Mean Field Theory of Critical Behavior

Yueyi Wang, Kristjan Haule

We develop a variational perturbation expansion around dynamical mean-field theory (DMFT) that systematically incorporates nonlocal correlations beyond the local correlations treated by DMFT. We apply this approach to investigate how the DMFT critical temperature is suppressed from its mean-field value and how the critical behavior near the finite-temperature phase transition evolves from the mean-field to the Heisenberg universality class. By identifying the symmetry breaking of paramagnetic diagrammatic expansions as a signature of the Néel transition, we accurately predict the Néel temperature of the three-dimensional cubic Hubbard model across all interaction strengths with low computational cost. Introducing a variational order parameter, our method can be applied to both paramagnetic and long-range ordered states, such as antiferromagnetic order. We compute magnetization and antiferromagnetic susceptibility, demonstrating minor corrections to DMFT solutions in the weak-coupling regime while revealing significant modifications to these properties in the intermediate correlation regime. From the analysis of critical exponents, we establish the emergence of Heisenberg critical behavior beyond the mean-field nature of DMFT.

C. Di Castro, Roberto Raimondi

Dieter Vollhardt, Krzysztof Byczuk, Marcus Kollar

Kurt Binder, D. P. Landau, D. M. Kroll

Critical and tricritical wetting transitions are studied in a simple-cubic nearest-neighbor Ising model. The magnetization ${m}_{1}$ in the surface layer and the susceptibilities ${\ensuremath{\chi}}_{11}=\frac{\ensuremath{\partial}{m}_{1}}{\ensuremath{\partial}{H}_{1}}$ and ${\ensuremath{\chi}}_{1}=\frac{\ensuremath{\partial}{m}_{1}}{\ensuremath{\partial}H}$, where ${H}_{1}$ is the surface field and $H$ the bulk field, are calculated by use of Monte Carlo methods. The critical behavior of these quantities is found to be consistent with simple mean-field behavior; the singularities predicted by various renormalization-group treatments are not observed.

E. Brézin

In four dimensions mean field theory is not valid since scaling is violated by powers of logarithms. Therefore the theory is not the ε = 0 limit of the 4 - ε dimensional theory. However ratios of critical amplitudes are in most cases correctly described by this limit. In some cases, it is nevertheless necessary to be cautious, and an example is provided of a ratio which does not coincide with its mean field value.