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 (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. This effect may be used experimentally to increase
TBEC of bosonic atoms in optical lattices.
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 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.
We develop a projection operator formalism for studying both the zero-temperature equilibrium phase diagram and the nonequilibrium dynamics of the Bose-Hubbard model. Our work, which constitutes an extension of that of Trefzger and Sengupta [Phys. Rev. Lett. 106, 095702 (2011)], shows that the method provides an accurate description of the equilibrium zero-temperature phase diagram of the Bose-Hubbard model for several lattices in two and three dimensions. We show that the accuracy of this method increases with the coordination number
z0 of the lattice and reaches to within
0.5 of quantum Monte Carlo data for lattices with
z0=6. We compute the excitation spectra of the bosons using this method in the Mott and the superfluid phases and compare our results with mean-field theory. We also show that the same method may be used to analyze the nonequilibrium dynamics of the model both in the Mott phase and near the superfluid-insulator quantum critical point where the hopping amplitude
J and the on-site interaction
U satisfy ${z}_{0}J/U\ensuremath{\ll}1$. In particular, we study the nonequilibrium dynamics of the model both subsequent to a sudden quench of the hopping amplitude
J and during a ramp from
Ji to
Jf characterized by a ramp time $\ensuremath{\tau}$ and exponent $\ensuremath{\alpha}$: $J(t)={J}_{i}+({J}_{f}\ensuremath{-}{J}_{i}){(t/\ensuremath{\tau})}^{\ensuremath{\alpha}}$. We compute the wave function overlap
F, the residual energy
Q, the superfluid order parameter $\ensuremath{\Delta}(t)$, the equal-time order parameter correlation function
C(t), and the defect formation probability
P for the above-mentioned protocols and provide a comparison of our results to their mean-field counterparts. We find that
Q,
F, and
P do not exhibit the expected universal scaling. We explain this absence of universality and show that our results for linear ramps compare well with the recent experimental observations.
Abstract This paper is a first step toward rigorously establishing the validity of mean-field approximations for the dynamics of quantum systems in a large dimensional limit. Our result is a derivation of the large d mean-field limit of the Bose–Hubbard model on a square lattice. The Bose–Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator phase transition. We prove a trace norm estimate between the one-lattice-site reduced density of the Schrödinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions.
We develop the nonequilibrium extension of bosonic dynamical mean-field theory and a Nambureal-time strong-coupling perturbative impurity solver. In contrast to Gutzwiller mean-field theory andstrong-coupling perturbative approaches, nonequilibrium bosonic dynamical mean-field theory capturesnot only dynamical transitions but also damping and thermalization effects at finite temperature. We applythe formalism to quenches in the Bose-Hubbard model, starting from both the normal and the Bose-condensed phases. Depending on the parameter regime, one observes qualitatively different dynamicalproperties, such as rapid thermalization, trapping in metastable superfluid or normal states, as well aslong-lived or strongly damped amplitude oscillations. We summarize our results in nonequilibrium“phase diagrams” that map out the different dynamical regimes.