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Physical review. B, Condensed matter

Z(4) model: Criticality and break-collapse method

Ananias M. Mariz, Constantino Tsallis, Paulo Fulco

Published Nov 1, 1985• 21 citations
Publisher

Abstract

Within a real-space renormalization-group (RG) framework, we study the criticality of the Z(4) ferromagnet on the square lattice. The phase diagram (exhibiting ferromagnetic, paramagnetic, and nematiclike phase) recovers all the available exact results, and possibly is of high precision everywhere. In particular, we establish the main asymptotic behaviors (bifurcation and Ising regions). In addition, we develop an operational procedure (break-collapse method) which considerably simplifies the exact calculation of arbitrary Z(4) two-terminal clusters (commonly appearing in RG approaches).

DOI
10.1103/physrevb.32.6055

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Physical review. B, Condensed matter•Feb 1, 1981•45 citations
Within a real-space renormalization-group (RG) framework, we study the criticality of the Z(4) ferromagnet on the square lattice. The phase diagram (exhibiting ferromagnetic, paramagnetic, and nematiclike phase) recovers all the available exact results, and possibly is of high precision everywhere. In particular, we establish the main asymptotic behaviors (bifurcation and Ising regions). In addition, we develop an operational procedure (break-collapse method) which considerably simplifies the exact calculation of arbitrary Z(4) two-terminal clusters (commonly appearing in RG approaches).
Within a real space renormalisation group (RG) scheme, the authors study the criticality of the ferromagnetic Z(4) model on an anisotropic square lattice. They use an RG cluster which has already proved to be very efficient for the Potts model on the same lattice. The establishment of the RG recurrence relations is greatly simplified through the break-collapse method. The phase diagram (exhibiting ferromagnetic, paramagnetic and nematic-like phases) recovers all the available exact results, and is believed to be of high precision everywhere. If the model is alternatively thought of as being associated with a particular hierarchical lattice rather than with the square lattice, then it is exact everywhere.
Within a real-space renormalization group (RG) which preserves two-site correlation functions, we study, on the square lattice, the criticality of the bond-diluted Z(q) ferromagnetic model. We generalize the `break-collapse method' which simplifies greatly the exact calculation of arbitrary Z(q) two-terminal clusters (commonly appearing in RG approaches) mainly for a large value of q. We reproduce, in the pure case, several known exact results. The structure of the phase diagrams, for all the values of q, is obtained with a good precision. The massless spin - wave-like phase, which evolves into the Kosterlitz - Touless phase for , occurs around (in agreement with the well known exact result). The structure of the phase diagrams in the diluted case is qualitatively similar to that obtained from the pure model. The massless spin - wave-like phase resists in an interval of concentration p which increases for the large values of q.
A real space renormalization group method is used to investigate the criticality (phase diagrams, critical expoentes and universality classes) of Z(4) model in two and three dimensions. The values of the interaction parameters are chosen in such a way as to cover the complete phase diagrams of the model, which presents the following phases: (i) Paramagnetic (P); (ii) Ferromagnetic (F); (iii) Antiferromagnetic (AF); (iv) Intermediate Ferromagnetic (IF) and Intermediate Antiferromagnetic (IAF). In the hierarquical lattices, generated by renormalization the phase diagrams are exact. It is also possible to obtain approximated results for square and simple cubic lattices. In the bidimensional case a self-dual lattice is used and the resulting phase diagram reproduces all the exact results known for the square lattice. The Migdal-Kadanoff transformation is applied to the three dimensional case and the additional phases previously suggested by Ditzian et al, are not found
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
In this work we apply two different real-space renormalization-group (RSRG) approaches to the anisotropic antiferromagnetic spin-1/2 Heisenberg model on the square lattice. Our calculations allow for an approximate evaluation of the T vs \ensuremath{\Delta} phase diagram: the results suggest the existence of a critical value of $\ensuremath{\Delta}>0,$ at which the critical temperature goes to zero, and the presence of reentrant behavior on the critical line between the ordered and disordered phases. This whole critical line is found to belong to the same universality class as the Ising model. Our results are in accordance with previous RSRG approaches but not with numerical simulations and spin-wave calculations.
The phase diagrams of general qqq-state planar spin models on a square lattice are studied for $q=4\ensuremath{-}7$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of Z(q)Z(q)Z(q) models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for q=6and7q=6 \mathrm{and} 7q=6and7 but are inconclusive for q=5q=5q=5. More accurate methods are needed to determine the phase structure of the Z(5)Z(5)Z(5) model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).