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arXiv:1304.4826 · Physical Review A

Leading-order auxiliary-field theory of the Bose-Hubbard model

John F. Dawson, Fred Cooper, Chih-Chun Chien, Bogdan Mihaila

Published Aug 12, 2013• 4 citations
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Abstract

We discuss the phase diagram of the Bose-Hubbard (BH) model in the leading-order auxiliary field (LOAF) theory. LOAF is a conserving nonperturbative approximation that treats on equal footing the normal and anomalous density condensates. The mean-field solutions in LOAF correspond to first-order and second-order phase transition solutions with two critical temperatures corresponding to a vanishing Bose-Einstein condensate ${T}_{c}$ and a vanishing diatom condensate ${T}^{\ensuremath{\star}}$. The second-order phase transition solution predicts the correct order of the transition in continuum Bose gases. For either solution, the superfluid state is tied to the presence of the diatom condensate related to the anomalous density in the system. In ultracold Bose atomic gases confined on a three-dimensional lattice, the critical temperature ${T}_{c}$ exhibits a quantum phase transition, where ${T}_{c}$ goes to zero at a finite coupling. The BH phase diagram in LOAF features a line of first-order transitions ending in a critical point beyond which the transition is second order while approaching the quantum phase transition. We identify a region where a diatom condensate is expected for temperatures higher than ${T}_{c}$ and less than ${T}_{0}$, the critical temperature of the noninteracting system. The LOAF phase diagram for the BH model compares qualitatively well with existing experimental data and results of ab initio Monte Carlo simulations.

DOI
10.1103/physreva.88.023607

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We derive the Josephson relation for a dilute Bose gas in the framework of an auxiliary-field (AF) resummation of the theory in terms of the normal- and anomalous-density condensates. In the leading order of the AF loop expansion (LOAF) we find two critical temperatures in the phase diagram: the critical temperature Tc{T}_{c}Tc​, where the atom Bose-Einstein condensate (BEC) appears first, and the temperature ${T}^{\ensuremath{\star}}>{T}_{c}$, which indicates the onset of superfluidity and the appearance of a diatom condensate in the system. In this context, the Josephson relation shows that the superfluid density is related to the square of a second-order parameter, the anomalous density, $\ensuremath{\langle}A\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\varphi}\ensuremath{\rangle}$, which can be thought of as a diatom condensate analogous to the Cooper-pair condensate discussed in the BCS approach to dilute Fermi gases. This is in contrast with the corresponding result in the Bose gas theory without a diatom condensate, which predicts that the superfluid density is proportional to the square of the usual atom BEC condensate, $\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\rangle}$. In the region between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ the anomalous-density Green's function contains a zero energy-momentum state corresponding to a Goldstone state. This is a two-particle state with U(1) charge two. Our findings are consistent with the prediction that in the temperature range between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ a fraction of the system is in the superfluid state in the absence of the usual atom BEC condensate. This situation is similar to the case of dilute Fermi gases, where the superfluid density is proportional to the square of the gap parameter. The Josephson relation relies on the existence of zero energy and momentum excitations showing the intimate relationship between superfluidity and the Goldstone theorem.
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Ultracold atoms provide clues to an important many-body problem regarding the dependence of the Bose-Einstein condensation transition temperature Tc{T}_{c}Tc​ on interactions. However, cold atoms are trapped in harmonic potentials and theoretical evaluations of the Tc{T}_{c}Tc​ shift of trapped interacting Bose gases are challenging. While previous predictions of the leading-order shift have been confirmed, more recent experiments exhibit higher-order corrections beyond available mean-field theories. By implementing two large-N\mathcal{N}N-based theories with the local density approximation (LDA), we extract next-order corrections of the Tc{T}_{c}Tc​ shift. The leading-order large-N\mathcal{N}N theory produces results quantitatively different from the latest experimental data. The leading-order auxiliary-field (LOAF) theory, containing both normal and anomalous density fields, captures the Tc{T}_{c}Tc​ shift accurately in the weak-interaction regime. However, the LOAF theory shows incompatible behavior with the LDA, and forcing the LDA leads to density discontinuities in the trap profiles. We present a phenomenological model based on the LOAF theory, which repairs the incompatibility and provides a prediction of the Tc{T}_{c}Tc​ shift in the stronger-interaction regime.
We discuss the equilibrium properties of dilute Bose gases using a nonperturbative formalism based on auxiliary fields related to the normal and anomalous densities. We show analytically that for a dilute Bose gas of weakly interacting particles at zero temperature, the leading-order auxiliary field (LOAF) approximation leads to well-known analytical results. Close to the critical point the LOAF predictions are the same as those obtained using an effective field theory in the large-NNN approximation. We also report analytical approximations for the LOAF results in the unitarity limit, which compare favorably with our numerical results. LOAF predicts that the equation of state for the Bose gas in the unitarity limit is E/(pV)=1E/(pV)=1E/(pV)=1, unlike the case of the Fermi gas when E/(pV)=3/2E/(pV)=3/2E/(pV)=3/2.
We develop the auxiliary field formalism corresponding to a dilute system of spin-1/2 fermions. This theory represents the Fermi counterpart of the Bose-Einstein condensation (BEC) theory developed recently by F. Cooper et al. [Phys. Rev. Lett. 105, 240402 (2010)] to describe a dilute gas of Bose particles. Assuming tunable interactions, this formalism is appropriate for the study of the crossover from the regime of Bardeen-Cooper-Schriffer (BCS) pairing to the regime of BEC in ultracold fermionic atom gases. We show that when applied to the Fermi case at zero temperature, the leading-order auxiliary field (LOAF) approximation gives the same equations as obtained in the standard BCS variational picture. At finite temperature, LOAF leads to the theory discussed by S\'a de Melo, Randeria, and Engelbrecht [Phys. Rev. Lett. 71, 3202 (1993); Phys. Rev. B 55, 15153 (1997)]. As such, LOAF provides a unified framework to study the interacting Fermi gas. The mean-field results discussed here can be systematically improved on by calculating the one-particle irreducible action corrections, order by order.
We investigate the temperature-dependent behavior emerging in the vicinity of the superfluid (SF) to Mott insulator (MI) transition of interacting bosons in a two-dimensional optical lattice, described by the Bose-Hubbard model. The equilibrium phase diagram at finite temperatures is computed by means of the cluster mean-field theory (CMF) where the effect of non-local correlations is analyzed systematically by finite-size scaling of the cluster size. The phase diagram exhibits a rich structure including a transition and a crossover of the SF and MI phases respectively to a normal fluid (NF) state at finite temperature. In order to characterize these phases, and the NF transition and crossover scales, we calculate, in addition to the condensate amplitude, the superfluid fraction, sound velocity and compressibility. The phase boundaries obtained by CMF with finite-size scaling agree quantitatively with quantum Monte Carlo (QMC) results as well as with experiments. The von Neumann entanglement entropy of a cluster exhibits critical enhancement near the SF-MI quantum critical point (QCP). We also discuss the behavior of the transition lines near this QCP at the particle-hole symmetric point located at the tip of a Mott lobe as well as away from particle-hole symmetry.