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

arXiv:1406.6219 · Journal of Statistical Mechanics Theory and Experiment

Mixed order transition and condensation in an exactly soluble one dimensional spin model

Amir Bar, David Mukamel

Published Nov 6, 2014• 46 citations
arXivHTMLPDF

Abstract

Mixed order phase transitions (MOT), which display discontinuous order parameter and diverging correlation length, appear in several seemingly unrelated settings ranging from equilibrium models with long-range interactions to models far from thermal equilibrium. In a recent paper [ 1 ], an exactly soluble spin model with long-range interactions that exhibits MOT was introduced and analyzed both by a grand canonical calculation and a renormalization group analysis. The model was shown to form a bridge between two classes of 1D models exhibiting MOT, namely between spin models with inverse distance square interactions and surface depinning models. In this paper, we elaborate on the calculations performed in [ 1 ]. We also analyze the model in the canonical ensemble, which yields a better insight into the mechanism of MOT. In addition, we generalize the model to include Potts and general Ising spins and also consider a broader class of interactions that decay with distance using a power law different from 2.

DOI
10.1088/1742-5468/2014/11/p11001

Related papers

Mixed-Order Phase Transition in a One-Dimensional Model

Amir Bar, David Mukamel

We introduce and analyze an exactly soluble one-dimensional Ising model with long range interactions that exhibits a mixed-order transition, namely a phase transition in which the order parameter is discontinuous as in first order transitions while the correlation length diverges as in second order transitions. Such transitions are known to appear in a diverse classes of models that are seemingly unrelated. The model we present serves as a link between two classes of models that exhibit a mixed-order transition in one dimension, namely, spin models with a coupling constant that decays as the inverse distance squared and models of depinning transitions, thus making a step towards a unifying framework.

arXiv:1309.4228•Physical Review Letters•Jan 7, 2014•81 citations

Mixed-order phase transition in a minimal, diffusion-based spin model

Agata Fronczak, Piotr Fronczak

In this paper we exactly solve, within the grand canonical ensemble, a minimal spin model with the hybrid phase transition. We call the model diffusion based because its Hamiltonian can be recovered from a simple dynamic procedure, which can be seen as an equilibrium statistical mechanics representation of a biased random walk. We outline the derivation of the phase diagram of the model, in which the triple point has the hallmarks of the hybrid transition: discontinuity in the average magnetization and algebraically diverging susceptibilities. At this point, two second-order transition curves meet in equilibrium with the first-order curve, resulting in a prototypical mixed-order behavior.

arXiv:1604.04244•Physical review. E•Jul 6, 2016•5 citations

The one-dimensional Potts model with long-range interactions: a renormalization group approach

Sergio Alejandro Cannas, A C N de Magalhães

The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.

Journal of Physics A Mathematical and General•May 21, 1997•46 citations

Phase transitions of the random-bond Potts chain with long-range interactions

Jean-Christian Anglès d’Auriac, Ferenc Iglói

We study phase transitions of the ferromagnetic $q$-state Potts chain with random nearest-neighbor couplings having a variance ${\mathrm{\ensuremath{\Delta}}}^{2}$ and with homogeneous long-range interactions, which decay with distance as a power ${r}^{\ensuremath{-}(1+\ensuremath{\sigma})}, \ensuremath{\sigma}>0$. In the large-$q$ limit the free-energy of random samples of length $L\ensuremath{\le}2048$ is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for $\ensuremath{\sigma}<{\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})\ensuremath{\le}0.5$, while the correlation length becomes divergent at the transition point for ${\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})<\ensuremath{\sigma}<1$. In the latter regime the average magnetization is continuous for small enough $\mathrm{\ensuremath{\Delta}}$, but for larger $\mathrm{\ensuremath{\Delta}}$---according to the numerical results---it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.

arXiv:1607.06968•Physical review. E•Dec 19, 2016•8 citations

Phase transitions in systems with non-additive long-range interactions

Takashi Mori

We consider spin systems with long-range interactions in a non-additive regime. When the non-additive scaling limit is employed, the energy and the entropy compete and the system exhibits some phase transitions. Such systems do not satisfy the additivity, which results in some unfamiliar properties related to phase transitions. In this paper, the concept of additivity and its consequences are explained, and recent progress on the statistical mechanics of long-range interacting systems is reviewed. It is shown that parameter space is clearly decomposed into three regions according to the stability of the uniform state predicted by the mean-field (MF) theory. Based on this parameter space decomposition, recent results on the exactness of the MF theory are explained. When the interaction is non-negative (ferromagnetic), the analysis of the MF theory is exact and a typical spin configuration is always uniform in the canonical ensemble. However, in the restricted canonical ensemble, i.e., the canonical ensemble with a restriction of the value of the magnetization, it is shown that the MF theory does not necessarily give the exact description of the system and phase transitions between the MF uniform states (MF phase) and the inhomogeneous states (non-MF phase) occur. A new finding is that when the interaction potential changes its sign depending on the distance, the non-MF phase appears even in the canonical ensemble.

arXiv:1310.3458•Journal of Statistical Mechanics Theory and Experiment•Oct 11, 2013•43 citations

Critical behaviour of the 1D q-state Potts model with long-range interactions

Zvonko Glumac, Katarina Uzelac

The critical behaviour of the one-dimensional q-state Potts model with long-range interactions decaying with distance r as r −(1+σ) has been studied in the wide range of parameters 0 < σ ≤ 1 and 1 16 ≤ q ≤ 64. A transfer matrix has been constructed for a truncated range of interactions for integer and continuous q, and finite range scaling has been applied. Results for the phase diagram and the correlation length critical exponent are presented. Short title: Critical behaviour of the 1D LR q-state Potts model Recently, finite range scaling (FRS) appeared as an efficient method to deal with various discrete one-dimensional models with long range (LR) interactions (Glumac and Uzelac, 1989, 1991). The aim of the present study is to apply it to the onedimensional q-state Potts model with LR interactions, defined by the hamiltonian

•Jan 1, 1993•21 citations
Mixed order phase transitions (MOT), which display discontinuous order parameter and diverging correlation length, appear in several seemingly unrelated settings ranging from equilibrium models with long-range interactions to models far from thermal equilibrium. In a recent paper [ 1 ], an exactly soluble spin model with long-range interactions that exhibits MOT was introduced and analyzed both by a grand canonical calculation and a renormalization group analysis. The model was shown to form a bridge between two classes of 1D models exhibiting MOT, namely between spin models with inverse distance square interactions and surface depinning models. In this paper, we elaborate on the calculations performed in [ 1 ]. We also analyze the model in the canonical ensemble, which yields a better insight into the mechanism of MOT. In addition, we generalize the model to include Potts and general Ising spins and also consider a broader class of interactions that decay with distance using a power law different from 2.
We introduce and analyze an exactly soluble one-dimensional Ising model with long range interactions that exhibits a mixed-order transition, namely a phase transition in which the order parameter is discontinuous as in first order transitions while the correlation length diverges as in second order transitions. Such transitions are known to appear in a diverse classes of models that are seemingly unrelated. The model we present serves as a link between two classes of models that exhibit a mixed-order transition in one dimension, namely, spin models with a coupling constant that decays as the inverse distance squared and models of depinning transitions, thus making a step towards a unifying framework.
In this paper we exactly solve, within the grand canonical ensemble, a minimal spin model with the hybrid phase transition. We call the model diffusion based because its Hamiltonian can be recovered from a simple dynamic procedure, which can be seen as an equilibrium statistical mechanics representation of a biased random walk. We outline the derivation of the phase diagram of the model, in which the triple point has the hallmarks of the hybrid transition: discontinuity in the average magnetization and algebraically diverging susceptibilities. At this point, two second-order transition curves meet in equilibrium with the first-order curve, resulting in a prototypical mixed-order behavior.
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.
We study phase transitions of the ferromagnetic qqq-state Potts chain with random nearest-neighbor couplings having a variance ${\mathrm{\ensuremath{\Delta}}}^{2}$ and with homogeneous long-range interactions, which decay with distance as a power ${r}^{\ensuremath{-}(1+\ensuremath{\sigma})}, \ensuremath{\sigma}>0$. In the large-qqq limit the free-energy of random samples of length $L\ensuremath{\le}2048$ is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for $\ensuremath{\sigma}<{\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})\ensuremath{\le}0.5$, while the correlation length becomes divergent at the transition point for ${\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})<\ensuremath{\sigma}<1$. In the latter regime the average magnetization is continuous for small enough $\mathrm{\ensuremath{\Delta}}$, but for larger $\mathrm{\ensuremath{\Delta}}$---according to the numerical results---it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.
We consider spin systems with long-range interactions in a non-additive regime. When the non-additive scaling limit is employed, the energy and the entropy compete and the system exhibits some phase transitions. Such systems do not satisfy the additivity, which results in some unfamiliar properties related to phase transitions. In this paper, the concept of additivity and its consequences are explained, and recent progress on the statistical mechanics of long-range interacting systems is reviewed. It is shown that parameter space is clearly decomposed into three regions according to the stability of the uniform state predicted by the mean-field (MF) theory. Based on this parameter space decomposition, recent results on the exactness of the MF theory are explained. When the interaction is non-negative (ferromagnetic), the analysis of the MF theory is exact and a typical spin configuration is always uniform in the canonical ensemble. However, in the restricted canonical ensemble, i.e., the canonical ensemble with a restriction of the value of the magnetization, it is shown that the MF theory does not necessarily give the exact description of the system and phase transitions between the MF uniform states (MF phase) and the inhomogeneous states (non-MF phase) occur. A new finding is that when the interaction potential changes its sign depending on the distance, the non-MF phase appears even in the canonical ensemble.
The critical behaviour of the one-dimensional q-state Potts model with long-range interactions decaying with distance r as r −(1+σ) has been studied in the wide range of parameters 0 < σ ≤ 1 and 1 16 ≤ q ≤ 64. A transfer matrix has been constructed for a truncated range of interactions for integer and continuous q, and finite range scaling has been applied. Results for the phase diagram and the correlation length critical exponent are presented. Short title: Critical behaviour of the 1D LR q-state Potts model Recently, finite range scaling (FRS) appeared as an efficient method to deal with various discrete one-dimensional models with long range (LR) interactions (Glumac and Uzelac, 1989, 1991). The aim of the present study is to apply it to the onedimensional q-state Potts model with LR interactions, defined by the hamiltonian