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Physical review. B, Condensed matter

Renormalization-group study of the Hamiltonian version of the Potts model

Jenő Sólyom, Pierre M. Pfeuty

Published Jul 1, 1981• 97 citations
Publisher

Abstract

The phase transition of the classical two-dimensional Potts model, in particular the order of the transition as the number of components $q$ increases, is studied by constructing renormalization-group transformations on the equivalent one-dimensional quantum problem. It is shown that the block transformation with two sites per cell indicates the existence of a critical ${q}_{c}$ separating the small-$q$ and large-$q$ regions with different critical behaviors. The physically accessible fixed point for $q>{q}_{c}$ is a discontinuity fixed point where the specific-heat exponent $\ensuremath{\alpha}=1$, and therefore the transition is of first order.

DOI
10.1103/physrevb.24.218

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Physical Review Letters•Sep 10, 1979•401 citations
The phase transition of the classical two-dimensional Potts model, in particular the order of the transition as the number of components qqq increases, is studied by constructing renormalization-group transformations on the equivalent one-dimensional quantum problem. It is shown that the block transformation with two sites per cell indicates the existence of a critical qc{q}_{c}qc​ separating the small-qqq and large-qqq regions with different critical behaviors. The physically accessible fixed point for q>qcq>{q}_{c}q>qc​ is a discontinuity fixed point where the specific-heat exponent \ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=1, and therefore the transition is of first order.
The one-dimensional Hamiltonian version of the two-dimensional Potts model is studied by the application of a self-dual renormalization-group transformation. Blocks containing up to 11 sites are considered and mapped onto new, equivalent Potts spins by keeping the lowest-lying levels only. The higher-lying states are also taken into account in a perturbational way for smaller cells. Unfortunately, this renormalization-group treatment cannot reproduce the first-order transition for large values of the number of states, and the critical exponents for the three- and four-state Potts model do not seem to converge to the conjectured results as larger and larger blocks are taken.
The renormalization-group transformation, applied earlier to the Hamiltonian Potts model to study the crossover from second-order to first-order transition as the number of states qqq increases, is extended by taking larger cells in the block transformation. The results improve systematically with increasing block size. The critical value of qqq above which the transition is of first order, seems to converge to 5 instead of the exactly known value qc=4{q}_{c}=4qc​=4. The classical equivalents of the new couplings, which drive the system to the first-order transition, are discussed.
We have used the Monte Carlo renormalization group method to study the order-disorder phase transition of the three and four dimensional, three-state Potts model. In both cases, the transition appears to be first order. However, no evidence for a discontinuity fixed point associated with these transitions was found. Instead, the existence of a nearby second order fixed point just beyond the first order transition could be inferred from the RG flow in the ordered as well as in the disordered state. These results imply that the thermodynamic quantities are analytically continuable into the metastable regions, and that apart from discontinuities caused by the intersection of the two free-energy branches, no additional singularities occur at the transition.
The surface and bulk properties of the two-dimensional Q>4Q>4Q>4 state Potts model in the vicinity of the first-order bulk transition point have been studied by exact calculations and by density-matrix renormalization-group techniques. For the surface transition the complete analytical solution of the problem is presented in the $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}$ limit, including the critical and tricritical exponents, magnetization profiles, and scaling functions. We have shown in an accurate numerical study that the universality class of the surface transition is independent of the value of Q>4.Q>4.Q>4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and disordered phases are identical at the transition point.
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.