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Physical review. B, Condensed matter

Gutzwiller projection for bosons

Daniel S. Rokhsar, B. Gabriel Kotliar

Published Nov 1, 1991• 263 citations
Publisher

Abstract

We apply Gutzwiller's variational method to the interacting Bose lattice gas. In contrast with the Fermi case, the Gutzwiller wave function for bosons can be treated with no further approximation in several limits. Furthermore, this wave function can be shown to be exact for large dimensionality, and also in arbitrary dimension for suitably chosen short-range interactions. At densities commensurate with the lattice, a superfluid-insulator transition is found. These results are compared with the fermionic case, and are applied to several interacting-boson systems.

DOI
10.1103/physrevb.44.10328

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We apply Gutzwiller's variational method to the interacting Bose lattice gas. In contrast with the Fermi case, the Gutzwiller wave function for bosons can be treated with no further approximation in several limits. Furthermore, this wave function can be shown to be exact for large dimensionality, and also in arbitrary dimension for suitably chosen short-range interactions. At densities commensurate with the lattice, a superfluid-insulator transition is found. These results are compared with the fermionic case, and are applied to several interacting-boson systems.
We study a model of strongly interacting lattice bosons with a Gutzwiller-type wave function that contains only on-site correlations. The variational energy and the condensate fraction associated with the variational wave function are exactly evaluated for both finite and infinite systems and compared with exact quantum Monte Carlo results in two dimensions. This ansatz for the wave function gives the correct qualitative picture of the phase diagram of this system; at commensurate densities, this system enters a Mott-insulator phase for large values of the interaction.
The properties of the Gutzwiller variational wave function, which is frequently used to study ground-state properties of correlated fermions with a short-range interaction, are investigated by use of a new, analytically tractable approach. As a first application several ground-state quantities are evaluated exactly in dimensionality d=1 for arbitrary band-filling and interaction strengths. The results allow for the first approximation-free assessment of the wave function. The method itself is applicable to arbitrary space dimensions.
We develop a variational wave function for the ground state of a one-dimensional bosonic lattice gas. The variational theory is initially developed for the quantum rotor model and later on extended to the Bose- Hubbard model. This theory is compared with quasi-exact numerical results obtained by Density Matrix Renormalization Group (DMRG) studies and with results from other analytical approximations. Our approach accurately gives local properties for strong and weak interactions, and it also describes the crossover from the superfluid phase to the Mott-insulator phase.
A versatile and numerically inexpensive method is presented allowing the accurate calculation of phase diagrams for bosonic lattice models. By treating clusters within the Gutzwiller theory, a surprisingly good description of quantum fluctuations beyond the mean-field theory is achieved approaching quantum Monte Carlo predictions for large clusters. Applying this powerful method to the Bose-Hubbard model, we demonstrate that it yields precise results for the superfluid to Mott-insulator transition in square, honeycomb, and cubic lattices. Due to the exact treatment within a cluster, the method can be effortlessly adapted to more complicated Hamiltonians in the fast progressing field of optical lattice experiments. This includes state- and site-dependent superlattices, large confined atomic systems, and disordered potentials, as well as various types of extended Hubbard models. Furthermore, the approach allows an excellent treatment of systems with arbitrary filling factors. We discuss the perspectives that allow for the computation of large, spatially varying lattices, low-lying excitations, and time evolution.
Various types of superfluid-insulator transitions are investigated for two-component lattice boson systems in two dimensions with on-site hard-core repulsion and the component-dependent intersite interaction. The mean-field phase diagram is obtained by the Gutzwiller-type variational technique. Various ground-state properties are also studied by the quantum Monte Carlo method. Our model exhibits two types of diagonal long-range orders; the density order around the density n =1/2 and the Ising-type component order near n =1. In contrast to the Gutzwiller results, the Monte Carlo results show that the continuous growth of the component correlation severely suppresses the superfluidity as well as the inverse of the effective mass in the critical region of the component order transition. We propose a mechanism of this mass enhancement observed even far from the Mott insulating filling n =1, when the Ising-type component order persists into n ≠1. Possible relevance of this type of mass enhancement in other systems is also discussed.