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arXiv:1812.02562 · Journal of Physics A Mathematical and Theoretical

Clusterization of weakly-interacting bosons in one dimension: an analytic study at zero temperature

Santi Prestipino, Alessandro Sergi, Ezio Bruno

Published Nov 6, 2018• 14 citations
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Abstract

Abstract We study a system of penetrable bosons on a line, focusing on the high-density/weak-interaction regime, where the ground state is, to a good approximation, a condensate. Under compression, the system clusterizes at zero temperature, i.e. particles gather together in separate, equally populated bunches. We compare predictions from the Gross–Pitaevskii (GP) equation with those of two distinct variational approximations of the single-particle state, written as either a sum of Gaussians or the square root of it. Not only the wave functions in the three theories are similar, but also the phase-transition density is the same for all. In particular, clusterization occurs together with the softening of roton excitations in GP theory. Compared to the latter theory, Gaussian variational theory has the advantage that the mean-field energy functional is written in (almost) closed form, which enables us to extract the phase-transition and high-density behaviors in fully analytic terms. We also compute the superfluid fraction of the clustered system, uncovering its exact behavior close, as well as very far away from, the transition.

DOI
10.1088/1751-8121/aaee94

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Abstract We study a system of penetrable bosons on a line, focusing on the high-density/weak-interaction regime, where the ground state is, to a good approximation, a condensate. Under compression, the system clusterizes at zero temperature, i.e. particles gather together in separate, equally populated bunches. We compare predictions from the Gross–Pitaevskii (GP) equation with those of two distinct variational approximations of the single-particle state, written as either a sum of Gaussians or the square root of it. Not only the wave functions in the three theories are similar, but also the phase-transition density is the same for all. In particular, clusterization occurs together with the softening of roton excitations in GP theory. Compared to the latter theory, Gaussian variational theory has the advantage that the mean-field energy functional is written in (almost) closed form, which enables us to extract the phase-transition and high-density behaviors in fully analytic terms. We also compute the superfluid fraction of the clustered system, uncovering its exact behavior close, as well as very far away from, the transition.
The properties of a macroscopic assembly of weakly repulsive bosons at zero temperature are well described by Gross-Pitaevskii mean-field theory. According to this formalism the system exhibits a quantum transition from superfluid to cluster supersolid as a function of pressure. We develop a thermodynamically rigorous treatment of the different phases of the system by adopting a variational formulation of the condensate wave function---represented as a sum of Gaussians---that is amenable to exact manipulations. Not only is this description quantitatively accurate, but it is also capable to predict the order (and sometimes even the location) of the transition. We consider a number of crystal structures in two and three dimensions and determine the phase diagram. Depending on the lattice, the transition from fluid to solid can be first order or continuous, a lower coordination entailing a milder transition. In two dimensions, crystallization would occur at the same pressure on three distinct lattices (square, honeycomb, and stripes), all providing metastable phases with respect to the triangular crystal. A similar scenario holds in three dimensions, where the simple-cubic and diamond crystals also share a common melting point; however, the stable crystal at low pressure is typically fcc. Upon compression and depending on the shape of the potential, the fcc crystal may transform into hcp. We conclude by sketching a theory of the solid-fluid interface and of quantum nucleation of the solid from the fluid.
We work out the ground-state diagram of weakly-repulsive penetrable bosons, using mean-field theory with a Gaussian ansatz on the single-particle wave function. Upon compression, the fluid transforms into a cluster supersolid, whose structure is characterized for various choices of the embedding space. In Euclidean space, the stable crystals are those with the most compact structure, i.e., triangular and fcc in two and three dimensions, respectively. For particles confined in a spherical surface, as the sphere radius increases we observe a sequence of transitions between different cluster phases, all having a regular or semiregular polyhedron as supporting frame for the clusters. The present results are relevant for the behavior of ultracold bosons weakly coupled to a Rydberg state.
We study a system of penetrable bosons embedded in a spherical surface. Under the assumption of weak interaction between the particles, the ground state of the system is, to a good approximation, a pure condensate. We employ thermodynamic arguments to investigate, within a variational ansatz for the single-particle state, the crossover between distinct finite-size ``phases'' in the parameter space spanned by the sphere radius and the chemical potential. In particular, for radii up to a few interaction ranges we examine the stability of the fluid phase with respect to a number of crystal-like arrangements having the symmetry of a regular or semiregular polyhedron. We find that, while quantum fluctuations keep the system fluid at low density, upon compression it eventually becomes inhomogeneous, i.e., particles gather together in clusters. As the radius increases, the nature of the high-density aggregate varies and we observe a sequence of transitions between different cluster phases (``solids''), whose underlying rationale is to maximize the coordination number of clusters, while ensuring at the same time the proper distance between each neighboring pair. We argue that, at least within our mean-field description, every cluster phase is supersolid.
The ground state of a two-dimensional (2D) system of Bose particles of spin zero, interacting via a repulsive Gaussian-core potential, has been investigated by means of quantum Monte Carlo simulations. The quantum phase diagram is qualitatively identical to that of 2D Yukawa bosons. While the system is a fluid at all densities for weak coupling, in the strong-coupling regime it transitions upon compression from a low-density superfluid to a crystal, and then into a reentrant superfluid phase. No evidence of a (supersolid) cluster crystal phase is seen.
Motivated by the fundamental question of the fate of interacting bosons in flat bands, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This type of band structure can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jim\'enez-Garcia, and I. B. Spielman, Nature (London) 471, 83 (2011)], in the regime where the lower-band dispersion has the form ${\ensuremath{\varepsilon}}_{\mathbit{k}}\ensuremath{\sim}{k}_{x}^{4}/4+{k}_{y}^{2}+...$, or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha, and C. Chin, Nat. Phys. 9, 769 (2013)]. We numerically compare the ground-state energies of the mean-field Bose-Einstein condensate (BEC) and various trial wave functions, where bosons avoid each other at short distances. We discover that, at low densities, several types of strongly correlated states have an energy per particle ($\ensuremath{\epsilon}$), which scales with density (nnn) as $\ensuremath{\epsilon}\ensuremath{\sim}{n}^{4/3}$, in contrast to $\ensuremath{\epsilon}\ensuremath{\sim}n$ for the weakly interacting Bose gas. These competing states include a Wigner crystal, quasicondensates described in terms of properly symmetrized fermionic states, and variational wave functions of Jastrow type. We find that one of the latter has the lowest energy among the states we consider. This Jastrow-type state has a strongly reduced, but finite, condensate fraction, and true off-diagonal long-range order, which suggests that the ground state of interacting bosons with quartic dispersion is a strongly correlated condensate reminiscent of superfluid helium-4. Our results show that even for weakly interacting bosons in higher dimensions, one can explore the crossover from a weakly coupled BEC to a strongly correlated condensate by simply tuning the single-particle dispersion or density.
We compute the superfluid density of a two-dimensional boson system with weak two-body repulsive interactions at zero temperature using one-loop perturbation theory in the weak coupling region. The boson fields are taken to be in continuum form in real space, and the interactions are approximated by a δ function of the distance between the bosons. We find that the one-loop fluctuations slightly decrease the superfluid density of a classical level. The superfluid density is approximated by the condensate density multiplied by the mass of the boson particle.