Many physical systems, including some examples of active matter, granular assemblies, and biological systems, show fluctuation-dominated phase ordering (FDPO), where macroscopic fluctuations coexist with long-range order. Most of these systems are out of equilibrium. By contrast, a recent work has analytically demonstrated that an equilibrium one-dimensional truncated inverse distance square Ising (TIDSI) model shows FDPO. The analytical results rely on a cluster representation of the model that we term TIDSI-CL and are governed by the ratio, c, of the long-range interaction strength to the critical temperature. We show that the allowed range of c is very narrow in the original TIDSI model while it is unbounded in TIDSI-CL. We perform Monte Carlo simulations for the TIDSI model and show consistency with the analytical results in the allowed range of c. The correlation length grows strongly on approaching the critical point, leading to a broad near-critical region. Within this region, α, which is the cusp exponent of the power-law decay of the scaled correlation function at criticality, changes to α^{eff}. We also investigate the coarsening dynamics of the model: The correlation function, domain size distribution, and aging behavior are consistent with the equilibrium properties upon replacing the system size, L, by the coarsening length, L(t). The mean largest cluster size shows logarithmic corrections due to finite L and waiting time, t_{w}. The aging autocorrelation function exhibits two different scaling forms, characterized by exponents β and γ, at short and long times compared to t_{w}, where β=α/2.
Many physical systems, including some examples of active matter, granular assemblies, and biological systems, show fluctuation-dominated phase ordering (FDPO), where macroscopic fluctuations coexist with long-range order. Most of these systems are out of equilibrium. By contrast, a recent work has analytically demonstrated that an equilibrium one-dimensional Truncated Inverse Distance Square Ising (TIDSI) model shows FDPO. The analytical results rely on a cluster representation of the model that we term TIDSI-CL and are governed by the ratio,
c, of the long-range interaction strength to the critical temperature. We show that the allowed range of
c is very narrow in the TIDSI model while it is unbounded in TIDSI-CL. We perform Monte-Carlo simulations for the TIDSI model and show consistency with the analytical results in the allowed range of
c. The correlation length grows strongly on approaching the critical point, leading to a broad near-critical region. Within this region,
α, which is the cusp exponent of the power-law decay of the scaled correlation function at criticality, changes to
αeff. We also investigate the coarsening dynamics of the model: the correlation function, domain size distribution, and aging behavior are consistent with the equilibrium properties upon replacing the system size,
L, with the coarsening length,
L(t). The mean largest cluster size shows logarithmic corrections due to finite
L and waiting time,
tw. The aging autocorrelation function exhibits two different scaling forms, characterized by exponents
β and
γ, at short and long times compared to
tw, where
β=α/2.
Abstract Mixed order transitions are those which show a discontinuity of the order parameter as well as a divergent correlation length at the transition (critical) point. We show that the behavior of the order parameter correlation function along the transition line of mixed order transitions can change from normal critical behavior with power law decay, to fluctuation-dominated phase ordering as a parameter is varied. The defining features of fluctuation-dominated order are anomalous fluctuations which remain large in the thermodynamic limit, and correlation functions which approach a finite value through a cusp singularity as the separation scaled by the system size approaches zero. We demonstrate that fluctuation-dominated order sets in along a portion of the transition line of an Ising model with truncated long-range interactions which was earlier shown to exhibit mixed order transitions. We also argue that this connection may hold more generally.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behaviour of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1/rd + σ, with σ = 0.75. The critical temperature, as well as the critical exponents, are evaluated from the power-law behaviour of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different truncated interaction ranges. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν) are in good agreement with renormalization group (RG) results and previously reported numerical data obtained under equilibrium conditions. On the other hand, the dynamic exponent of the initial increase of the magnetization (θ) was close to RG predictions. However, the dynamic exponent z of the time correlation length is slightly different to the RG results probably due to the fact that it may depend on the specific dynamics used or because the two-loop expansion used in the RG analysis may be insufficient.
The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T f =0 and T f =T c . The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t 1 2/, for T f =0, and the 'nonequilibrium correlation length', xi (t) approximately t 1 z/, for T f =T c . The equal-time correlation function has the expected scaling forms f(r/L(t)) and r -(d-2+ eta ) f c (r/ xi (t)) for T f =0 and T c respectively. The scaling function f c (x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S i (t))S i (O)) approximately L(t) - . Results for a quench from the equilibrium critical state to T f =0 are consistent with theoretical predictions.
A phenomenological theory of the ordered phase of short-range Ising spin glasses is developed in terms of droplet excitations and presented in detail. These excitations have free energies with a broad distribution whose characteristic magnitude grows with length scale L as ${L}^{\ensuremath{\theta}}$. A small fraction of droplets of all scales are thermally active; these dominate much of the physics. The mean-square correlation functions are found to decay with distance as 1/${r}^{\ensuremath{\theta}}$ for all T
Tc and the autocorrelations decay logarithmically with time because of large activation barriers for creation and annihilation of the droplet excitations. A renormalization procedure is sketched in order to define excitations at positive temperature. It is found that the long-distance equilibrium correlation functions are extremely sensitive to small temperature changes, yielding breakdown of certain relations between fluctuations and thermodynamic derivatives. The behavior near to the critical temperature is discussed and some of the ideas are extended to systems with power-law interactions and to spin glasses with X-Y or Heisenberg symmetry. The inequality \ensuremath{\theta}\ensuremath{\le}(d-1)/2 is also derived.
The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.