Virtual walks in the Ising model: finite time scaling

Abstract

The dynamics of the spins in the Ising model are analyzed using a virtual walk scenario. The system is quenched from a very high temperature to a lower one using the Glauber scheme in one and two dimensions. A walk is associated with each spin which evolves according to the current state of the spin. The probability distribution of the displacement is calculated that shows a distinct change as the temperature is increased. The average displacement as a function of time shows a non-equilibrium region stretched over a much longer time interval compared to the bulk magnetization. Nevertheless, one can still detect a time dependent critical point determined by two different methods. In addition, we introduce a virtual walk constructed from the local energy of individual spins. Finite time scaling of the different quantities estimated in two dimensions show excellent consistency with the values of the known critical exponents.

Figures (11)

• Figure 1: Data collapse of $P(x,t)$ at three different times shown for $T = 0$ shown for (a) one dimension and (b) two dimensions. Insets show U-shaped zero temperature probability distribution at different times (t) where the peaks at $x=\pm t$ and $P(x=\pm t,t)$ curves are fitted in the form of $\sim t^{-\theta}$ with $\theta_{1d} = 0.375$ and $\theta_{2d}\approx 0.2$.
• Figure 2: Crossover behavior of the probability distribution $P(x,t)$, from a double-peaked to single-peaked form as time increases, is shown for $T = 0.45,0.55$. The ratio $r = \frac{P(x=0)}{P(x=x_{m})}$ fitted with exponential form $e^{a^{\prime}(t-t_c)}$ for $t=t_c^{-}$ with characteristic time scale $1/a^{\prime} \approx 1370,370$ and $t_c \approx 6930,1399$ at $T=0.45,0.55$ respectively and $r$ remains $1$ at $t\geq t_c$. Here $t_c$ is the crossover time at which the double peaked structure changes to single peaked structure. Inset shows the variation of crossover time $t_c$ with $\frac{1}{T}$ in log-linear scale. $t_c$ has a exponential divergence at $T=0$ and then it decreases with increasing temperature exponentially fast as $\sim e^{k/T}$ with $k \approx 4.08$.
• Figure 3: Data collapse of the rescaled variance $\sigma_x^2/t^{2}$ plotted against the scaled variable $t(1-\tanh(2/T))$ for the $1D$ spin walk, confirming the finite time scaling form given in Eq. (\ref{['1D FTS']}). The data corresponds to fixed observation times $t=30000,40000,50000$ and $60000$. In inset fluctuation per unit time $\sigma_x^2/t$, plotted as a function of inverse temperature $1/T$ in log-linear scale and fitted with the analytical form given in Eq. (\ref{['1D spin walk fluctuation']}).
• Figure 4: Crossover behavior of the probability distribution $P(x,t)$, from a double-peaked to single-peaked form as temperature increases, is shown for $t = 25000$.
• Figure 5: Plot of $r = \frac{P(x=0)}{P(x=x_{m})}$ vs $T$ fitted with exponential form $e^{\mu(T-T_c)}$ at $T=T_c^{-}$ with characteristic inverse temperature scale $\mu \approx 96.5,124,170.2$ and $T_c \approx 2.29,2.285,2.28$ at $t = 15000,30000,50000$ respectively. $r$ remains $1$ at $T\geq T_c$.