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Physical Review Letters

Criterion for Distinguishing between Continuous and Discontinuous Transitions in Spin and Gauge Systems

Roberto Livi, Amos Maritan, Stefano Ruffo, Attilio L. Stella

Published Feb 14, 1983• 7 citations
Publisher

Abstract

A conjecture, based on a comparison of different mean-field-like approximations, is proposed for determining the first- or second-order character of phase transitions. The $q$-state Potts model and several gauge theories [${Z}_{2}$, $\mathrm{U}(N)$, $\mathrm{SU}(N)$, SO(3)] on a lattice are considered. For ${Z}_{2}$ gauge theory, in particular, a critical dimensionality ${d}_{c}=3.66$ is predicted, whereas for the Potts model ${q}_{c}=2.75$ in $d=3$, ${q}_{c}=2.4$ in $d=4$, and ${q}_{c}\ensuremath{\propto}\mathrm{exp}[\frac{1}{(d\ensuremath{-}1)}]$ for $d\ensuremath{\rightarrow}1$ are obtained.

DOI
10.1103/physrevlett.50.459

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A generalization of the Niemeijer-van Leeuwen renormalization-group transformation treating disordered cells of spins as vacancies is introduced and applied to the two-dimensional qqq-state Potts model. Finite-lattice approximations yield the changeover from continuous to first-order phase transitions predicted by Baxter but not observed in previous renormalization-group calculations. Exact results are conjectured for the tricritical exponents of dilute Potts models.
A quantum model with nnn-spin interactions is studied in the ground state. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ the model is solved exactly by mean-field theory and undergoes a first-order transition. The transition is expected to become second order for n<ncn<{n}_{c}n<nc​. With exploitation of self-duality and suitably improved mean-field approximations, a recently proposed criterion leads to the conjecture nc=3{n}_{c}=3nc​=3 in one dimension. For the two-dimensional case the transition appears to be definitely of first order for n=3(2<nc<3)n=3 (2<{n}_{c}<3)n=3(2<nc​<3).
A new scheme of the coherent-anomaly method (CAM) is proposed to study critical phenomena in the models for which a mean-field description gives spurious first-order phase transition. A canonical series of mean-field-type approximations are constructed so that the spurious discontinuity should vanish asymptotically as the approximate critical temperature approachs the true value. The true value of the critical exponents β and γ are related to the coherent-anomaly exponents defined among the classical approximations. The formulation is demonstrated in the two-dimensional q -state Potts models for q =3 and 4. The result shows that the present method enables us to estimate the critical exponents with high accuracy by using the date of the cluster-mean-field approximations.
Quantum phase transitions of a q-state Potts model in fractal lattices are studied using a continuous-time quantum Monte Carlo simulation technique. For small values of q, the transition is found to be second order and critical exponents of the quantum critical point are calculated. The dynamic critical exponent z is found to be greater than one for all fractals studied, which is in contrast to integer-dimensional regular lattices. When q is greater than a certain value q_{c}, the phase transition becomes first order, where q_{c} depends on the lattice. Further analysis shows that the characteristics of phase transitions are more sensitive to the average number of nearest neighbors than the Hausdorff dimension or the order of ramification.
We report the numerical study, using Wilson's $\ensuremath{\eta}=0$ approximate renormalization-group recursion formula, of a continuous-spin generalization of the three-component Potts model. Previous numerical studies, based on series expansions of the partition function, have indicated strongly that the model has a phase transition of second order. This is in conflict with Landau theory which predicts that the transition be of first order. We find, in agreement with Landau theory, that our model exhibits a first-order phase transition having finite zero-field spontaneous magnetization at the transition point.
The Monte Carlo method is used to study phase transitions in the two-dimensional Potts model with the number of states spin q=3 on square and hexagonal lattices. Considered systems with linear dimensions Lx L=N, L=21/102. The obtained numerical data indicate that in the considered In the Potts model, a second-order phase transition is observed in accordance with the analytical theory. The Binder cumulants method of the fourth order determines the values of the critical points of the Potts model on different lattices. Keywords: Phase transitions, Potts model, Monte Carlo, thermodynamic parameters, critical temperature.