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Process-chain approach and its application to bosonic lattice systems

Dennis Hinrichs, Niklas Teichmann, Martin Holthaus

Published Mar 12, 2010• 0 citations
OpenAlex

Abstract

We study the transition from a Mott insulator to a superfluid in the Bose-Hubbard model at zero temperature for different lattice types. Converting Kato's perturbation series into an efficient numerical algorithm capable of reaching high orders, we obtain accurate critical parameters for any filling factor g and dimensionality d. Furthermore we clarify the g- and the d-dependence of the critical value and devise a method for extracting the critical value from the correlation function.

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We study the transition from a Mott insulator to a superfluid in the
Bose-Hubbard model at zero temperature for different lattice types. Converting Kato's perturbation series into an
efficient numerical algorithm capable of reaching high orders, we
obtain accurate critical parameters for any filling factor g and dimensionality d.
Furthermore we clarify the g- and the d-dependence of the critical value
and devise a method for extracting the critical value from the correlation function.
We study the transition from a Mott insulator to a superfluid in both the two- and the three-dimensional Bose-Hubbard model at zero temperature, employing the method of the effective potential. Converting Kato's perturbation series into an algorithm capable of reaching high orders, we obtain accurate critical parameters for any integer filling factor. Our technique allows us to monitor both the approach to the mean-field limit by considering spatial dimensionalities d>3d>3d>3 and to the quantum rotor limit of high filling, which refers to an array of Josephson junctions.
We carry out a perturbative analysis, of high order in the tunneling parameter, of the ground state of the homogeneous Bose-Hubbard model in the Mott insulator phase. This is made possible by a diagrammatic process-chain approach, derived from Kato's representation of the many-body perturbation series, which can be implemented numerically in a straightforward manner. We compute ground-state energies, atom-atom correlation functions, density-density correlations, and occupation number fluctuations, for one-, two-, and three-dimensional lattices with arbitrary integer filling. A phenomenological scaling behavior is found which renders the data almost independent of the filling factor. In addition, the process-chain approach is employed for calculating the boundary between the Mott insulator phase and the superfluid phase with high accuracy. We also consider systems with dimensionalities d>3d>3d>3, thus monitoring the approach to the mean-field limit. The versatility of the method suggests further applications to other systems which are less well understood.
We investigate systems of bosonic particles at zero temperature in triangular and hexagonal optical lattice potentials in the framework of the Bose-Hubbard model. Employing the process-chain approach, we obtain accurate values for the boundaries between the Mott insulating phase and the superfluid phase. These results can serve as reference data for both other approximation schemes and upcoming experiments. Since arbitrary integer filling factors g are amenable to our technique, we are able to monitor the behavior of the critical hopping parameters with increasing filling. We also demonstrate that the g -dependence of these exact parameters is described almost perfectly by a scaling relation inferred from the mean-field approximation.
We study the quantum phase transition between the superfluid and the Mott insulator in the one-dimensional (1D) Bose-Hubbard model. Using the time-evolving block-decimation method, we numerically calculate the tunneling splitting of two macroscopically distinct states with different winding numbers. From the scaling of the tunneling splitting with respect to the system size, we determine the critical point of the superfluid-to-Mott-insulator transition for arbitrary integer filling factors. We find that the critical values versus the filling factor in 1D, 2D, and 3D are well approximated by a simple analytical function. We also discuss the condition for determining the transition point from a perspective of the instanton method.
The values of the insulator gap Δ in one-dimensional systems of interacting bosons described by the Hubbard Hamiltonian are calculated at low temperatures by the quantum world-line Monte Carlo algorithm. The dependence of Δ on the size of the system, the temperature, and the parameters of the model is investigated. It is shown that a chain with N a =50 sites is already sufficient to estimate the thermodynamic value of the critical quantity ( t/U ) c for which a transition from the insulator into the superfluid state occurs in a commensurate system. To within the computational error, this value, ( t/U ) c =0.300±0.005, agrees with the value ( t/U ) c =0.304±0.002 obtained previously by the combined “exact diagonalization + renormalization-group analysis” method. The characteristic Kosterlitz-Thouless behavior of the insulator gap is demonstrated near the critical region: Δ∼exp[− b (1− t/t c ) −1/2 ].