Abstract
We derive the $\mathrm{Bogoliubov}+U$ formalism to study the thermodynamical properties of the Bose-Hubbard model. The framework can be viewed as the zero-frequency limit of bosonic dynamical mean-field theory (B-DMFT), but equally well as an extension of the mean-field decoupling approximation in which pair creation and annihilation of depleted particles is taken into account. The self-energy on the impurity site is treated variationally, minimizing the grand potential. The theory containing just three parameters that are determined self-consistently reproduces the $T=0$ phase diagrams of the three-dimensional and two-dimensional Bose-Hubbard model with an accuracy of $1%$ or better. The superfluid to normal transition at finite temperature is also reproduced well and only slightly less accurately than in B-DMFT.
Highlights
The properties of cold atomic gases trapped in an optical lattice can be tuned and controlled very precisely, providing a powerful tool for the simulation of the iconical lowenergy effective Hamiltonians of condensed-matter models [1]
We compare the results with mean-field theory, path integral Monte Carlo simulations with worm-type updates (QMC) from Ref. [28], and bosonic dynamical mean-field theory (B-DMFT) results from Ref. [10]
The results are identical with the B-DMFT results and agree within a percent with the quantum Monte Carlo (QMC) data both for the 3D and the 2D cases
Summary
The properties of cold atomic gases trapped in an optical lattice can be tuned and controlled very precisely, providing a powerful tool for the simulation of the iconical lowenergy effective Hamiltonians of condensed-matter models [1]. We filter out the ingredients of B-DMFT that are indispensable for its accuracy and arrive at a simpler formalism This is the Bogoliubov+U theory (B + U ), which makes use of a simplified effective impurity Hamiltonian, similar to the action of extended mean-field theory, which was recently developed in the high-energy community [14,15] but differs conceptually from our formalism. It is different from the variational cluster approximation (VCA) by considering nonzero values of pair creation and annihilation of depleted particles [16] It is the simplest accurate extension of the weakly interacting Bose gas theory [17] to lattice systems with a superfluid to Mott insulator transition. VII we conclude and present a short outlook about future applications of the B + U formalism
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