The Mean-Field Approximation, Scaling and Critical Exponents

Fu Cheng Yu

By using the fractal series (the exponents are equal to arbitrary real number) and taking term M 2+0.75 instead of M 2 in the Landau ′s free energy formula, the improved mean field approximation of continuous phase transition beneath critical point was presented. The derived critical exponents can satisfy the two scaling equations and the accurate critical exponents were given as follows: α =0.13, β =0.34, γ =1.19, δ =4.5.

Anastasios Malakis, S.S. Martinos

It is shown that 'mean-field finite-size scaling theory' can be adjusted and applied to sequences of approximations that do not obey the well known power laws of finite-size scaling. The proposed modification of the theory seems to be well directed to the sequence of Baxter's variational approximations. This sequence of approximations shows a novel feature according to which exponential laws appear in places where power laws should be expected, but the modified scaling theory still yields relations from which non-classical critical exponents can be estimated. Thus, two techniques for the estimation of the critical exponent beta are suggested from the finite-order variational approximations. These techniques are applied to the zero-field Ising model on the square lattice using the systematic series of the Baxter-Tsang systems and excellent estimates of beta are obtained correct up to 13 significant figures.

Uday Sood, Martı́n Kruczenski

Abstract We consider the Bose–Hubbard model in two and three spatial dimensions and numerically compute the quantum circuit complexity of the ground state in the Mott insulator and superfluid phases using a mean field approximation with additional quadratic fluctuations. After mapping to a qubit system, the result is given by the complexity associated with a Bogoliubov transformation applied to the reference state taken to be the mean field ground state. In particular, the complexity has peaks at the O (2) critical points where the system can be described by a relativistic quantum field theory. Given that we use a Gaussian approximation, near criticality the numerical results agree with a free field theory calculation. To go beyond the Gaussian approximation we use general scaling arguments that imply that, as we approach the critical point t → t c , there is a non-analytic behavior in the complexity c 2 ( t ) of the form | c 2 ( t ) − c 2 ( t c )| ∼ | t − t c | νd , up to possible logarithmic corrections. Here d is the number of spatial dimensions and ν is the usual critical exponent for the correlation length ξ ∼ | t − t c | − ν . As a check, for d = 2 this agrees with the numerical computation if we use the Gaussian critical exponent ν = 1 2 . Finally, using AdS/CFT methods, we study higher dimensional examples and confirm this scaling argument with non-Gaussian exponent ν for strongly interacting theories that have a gravity dual.

Pierre Cénédèse, J. M. Sanchez, R. Kikuchic