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Chinese Journal of Physics

Mean-field approximation of the O(2) model on 3D lattice Bose systems

Khaled M. Barakat

Published May 9, 2023• 0 citations
Publisher

Abstract

New results are presented for bosons defined on the three-dimensional (3D) cubic lattice at very low temperatures. Calculations are made within the mean-filed (or Hartree) theory applied to the so-called O(2) model. The analysis is used to calculate the zero-temperature phase diagram and determine the critical coupling strength λ for a quantum phase transition. Properties of the zero temperature quantum state, such as: the energy per site, the order parameter, the first possible excitation, and the test single-site phase wave function, are displayed and discussed. Their behavior predicts a transition from an ordered (superfluid) to a disordered (normal/insulating) phase at a critical value of the coupling constant λc = 6.

DOI
10.1016/j.cjph.2023.05.011

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New results are presented for bosons defined on the three-dimensional (3D) cubic lattice at very low temperatures. Calculations are made within the mean-filed (or Hartree) theory applied to the so-called O(2) model. The analysis is used to calculate the zero-temperature phase diagram and determine the critical coupling strength λ for a quantum phase transition. Properties of the zero temperature quantum state, such as: the energy per site, the order parameter, the first possible excitation, and the test single-site phase wave function, are displayed and discussed. Their behavior predicts a transition from an ordered (superfluid) to a disordered (normal/insulating) phase at a critical value of the coupling constant λc = 6.
Bosons in a periodic lattice with on-site disorder at low but nonzero temperatures are considered within a mean-field theory. The criteria used for the definition of the superfluid, Mott insulator and Bose glass are analysed. Since the compressibility never vanishes at nonzero temperatures, it cannot be used as a general criterion. We show that the phases are unambiguously distinguished by the superfluid density and the density of states of the low-energy excitations. The phase diagram of the system is calculated. It is shown that even a tiny temperature leads to a significant shift of the boundary between the Bose glass and superfluid.
We perform a comparative study of the finite temperature behavior of ultracold Bose atoms in optical lattices by the slave fermion and the slave boson approaches to the Bose-Hubbard model. The phase diagram of the system is presented. Although both approaches are equivalent without approximations, the mean field theory based on the slave fermion technique is quantitatively more appropriate. Conceptually, the slave fermion approach automatically excludes the double occupancy of two identical fermions on the same lattice site. By comparing to known results in limiting cases, we find the slave fermion approach better than the slave boson approach. For example, in the non-interacting limit, the critical temperature of the superfluid-normal liquid transition calculated by the slave fermion approach is closer to the well-known ideal Bose gas result. At zero-temperature limit of the critical interaction, strength from the slave fermion approach is also closer to that from the direct calculation using a zero-temperature mean field theory.
We formulate the collective quantum field theory for three-dimensional bosonic optical lattices and evaluate its consequences in a mean-field approximation to two collective fields, proposed by Cooper et al (2011 Phys. Rev. A 83 053622), and in a lowest-order variational perturbation theory. It is shown that the present mean-field approximation predicts some essential features of the experimentally observed dependence of the critical temperature on the coupling strength and a second-order quantum phase transition. In contrast to a recent prediction for atomic gases given by Cooper et al , our finding indicates no superfluid state with zero condensate fraction.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
The Bose–Hubbard model describing interacting bosons in an optical lattice is reduced to a simple spin-1 XY model with single-ion anisotropy in the vicinity of the Mott phase. We propose a mean-field theory based on a constraint SU(3) pseudo-boson representation on the effective model to study the properties of the superfluid-Mott-insulator phase transition. By calculating the elementary excitation spectra and the average particle number fluctuation in the Brillouin zone center, we find that the energy gaps vanish continuously around ( J XY / J z ) c ≈0.175 and ( J XY / J z ) c ≈0.094 for 2D and 3D cubic lattices respectively, where the superfluid order parameters come up from zero and the Mott insulator state changes into a superfluid state.
The Bose glass (BG) phase is the Griffiths region of the disordered Bose–Hubbard model (BHM), characterized by finite, quasi-superfluid clusters within a Mott insulating background. We propose to utilize this characterization to identify the complete zero-temperature phase diagram of the disordered BHM in d ⩾ 2 dimensions by analysing the geometric properties of what we call superfluid (SF) clusters, which are defined to be clusters of sites with non-integer expectation values for the local boson occupation number. The Mott insulator phase then is the region in the phase diagram where no SF clusters exist, and the SF phase the region where SF clusters percolate—the BG phase is inbetween: SF clusters exist, but do not percolate. This definition is particularly useful in the context of local mean field (LMF) or Gutzwiller–Ansatz calculations, where we show that an identification of the phases on the basis of global quantities such as the averaged SF order parameter and the compressibility is misleading. We apply the SF cluster analysis to the LMF ground states of the two-dimensional disordered BHM to produce its phase diagram and find (a) an excellent agreement with the phase diagram predicted on the basis of quantum Monte Carlo simulations for the commensurate density n = 1 and (b) large differences to stochastic mean field and other mean field predictions for fixed disorder strength. The relation of the percolation transition of the SF clusters with the onset of non-vanishing SF stiffness indicating the BG to SF transition is discussed.