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arXiv:cond-mat/0511294 · Journal of Physics Condensed Matter

Variational cluster perturbation theory for Bose–Hubbard models

W Koller, N. Dupuis

Published Sep 29, 2006• 39 citations
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Abstract

We discuss the application of the variational cluster perturbation theory (VCPT) to the Mott-insulator-to-superfluid transition in the Bose–Hubbard model. We show how the VCPT can be formulated in such a way that it gives a translation invariant excitation spectrum—free of spurious gaps—despite the fact that it formally breaks translation invariance. The phase diagram and the single-particle Green function in the insulating phase are obtained for one-dimensional systems. When the chemical potential of the cluster is taken as a variational parameter, the VCPT reproduces the dimensional dependence of the phase diagram even for one-site clusters. We find a good quantitative agreement with the results of the density-matrix renormalization group when the number of sites in the cluster becomes of order 10. The extension of the method to the superfluid phase is discussed.

DOI
10.1088/0953-8984/18/41/019

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We discuss the application of the variational cluster perturbation theory (VCPT) to the Mott-insulator-to-superfluid transition in the Bose–Hubbard model. We show how the VCPT can be formulated in such a way that it gives a translation invariant excitation spectrum—free of spurious gaps—despite the fact that it formally breaks translation invariance. The phase diagram and the single-particle Green function in the insulating phase are obtained for one-dimensional systems. When the chemical potential of the cluster is taken as a variational parameter, the VCPT reproduces the dimensional dependence of the phase diagram even for one-site clusters. We find a good quantitative agreement with the results of the density-matrix renormalization group when the number of sites in the cluster becomes of order 10. The extension of the method to the superfluid phase is discussed.
Convergence properties of the variational cluster approach with respect to the variational parameter space, cluster size, and boundary conditions of the reference system are investigated and discussed for bosonic many-body systems. Specifically, the variational cluster approach is applied to the one-dimensional Bose-Hubbard model, which exhibits a quantum phase transition from Mott to superfluid phase. In order to benchmark the variational cluster approach, results for the phase boundary delimiting the first Mott lobe are compared with essentially exact density matrix renormalization group data. Furthermore, static quantities, such as the ground state energy and the one-particle density matrix are compared with high-order strong coupling perturbation theory results. For reference systems with open boundary conditions the variational parameter space is extended by an additional variational parameter which allows for a more uniform particle density on the reference system and thus drastically improves the results. It turns out that the variational cluster approach yields accurate results with relatively low-computational effort for both the phase boundary as well as the static properties of the one-dimensional Bose-Hubbard model, even at the tip of the first Mott lobe where correlation effects are most pronounced.
We use strong-coupling perturbation theory, the variational cluster approach (VCA), and the dynamical density-matrix renormalization group (DDMRG) method to investigate static and dynamical properties of the one-dimensional Bose--Hubbard model in both the Mott-insulating and superfluid phases. From the von Neumann entanglement entropy we determine the central charge and the transition points for the first two Mott lobes. Our DMRG results for the ground-state energy, momentum distribution function, boson correlation function decay, Mott gap, and single-particle spectral function are reproduced very well by the strong-coupling expansion to fifth order, and by VCA with clusters up to 12 sites as long as the ratio between the hopping amplitude and onsite repulsion, t/Ut/Ut/U, is smaller than 0.15 and 0.25, respectively. In addition, in the superfluid phase VCA captures well the ground-state energy and the sound velocity of the linear phonon modes. This comparison provides an authoritative estimate for the range of applicability of these methods. In strong-coupling theory for the Mott phase, the dynamical structure factor is obtained from the solution of an effective single-particle problem with an attractive potential. The resulting resonances show up as double-peak structures close to the Brillouin zone boundary. These high-energy features also appear in the superfluid phase which is characterized by a pronounced phonon mode at small momenta and energies, as predicted by Bogoliubov and field theory. In one dimension, there are no traces of an amplitude mode in the dynamical single-particle and two-particle correlation functions.
The variational cluster approach (VCA) is applied to the one-dimensional Hubbard model at zero temperature using clusters (chains) of up to ten sites with full diagonalization and the Lanczos method as cluster solver. Within the framework of the self-energy-functional theory (SFT), different cluster reference systems with and without bath degrees of freedom, in different topologies and with different sets of variational parameters, are considered. Static and one-particle dynamical quantities are calculated for half-filling as a function of UUU as well as for fixed UUU as a function of the chemical potential to study the interaction- and filling-dependent metal-insulator (Mott) transition. The recently developed QQQ-matrix technique is used to compute the SFT grand potential. For benchmarking purposes we compare the VCA results with exact results available from the Bethe ansatz, with essentially exact dynamical density-matrix renormalization-group data, with (cellular) dynamical mean-field theory and full diagonalization of isolated Hubbard chains. Several issues are discussed including convergence of the results with cluster size, the ability of cluster approaches to access the critical regime of the Mott transition, efficiency in the optimization of correlated-site vs bath-site parameters, and of multidimensional parameter optimization. We also study the role of bath sites for the description of excitation properties and as charge reservoirs for the description of filling dependencies. The VCA turns out to be a computationally cheap method which is competitive with established cluster approaches.
We extend the variational cluster approach to deal with strongly correlated lattice bosons in the superfluid phase. To this end, we reformulate the approach within a pseudoparticle formalism, whereby cluster excitations are described by particlelike excitations. The approximation amounts to solving a multicomponent noninteracting bosonic system by means of a multimode Bogoliubov approximation. A source-and-drain term is introduced in order to break U(1) symmetry at the cluster level. We provide an expression for the grand potential, the single-particle normal and anomalous Green's functions, the condensate density, and other static quantities. As a first nontrivial application of the method we choose the two-dimensional Bose-Hubbard model and evaluate results in both the Mott and the superfluid phases. Our results show an excellent agreement with quantum Monte Carlo calculations.
Abstract The cluster mean-field with density matrix renormalization (CMFT + DMRG) method which combines the simplicity of the mean-field theory and the numerical power of the density-matrix renormalization group method is applied to understand the quantum phases of the one-dimensional Bose–Hubbard models. We show that the CMFT + DMRG method is an effective numerical technique with moderate computational resources to determine relevant order parameters and correlation functions of large one-dimensional systems. We apply the CMFT + DMRG for the Bose Hubbard and extended Bose Hubbard models to account for the superfluid, Mott insulator, and density wave phases in these models. Our results are in good agreement with the known phase diagram of these models, demonstrating the efficacy of this method.
The phase transitions in the Bose-Hubbard model are investigated.A single-particle Green's function is calculated in the random phase approximation and the formalism of the Hubbard operators is used.The regions of existence of the superfluid and Mott insulator phases are established and the (µ, t) (the chemical potentialtransfer parameter) phase diagrams are built.The effect of temperature change on this transition is analyzed and the phase diagram in the (T, µ) plane is constructed.The role of thermal activation of the ion hopping is investigated by taking into account the temperature dependence of the transfer parameter.The reconstruction of the Mott-insulator lobes due to this effect is analyzed.