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arXiv:cond-mat/9405053

Slave Boson Formulation for Interacting Boson Systems and the Superfluid-Insulator Transition

Raymond Frésard

Published May 20, 1994• 1 citation
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Abstract

A new formulation for the study of interacting bosons on a lattice is introduced. This approach is used to give analytical expressions for the Mott insulating lobes in the phase diagram and to calculate the density-density correlation function. It is also shown that, at mean-field level, this newly introduced slave boson theory coincides with mean-field theory of a suitably introduced order parameter.

DOI
10.48550/arxiv.cond-mat/9405053

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A new formulation for the study of interacting bosons on a lattice is introduced. This approach is used to give analytical expressions for the Mott insulating lobes in the phase diagram and to calculate the density-density correlation function. It is also shown that, at mean-field level, this newly introduced slave boson theory coincides with mean-field theory of a suitably introduced order parameter.
By using slave particle (slave boson and slave fermion) techniques on the Bose-Hubbard model, we study the finite temperature properties of ultracold Bose gases in optical lattices. The phase diagrams at finite temperature are depicted by including different types of slave particles and the effect of the finite types of slave particles is estimated. The superfluid density is evaluated using the Landau second order phase transition theory. The atom density, excitation spectrum, and dispersion curve are also computed at various temperatures, and how the Mott-insulator evolves as the temperature increases is demonstrated. For most quantities to be calculated, we find that there are no qualitative differences in using the slave boson or the slave fermion approaches. However, when studying the stability of the mean field state, we find that in contrast to the slave fermion approach, the slave boson mean field state is not stable. Although the slave boson mean field theory gives a qualitatively correct phase boundary, it corresponds to a local maximum of Landau free energy and cannot describe the second order phase transition because the coefficient a4{a}_{4}a4​ of the fourth order term is always negative in the free energy expansion.
The bosonic Hubbard model is studied via a simple mean-field theory. At zero temperature, in addition to yielding a phase diagram that is qualitatively correct, namely a superfluid phase for non-integer fillings and a Mott transition from a superfluid to an insulating phase for integer fillings, this theory gives results that are in good agreement with Monte Carlo simulations. In particular, the superfluid fraction obtained as a function of the interaction strength U for both integer and non-integer fillings is close to the simulation results. In all phases the excitation spectra are obtained by using the random phase approximation (RPA): the spectrum has a gap in the insulating phase and is gapless (and linear at small wave vectors) in the superfluid phase. Analytic results are presented in the limits of large U and small superfluid density. Finite-temperature phase diagrams and the Mott-insulator-normal-phase crossover are also described.
We use the Kotliar-Ruckenstein slave boson formalism to study the phase diagram of the doped antiferromagnetic Mott insulator at zero and finite temperatures. We obtain the commensurate magnetic phase diagram as a function of doping and temperature. At higher doping and temperatures we find a paramagnetic metallic phase. Reducing doping and temperature, we find antiferromagnetic phases with both metallic and insulating solutions. We calculate the electronic density of states in various regions of the phase diagram. We use our results for the electronic compressibility and magnetic susceptibility to analyze the stability of the various phases obtained within this mean field formalism. We find various parameter regimes where the method fails to provide physical solutions. We argue that those regions correspond to more exotic phases that cannot be described within the current mean field theory.
We perform a comparative study of the finite temperature behavior of ultracold Bose atoms in optical lattices by the slave fermion and the slave boson approaches to the Bose-Hubbard model. The phase diagram of the system is presented. Although both approaches are equivalent without approximations, the mean field theory based on the slave fermion technique is quantitatively more appropriate. Conceptually, the slave fermion approach automatically excludes the double occupancy of two identical fermions on the same lattice site. By comparing to known results in limiting cases, we find the slave fermion approach better than the slave boson approach. For example, in the non-interacting limit, the critical temperature of the superfluid-normal liquid transition calculated by the slave fermion approach is closer to the well-known ideal Bose gas result. At zero-temperature limit of the critical interaction, strength from the slave fermion approach is also closer to that from the direct calculation using a zero-temperature mean field theory.
We derive an approximate analytical solution of the self-consistency equations of the bosonic dynamical mean-field theory (B-DMFT) in the strong-coupling limit. The approach is based on a linked-cluster expansion in the hybridization function of normal bosons around the atomic limit. The solution is used to compute the phase diagram of the bosonic Hubbard model for different lattices. We compare our results with numerical solutions of the B-DMFT equations and numerically exact methods, respectively. The very good agreement with those numerical results demonstrates that our approach captures the essential physics of correlated bosons both in the Mott insulator and in the superfluid phase. Close to the transition into the superfluid phase the momentum distribution function at zero momentum is found to be strongly enhanced already in the normal phase. The linked-cluster expansion also allows us to compute dynamical properties such as the spectral function of bosons. The evolution of the spectral function across the transition from the normal to the superfluid phase is seen to be characteristically different for the interaction driven and density driven transition, respectively.
The grand canonical system of hard-core bosons, subject to thermal fluctuations, is studied on a lattice. Starting from the slave-boson representation we derive an effective Ginzburg-Landau functional for the superfluid order parameter. The order parameter field is a collective field constructed from the slave-boson fields for empty and singly occupied sites. We find two phase transitions; one from a normal fluid at high density to a superfluid and another one from the superfluid to a normal fluid at low density.