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arXiv:cond-mat/0110314 · Physical review. B, Condensed matter

Analytical and numerical study of hardcore bosons in two dimensions

Karim Bernardet, G. G. Batrouni, J.L. Meunier, Guido Schmid, Matthias Troyer, Ansgar Dorneich

Published Feb 28, 2002• 89 citations
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Abstract

We study various properties of bosons in two dimensions interacting only via on-site hardcore repulsion. In particular, we use the lattice spin-wave approximation to calculate the ground-state energy, the density, the condensate density, and the superfluid density in terms of the chemical potential. We also calculate the excitation spectrum, \ensuremath{\omega}(k). In addition, we perform high precision numerical simulations using the stochastic series expansion algorithm. We find that the spin-wave results describe extremely well the numerical results over the whole density range 0\ensuremath{\leqslant}\ensuremath{\rho}\ensuremath{\leqslant}1. We also compare the lattice spin-wave results with continuum results obtained by summing the ladder diagrams at low density. We find that for \ensuremath{\rho}\ensuremath{\leqslant}0.1 there is good agreement, and that the difference between the two methods vanishes as ${\ensuremath{\rho}}^{2}$ for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\rho}}}$0. This offers the possibility of obtaining precise continuum results by taking the continuum limit of the spin-wave results for all densities. Finally, we study the finite-temperature phase transition numerically for the entire density range, and compare with low density predictions.

DOI
10.1103/physrevb.65.104519

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We study various properties of bosons in two dimensions interacting only via on-site hardcore repulsion. In particular, we use the lattice spin-wave approximation to calculate the ground-state energy, the density, the condensate density, and the superfluid density in terms of the chemical potential. We also calculate the excitation spectrum, \ensuremath{\omega}(k). In addition, we perform high precision numerical simulations using the stochastic series expansion algorithm. We find that the spin-wave results describe extremely well the numerical results over the whole density range 0\ensuremath{\leqslant}\ensuremath{\rho}\ensuremath{\leqslant}1. We also compare the lattice spin-wave results with continuum results obtained by summing the ladder diagrams at low density. We find that for \ensuremath{\rho}\ensuremath{\leqslant}0.1 there is good agreement, and that the difference between the two methods vanishes as \ensuremathρ2{\ensuremath{\rho}}^{2}\ensuremathρ2 for \ensuremathρ\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\rho}}}\ensuremathρ\ensuremath→​0. This offers the possibility of obtaining precise continuum results by taking the continuum limit of the spin-wave results for all densities. Finally, we study the finite-temperature phase transition numerically for the entire density range, and compare with low density predictions.
Motivated by excellent comparisons of spin wave theory (SWT) with Quantum Monte Carlo (QMC) calculations of non-interacting hardore bosons (Bernardet et al.) and superlattices (Hen et al.), we develop an SWT for interacting hardcore bosons on a d-dimensional cubic lattice at zero temperature and compare thermodynamical properties with results of Stochastic Series Expansions (SSE): there is good comparison between SWT and SSE results for the simple cubic lattice at small repulsive strengths but differences arise between the results of the two approaches for the square lattice at strong repulsions. The boundaries of the three phases - Mott insulating (MI), superfluid (SF) and Néel solid (NS) - can be readily estimated from calculations of particle density, signalling either a first order (SF NS) or second order transition (MI SF). Comparison is made with results of the Bethe ansatz in one dimension and QMC in two dimensions.
We study the ground-state properties of trapped inhomogeneous systems of hardcore bosons in two- and three-dimensional lattices. We obtain results both numerically, using quantum Monte Carlo techniques, and via several analytical approximation schemes, such as the Gutzwiller mean-field approach, a cluster mean-field method, and a spin-wave analysis which takes quantum fluctuations into account. We first study the homogeneous case, for which simple analytical expressions are obtained for all observables of interest, and compare the results with the numerical ones. We obtain the equation of state of the system along with other thermodynamic properties such as the free energy, kinetic energy, superfluid density, condensate density, and compressibility. In the presence of a trap, there is in general a spatial coexistence of superfluid and insulating domains. We show that the spin-wave-based method reproduces the quantum Monte Carlo results for global as well as for local quantities with a high degree of accuracy. We also discuss the validity of the local density approximation. Our analysis can be used to describe bosons in optical lattices where the onsite interaction UUU is much larger than the hopping amplitude ttt.
We study the superfluid to Mott-insulator transition of hardcore bosons in commensurate superlattices in two and three dimensions. We focus on the special case where the superlattice has period two and the system is at half-filling. We obtain numerical results by using the stochastic series expansion algorithm, and compute various properties of the system, such as the ground-state energy, the density of bosons in the zero-momentum mode, the superfluid density, and the compressibility. We employ finite-size scaling to extrapolate the thermodynamic limit, and find the critical points of the phase transition. We also explore the extent to which several approximate solutions such as mean-field theory, with and without spin-wave corrections, can help one gain analytical insight into the behavior of the system in the vicinity of the phase transition.
We analyse the ground-state quantum phase diagram of hardcore Bosons interacting with repulsive dipolar potentials. The bosons dynamics is described by the extended-Bose-Hubbard Hamiltonian on a two-dimensional lattice. The ground state results from the interplay between the lattice geometry and the long-range interactions, which we account for by means of a classical spin mean-field approach limited by the size of the considered unit cells. This extended classical spin mean-field theory accounts for the long-range density-density interaction without truncation. We consider three different lattice geometries: square, honeycomb, and triangular. In the limit of zero hopping the ground state is always a devil’s staircase of solid (gapped) phases. Such crystalline phases with broken translational symmetry are robust with respect to finite hopping amplitudes. At intermediate hopping amplitudes, these gapped phases melt, giving rise to various lattice supersolid phases, which can have exotic features with multiple sublattice densities. At sufficiently large hoppings the ground state is a superfluid. The stability of phases predicted by our approach is gauged by comparison to the known quantum phase diagrams of the Bose-Hubbard model with nearest-neighbour interactions as well as quantum Monte Carlo simulations for the dipolar case on the square and triangular lattice. Our results are of immediate relevance for experimental realisations of self-organised crystalline ordering patterns in analogue quantum simulators, e.g., with ultracold dipolar atoms in an optical lattice.
We obtain the complete phase diagram of the hardcore Bose-Hubbard model in the presence of a period-two superlattice in two and three dimensions. First we acquire the phase boundaries between the superfluid phase and the `trivial' insulating phases of the model (the completely-empty and completely-filled lattices) analytically. Next, the boundary between the superfluid phase and the half-filled Mott-insulating phase is obtained numerically, using the stochastic series expansion (SSE) algorithm followed by finite-size scaling. We also compare our numerical results against the predictions of several approximation schemes, including two mean-field approaches and a fourth-order strong-coupling expansion (SCE), where we show that the latter method in particular is successful in producing an accurate picture of the phase diagram. Finally, we examine the extent to which several approximation schemes, such as the random phase approximation and the strong-coupling expansion, give an accurate description of the momentum distribution of the bosons inside the insulating phases.
In this work,we theoretically study hard-core bosons on a two-dimensional square optical superlattice at T = 0.First of all,we present the mean field phase diagram of this model in terms of the chemical potential μ and the alternating potential strength △.Besides a superfluid(SF) phase at △ = 0 and a charge density wave(CDW)phase in the large △ at half filling,we demonstrate that a supersolid(SS) phase emerges in the moderate △.Then,we focus on the μ = 0,e.g.,half filling case,using large-S semi-classical spin-wave approximation to study the SS to CDW quantum phase transition.In particular,we calculate the ground-state energy and the superfluid density at the level of1/S correction.We then compare the spin-wave results with the large scale quantum Monte Carlo(QMC) simulations using the cluster stochastic series expansion(CSSE) algorithm,and find that while the spin wave method is intuitive with clear physical pictures,the quantum critical point is quite different from that of numerical results which is believed to be accurate.We suggest that as simple as it is,this model still exhibits strong quantum fluctuations near the quantum critical point beyond the power of semiclassical spin-wave approach.