Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMetastability in the Potts model: exact results in the large q limit
130
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the metastable equilibrium properties of the Potts model with heat-bath transition rates using a novel expansion. The method is especially powerful for large number of state spin variables and it is notably accurate in a rather wide range of temperatures around the phase transition.
rate research
Read More
We study the low temperature quench dynamics of the two-dimensional Potts model in the limit of large number of states, q >> 1. We identify a q-independent crossover temperature (the pseudo spinodal) below which no high-temperature metastability stops the curvature driven coarsening process. At short length scales, the latter is decorated by freezing for some lattice geometries, notably the square one. With simple analytic arguments we evaluate the relevant time-scale in the coarsening regime, which turns out to be of Arrhenius form and independent of q for large q. Once taken into account dynamic scaling is universal.
The $q$-state Potts model has stood at the frontier of research in statistical mechanics for many years. In the absence of a closed-form solution, much of the past efforts have focused on locating its critical manifold, trajectory in the parameter ${q, e^J}$ space where $J$ is the reduced interaction, along which the free energy is singular. However, except in isolated cases, antiferromagnetic (AF) models with $J<0$ have been largely neglected. In this paper we consider the Potts model with AF interactions focusing on deducing its critical manifold in exact and/or closed-form expressions. We first re-examine the known critical frontiers in light of AF interactions. For the square lattice we confirm the Potts self-dual point to be the sole critical point for $J>0$. We also locate its critical frontier for $J<0$ and find it to coincide with a solvability condition observed by Baxter in 1982. For the honeycomb lattice we show that the known critical point holds for {all} $J$, and determine its critical $q_c = frac 1 2 (3+sqrt 5) = 2.61803$ beyond which there is no transition. For the triangular lattice we confirm the known critical point to hold only for $J>0$. More generally we consider the centered-triangle (CT) and Union-Jack (UJ) lattices consisting of mixed $J$ and $K$ interactions, and deduce critical manifolds under homogeneity hypotheses. For K=0 the CT lattice is the diced lattice, and we determine its critical manifold for all $J$ and find $q_c = 3.32472$. For K=0 the UJ lattice is the square lattice and from this we deduce both the $J>0$ and $J<0$ critical manifolds and find $q_c=3$ for the square lattice. Our theoretical predictions are compared with recent tensor-based numerical results and Monte Carlo simulations.
We calculate the partition function of the $q$-state Potts model on arbitrary-length cyclic ladder graphs of the square and triangular lattices, with a generalized external magnetic field that favors or disfavors a subset of spin values ${1,...,s}$ with $s le q$. For the case of antiferromagnet spin-spin coupling, these provide exactly solved models that exhibit an onset of frustration and competing interactions in the context of a novel type of tensor-product $S_s otimes S_{q-s}$ global symmetry, where $S_s$ is the permutation group on $s$ objects.
The surface and bulk properties of the two-dimensional Q > 4 state Potts model in the vicinity of the first order bulk transition point have been studied by exact calculations and by density matrix renormalization group techniques. For the surface transition the complete analytical solution of the problem is presented in the $Q to infty$ limit, including the critical and tricritical exponents, magnetization profiles and scaling functions. According to the accurate numerical results the universality class of the surface transition is independent of the value of Q > 4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and in the disordered phases are identical at the transition point.
We studied the non-equilibrium dynamics of the q-state Potts model in the square lattice, after a quench to sub-critical temperatures. By means of a continuous time Monte Carlo algorithm (non-conserved order parameter dynamics) we analyzed the long term behavior of the energy and relaxation time for a wide range of quench temperatures and system sizes. For q>4 we found the existence of different dynamical regimes, according to quench temperature range. At low (but finite) temperatures and very long times the Lifshitz-Allen-Cahn domain growth behavior is interrupted with finite probability when the system stuck in highly symmetric non-equilibrium metastable states, which induce activation in the domain growth, in agreement with early predictions of Lifshitz [JETP 42, 1354 (1962)]. Moreover, if the temperature is very low, the system always gets stuck at short times in a highly disordered metastable states with finite life time, which have been recently identified as glassy states. The finite size scaling properties of the different relaxation times involved, as well as their temperature dependency are analyzed in detail.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
