Phases and Phase Transitions in Less Than Three Dimensions

Paul Fendley, Oleg Tchernyshyov

Paul Fendley, Oleg Tchernyshyov

Phase transitions can occur in one-dimensional classical statistical mechanics at non-zero temperature when the number of components N of the spin is infinite. We show how to solve such magnets in one dimension for any N, and how the phase transition develops at N = ∞. We discuss SU(N) and Sp(N) magnets, where the transition is second-order. In the new high-temperature phase, the correlation length is zero. We also show that for the SU(N) magnet on exactly three sites with periodic boundary conditions, the transition becomes first order. 1

Michael N. Barber

Dietrich Stauffer

One of the most illuminating ways of modelling phase transitions is to define a set of points on a lattice, specify a spin at each point and a set of rules for their mutual interactions, and use its behaviour to predict fundamental quantities such as the critical point. Hitherto these models have suffered severe limitations in three dimensions, but now several groups have shown that computer simulations are succeeding where other calculations were less accurate.

G. V. Klimachev

W. D. Langer, Michael Plischke, Daniel Charles Mattis

We solve the Bubbard model, for one electron per atom in a simple cubic structure, using one-particle Green's functions. We determine the accuracy of this calculation to be good by comparison with an exact solution by Lieb and Wu of a one-dimensional limiting case. When the Coulomb interaction $U$ exceeds about 0.27 of the bandwidth in three dimensions we find there are two critical temperatures: ${T}_{\mathrm{N}}$, the N\'eel order-disorder transition temperature ($\ensuremath{\propto}{U}^{\ensuremath{-}1}$ at large $U$) and a higher critical temperature ${T}_{M}$, at which the atoms lose all vestige of localized moments and at which the insulator-metal transition occurs (${T}_{M}\ensuremath{\propto}U$ at large $U$). For $U$ less than 0.27 of the bandwidth only ${T}_{M}$ exists.