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Frontiers in Complex Systems

Traces of tricritical dynamics beyond SSB in finite-size systems undergoing second-order phase transition: the case of the 3D Ising model

Yiannis Contoyiannis, Stelios M. Potirakis, Stavros G. Stavrinides, Michael P. Hanias, Pericles Papadopoulos, N. L. Matiadou

Published Aug 1, 2025• 0 citations
Publisher

Abstract

In finite-size thermal systems that exhibit second-order phase transition, the fluctuations of the order parameter ϕn obey type I intermittent dynamics at their pseudocritical temperature Tpc . Moreover, as recently demonstrated, spontaneous symmetry breaking (SSB) is gradually completed as temperature is reduced until reaching an SSB completion temperature, TSSB . Within this temperature zone, ϕn obey the dynamics of critical intermittency. This behavior has also been observed in pre-seismic fracture-induced electromagnetic emissions (FEME) of the MHz band—a real-world finite-size system undergoing a second-order phase transition. Interestingly, MHz FEME has recently been found to consistently present indications of tricritical dynamics after the SSB. We examine here whether this could also be true for a finite-size thermal system. We conduct a numerical experiment for the 3D Ising model at different temperatures by gradually reducing temperature beyond SSB and analyze order parameter fluctuations using the method of critical fluctuations (MCF) and a recently introduced wavelet-based method for detecting scaling behavior in noisy experimental data. Our results reveal that power-laws still exist within a very narrow zone of temperatures right after SSB completion for the 3D Ising model. These power-laws are shown to be compatible with another form of intermittency that determines the dynamics of the order parameter fluctuations close to the Griffiths tricritical point. As a possible interpretation of this finding, we suggest that our results imply that 3D Ising presents, just below TSSB , an imprint approaching the Griffiths tricritical point from the second-order phase transition line.

DOI
10.3389/fcpxs.2025.1612998

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In finite-size thermal systems that exhibit second-order phase transition, the fluctuations of the order parameter ϕn obey type I intermittent dynamics at their pseudocritical temperature Tpc . Moreover, as recently demonstrated, spontaneous symmetry breaking (SSB) is gradually completed as temperature is reduced until reaching an SSB completion temperature, TSSB . Within this temperature zone, ϕn obey the dynamics of critical intermittency. This behavior has also been observed in pre-seismic fracture-induced electromagnetic emissions (FEME) of the MHz band—a real-world finite-size system undergoing a second-order phase transition. Interestingly, MHz FEME has recently been found to consistently present indications of tricritical dynamics after the SSB. We examine here whether this could also be true for a finite-size thermal system. We conduct a numerical experiment for the 3D Ising model at different temperatures by gradually reducing temperature beyond SSB and analyze order parameter fluctuations using the method of critical fluctuations (MCF) and a recently introduced wavelet-based method for detecting scaling behavior in noisy experimental data. Our results reveal that power-laws still exist within a very narrow zone of temperatures right after SSB completion for the 3D Ising model. These power-laws are shown to be compatible with another form of intermittency that determines the dynamics of the order parameter fluctuations close to the Griffiths tricritical point. As a possible interpretation of this finding, we suggest that our results imply that 3D Ising presents, just below TSSB , an imprint approaching the Griffiths tricritical point from the second-order phase transition line.
We show that the spontaneous symmetry breaking can be defined also for finite systems based on the properly defined jump probability between the ground states in the 2d and 3d Ising models on a square and a cubic lattice respectively. Our analysis reveals the existence of an interval in the temperature (control parameter) space within which the spontaneous symmetry breaking takes place. The upper limit of this region is the pseudocritical point where the symmetric vacuum bifurcates in energetically degenerate non-symmetric vacua, initiating the spontaneous symmetry breaking process. The lower limit, identified as the temperature value at which the spontaneous symmetry breaking is completed, is characterized by maximal characteristic time for the decay of magnetization (order parameter) auto-correlations. We argue that this anomalous enhancement of auto-correlations is attributed to the transition from type I to on-off intermittency in the order parameter dynamics. Possible phenomenological implications of this behaviour are briefly discussed.
Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.
A numerical simulation using the chaotic Dynamics of intermittency at a finite size Z(3) spin system in a 3D lattice reveals: (a) the existence of a second order phase transition with a zone hysteresis characterized from resonances phenomena (b) An hybrid appearance of mean-field universality class and 3D Ising model universality class , all these inside the zone hysteresis (c) a weak first order phase transition through a tricritical crossover. So, a complicated behavior in Z(3) symmetry exists.
Abstract We present a dynamical numerical study on a three-dimensional (3D) Gaussian distribution random-field Ising model in a wide range of disorder strengths by means of the finite-time dynamics method. By validating the Rushbrooke scaling law and analyzing the variation of exponents with disorder amplitudes, the asymptotic critical regime controlled by a hysteresis fixed point is identified with exponents , , and . The phase transition is found to be continuous and there is not a tricritical point in this version of Gaussian distribution. Moreover, our results support that the hysteresis and zero-temperature fixed points belong to a different universality class.
We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat Cv based on the finite-size scaling of its maximal values Cvmax depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of Cv. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of Cvmax in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.
Abstract In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent ν ∼ 1.3 is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent α ∼ − 1.9 . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.