Correlated bosons on a lattice: Dynamical mean-field theory for Bose-Einstein condensed and normal phases

Abstract

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 d or coordination number Z of the lattice. In contrast to its fermionic counterpart the construction of the B-DMFT requires different scalings of the hopping amplitudes with Z depending on whether the bosons are in their normal state or in the Bose-Einstein condensate. 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 Bose-Einstein condensate (BEC) transition temperature TBEC. This effect may be used experimentally to increase TBEC of bosonic atoms in optical lattices.

Bose-Bose mixtures with synthetic spin-orbit coupling in optical lattices

“…Within this approach, the physics on each lattice site is determined from a local effective action obtained by integrating out all other degrees of freedom in the lattice model, excluding the lattice site considered. The local effective action is then represented by an Anderson impurity model [24][25][26][27]. We use exact diagonalization (ED) [30,31] of the effective Anderson Hamiltonian with a finite number of bath orbitals to solve the local action.…”

“…from the strong coupling deep MI regime all the way to the SF at weak coupling, we apply BDMFT [24][25][26][27], which is non-perturbative and can capture the local quantum fluctuations exactly. For exploring various possible exotic magnetic or superfluid phases which break the translational symmetry of the lattice, here we specifically employ real-space BDMFT (R-BDMFT) [29], which generalizes BDMFT to a positiondependent self-energy and captures inhomogeneous quantum phases.…”

“…In this work, we investigate these fundamental issues within the non-perturbative theoretical framework of Bosonic Dynamical Mean Field Theory (BDMFT) [24][25][26][27]. We find that the exotic spin-textures are robust throughout the whole Mott regime, see Fig.…”

Dynamical Mean-Field Theory

“…In the normal phase the hopping amplitude needs to be scaled as in the fermionic case ("quantum scaling"), i.e., as t ij = t * ij / √ Z, while in the condensed phase a classical scaling t ij = t * ij /Z is required. Such a scaling of the hopping amplitudes cannot be performed on the level of the Hamiltonian, but is possible in the effective action [139]. The bosonic DMFT ("B-DMFT") derived thereby treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling.…”

“…It not only reproduces all exactly solvable limits, such as the non-interacting (U = 0) and the atomic (t ij = 0) limit, but also well-known approximation schemes for interacting bosons. For example, by neglecting all terms containing the hybridization function in the local action one obtains the mean-field theory of Fisher et al [138]; for a detailed discussion see [139].…”