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arXiv:0809.0706 · Physical Review A

Effective action approach for quantum phase transitions in bosonic lattices

Barry Bradlyn, F. E. A. dos Santos, Axel Pelster

Published Jan 15, 2009• 34 citations
arXivHTMLPDF

Abstract

Based on standard field-theoretic considerations, we develop an effective action approach for investigating quantum phase transitions in lattice Bose systems at arbitrary temperature. We begin by adding to the Hamiltonian of interest a symmetry breaking source term. Using time-dependent perturbation theory, we then expand the grand-canonical free energy as a double power series in both the tunneling and the source term. From here, an order parameter field is introduced in the standard way and the underlying effective action is derived via a Legendre transformation. Determining the Ginzburg-Landau expansion to first order in the tunneling term, expressions for the Mott insulator-superfluid phase boundary, condensate density, average particle number, and compressibility are derived and analyzed in detail. Additionally, excitation spectra in the ordered phase are found by considering both longitudinal and transverse variations of the order parameter. Finally, these results are applied to the concrete case of the Bose-Hubbard Hamiltonian on a three-dimensional cubic lattice, and compared with the corresponding results from mean-field theory. Although both approaches yield the same Mott insulator--superfluid phase boundary to first order in the tunneling, the predictions of our effective action theory turn out to be superior to the mean-field results deeper into the superfluid phase.

DOI
10.1103/physreva.79.013615

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Based on standard field-theoretic considerations, we develop an effective action approach for investigating quantum phase transitions in lattice Bose systems at arbitrary temperature. We begin by adding to the Hamiltonian of interest a symmetry breaking source term. Using time-dependent perturbation theory, we then expand the grand-canonical free energy as a double power series in both the tunneling and the source term. From here, an order parameter field is introduced in the standard way and the underlying effective action is derived via a Legendre transformation. Determining the Ginzburg-Landau expansion to first order in the tunneling term, expressions for the Mott insulator-superfluid phase boundary, condensate density, average particle number, and compressibility are derived and analyzed in detail. Additionally, excitation spectra in the ordered phase are found by considering both longitudinal and transverse variations of the order parameter. Finally, these results are applied to the concrete case of the Bose-Hubbard Hamiltonian on a three-dimensional cubic lattice, and compared with the corresponding results from mean-field theory. Although both approaches yield the same Mott insulator--superfluid phase boundary to first order in the tunneling, the predictions of our effective action theory turn out to be superior to the mean-field results deeper into the superfluid phase.
We study the phase transition from Mott insulator (MI) to Bose glass (BG) of a disordered Bose-Hubbard model within the framework of Ginzburg-Landau effective action approach. By treating MI as unperturbed ground state and performing a systematic expansion with respect to tunneling matrix element, we extend such a field-theoretic method into the disordered lattice Bose systems. To the lowest order, a second order phase transition is confirmed to happen here and the corresponding phase boundary equation coincides with the previous mean-field approximation result. Keeping all the terms second order in hopping parameter, we obtain the beyond mean-field results of MI-BG phase boundary of 2D and 3D disordered Bose-Hubbard models. Our analytic predictions are in agreement with recent semianalytic results.
We present a strong-coupling expansion of the Bose-Hubbard model which describes both the superfluid and the Mott phases of ultracold bosonic atoms in an optical lattice. By performing two successive Hubbard-Stratonovich transformations of the intersite hopping term, we derive an effective action which provides a suitable starting point to study the strong-coupling limit of the Bose-Hubbard model. This action can be analyzed by taking into account Gaussian fluctuations about the mean-field approximation as in the Bogoliubov theory of the weakly interacting Bose gas. In the Mott phase, we reproduce results of previous mean-field theories and also calculate the momentum distribution function. In the superfluid phase, we find a gapless spectrum and compare our results with the Bogoliubov theory.
We investigate how strongly interacting bosons, described by the Bose-Hubbard Hamiltonian, transition from the Mott insulator to the superfluid phase and, ultimately, to a regime of vanishing interaction where the system smoothly transitions to an ensemble of free bosons undergoing Bose-Einstein condensation. Our study focuses on bosons residing on a three-dimensional simple cubic lattice, with tight-binding interactions characterized by the kinetic energy ttt between neighboring lattice sites and on-site repulsion UUU. To achieve this, we develop a formalism based on mapping the original quantum mechanical Hamiltonian onto an effective quantum spherical model. This approach is nonperturbative in interaction energy and spans the entire range of the interaction parameter UUU. Our findings are presented through a series of multiparameter phase diagrams that include temperature, interaction energy, chemical potential, and particle density, highlighting the corresponding critical surfaces and lines. In this context, we address the issue of quantum critical points at fixed densities, which appear as endpoints in the temperature-interaction phase diagram. We also compare our theoretical results, where applicable, with other approaches, including quantum Monte Carlo simulations.
In recent years the quantum simulation of condensed-matter physics problems has resulted from ex- citing experimental progress in the realm of ultracold atoms and molecules in optical lattices. In this thesis we analyze theoretically a spinor Bose gas loaded into a three-dimensional cubic optical lattice. In order to account for different superfluid phases of spin-1 bosons with a linear Zeeman effect, we work out a Ginzburg-Landau theory for the underlying spin-1 Bose-Hubbard model. To this end we add artificial symmetry- breaking currents to the spin-1 Bose-Hubbard Hamiltonian in order to break the global U (1) symmetry. With this we determine a diagrammatic expansion of the grand-canonical free energy up to fourth order in the symmetry-breaking currents and up to the leading non-trivial order in the hopping strength which is of first order. As a cross-check we demonstrate that the resulting grand- canonical free energy allows to recover the mean-field theory. Applying a Legendre transfor- mation to the grand-canonical free energy, where the symmetry-breaking currents are transformed to order parameters, we obtain the effective Ginzburg-Landau action. With this we calculate in detail at zero temperature the Mott insulator-superfluid quantum phase boundary as well as condensate and particle number density in the superfluid phase. We find that both mean-field and Ginzburg-Landau theory yield the same quantum phase transition between the Mott insulator and superfluid phases, but the range of validity of the mean-field theory turns out to be smaller than that of the Ginzburg-Landau theory. Due to this finding we expect that the Ginzburg-Landau theory gives better results for the superfluid phase and, thus, we restrict ourselves to extremize only the effective Ginzburg-Landau action with respect to the order parameters. With- out external magnetic field the superfluid phase is a polar (ferromagnetic) state for anti-ferromagnetic (ferromagnetic) interactions, i.e. only the hyperfine spin 0 (1) is macroscopically occupied, in accor- dance with previous mean-field results. On the other hand, in the presence of the external magnetic field for ferromagnetic interaction, the superfluid phase does not change as the minimization of the energy implies the maximal spin value. However, when an anti-ferromagnetic interaction competes with the linear Zeeman effect, we can distinguish various ferromagnetic and anti-ferromagnetic su- perfluid phases within the range of validity of the Ginzburg-Landau theory. Increasing the external magnetic field yields a breaking of spin singlet pairs and a subsequent alignment of spins, thus anti- ferromagnetic phases decrease until only a ferromagnetic superfluid phase prevails. In addition, we find that the superfluid-Mott insulator phase transition is always of second order for both ferromag- netic and anti- ferromagnetic interactions. However, the transitions between different superfluid phases for an anti-ferromagnetic interaction can be both of first and second order depending on whether the respective macroscopic occupation of hyperfine spin states changes discontinuously or continuously. The established Ginzburg-Landau theory for spin-1 bosons in optical lattices will certainly be the basis for many further applications as, for instance, time-of-flight absorption pictures or collective excitations, which are of experimental importance.
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.
We work out two different analytical methods for calculating the boundary of the Mott-insulator--superfluid (MI-SF) quantum phase transition for scalar bosons in cubic optical lattices of arbitrary dimension at zero temperature which improve upon the seminal mean-field result. The first one is a variational method, which is inspired by variational perturbation theory, whereas the second one is based on the field-theoretic concept of effective potential. Within both analytical approaches we achieve a considerable improvement of the location of the MI-SF quantum phase transition for the first Mott lobe in excellent agreement with recent numerical results from quantum Monte Carlo simulations in two and three dimensions. Thus, our analytical results for the whole quantum phase diagram can be regarded as being essentially exact for all practical purposes.