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arXiv:0906.4732 · Annalen der Physik

Mixtures of correlated bosons and fermions: Dynamical mean‐field theory for normal and condensed phases

Krzysztof Byczuk, D. Vollhardt

Published Aug 17, 2009• 14 citations
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Abstract

Abstract We derive a dynamical mean‐field theory for mixtures of interacting bosons and fermions on a lattice (BF‐DMFT). The BF‐DMFT is a comprehensive, thermodynamically consistent framework for the theoretical investigation of Bose‐Fermi mixtures and is applicable for arbitrary values of the coupling parameters and temperatures. It becomes exact in the limit of high spatial dimensions d or coordination number Z of the lattice. In particular, the BF‐DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. Using the BF‐DMFT we investigate two different interaction models of correlated lattice bosons and fermions, one where all particles are spinless (model I) and one where fermions carry a spin one‐half (model II). In model I the local, repulsive interaction between bosons and fermions can give rise to an attractive effective interaction between the bosons. In model II it can also lead to an attraction between the fermions.

DOI
10.1002/andp.20095210904

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Abstract We derive a dynamical mean‐field theory for mixtures of interacting bosons and fermions on a lattice (BF‐DMFT). The BF‐DMFT is a comprehensive, thermodynamically consistent framework for the theoretical investigation of Bose‐Fermi mixtures and is applicable for arbitrary values of the coupling parameters and temperatures. It becomes exact in the limit of high spatial dimensions d or coordination number Z of the lattice. In particular, the BF‐DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. Using the BF‐DMFT we investigate two different interaction models of correlated lattice bosons and fermions, one where all particles are spinless (model I) and one where fermions carry a spin one‐half (model II). In model I the local, repulsive interaction between bosons and fermions can give rise to an attractive effective interaction between the bosons. In model II it can also lead to an attraction between the fermions.
We formulate a bosonic dynamical mean-field theory (B-DMFT) which provides a comprehensive, thermodynamically consistent framework for the theoretical investigation of correlated lattice bosons. The B-DMFT is applicable for arbitrary values of the coupling parameters and temperature and becomes exact in the limit of high spatial dimensions ddd or coordination number ZZZ of the lattice. In contrast to its fermionic counterpart, the construction of the B-DMFT requires different scalings of the hopping amplitudes with ZZZ depending on whether the bosons are in their normal state or in the Bose-Einstein condensate (BEC). A detailed discussion of how this conceptual problem can be overcome by performing the scaling in the action rather than in the Hamiltonian itself is presented. The B-DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. It reproduces all previously investigated limits in parameter space, such as the Beliaev-Popov and Hartree-Fock-Bogoliubov approximations, and generalizes the existing mean-field theories of interacting bosons. The self-consistency equations of the B-DMFT are those of a bosonic single-impurity coupled to two reservoirs corresponding to bosons in the condensate and in the normal state, respectively. We employ the B-DMFT to solve a model of itinerant and localized, interacting bosons analytically. The local correlations are found to enhance the condensate density and the BEC transition temperature TBEC{T}_{\text{BEC}}TBEC​. This effect may be used experimentally to increase TBEC{T}_{\text{BEC}}TBEC​ of bosonic atoms in optical lattices.
We give a detailed discussion of the recently developed generalized dynamical mean-field theory (GDMFT) for a mixture of bosonic and fermionic particles. We show that this method is nonperturbative and exact in infinite dimensions, and reliably describes the full range from weak to strong coupling. Like in conventional dynamical mean-field theory, the small parameter is 1/z1/z1/z, where zzz is the lattice coordination number. We apply the GDMFT scheme to a mixture of spinless fermions and bosons in an optical lattice. We investigate the possibility of a supersolid phase, focusing on the case of 1/2 filling for the fermions and 3/2 filling for the bosons.
We discuss the recently developed bosonic dynamical mean-field theory (B-DMFT) framework, which maps a bosonic lattice model onto the self-consistent solution of a bosonic impurity model with coupling to a reservoir of normal and condensed bosons. The effective impurity action is derived in several ways: (i) as an approximation to the kinetic energy functional of the lattice problem, (ii) using a cavity approach and (iii) using an effective medium approach based on adding a one-loop correction to the self-consistently defined condensate. To solve the impurity problem, we use a continuous-time Monte Carlo algorithm based on the sampling of a perturbation expansion in the hybridization functions and the condensate wave function. As applications of the formalism, we present finite-temperature B-DMFT phase diagrams for the bosonic Hubbard model on a three-dimensional (3D) cubic and a 2D square lattice, the condensate order parameter as a function of chemical potential, critical exponents for the condensate, the approach to the weakly interacting Bose gas regime for weak repulsions and the kinetic energy as a function of temperature.
The dynamical mean-field theory (DMFT), which is successful in the study of strongly correlated fermions, was recently extended to boson systems [K. Byczuk and D. Vollhardt, Phys. Rev. B 77, 235106 (2008)]. In this paper, we employ the bosonic DMFT to study the Bose-Hubbard model which describes on-site interacting bosons in a lattice. Using exact diagonalization as the impurity solver, we get the DMFT solutions for the Green's function, the occupation density, as well as the condensate fraction on a Bethe lattice. Various phases are identified: the Mott insulator, the Bose-Einstein condensed (BEC) phase, and the normal phase. At finite temperatures, we obtain the crossover between the Mott-type regime and the normal phase, as well as the BEC-to-normal phase transition. Phase diagrams on the $\ensuremath{\mu}/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane and on the $T/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane are produced ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}$ is the scaled hopping amplitude). We compare our results with the previous ones, and discuss the implication of these results to experiments.
We study the low-energy properties of the spinful Bose-Fermi Hubbard model on a simple cubic lattice for attractive fermion-fermion and repulsive boson-boson interactions at half-filling for the fermions and unit-filling for the bosons. Recent DMFT results predict a variety of phases, including superfluids, charge density waves, mixed states and supersolids. We develop a self-consistent mean-field scheme which allows for all orders to appear and determine the ground state of the system by comparing the corresponding energies. In the double superfluid phase, we consider possibilities for exotic extended s-, p- and d-wave boson-assisted superfluidity.
We theoretically analyze Fermi-Bose mixtures consisting of light fermions and heavy bosons that are loaded into optical lattices (ignoring the trapping potential). To describe such mixtures, we consider the Fermi-Bose version of the Falicov-Kimball model on a periodic lattice. This model can be exactly mapped onto the spinless Fermi-Fermi Falicov-Kimball model at zero temperature for all parameter space as long as the mixture is thermodynamically stable. We employ dynamical mean-field theory to investigate the evolution of the Fermi-Bose Falicov-Kimball model at higher temperatures. We calculate spectral moment sum rules for the retarded Green's function and self-energy, and use them to benchmark the accuracy of our numerical calculations, as well as to reduce the computational cost by exactly including the tails of infinite summations or products. We show how the occupancy of the bosons, single-particle many-body density of states for the fermions, momentum distribution, and the average kinetic energy evolve with temperature. We end by briefly discussing how to experimentally realize the Fermi-Bose Falicov-Kimball model in ultracold atomic systems.