Perturbative calculation of critical exponents for the Bose–Hubbard model

Abstract

We develop a strategy for calculating critical exponents for the Mott insulator-to-superfluid transition shown by the Bose-Hubbard model. Our approach is based on the field-theoretic concept of the effective potential, which provides a natural extension of the Landau theory of phase transitions to quantum critical phenomena. The coefficients of the Landau expansion of that effective potential are obtained by high-order perturbation theory. We counteract the divergency of the weak-coupling perturbation series by including the seldom considered Landau coefficient a6 into our analysis. Our preliminary results indicate that the critical exponents for both the condensate density and the superfluid density, as derived from the two-dimensional Bose-Hubbard model, deviate by less than 1% from the best known estimates computed so far for the three-dimensional XY universality class.

Tuning the quantum phase transition of bosons in optical lattices via periodic modulation of thes-wave scattering length

“…Thus, a period driving of the interaction allows to tune dynamically the hopping within an optical lattice. Furthermore, these findings corroborate the hypothesis that the Bose-Hubbard model with a periodically driven s-wave scattering length and the usual Bose-Hubbard model belong to the same universality class from the point of view of critical phenomena [58,59] and, thus, should have the same critical exponents [1,60,61]. …”

Improving mean-field theory for bosons in optical lattices via degenerate perturbation theory

“…Thus, we conclude that the approximations within the mean-field approach are too strong to result in any difference between the condensate density and the superfluid density. In order to improve this, one must not apply the mean-field theory, but use some other method to deal with the system, like the field-theoretic method, where a Legendre transform of the grand-canonical free energy [18,24,34] is used.…”

Hypergeometric analytic continuation of the strong-coupling perturbation series for the 2d Bose-Hubbard model

“…We have applied this strategy to the series (16) for the two-dimensional Bose-Hubbard model with 0 ≤ µ/U ≤ 4, having at disposal its coefficients α (ν) 2 up to ν max = 10 [22][23][24]. Figure 2 again displays the ratios α to the binomial series hypothesis (24), and from the best fit…”

“…Although this series requires a small parameter J/U it is referred to as a strong-coupling expansion [11], since it should converge in the strongly correlated Mott regime. We have evaluated its coefficients α (ν) 2 numerically up to the order ν max = 10 in J/U [22][23][24], making use of the process-chain approach as devised in general form by Eckardt [25]. This technique, which has been recognized as an extremely powerful method [26], is based on Kato's non-recursive formulation of the Rayleigh-Schrödinger perturbation series [27].…”