[Paper Review] A phase-space method for the Bose-Hubbard model

Poonam Jain, C. W. Gardiner|arXiv (Cornell University)|Apr 27, 2004

Cold Atom Physics and Bose-Einstein Condensates14 references3 citations

TL;DR

This paper introduces a phase-space method based on the Q-function representation to approximate the ground state of the Bose-Hubbard model, using mean-field approximations for one and two sites. It shows that quantum constraints—particularly uncertainty relations—dominate the phase diagram, yielding good qualitative agreement with exact solutions, though the two-site model fails to correctly predict the Mott insulator transition due to parameterization limits.

ABSTRACT

We present a phase-space method for the Bose-Hubbard model based on the Q-function representation. In particular, we consider two model Hamiltonians in the mean-field approximation; the first is the standard "one site" model where quantum tunneling is approximated entirely using mean-field terms; the second "two site" model explicitly includes tunneling between two adjacent sites while treating tunneling with other neighbouring sites using the mean-field approximation. The ground state is determined by minimizing the classical energy functional subject to quantum mechanical constraints, which take the form of uncertainty relations. For each model Hamiltonian we compare the ground state results from the Q-function method with the exact numerical solution. The results from the Q-function method, which are easy to compute, give a good qualitative description of the main features of the Bose-Hubbard model including the superfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem and show there are some limitations of the method particularly in the weak lattice regime.

Motivation & Objective

• To develop a computationally efficient method for approximating the ground state of the Bose-Hubbard model across both superfluid and Mott insulator phases.
• To investigate the role of quantum mechanical constraints—especially uncertainty relations—in determining the phase structure of the model.
• To compare the Q-function phase-space method with exact numerical solutions for one- and two-site models under mean-field approximations.
• To assess the limitations of the two-site Q-function approach in capturing the Mott insulator phase transition.
• To explore whether extending the method beyond two sites could improve agreement with numerically exact results like DMRG and QMC.

Proposed method

• The Q-function representation is used to reparameterize the Bose-Hubbard Hamiltonian in terms of Gaussian variables, enabling a classical-like phase-space description.
• The ground state is found by minimizing the classical energy functional subject to quantum constraints derived from uncertainty relations: one on number variance and another linking number and phase fluctuations.
• Two model Hamiltonians are studied: a one-site model with full mean-field treatment of tunneling, and a two-site model that explicitly includes intersite tunneling while using mean-field for other neighbors.
• The one-site model is solved exactly via variational optimization over number variance and mean-field amplitude, equivalent to the Gutzwiller ansatz.
• The two-site model is solved both via the Q-function method and exact numerical minimization under lattice symmetry assumptions.
• Results are compared with numerically exact methods such as DMRG and Quantum Monte Carlo, particularly for one-dimensional lattices.

Experimental results

Research questions

• RQ1 Can a phase-space method based on the Q-function accurately describe the superfluid to Mott insulator transition in the Bose-Hubbard model?
• RQ2 How do quantum mechanical uncertainty constraints influence the ground state structure in the one- and two-site mean-field approximations?
• RQ3 Why does the two-site Q-function method fail to correctly predict the Mott insulator phase despite good qualitative agreement?
• RQ4 How does the predicted critical interaction strength $\overline{U}_c$ for the phase transition compare between the Q-function method and exact numerical methods?
• RQ5 To what extent does extending the two-site model to include more sites improve agreement with exact results like DMRG and QMC?

Key findings

• The Q-function method provides a qualitatively accurate description of the superfluid to Mott insulator transition, with results typically within 5% of exact solutions for the one-site model.
• The one-site Q-function approach agrees well with exact numerical solutions, achieving similar results in a fraction of the computational time.
• The two-site Q-function method fails to correctly predict the Mott insulator phase, as it does not yield a vanishing mean-field at the transition, unlike the exact two-site solution.
• The exact two-site solution shows a well-defined phase transition with a vanishing mean-field for commensurate filling, consistent with the Mott insulator phase.
• The critical interaction strength $\overline{U}_c$ for the transition is lower in the two-site exact solution than in the one-site model, with the shift being most pronounced in one dimension.
• The results indicate that quantum constraints—particularly the lower bound on number variance—play a dominant role in determining the ground state, especially at the onset of the Mott insulator phase.