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Physical Review A

Cell Model of Quantum Liquids and Solids. I. Crystalline State

William J. Mullin

Published Sep 1, 1971• 12 citations
Publisher

Abstract

A second-quantized cell-model Hamiltonian is derived to provide a model for quantum crystals and liquids. The boson crystal is treated in this article. The cells are divided into two sublattices: (i) regular, which are usually occupied, and (ii) interstitial, which are usually empty. The particle hard cores are simulated by assuming Fermi commutation relations for operators referring to a single cell; this allows a discussion in terms of a spin-analog Hamiltonian which is diagonalized in the spin-wave approximation. Because the Hamiltonian includes a term which allows tunneling between regular and interstitial sites, the ground state includes a description of zero-point motion and exchange via virtual intermediate interstitial occupation. Excited states include nonlocalized vacancies, interstitials, and vacancy-interstitial pairs. Phonon states are not included in this analysis. The model exhibits a phase transition to the Bose condensed state which is examined briefly.

DOI
10.1103/physreva.4.1247

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A second-quantized cell-model Hamiltonian is derived to provide a model for quantum crystals and liquids. The boson crystal is treated in this article. The cells are divided into two sublattices: (i) regular, which are usually occupied, and (ii) interstitial, which are usually empty. The particle hard cores are simulated by assuming Fermi commutation relations for operators referring to a single cell; this allows a discussion in terms of a spin-analog Hamiltonian which is diagonalized in the spin-wave approximation. Because the Hamiltonian includes a term which allows tunneling between regular and interstitial sites, the ground state includes a description of zero-point motion and exchange via virtual intermediate interstitial occupation. Excited states include nonlocalized vacancies, interstitials, and vacancy-interstitial pairs. Phonon states are not included in this analysis. The model exhibits a phase transition to the Bose condensed state which is examined briefly.
We consider a two-species hard-core boson Hubbard model for a supersolid, where two types of bosons represent vacancies and interstitials doped into a commensurate crystal. The on-site interspecies interaction can create bound states of vacancies and interstitials facilitating vacancy condensation at lower energies than in a single-species model, as suggested in an earlier mean-field study. Here, we carry out quantum Monte Carlo simulations to study possible supersolid phases of the model, which correspond to superfluid phases of the vacancies or interstitials. At low temperatures, we find three distinct superfluid phases. The extent of the phases and the nature of the phase transitions are discussed in comparison to mean-field theory.
The Fermi liquid is perhaps the most familiar quantum many-body state of solid state physics; we met it briefly in Section 16.2.2. It is the generic state of fermions at nonzero density, and is found in all metals. Its basic characteristics can already be understood in a simple free fermion picture. Noninteracting fermions occupy the lowest energy single-particle states, consistent with the exclusion principle. This leads to the fundamental concept of the Fermi surface: a surface in momentum space separating the occupied and empty single fermion states. The lowest energy excitations then consist of quasiparticle excitations which are particle-like outside the Fermi surface, and hole-like inside the Fermi surface. Landau's Fermi liquid theory is a careful justification for the stability of this simple picture in the presence of interactions between fermions. Just as we found in Chapters 5 and 7 for the quantum Ising and rotor models, interaction corrections modify the wavefunction of the quasiparticle and so introduce a quasiparticle residue A ; however, they do not destabilize the integrity of the quasiparticle, as we review in Section 18.1. The purpose of this chapter is to describe two paradigms of symmetry breaking quantum transitions in Fermi liquids. In the first class, studied in Section 18.2, the broken symmetry is related to the point-group symmetry of the crystal, while translational symmetry is preserved; consequently, the order parameter resides at zero wavevector. In the second class, studied in Section 18.3, the order parameter is at a finite wavevector, and so translational symmetry is also broken.
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.
The Bose-Hubbard model of a twofold degenerate Bose gas is studied in an optical lattice with one particle per site and virtual tunneling to empty and doubly occupied sites. An effective Hamiltonian for this system is derived within a continued-fraction approach. The ground state of the effective model is studied in mean-field approximation for a modulated optical lattice. A dimerized mean-field state gives a Mott insulator whereas the lattice without modulations develops long-range correlated phase fluctuations due to a Goldstone mode. This result is discussed in comparison with the superfluid and the Mott-insulating state of a single-component hard-core Bose gas.
An approach is presented toward validating the assumption that the ground state of bosons with repulsive interactions at low densities is characterized by macroscopic occupation of the zero momentum level. We use a cell model which affords a simple description of the high-density region, where fluctuations in number density are small and where no single-particle level is macroscopically occupied. As the density decreases, fluctuations increase, and we reach a critical density at which the small fluctuation approximation becomes unstable with respect to plane wave states of zero momentum. At this critical density, the single-particle energy gap disappears, and the dependence of excitation energy on momentum changes from quadratic to linear, for small values of momentum.
We present a phase-space method for the Bose–Hubbard model based on the Q-function representation. In particular, we consider two model Hamiltonians in the mean-field approximation; the first is the standard 'one-site' model where quantum tunnelling is approximated entirely using mean-field terms; the second 'two-site' model explicitly includes tunnelling between two adjacent sites while treating tunnelling with other neighbouring sites using the mean-field approximation. The ground state is determined by minimizing the classical energy functional subject to quantum mechanical constraints, which take the form of uncertainty relations. For each model Hamiltonian, we compare the ground state results from the Q-function method with the exact numerical solution. The results from the Q-function method, which are easy to compute, give a good qualitative description of the main features of the Bose–Hubbard model including the superfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem and show there are some limitations of the method particularly in the weak lattice regime.