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

Physical Review Letters

Essential Singularities at First-Order Phase Transitions

W. Klein, D. J. Wallace, Royce K. P. Zia

Published Sep 13, 1976• 57 citations
Publisher

Abstract

Renormalization group ideas are applied to the problem of first-order phase transitions in Ising-like models. They support the existence of essential singularities as the coexistence region is approached, at all temperatures below the critical temperature.
DOI
10.1103/physrevlett.37.639

Related papers

Renormalization-group calculation of first- and second-order phase transitions in the Potts model

Erling Pytte

A renormalization-group calculation has been carried out in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions to lowest order in $\ensuremath{\epsilon}$, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for $p>2$. For $2<p<\frac{10}{3}$ the transition is first order in spite of the fact that there is an accessible stable fixed point for all $p<\frac{10}{3}$. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of ($p\ensuremath{-}2$). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature ${T}_{c}$, due to instantons, which is of the same form as that of the Ising model at $H=0$.

Physical review. B, Condensed matter•Nov 1, 1980•22 citations

Renormalization-Group Approach to the Solution of General Ising Models

Michael Nauenberg, Bernard Nienhuis

It is shown how the renormalization-group ideas of Wilson can be applied to obtain the complete thermodynamic functions for general Ising models. The most singular term near the critical transition is given explicitly in terms of the renormalization-group transformations. As an example an approximate numerical solution is given to the original Ising model.

Physical Review Letters•Dec 30, 1974•142 citations

First-order phase transitions and linear renormalization group

Krishnappa Subbarao

The description of first-order phase transitions using the linear renormalization group is illustrated with the two-dimensional Ising model. Different linear renormalization groups should be used to describe the system in the neighborhood of criticality and away from it. In the latter case, the renormalization-group trajectories are lines of constant spin expectation value, which map a system below critical temperature in a small magnetic field to a system of free spins at a finite magnetic field. A finite-lattice approximation is also presented. The equation of state and several thermodynamic quantities are obtained; a smooth matching of the behavior near and away from criticality occurs naturally.

Physical review. B, Solid state•May 1, 1976•12 citations

Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line

Jaron Kent-Dobias, James P. Sethna

Renormalization group ideas and results from critical droplet theory are used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at the abrupt transition. Analogous forms for the magnetization and susceptibility in two dimensions are fit to numeric data and show good agreement, especially when their nonsingular behavior is modified to match existing numeric results.

arXiv:1707.03791•Jul 12, 2017•0 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.

arXiv:cond-mat/9711308•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1999•15 citations

First-Order Phase Transitions and the Three-State Potts Model

Henk W. J. Blöte, Robert H. Swendsen

We have used the Monte Carlo renormalization-group method to study the three-state Potts model in three and four dimensions. In both cases, we find a first-order transition without an associated discontinuity fixed point. The transition in three dimensions is "almost second order" in the sense that some evidence was found for the existence of second-order fixed points associated with singularities in the metastable region just beyond the first-order transition.

Physical Review Letters•Sep 10, 1979•173 citations
A renormalization-group calculation has been carried out in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions to lowest order in $\ensuremath{\epsilon}$, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for p>2p>2p>2. For 2<p<1032<p<\frac{10}{3}2<p<310​ the transition is first order in spite of the fact that there is an accessible stable fixed point for all p<103p<\frac{10}{3}p<310​. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of ($p\ensuremath{-}2$). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature Tc{T}_{c}Tc​, due to instantons, which is of the same form as that of the Ising model at H=0H=0H=0.
The description of first-order phase transitions using the linear renormalization group is illustrated with the two-dimensional Ising model. Different linear renormalization groups should be used to describe the system in the neighborhood of criticality and away from it. In the latter case, the renormalization-group trajectories are lines of constant spin expectation value, which map a system below critical temperature in a small magnetic field to a system of free spins at a finite magnetic field. A finite-lattice approximation is also presented. The equation of state and several thermodynamic quantities are obtained; a smooth matching of the behavior near and away from criticality occurs naturally.
Renormalization group ideas and results from critical droplet theory are used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at the abrupt transition. Analogous forms for the magnetization and susceptibility in two dimensions are fit to numeric data and show good agreement, especially when their nonsingular behavior is modified to match existing numeric results.
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.