Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Simple renormalisation group method for calculating thermal equations of states

A. O. Caride, Constantino Tsallis

Published Jul 11, 1987• 11 citations
Publisher

Abstract

Renormalisation group (RG) techniques are currently used to derive relevant quantities in the vicinity of the critical point. The authors present a real space RG procedure which directly yields the order parameter for all values of the external parameters (e.g. temperature). It is as simple as a mean-field calculation, although it provides non-trivial results, which can be systematically improved. The method is successfully illustrated on the square lattice Potts ferromagnet. The whole approach suggests that the order parameter on a hierarchical lattice is, on every site, proportional to its coordination number.

DOI
10.1088/0305-4470/20/10/008

Related papers

Equations of state of the Potts ferromagnet in an anisotropic square lattice: Renormalization-group approach

Anna Chame, Constantino Tsallis, Uriel M. S. Costa

We extend to the anisotropic case (not necessarily equal coupling constants along the various crystalline directions) the real-space renormalization-group procedure recently introduced by Caride and Tsallis, which directly yields the order parameter for arbitrary temperatures. The method does not demand the calculation of the thermodynamic energy; it is as simple as a mean-field calculation although it provides nontrivial results, which can be systematically improved. We apply the procedure to the q-state Potts ferromagnet in an anisotropic square lattice, whose exact equation of state is still unknown. The results compare satisfactorily with available exact results.

Physical review. B, Condensed matter•May 1, 1988•5 citations

Real Space Renormalisation Group Approach

M. A. Contínentino

Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).

Cambridge University Press eBooks•Jan 1, 2017•0 citations

Phase Transitions and Critical Phenomena: Renormalisation Group Method

Deepak Kumar

These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.

Texts and readings in physical sciences•Jan 1, 2001•0 citations

Three-state Potts model with antiferromagnetic interactions: a MFRG approach

M C Marques

The mean-field renormalisation group method is used to study the critical temperature of the three-state Potts antiferromagnet in 3D lattices, as well as the three-state Potts model on the square lattice with ferromagnetic (along x) and antiferromagnetic (along y) interactions, and its quantum version, the 1D Potts antiferromagnet in a transverse field. Comparison is made with other results.

Journal of Physics A Mathematical and General•Feb 21, 1988•15 citations

Phase diagram of the Z(4) ferromagnet in an anisotropic square lattice

Constantino Tsallis, Jean Souletie

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.

Journal of Physics A Mathematical and General•Jun 21, 1986•5 citations

Determination of critical temperature for three-dimensional Ising systems

W. Reich

A real space renormalisation group method has been applied to find the critical temperature of the Ising model on cubic and octahedral lattices. The result obtained for a cubic lattice is in good agreement with that obtained from high-temperature series expansion and is better than the results of other papers. A modification of the transformation tested for a simple cubic lattice allowed the authors to find the critical temperature for an octahedral lattice. The latter has not so far been studied by any version of the real space renormalisation group method.

Journal of Physics A Mathematical and General•Dec 21, 1984•10 citations
Renormalisation group (RG) techniques are currently used to derive relevant quantities in the vicinity of the critical point. The authors present a real space RG procedure which directly yields the order parameter for all values of the external parameters (e.g. temperature). It is as simple as a mean-field calculation, although it provides non-trivial results, which can be systematically improved. The method is successfully illustrated on the square lattice Potts ferromagnet. The whole approach suggests that the order parameter on a hierarchical lattice is, on every site, proportional to its coordination number.
We extend to the anisotropic case (not necessarily equal coupling constants along the various crystalline directions) the real-space renormalization-group procedure recently introduced by Caride and Tsallis, which directly yields the order parameter for arbitrary temperatures. The method does not demand the calculation of the thermodynamic energy; it is as simple as a mean-field calculation although it provides nontrivial results, which can be systematically improved. We apply the procedure to the q-state Potts ferromagnet in an anisotropic square lattice, whose exact equation of state is still unknown. The results compare satisfactorily with available exact results.
Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
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.
A real space renormalisation group method has been applied to find the critical temperature of the Ising model on cubic and octahedral lattices. The result obtained for a cubic lattice is in good agreement with that obtained from high-temperature series expansion and is better than the results of other papers. A modification of the transformation tested for a simple cubic lattice allowed the authors to find the critical temperature for an octahedral lattice. The latter has not so far been studied by any version of the real space renormalisation group method.