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arXiv:1707.07004 · Physical review. B./Physical review. B

Universal upper bounds on the Bose-Einstein condensate and the Hubbard star

Felix Tennie, Vlatko Vedral, Christian Schilling

Published Aug 2, 2017• 14 citations
arXivHTMLPDF

Abstract

For $N$ hard-core bosons on an arbitrary lattice with $d$ sites and independent of additional interaction terms we prove that the hard-core constraint itself already enforces a universal upper bound on the Bose-Einstein condensate given by ${N}_{\text{max}}=(N/d)(d\ensuremath{-}N+1)$. This bound can only be attained for one-particle states $|\ensuremath{\varphi}\ensuremath{\rangle}$ with equal amplitudes with respect to the hard-core basis (sites) and when the corresponding $N$-particle state $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ is maximally delocalized. This result is generalized to the maximum condensate possible within a given sublattice. We observe that such maximal local condensation is only possible if the mode entanglement between the sublattice and its complement is minimal. We also show that the maximizing state $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ is related to the ground state of a bosonic ``Hubbard star'' showing Bose-Einstein condensation.

DOI
10.1103/physrevb.96.064502

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For NNN hard-core bosons on an arbitrary lattice with ddd sites and independent of additional interaction terms we prove that the hard-core constraint itself already enforces a universal upper bound on the Bose-Einstein condensate given by Nmax=(N/d)(d\ensuremath−N+1){N}_{\text{max}}=(N/d)(d\ensuremath{-}N+1)Nmax​=(N/d)(d\ensuremath−N+1). This bound can only be attained for one-particle states ∣\ensuremathφ\ensuremath⟩|\ensuremath{\varphi}\ensuremath{\rangle}∣\ensuremathφ\ensuremath⟩ with equal amplitudes with respect to the hard-core basis (sites) and when the corresponding NNN-particle state ∣\ensuremathΨ\ensuremath⟩|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}∣\ensuremathΨ\ensuremath⟩ is maximally delocalized. This result is generalized to the maximum condensate possible within a given sublattice. We observe that such maximal local condensation is only possible if the mode entanglement between the sublattice and its complement is minimal. We also show that the maximizing state ∣\ensuremathΨ\ensuremath⟩|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}∣\ensuremathΨ\ensuremath⟩ is related to the ground state of a bosonic ``Hubbard star'' showing Bose-Einstein condensation.
Abstract We propose a model for hard-core bosons in a lattice which allows to achieve the optimal occupation number predicted by Tennie et al (2017 Phys. Rev. B 96 064502) for a finite number of sites. The model is based on an extension of the Hamiltonian of the so-called Hubbard star, whose quantum properties are studied by means of quantum information descriptors such as the von Neumann entropy and the mutual information. These metrics are analyzed as a function of the one- and two-particle reduced density matrices, allowing to explore the relationship between condensation and entanglement by means of a control parameter that, under a given limit, connects our findings with previous results. All developments comprised in this article have been derived by analytical methods.
The Bose-Hubbard model is a system of interacting bosons that live on the vertices of a graph. The particles can move between adjacent vertices and experience a repulsive on-site interaction. The Hamiltonian is determined by a choice of graph that specifies the geometry in which the particles move and interact. We prove that approximating the ground energy of the Bose-Hubbard model on a graph at fixed particle number is QMA-complete. In our QMA-hardness proof, we encode the history of an n-qubit computation in the subspace with at most one particle per site (i.e., hard-core bosons). This feature, along with the well-known mapping between hard-core bosons and spin systems, lets us prove a related result for a class of 2-local Hamiltonians defined by graphs that generalizes the XY model. By avoiding the use of perturbation theory in our analysis, we circumvent the need to multiply terms in the Hamiltonian by large coefficients.
We study the extended Bose-Hubbard model on a two-dimensional honeycomb lattice by using large-scale quantum Monte Carlo simulations. We present the ground-state phase diagrams for both the hard-core and the soft-core bosons. For the hard-core case, the transition between the $\ensuremath{\rho}=1∕2$ solid and the superfluid is first order, and the supersolid state is unstable toward phase separation. For the soft-core case, due to the presence of multiple occupation, a stable particle-induced supersolid (SS-p) phase emerges when $1∕2<\ensuremath{\rho}<1$. The transition from the solid at $\ensuremath{\rho}=1∕2$ to the SS-p phase is second order with the superfluid density scaling as ${\ensuremath{\rho}}_{s}\ensuremath{\sim}\ensuremath{\rho}\ensuremath{-}1∕2$. The SS-p phase has the same diagonal order as the solid at $\ensuremath{\rho}=1∕2$. As the chemical potential increases further, the SS-p phase turns into a solid where two bosons occupy each site of one sublattice through a first-order transition. We also calculate the critical exponents of the transition between the $\ensuremath{\rho}=1∕2$ solid and superfluid at the Heisenberg point for the hard-core case. We find the dynamical critical exponent z=0.15z=0.15z=0.15, which is smaller than results obtained on smaller lattices. This indicates that zzz approaches zero in the thermodynamic limit, and thus the transition is also first order even at the Heisenberg point.
We consider a homogeneous Bose gas in the Gross-Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose-Einstein condensation. Recently, an upper bound for the grand canonical free energy was proved in Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024) capturing two novel contributions. First, the free energy of the interacting condensate is given in terms of an effective theory describing the probability distribution of the number of condensed particles. Second, the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. We extend this result to a more general class of interaction potentials, including interactions with a hard core. Our proof follows a different approach than the one in Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024): We model microscopic correlations between the particles by a Jastrow factor and exploit a cancellation in the computation of the energy that emerges due to the different length scales in the system.
We investigate the properties of the ground state of a system of interacting bosons on regular lattices with coordination number k≥2k\geq 2k≥2. The interaction is a pure, infinite, on-site repulsion. Our concern is to give an improved upper bound on the ground state energy per site. For a density ρρρ a trivial upper bound is known to be −kρ(1−ρ)-kρ(1-ρ)−kρ(1−ρ). We obtain a smaller variational bound within a reasonably large family of trial functions. The estimates make use of a large deviation principle for the energy of the Ising model on the same lattice.
We study the relationship between long cycles and Bose-Einstein condensation (BEC) in the case of several models. A convenient expression for the density of particles on cycles of length qqq is obtained, in terms of qqq unsymmetrised particles coupled with a boson field. Using this formulation we reproduce known results on the Ideal Bose Gas, Mean-Field and Perturbed Mean-Field Models, where the condensate density exactly equals the long cycle density. Then we consider the Infinite-Range-Hopping Bose-Hubbard model, with and without hard-cores. For the hard-core case, we find we can disregard the hopping contribution of the q unsymmetrised particles to obtain an exact expression for the density of particles on long cycles. It is shown that only the cycle of length one contributes to the cycle density. We conclude that while the existence of a non-zero long cycle density coincides with the occurrence of Bose-Einstein condensation, the respective densities are not necessarily equal. For the case of a finite on-site repulsion, we obtain an expression for the cycle density by neglecting the qqq unsymmetrised hop. Inspired by the Approximating Hamiltonian method we conjecture a simplified expression for the short cycle density as a ratio of single-site partition functions. In the absence of condensation we prove that this simplification is exact and find that in this case the long-cycle density vanishes. In the presence of condensation we justify this simplification when a gauge-symmetry breaking term is introduced in the Hamiltonian. Assuming our conjecture is correct, we compare numerically the long-cycle density with the condensate and again find that though they coexist, in general they are not equal.