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

Renormalization group and phase transitions

Belal Ehsan Baaquie

Cambridge University Press eBooksAug 22, 20180 citations

Blume-Emery-Griffiths-Potts model in two dimensions: Phase diagram and critical properties from a position-space renormalization group

A. Nihat Berker, Michael Wortis

The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear ( $J$ ) and biquadratic ( $K$ ) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in $J$ , $K$ , $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field ( $H$ ) and another odd interaction ( $L$ ) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear ($J$) and biquadratic ($K$) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in $J$, $K$, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field ($H$) and another odd interaction ($L$) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.

Physical review. B, Solid stateDec 1, 1976371 citations

Phase transitions and the renormalization group

Ivo Sachs, Siddhartha Sen, James Sexton

Cambridge University Press eBooksMay 11, 200694 citations

Studies of phase transitions with the renormalization-group method

武辰巳

Jan 1, 19780 citations

First-order phase transitions, the Maxwell construction, and the momentum-space renormalization group

Alberto Parola, Davide Pini, L. Reatto

The behavior of a nonperturbative momentum-space renormalization group (RG) is analyzed both above and below the critical temperature. The case of a scalar order parameter and of the Ising model is studied in detail by analytical and numerical means. It is shown that this RG transformation is always well defined even inside the coexistence curve. van der Waals loops are suppressed by long-wavelength fluctuations which enforce the convexity of the free energy. The RG description emerging from this study is then compared with exact results and other approximate theories.The behavior of a nonperturbative momentum-space renormalization group (RG) is analyzed both above and below the critical temperature. The case of a scalar order parameter and of the Ising model is studied in detail by analytical and numerical means. It is shown that this RG transformation is always well defined even inside the coexistence curve. van der Waals loops are suppressed by long-wavelength fluctuations which enforce the convexity of the free energy. The RG description emerging from this study is then compared with exact results and other approximate theories.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsNov 1, 199338 citations

The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (

J

) and biquadratic (

K

) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in

J

K

, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (

H

) and another odd interaction (

L

) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.