Exact solution of the Bose-Hubbard model on the Bethe lattice

Abstract

The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent state path integral, leads in the large connectivity limit to the mean field treatment of Fisher et al. [Phys. Rev. B {\bf 40}, 546 (1989)] at the leading order, and to the bosonic Dynamical Mean Field Theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B {\bf 77}, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.

27 pages, 16 figures, minor corrections

References in corpus (21)

• Many-Body Physics with Ultracold Gases
• Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms
• Double species condensate with tunable interspecies interactions
• Supersolid hardcore bosons on the triangular lattice
• Monte Carlo study of two-dimensional Bose-Hubbard model
• Mott insulator to superfluid transition in the Bose-Hubbard model: a strong-coupling approach
• Boson Mott insulators at finite temperatures
• Correlated bosons on a lattice: Dynamical mean-field theory for Bose-Einstein condensed and normal phases
• Strong-coupling expansion for the momentum distribution of the Bose Hubbard model with benchmarking against exact numerical results
• Spectral weight redistribution in strongly correlated bosons in optical lattices
• On the path integral representation for quantum spin models and its application to the quantum cavity method and to Monte Carlo simulations
• Cavity method for quantum spin glasses on the Bethe lattice
• The Quantum Transverse Field Ising Model on an Infinite Tree from Matrix Product States
• Supersolid phase induced by correlated hopping in spin-1/2 frustrated quantum magnets
• Process chain approach to the Bose-Hubbard model: Ground-state properties and phase diagram
• Mean-field phase diagram of disordered bosons in a lattice at non-zero temperature
• Stochastic Mean-Field Theory for the Disordered Bose-Hubbard Model
• Vacancy supersolid of hard-core bosons on the square lattice
• On-site number statistics of ultracold lattice bosons
• C^*-algebraic approach to the Bose-Hubbard model
• Statistical Mechanics of the Quantum K-Satisfiability problem

Cited by in corpus (31)

• The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
• Dynamical mean field solution of the Bose-Hubbard model
• Efficient simulation of infinite tree tensor network states on the Bethe lattice
• Dynamical mean-field theory for bosons
• Dynamical Mean Field Theory for the Bose-Hubbard Model
• Stochastic Mean-Field Theory: Method and Application to the Disordered Bose-Hubbard Model at Finite Temperature and Speckle Disorder
• Simulation of three-dimensional quantum systems with projected entangled-pair states
• Quantum phase transitions of interacting bosons on hyperbolic lattices
• Bose-Einstein condensation in quantum glasses
• Heisenberg antiferromagnet on Cayley trees: low-energy spectrum and even/odd site imbalance
• Phase diagram of the Bose-Hubbard Model on Complex Networks
• The quantum Biroli-Mézard model: glass transition and superfluidity in a quantum lattice glass model
• There are no Goldstone bosons on the Bethe lattice
• Disordered driven coupled cavity arrays: Non-equilibrium stochastic mean-field theory
• Mixtures of correlated bosons and fermions: Dynamical mean-field theory for normal and condensed phases
• Sign problem in the Bethe approximation
• Thermal Tensor Network Simulations of the Heisenberg Model on the Bethe Lattice
• Critical Phenomena in Hyperbolic Space
• Strong-coupling solution of the bosonic dynamical mean-field theory
• Effects of long range hopping in the Bose-Hubbard model
• The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid
• Cavity approach to variational quantum mechanics
• Inner non-Hermitian skin effect on Bethe lattice
• Local Density of the Bose Glass Phase
• Solution of Disordered Microphases in the Bethe approximation
• Thermodynamics of the Hubbard Model on the Bethe Lattice
• Statistical mechanics of classical and quantum computational complexity
• Quantum Criticality for Extended Nodes on a Bethe Lattice in the Large Connectivity Limit
• Avoiding critical slowdown in models with SALR interactions
• Fluctuation driven transitions in localized insulators: Intermittent metallicity and path chaos precede delocalization
• Topological states and flat bands in exactly solvable decorated Cayley trees