Critical Phenomena in Two-Dimensional Systems.

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of ${V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}$ , where ${q}_{X}$ is the coordination number, ${{K}_{C}}^{X}$ is the dimensionless inverse critical temperature, and ${n}_{X}$ , ${g}_{X}$ , and ${l}_{X}$ are the universality scale factors. ${V}_{X}$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{{V}_{X}}{{l}_{X}}$ is an exact invariant.An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of ${V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}$, where ${q}_{X}$ is the coordination number, ${{K}_{C}}^{X}$ is the dimensionless inverse critical temperature, and ${n}_{X}$, ${g}_{X}$, and ${l}_{X}$ are the universality scale factors. ${V}_{X}$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{{V}_{X}}{{l}_{X}}$ is an exact invariant.

Physical Review LettersMar 31, 19753 citations

CRITICAL PHENOMENA NEAR TWO DIMENSIONS

WORLD SCIENTIFIC eBooksJun 1, 20050 citations

Universal role of correlation entropy in critical phenomena

Shi-Jian Gu, Chang-Pu Sun, Hai-Qing Lin

In statistical physics, if we divide successively an equilibrium system into two parts, we will face a situation that, to a certain length ξ, the physics of a subsystem is no longer the same as the original one. The extensive property of the thermal entropy S (A ∪ B) = S (A) + S (B) is then violated. This observation motivates us to introduce a concept of correlation entropy between two points, as measured by the mutual information in information theory, to study the critical phenomena. A rigorous relation is established to display some drastic features of the non-vanishing correlation entropy of a subsystem formed by any two distant particles with long-range correlation. This relation actually indicates a universal role played by the correlation entropy for understanding the critical phenomena. We also verify these analytical studies in terms of two well-studied models for both the thermal and quantum phase transitions: the two-dimensional Ising model and the one-dimensional transverse-field Ising model. Therefore, the correlation entropy provides us with a new physical intuition of the critical phenomena from the point of view of information theory.In statistical physics, if we divide successively an equilibrium system into two parts, we will face a situation that, to a certain length ξ, the physics of a subsystem is no longer the same as the original one. The extensive property of the thermal entropy S (A ∪ B) = S (A) + S (B) is then violated. This observation motivates us to introduce a concept of correlation entropy between two points, as measured by the mutual information in information theory, to study the critical phenomena. A rigorous relation is established to display some drastic features of the non-vanishing correlation entropy of a subsystem formed by any two distant particles with long-range correlation. This relation actually indicates a universal role played by the correlation entropy for understanding the critical phenomena. We also verify these analytical studies in terms of two well-studied models for both the thermal and quantum phase transitions: the two-dimensional Ising model and the one-dimensional transverse-field Ising model. Therefore, the correlation entropy provides us with a new physical intuition of the critical phenomena from the point of view of information theory.

arXiv:quant-ph/0605164Journal of Physics A Mathematical and TheoreticalDec 19, 200760 citations

Conformal Field Theory and Critical Phenomena in Two Dimensional Systems

Al.B. Zamolodchikov, Al.B. Zamolodchikov

Jan 31, 198949 citations

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of

{V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}

, where

{q}_{X}

is the coordination number,

{{K}_{C}}^{X}

is the dimensionless inverse critical temperature, and

{n}_{X}

{g}_{X}

, and

{l}_{X}

are the universality scale factors.

{V}_{X}

is an exact invariant for the two-dimensional

S=\frac{1}{2}

Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however,

\frac{{V}_{X}}{{l}_{X}}

is an exact invariant.