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Approximate renormalization group transformation in the theory of phase transitions

I. O. Vakarchuk, Yu. K. Rudavskii

Theoretical and Mathematical PhysicsApr 1, 19824 citations

Phase Transitions in Low-Dimensional Systems and Renormalization Group Theory

A. P. Young

Springer eBooksJan 1, 19800 citations

Quantum field theory and phase transitions: universality and renormalization group

Jean Zinn‐Justin

Jan 1, 20030 citations

Renormalization group for renormalization-group equations toward the universality classification of infinite-order phase transitions

Chigak Itoi, Hisamitsu Mukaida

We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.

arXiv:cond-mat/9711308Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsOct 1, 199915 citations

Phase transitions and the renormalization group

Ivo Sachs, Siddhartha Sen, James Sexton

Cambridge University Press eBooksMay 11, 200694 citations

Phase Transitions and Critical Phenomena: Renormalisation Group Method

Deepak Kumar

This chapter describes the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, the general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Phase transitions occur in macroscopic systems when one or more external parameters like temperature, pressure, electrical field, magnetic field, etc are varied. The chapter demonstrates the utility of the alternative formulations of thermodynamics and statistical mechanics in the context of phase transitions and shows how their use clarifies and unifies the theory of phase transitions. It deals with the basic thermodynamic description of a single component fluid. To understand different phases and associated transitions, one studies how the couplings flow in the parameter space under repeated Renormalization Group (RG) transformations. The flows around the fixed point are easily studied by linearizing the RG equations around the fixed points and determining the eigenvalues and eigenvectors of the linearized RG transformations.This chapter describes the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, the general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Phase transitions occur in macroscopic systems when one or more external parameters like temperature, pressure, electrical field, magnetic field, etc are varied. The chapter demonstrates the utility of the alternative formulations of thermodynamics and statistical mechanics in the context of phase transitions and shows how their use clarifies and unifies the theory of phase transitions. It deals with the basic thermodynamic description of a single component fluid. To understand different phases and associated transitions, one studies how the couplings flow in the parameter space under repeated Renormalization Group (RG) transformations. The flows around the fixed point are easily studied by linearizing the RG equations around the fixed points and determining the eigenvalues and eigenvectors of the linearized RG transformations.

CRC Press eBooksApr 24, 20190 citations

The Spectrum of Eigenvalues in the Renormalization Group Theory of Phase Transitions

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The Spectrum of Eigenvalues in the Renormalization Group Theory of Phase Transitions

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