bearing a resemblance to the case of ion Coulomb crystals. We show that a mixture of two antiparallel dipolar condensates allows for the creation of potentially large, self-bound crystals, which, resembling ionic crystals in solid-state physics, are maintained by the mutual dipolar attraction between the components, with no need of transversal conﬁnement. This opens intriguing possibilities, including three-dimensionally self-bound droplet-ring structures, stripe / labyrinthic patterns, and self-bound crystals of droplets surrounded by an interstitial superﬂuid, resembling the case of superﬂuid Helium in porous media.

Recent experiments have created supersolids of dipolar quantum droplets.The resulting crystals lack, however, a genuine cohesive energy and are maintained by the presence of an external confinement, bearing a resemblance to the case of ion Coulomb crystals.We show that a mixture of two antiparallel dipolar condensates allows for the creation of potentially large, self-bound crystals which, resembling ionic crystals in solid-state physics, are maintained by the mutual dipolar attraction between the components, with no need of transversal confinement.This opens intriguing novel possibilities, including three-dimensionally self-bound droplet-ring structures, stripe/labyrinthic patterns, and self-bound crystals of droplets surrounded by an interstitial superfluid, resembling the case of superfluid Helium in porous media.
Solid-state crystals are held together by the interplay between different forms of attractive and repulsive interactions between their constituents [1].This interplay results in a finite cohesive or binding energy, defined as the energy that must be added to the crystal to separate its components infinitely apart.In the presence of an external confinement, crystals may form even in the absence of genuine cohesion.A prominent example is provided by trapped ions, which form crystals due to the combination of repulsive Coulomb interactions and external confinement [2].There is, however, no cohesive energy, and ion Coulomb crystals unravel in the absence of the trap.This feature is shared by recently created crystals of quantum droplets in dipolar Bose-Einstein condensates [3,4].Self-bound droplets, elongated along the dipole direction, result from the quasi-cancellation of contact and dipolar interactions, and the stabilizing effect of quantum fluctuations [5][6][7][8].In the presence of confinement along the dipole direction, energy is minimized by the creation of multiple droplets, which, in the presence of an external confinement perpendicular to the dipole orientation (transversal trap), arrange forming a crystal [9] that may present supersolid properties [10][11][12][13][14][15][16][17].Similar to the case of ions in Coulomb crystals, droplets repel each other.There is hence no genuine cohesive energy of the droplet crystal (or of any other possible density pattern [18][19][20][21]).The transversal trap is crucial to keep it bound.
Recent experiments have created a mixture of two dipolar components [22][23][24].These mixtures are expected to present rich physics due to the competition between intra-and inter-component contact and dipolar interactions, including immiscible droplets [25,26], doping-induced droplet nucleation [24,27], two-fluid supersolidity [27], and the formation of alternating-domain supersolids [28][29][30].Interestingly, the dipoles of the two components may be antiparallel, and hence the inter-and intra-component interactions may have opposite sign [29] (Fig. 1(a)).In this letter, we investigate crystal formation in an antiparallel dipolar mixture (ADM).As for a parallel one [25,26], in the absence of any confinement, an ADM may form an immiscible three-dimensionally self-bound mixture, although with a markedly different topology in which one of the components may eventually form a ring around a droplet of the other.The presence of confinement along the dipole direction results in crystal formation.In stark contrast to both single-component dipolar condensates and parallel binary mixtures, in an ADM the crystal has a genuine cohesive energy, remaining self-bound in the absence of a transversal trap due to the mutual attraction between the components.This resembles the case of ionic crystals in solid-state physics, where ions of opposite charge arrange in an intertwined crystalline structure bound by their mutual electrostatic interaction [1].However, the resulting self-bound ADM is not given by two intertwined droplet arrays.Symmetric ADMs with similar intra-component interaction strengths form self-bound stripe/labyrinthic density patterns.In contrast, in sufficiently asymmetric ADMs, one of the components forms an incoherent droplet crystal with an approximate triangular structure, whereas the second one remains superfluid and fills the lattice interstitials, resembling to some extent superfluid Helium in porous media [31].
Model.-We consider a bosonic ADM, with dipoles oriented, respectively, along and antiparallel to the z axis.The components may belong to the same species or to two different ones.In order to illustrate the possible physics, we consider a dysprosium mixture, with magnetic dipoles µ 1 = 10 µ B and µ 2 = −10 µ B , with µ B the Bohr magneton.Short-range interactions are characterized by the intra-and inter-component scattering lengths: a 11 , a 22 , and a 12 .The physics of the mixture is well described by the extended Gross-Pitaevskii equation [25,26]: where Ψ σ (r, t) is the condensate wavefunction of component σ = 1, 2, n σ = |Ψ σ | 2 , and g σσ = 4π 2 a σσ /m, with m the mass of the bosons.The atoms are confined, if at all, only along the z axis by a potential V trap (r) = 1 2 mω 2 z z 2 .The dipole-dipole interaction is given by the potential V σσ dd (r) = µ0µσµ σ 4πr 3 1 − 3 cos 2 θ , with θ the angle sustained by the z axis and r.The effect of quantum fluctuations is provided by the Lee-Huang-Yang (LHY) term µ LHY,σ [n 1,2 (r, t)] = δE LHY /δn σ , where (2) is the LHY energy correction, with (3) and η σσ = g σσ + g d σσ (3 cos 2 θ k − 1), being g d σσ = µ 0 µ σ µ σ /3 and θ k the angle sustained by k with the z axis.Since the dipole moments of the components are antiparallel, the inter-component dipolar potential is repulsive (attractive) when the components are placed headwith-tail (side-by-side) (see Fig. 1(a)).As a result, the dipolar interaction strongly favors immiscibility, and a very large and negative a 12 is needed to drive the system miscible.In the following, we consider a 12 = 150 a 0 , but the actual value is irrelevant as long as the intercomponent overlapping remains negligible.
Three-dimensionally self-bound ADM.-We first consider the case of fully unconfined mixtures (ω z = 0).As for parallel dipolar mixtures [25,26], an immiscible ADM may present a three-dimensionally self-bound solution, but of a markedly different nature.This is best understood in the impurity limit (N 1 N 2 ).Let us assume that component 1 forms a self-bound droplet with density n 1 (r).The droplet exerts a potential V 1→2 dd (r) = d 3 r V 12 dd (r − r )n 1 (r ) on component 2, which, as seen in Fig. 1(b), is characterized by a marked minimum at a given radius ρ 0 , well outside the droplet.Particles in component 2 are trapped in this mexican-hat potential.
In a more balanced mixture, the argument remains valid, but component 2 also induces a similar potential V 2→1 dd (r) on component 1. Hence the two components confine each other mutually on the xy plane, resulting in self-bound ADMs, as illustrated in Fig. 2 for N 1,2 = N/2, a 11 = 50 a 0 , and different values of a 22 and N .For asymmetric intra-component interactions a 11 < a 22 , component 1 remains a compact droplet, whereas the second component accommodates on the ring potential around the droplet.For low enough a 22 , the energy is minimized by the formation of a single droplet in component 2, which for growing N and a 22 spreads around the mexican-hat minimum until eventually forming a ringlike configuration.For intermediate a 22 values, there is a second possible topology with two droplets of component 2 placed at opposite sides of the annular potential.
Self-bound droplet crystals.-Whenω z = 0, increasing the particle number N results in more elongated solutions along the z direction.As for singlecomponent (scalar) dipolar condensates [3], this elongation is frustrated in the presence of a trap along z (ω z > 0).In scalar condensates, this frustration results in the formation of multiple droplets.Although the droplets repel each other, the presence of a transversal trap on the xy plane allows for the creation of 2D droplet crystals [16,17].These crystals have however no intrinsic cohesion, and hence unravel in the absence of the xy confinement.
Remarkably, this is not the case in an ADM, as illustrated in Fig. 3 for a balanced mixture N 1 = N 2 and asymmetric intra-component interactions, a 11 = 50 a 0 and a 22 = 70 a 0 .For a low-enough ω z , the three- dimensional solution (with a single droplet in component 1) remains valid (Fig. 3(b)).For an N -dependent critical ω z the droplet splits into two.Each one of them exerts a mexican-hat potential on the second component, which gets trapped in the combined energy minimum.At the same time, crucially, the second component glues the two droplets together, forming a self-bound ADM (Fig. 3(c)).As shown in Fig. 3(a), and illustrated for particular cases in Figs.3(d-g), further increasing ω z results in a growing number of droplets of component 1 surrounded by a bath of component 2. In a scalar condensate, each droplet requires a minimal atom number to remain selfbound (otherwise kinetic energy unbinds it), drastically limiting the total number of droplets.In contrast, in an ADM, droplets remain confined by the inter-component interaction, allowing for droplets with a much smaller number of atoms [28,29].As a result, increasing ω z results in 2D crystals with much more droplets compared to scalar condensates with the same total number of atoms.
We should emphasize that our results, based on imaginary-time evolution of Eq. ( 1) with random initial conditions, reveal many possible solutions with very similar energy, which differ in the exact number and arrangement of the droplets (see [32]).We hence expect a significant experimental shot-to-shot variability, similar to that recently observed in experiments on 2D supersolids [16].
Interstitial superfluid.-Due to the lack of overlapping, the droplets are mutually incoherent.In contrast, the component filling the crystal interstitials forms a superfluid that resembles, to some extent, the case of Helium in a porous medium (although, in contrast to that scenario, droplets of component 1 do not form a rigid structure).The approximately triangular crystalline structure of the droplets is inherited as well by the interstitial component 2, which builds hence a peculiar form of supersolid.The coherence and spatial density modulation of component 2 may be revealed in time- of-flight measurements.Figure 4 shows the momentum distribution ñ2 (k x , k y ) in the k z = 0 plane.The approximate triangular structure (Fig. 4(a)) results in an hexagonal pattern in the ñ2 distribution (Fig. 4(c)), although the above-mentioned variability of the exact droplet arrangement may result in a significant shot-dependent distortion (see Figs. 4(b,d)).Note as well that, due to the lack of any confinement on the xy plane, the patterns spontaneously break the polar symmetry and hence experience a random rotation from shot to shot.In any case, as expected from the theory of roton immiscibility [29,33], the inter-droplet distance R is fixed by the oscillator length a z = /mω z .For the case of Fig. 3, R 3 a z for all values of N and ω z .This periodicity becomes evident from the average of the momentum distribution over many realizations, which shows a marked ring at 1/R (see Fig. 4(e)).
Crystal sublimation.-Fora fixed total number of particles, the cohesive energy decreases when the droplet number grows, since lowering the density reduces the inter-component dipolar attraction.Eventually, at a critical frequency ω cr z , the crystal unbinds, and both components evaporate.The critical frequency (ω cr z /2π 1400 Hz for the case on Fig. 3) is approximately determined as that for which the energy per particle reaches ω z /2, corresponding to an infinitely spread solution on the xy plane.Interestingly, when ω z approaches ω cr z , mutual attraction may still be enough to maintain a stable crystal, but insufficient to bind the whole interstitial superfluid, which hence partially evaporates (see [32] for a more detailed discussion).
Self-bound stripe/labyrinthic patterns.-Upto this point, we have considered a mixture with markedly asymmetric intra-component interactions.Interestingly, when a 11 a 22 , the mixture arranges in a different form of self-bound pattern (note that a 11 = a 22 if we consider a mixture of two maximally stretched magnetic states of the same atomic species).This is illustrated by the phase diagram of Fig. 5 (a), obtained for ω z /2π = 1200 Hz and N 1,2 = 5 × 10 4 .For sufficiently large |a 11 − a 22 |, we obtain the above-mentioned droplet crystal (Fig. 5(b)), which, as mentioned above, presents partial evaporation of the interstitial component in the vicinity of the unbinding threshold.In contrast, when a 11 a 22 the mixture arranges in a labyrinthic phase, with a large shot-toshot variability, formed by stripes with different orientations (Fig. 5(c)).For lower trap frequencies, the groundstate configuration is given by a well-defined stripe crystal (Fig. 5(d)).Note that in the labyrinthic/stripe phase both components form mutually incoherent domains.
Summary and Outlook.-Antiparalleldipolar mixtures allow for the formation of crystals with a genuine cohesive energy that remain self-bound in the absence of a transversal trap.The mutual confinement stems from the attractive inter-component interactions, and results in incoherent stripe/labyrinthic crystals in mixtures with symmetric intra-component interactions, and self-bound droplet crystals in asymmetric mixtures.The latter are particularly interesting, since while one component forms an approximately triangular array of incoherent droplets, the other component builds a superfluid in the interstitials, forming a peculiar form of supersolid that may be readily probed using time-of-flight measurements.Although we have considered the particular example of a dysprosium mixture, our results generally apply to other antiparallel magnetic or electric dipolar mixtures, including those of polar molecules.
The possibility of creating self-bound dipolar crystals opens intriguing perspectives for future studies, including the character of lattice excitations, which may remain self-bound or result in phonon evaporation (resembling droplet evaporation in non-dipolar mixtures [5]), the probing (e.g. by vortex formation) of the superfluidity of the interstitial component, as well as in general the exploration of the dynamics of self-bound crystals.This work has been supported by Grant No. PID2020-114626GB-I00 (Ministerio de Ciencia e Innovación), by the European Union Regional Development Fund within the ERDF Operational Program of Catalunya (project QUASICAT/QuantumCat), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2123 QuantumFrontiers-390837967, and FOR 2247.M. A. is supported by FPI Grant PRE2018-084091.

FIG. 1 .
FIG. 1.(a) In an ADM, intra-component interactions are attractive (yellow arrows) when the particles are head-to-tail, and repulsive when they are side-by-side, whereas the opposite is true for the inter-component ones.(b) Dipolar interaction V 1→2 dd (r) = 2µ 0 µ 2 1 3πl 2 z

FIG. 2 .
FIG.2.Three-dimensionally self-bound ADMs.Groundstate configuration as a function of the total atom number N and of a22, for a11 = 50 a0 and N1,2 = N/2.Whereas component 1 always forms a single elongated droplet, component 2 may acquire different topologies, which we characterize using the separation ∆rCM between the center of masses of the two components (color code).The different topologies are illustrated in the insets, where we depict the column density (integrated over z) of the components, with red (blue) indicating component 1 (2).

FIG. 3 .
FIG. 3. Self-bound droplet crystals.(a) Phase diagram as a function of the atom number N and the trap frequency fz = ωz/2π, for a11 = 50 a0 and a22 = 70 a0.Colors correspond to configurations with a different number of droplets ND in component 1. Figures (b-g) show the column magnetization (integrated along z) of the lowest-energy solution for selected cases, indicated with the corresponding symbol in Fig. (a).Red (blue) regions are populated by component 1 (2).