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:1604.04244 · Physical review. E

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

Agata Fronczak, Piotr Fronczak

Published Jul 6, 2016• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.94.012103

Related papers

Minimal exactly solved model with the extreme Thouless effect

Agata Fronczak, Piotr Fronczak, A. Krawiecki

We present and analyze a minimal exactly solved model that exhibits a mixed-order phase transition known in the literature as the Thouless effect. Such hybrid transitions do not fit into the modest classification of thermodynamic transitions and, as such, they used to be overlooked or incorrectly identified in the past. The recent series of observations of such transitions in many diverse systems suggest that a new taxonomy of phase transitions is needed. The spin model we present due to its simplicity and possible experimental designs could bring us to this goal. We find the Hamiltonian of the model from which partition function is easily calculated. Thermodynamic properties of the model, i.e., discontinuous magnetization and diverging susceptibility, are discussed. Finally, its generalizations and further research directions are proposed.

arXiv:1511.03859•Physical review. E•Jan 14, 2016•8 citations

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

Amir Bar, David Mukamel

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.

arXiv:1406.6219•Journal of Statistical Mechanics Theory and Experiment•Nov 6, 2014•46 citations

Dynamic critical behavior of the mixed-spin (1, 3/2, 2) trilayer Ising system

Muhammed Derviş Baran, Ümüt Temizer

The dynamic magnetic behavior of the mixed-spin Ising trilayer structure, composed of a ferrimagnetically A-layer with spin-1, B-layer with spin-3/2 and a C-layer with spin-2, is investigated by using the mean-field approximation with Glauber-type stochastic dynamics. The dynamic equations describing the time-dependencies of the system are derived from the Master equation and the Glauber transition rates. The time variations of the average magnetizations and the behaviors of the dynamic magnetizations as a function of the temperature and magnetic field are investigated to find the phases and phase transition points, respectively. The dynamic phase diagrams are presented on different planes, and they exhibit several ordered phases and coexistence phase regions besides the disordered phase. Our model highlights first- and second-order phase transitions as well as the emergence of tricritical point, triple point, quadruple point, critical end point, double critical end point and multicritical point.

Phase Transitions•Apr 3, 2026•0 citations

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

Phase transitions in a three-dimensional kinetic spin-1/2 Ising model with random field: Effective-field-theory study

Emanuel Costabile, J. Ricardo de Sousa

The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.

Physical Review E•Jan 13, 2012•15 citations

Multiple point interactions in lattice gas models far from equilibrium

Matthias Paeßens

In this work non-equilibrium phenomena in statistical models with more than nearest neighbor interactions are examined. The investigated models belong to three different types of non-equilibrium, i.e. systems undergoing aging, absorbing phase transitions and systems being intrinsically out of equilibrium. The behavior of the mean spherical model after a temperature quench from the disordered phase into the ordered phase in an external magnetic field is calculated. For a sufficiently small external field, the kinetics of the magnetization reversal transition from the metastable to the ground state is comparable to the aging behavior in the case without external field. For an oscillating external magnetic field the absence of a dynamic phase transition is demonstrated. The bosonic pair contact process with diffusion shows unusual behavior of the second moment of the probability distribution for which a phase transition from a constant asymptotic value to exponential divergence is found. The control parameter of this transition is the ratio of reaction and diffusion rates which gives insight into the role of diffusion in this process. The divergence of the second moment can lead to a negative correlation time while the correlation length is positive. Coupling conservative diffusion system to reaction dynamics leads to interesting behavior if the rates are tuned adequately. The general conditions for ergodicity breaking in those systems are derived and an example is constructed. For this example hysteresis is demonstrated. The behavior, including the exact steady state phase diagram, of this non-equilibrium many particle system is well described by an effective equilibrium one-particle problem. The motion of domain walls in reaction diffusion systems is investigated for three site interactions. A variety of models is found and they show phenomena like double shocks or additional symmetries not possible for two-site interaction models. While the two-site interaction models leading to exact shock measures are connected by the transformability to free fermion systems, this is not valid for the three site interaction models.

•Jan 1, 2004•0 citations
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.
We present and analyze a minimal exactly solved model that exhibits a mixed-order phase transition known in the literature as the Thouless effect. Such hybrid transitions do not fit into the modest classification of thermodynamic transitions and, as such, they used to be overlooked or incorrectly identified in the past. The recent series of observations of such transitions in many diverse systems suggest that a new taxonomy of phase transitions is needed. The spin model we present due to its simplicity and possible experimental designs could bring us to this goal. We find the Hamiltonian of the model from which partition function is easily calculated. Thermodynamic properties of the model, i.e., discontinuous magnetization and diverging susceptibility, are discussed. Finally, its generalizations and further research directions are proposed.
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.
The dynamic magnetic behavior of the mixed-spin Ising trilayer structure, composed of a ferrimagnetically A-layer with spin-1, B-layer with spin-3/2 and a C-layer with spin-2, is investigated by using the mean-field approximation with Glauber-type stochastic dynamics. The dynamic equations describing the time-dependencies of the system are derived from the Master equation and the Glauber transition rates. The time variations of the average magnetizations and the behaviors of the dynamic magnetizations as a function of the temperature and magnetic field are investigated to find the phases and phase transition points, respectively. The dynamic phase diagrams are presented on different planes, and they exhibit several ordered phases and coexistence phase regions besides the disordered phase. Our model highlights first- and second-order phase transitions as well as the emergence of tricritical point, triple point, quadruple point, critical end point, double critical end point and multicritical point.
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.
The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.
In this work non-equilibrium phenomena in statistical models with more than nearest neighbor interactions are examined. The investigated models belong to three different types of non-equilibrium, i.e. systems undergoing aging, absorbing phase transitions and systems being intrinsically out of equilibrium. The behavior of the mean spherical model after a temperature quench from the disordered phase into the ordered phase in an external magnetic field is calculated. For a sufficiently small external field, the kinetics of the magnetization reversal transition from the metastable to the ground state is comparable to the aging behavior in the case without external field. For an oscillating external magnetic field the absence of a dynamic phase transition is demonstrated. The bosonic pair contact process with diffusion shows unusual behavior of the second moment of the probability distribution for which a phase transition from a constant asymptotic value to exponential divergence is found. The control parameter of this transition is the ratio of reaction and diffusion rates which gives insight into the role of diffusion in this process. The divergence of the second moment can lead to a negative correlation time while the correlation length is positive. Coupling conservative diffusion system to reaction dynamics leads to interesting behavior if the rates are tuned adequately. The general conditions for ergodicity breaking in those systems are derived and an example is constructed. For this example hysteresis is demonstrated. The behavior, including the exact steady state phase diagram, of this non-equilibrium many particle system is well described by an effective equilibrium one-particle problem. The motion of domain walls in reaction diffusion systems is investigated for three site interactions. A variety of models is found and they show phenomena like double shocks or additional symmetries not possible for two-site interaction models. While the two-site interaction models leading to exact shock measures are connected by the transformability to free fermion systems, this is not valid for the three site interaction models.