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arXiv:1406.0890 · Physical Review E

Analysis and resolution of the ground-state degeneracy of the two-component Bose-Hubbard model

Wei Wang, Vittorio Penna, Barbara Capogrosso-Sansone

Published Aug 15, 2014• 9 citations
arXivHTMLPDF

Abstract

We study the degeneracy of the ground-state energy E of the two-component Bose-Hubbard model and of the perturbative correction E(1). We show that the degeneracy properties of E and E(1) are closely related to the connectivity properties of the lattice. We determine general conditions under which E is nondegenerate. This analysis is then extended to investigate the degeneracy of E(1). In this case, in addition to the lattice structure, the degeneracy also depends on the number of particles present in the system. After identifying the cases in which E(1) is degenerate and observing that the standard (degenerate) perturbation theory is not applicable, we develop a method to determine the zeroth-order correction to the ground state by exploiting the symmetry properties of the lattice. This method is used to implement the perturbative approach to the two-component Bose-Hubbard model in the case of degenerate E(1) and is expected to be a valid tool to perturbatively study the asymmetric character of the Mott insulator to superfluid transition between the particle and hole side.

DOI
10.1103/physreve.90.022116

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We study the degeneracy of the ground-state energy E of the two-component Bose-Hubbard model and of the perturbative correction E(1). We show that the degeneracy properties of E and E(1) are closely related to the connectivity properties of the lattice. We determine general conditions under which E is nondegenerate. This analysis is then extended to investigate the degeneracy of E(1). In this case, in addition to the lattice structure, the degeneracy also depends on the number of particles present in the system. After identifying the cases in which E(1) is degenerate and observing that the standard (degenerate) perturbation theory is not applicable, we develop a method to determine the zeroth-order correction to the ground state by exploiting the symmetry properties of the lattice. This method is used to implement the perturbative approach to the two-component Bose-Hubbard model in the case of degenerate E(1) and is expected to be a valid tool to perturbatively study the asymmetric character of the Mott insulator to superfluid transition between the particle and hole side.
The objective of this paper is the theoretical description of the Mott insulator to superfluid quantum phase transition of a Bose gas in an optical lattice. In former works the Rayleigh-Schr\"odinger perturbation theory was used within a mean-field approach, which yields partially nonphysical results since the degeneracy between two adjacent Mott lobes is not taken into account. In order to correct such nonphysical results we apply the Brillouin-Wigner perturbation theory to the mean-field approximation of the Bose-Hubbard model. Detailed explanations of how to use the Brillouin-Wigner theory are presented, including a graphical approach that allows one to efficiently keep track of the respective analytic terms. To prove the validity of this computation, the results are compared with other works. Besides the analytic calculation of the phase boundary from Mott insulator to superfluid phase, the condensate density is also determined by simultaneously solving two algebraic equations. The analytical and numerical results turn out to be physically meaningful and can cover a region of system parameters inaccessible until now. Our results are of particular interest provided a harmonic trap is added to the former calculations in a homogeneous system, in view of describing an experiment within the local-density approximation. Thus, the paper represents an essential preparatory work for determining the experimentally observed wedding cake structure of particle-density profile at both finite temperature and finite hopping.
Bosonic atoms confined in optical lattices can exist in two different phases, Mott insulator and superfluid, depending on the strength of the system parameters, such as the on-site interaction between particles and the hopping parameter. This work is motivated by the fact that nondegenerate perturbation theory applied to the mean-field approximation of the Bose-Hubbard Hamiltonian at both zero and finite temperature fails to give consistent results in the vicinity of the Mott insulator--superfluid phase transition, e.g., the order parameter calculated via nondegenerate perturbation theory reveals an unphysical behavior between neighboring Mott lobes, which is an explicit consequence of degeneracy problems that artificially arise from such a treatment. This problem also appears in other bosonic systems that present similar Mott-lobe structures. Therefore, in order to fix this problem, we propose a finite-temperature degenerate perturbation theory approach based on a projection operator formalism which ends up solving such degeneracy problems in order to obtain physically consistent results for the order parameter near the phase transition.
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 Bose-Hubbard model of a twofold degenerate Bose gas is studied in an optical lattice with one particle per site and virtual tunneling to empty and doubly occupied sites. An effective Hamiltonian for this system is derived within a continued-fraction approach. The ground state of the effective model is studied in mean-field approximation for a modulated optical lattice. A dimerized mean-field state gives a Mott insulator whereas the lattice without modulations develops long-range correlated phase fluctuations due to a Goldstone mode. This result is discussed in comparison with the superfluid and the Mott-insulating state of a single-component hard-core Bose gas.
We present rigorous and universal results for the ground states of the f=2 spinor Bose-Hubbard model. The model includes three two-body on-site interaction terms, two of which are spin dependent while the other one is spin independent. We prove that, depending only on the coefficients of the two spin-dependent terms, the ground state exhibits maximum or minimum total spin or SU(5) ferromagnetism. Exact ground-state degeneracies and the forms of ground-state wave function are also determined in each case. All these results are valid regardless of dimension, lattice structure, or particle number. Our approach takes advantage of the symmetry of the Hamiltonian and employs mathematical tools including the Perron-Frobenius theorem and the Lie algebra so(5).
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.