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.
Abstract Godrèche, in Comment on ‘Fluctuation dominated phase ordering at a mixed order transition’ [2021 J. Phys. A: Math. Theor. 54 038001], has commented on our recent paper Fluctuation dominated phase ordering at a mixed order transition (2019 J. Phys. A: Math. Theor. 52 254001). This comment concerns the prefactor of the cusp-like small-argument singularity of the scaled spin-spin correlation function at criticality. We remark that the approach used in our paper is adequate for computing the cusp exponent, which is what is really needed to establish fluctuation-dominated phase ordering. Computing the precise value of the prefactor of the cusp singularity is irrelevant for this purpose, or for the physics behind the relation between the mixed order phase transition and the fluctuation-dominated phase ordering—the understanding of which was the main purpose of our paper.
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.
Harris criterion In studies of the phase-transition phenomena, the systems considered before were assumed to be perfectly homogeneous. In real physical systems, however, some defects or impurities are always present. Therefore, it is natural to consider what effect impurities might have on the phase-transition phenomena. As we have seen in the previous chapter, the thermodynamics of the second-order phase transition is dominated by large-scale fluctuations. The dominant scale, or the correlation length, R c ∼ | T / T C – l| – v grows as T approaches the critical temperature T c , where it becomes infinite. The large-scale fluctuations lead to singularities in the thermodynamical functions as |τ| ≡ | T / T c – 1| → 0. These singularities are the main subject of the theory. If the concentration of impurities is small, their effect on the critical behavior remains negligible so long as R c is not too large, i.e. for T not too close to T c . In this regime the critical behavior will be essentially the same as in the perfect system. However, as |τ| → 0 ( T → T c ) and R c becomes larger than the average distance between impurities, their influence can become crucial. As T c is approached the following change of length scale takes place. First, the correlation length of the fluctuations becomes much larger than the lattice spacing, and the system ‘forgets’ about the lattice. The only relevant scale that remains in the system in this regime is the correlation length R c (τ).
Abstract We study the critical order parameter fluctuations of the absorbing-state phase-transition exhibited by branching and annihilating random walkers performing anomalous diffusion in a linear chain. The diffusion process is considered to follow a power-law distribution of jump lengths with a typical decay exponent α . We focus in the case of parity conserving dynamics for which deviations from the usual directed percolation universality class have been previously demonstrated even for the limiting cases of normal diffusion. Anomalous diffusion induces a continuous change of the critical exponents. By performing a finite-size scaling analysis of simulation data, we show that the critical order parameter moment ratio also varies continuously with α . We unveil that the critical order parameter distribution evolves from a nearly Gaussian to an exponential form as the range of the jump distribution is increased up to the limit α = 5 / 2 on which the active state predominates for any finite branching probability.
First order quantum phase transitions (1QPTs) are signalled, in the thermodynamic limit, by discontinuous changes in the ground state properties. These discontinuities affect expectation values of observables, including spatial correlations. When a 1QPT is crossed in the vicinity of a second order one, due to the correlation length divergence of the latter, the corresponding ground state is modified and it becomes increasingly difficult to determine the order of the transition when the size of the system is finite. Here we show that, in such situations, it is possible to apply finite size scaling (FSS) to entanglement measures, as it has recently been done for the order parameters and the energy gap, in order to recover the correct thermodynamic limit (Campostrini et al 2014 Phys. Rev. Lett. 113 070402). Such a FSS can unambiguously discriminate between first and second order phase transitions in the vicinity of multicritical points even when the singularities displayed by entanglement measures lead to controversial results.