Resonance Maximizes Mean Waiting Time in Finite-Size Continuous Phase Transitions

Researchers have long sought to understand how continuous phase transitions occur in finite-size systems, particularly the mechanisms governing the shift from symmetry to broken symmetry. Yiannis F. Contoyiannis and Stelios M. Potirakis, both from the Department of Electrical and Electronics Engineering at the University of West Attica, alongside their colleagues, now demonstrate a surprising resonance phenomenon within the hysteresis zone preceding spontaneous symmetry breaking. Their work reveals this resonance as a peak in mean waiting time as temperature varies, potentially linked to the existence of exotic particles like tachyons or kink solitons. Significantly, this research proposes that this resonance marks a fundamental transition , a passage from a classical to a quantum world , for binary systems such as the three-dimensional Ising model, offering insights applicable to fermion-antifermion and even matter-antimatter systems.

Researchers found that these particles and quasiparticles appear to be intrinsically linked to the observed resonance, suggesting a deeper connection between the resonance phenomenon and the underlying physics of the phase transition. This work establishes a novel perspective on SSB, moving beyond the traditional view of a sharp transition and highlighting the importance of the hysteresis zone as a region of complex dynamics. The resonance, therefore, isn’t simply a byproduct of the transition, but a key feature delineating a shift in the system’s fundamental behaviour.

This research introduces the compelling idea that the identified resonance acts as a boundary separating a classical phase from a quantum thermal fluctuation phase. The study meticulously examined a finite cubic lattice of size 20x20x20, determining a pseudocritical temperature of 4.545 for the 3D Ising model with interaction strength Jij equal to 1. Furthermore, the research builds upon previous work demonstrating the emergence of a temperature interval, the hysteresis zone, in second-order phase transitions of finite-size systems. The current investigation delves into the dynamics within this zone, specifically focusing on how the critical character of the system is altered. This detailed analysis of the critical exponent governing thermal fluctuations provides a quantitative understanding of this degradation.

Metropolis Algorithm Simulating 3D Ising Model Systems

The research team employed the Metropolis algorithm to generate equilibrium configurations of three-dimensional Ising models, simulating a cubic lattice of size 203. This algorithm sampled configurations at fixed temperature according to the Boltzmann statistical weight, utilising the Hamiltonian H= −∑Jijsisj ⟨i,j⟩, where si represents spin variables and Jij denotes interaction strength. The study determined the pseudocritical temperature, Tpc, to be 4.545 (with Jij= 1) for this lattice size, defining a single lattice sweep as the algorithmic unit of time. Experiments involved recording the mean magnetization, M, as the order parameter, generating a time series of fluctuations over Niter = 200,000 sweeps.

To analyse system dynamics, researchers moved beyond simple histograms and pioneered a phase-space representation of the order parameter. This technique adapted the traditional statistical mechanics phase-space diagram (momentum p versus position x) by replacing position with the magnetization field, M, and momentum with its temporal derivative. Phase-space points were calculated as ( Mn+1−Mn Δτ ), where Δτ= 1 corresponded to one lattice sweep, effectively mapping the trajectory of the order parameter in a novel way. The work further developed a critical map to describe the dynamics of order parameter fluctuations at the critical state: φn+1 = φn+ u∙φnz+ εn.

Here, φn represents the nth sample of the scaled order parameter, u is a coupling parameter, z is a characteristic exponent related to the critical isothermal exponent δ by z= δ+ 1, and εn represents non-universal stochastic noise. This map describes type-I intermittency, characterised by laminar lengths interrupted by bursts of high fluctuation. Crucially, the team connected these laminar lengths to waiting times, the number of consecutive time series values within a defined laminar region, and demonstrated that the distribution of waiting times reveals the critical state of the system if it follows a scale-free, power-law function. This precise calculation of waiting times, facilitated by the phase-space representation and critical map, enabled the identification of the resonance and its connection to the classical-to-quantum transition.

Mean waiting time resonance precedes symmetry breaking

The team measured time series of 300,000 to 400,000 points to achieve a satisfactory level of accuracy, approximating physical reality more closely than infinite iterations. Researchers found that the distribution of magnetization values, denoted as M, is zero in the critical state, aligning with the expectation that the order parameter must be zero in a perfectly symmetric state. Data shows that the position of the end of the laminar region, denoted φL, was determined by shifting a blue line until it resulted in a laminar lengths distribution closely resembling a power law. The distribution of laminar lengths, as depicted in Fig0.5b, exhibited a curvature in its tail beyond the first 30 points, deviating from the linearity observed in the initial segment of the log-log plot.

This curvature necessitated the implementation of a new wavelet-based method for accurate exponent estimation, particularly for long scales (L≫1), as traditional fitting methods proved unreliable due to noise and poor statistics in the tails of the distributions. Tests prove that the wavelet analysis, utilising the Haar wavelet basis ψj,k(x) = 2j 2ψ(2jx−k), effectively addresses noise in the signal and allows for calculations focused on L≫1. The algorithm calculates coefficients dj,k, with the coarse-graining description ignoring signal noise, enabling analysis of any discrete distribution P(i). Measurements confirm the algorithm’s ability to determine whether a distribution embeds a power-law behavior, calculate the corresponding exponent p, and provide a fitting method robust against noise, especially at high laminar lengths.

Results demonstrate the definition of quantities λ and R, calculated using wavelet coefficients and the distribution P(L), to assess the convergence towards a power-law. The team quantified the proximity to a power-law by calculating Qλ, aiming for values close to zero, and then determined the exponent p by solving Eq. (12) numerically for a given mean value of R. The breakthrough delivers a method for identifying critical states, with p∈[1,2) indicating a system in criticality and Qλ order(10−3) confirming an exact power-law distribution.

Hysteresis Resonance Links to Tachyons and Solitons

This resonance occurs during the transition from a symmetric to a broken-symmetry state, a crucial process in spontaneous symmetry breaking. Through a detailed analysis of laminar lengths and the development of a quantitative approach using parameters Qλ and R, the researchers demonstrated that a time series at a specific temperature (T=4.52) is indeed in a critical state, evidenced by an exponent p falling within the range of 1 to 2. The authors acknowledge that their method relies on specific criteria for determining the convergence region and the selection of Δmax, potentially introducing some degree of arbitrariness. They also highlight that the power-law test function is only applicable when Qλ is sufficiently close to zero, limiting its use in cases where the distribution deviates significantly from a perfect power law.