Relaxation time for competing short- and long-range interactions in the model A dynamic universality class

TL;DR

This work establishes the nonequilibrium critical relaxation of the one-dimensional Nagle–Kardar model by deriving a two-variable Fokker-Planck equation for the magnetization density m and defect fraction s from microscopic Glauber dynamics and validating it against simulations. Center-manifold reduction shows that dm/dt scales cubically with m on the critical line and quartically at the tricritical point, yielding a dynamical exponent z = 2 and exponents λ_m = 1/2 (critical line) and λ_m = 1/4 (tricritical point), consistent with Model A dynamics for a non-conserved order parameter. The analysis also reveals a mean-field upper critical dimension du of 4 on the critical line and 3 at the tricritical point, with corresponding finite-size scaling Δ_m exponents Δ_m = 1/4 and Δ_m = 1/6, respectively. Additionally, large-deviation methods show an Arrhenius-type scaling for the average first-passage time between metastable states, indicating exponential growth with system size. Collectively, the results place the macroscopic dynamics of the Nagle–Kardar model in the Model A universality class, while extending finite-size scaling concepts to systems with competing short- and long-range interactions. All mathematical expressions are given and derived within the FP/Langevin framework and corroborated by exact finite-size counts and numerical simulations.

Abstract

We study the relaxation dynamics at criticality in the one-dimensional spin-$1/2$ Nagle-Kardar model, where short- and long-range interactions can compete. The phase diagram of this model shows lines of first and second-order phase transitions, separated by a tricritical point. We consider Glauber dynamics, focusing on the slowing-down of the magnetization $m$ both along the critical line and at the tricritical point. Starting from the master equation and performing a coarse-graining procedure, we obtain a Fokker-Planck equation for $m$ and the fraction of defects. Using central manifold theory, we analytically show that $m$ decays asymptotically as $t^{-1/2}$ along the critical line, and as $t^{-1/4}$ at the tricritical point. This result implies that the dynamical critical exponent is $z=2$, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamic universality class of purely relaxational dynamics with a non-conserved order parameter (model A). Large deviation techniques enable us to show that the average first passage time between local equilibrium states follows an Arrhenius law.

Figures (6)

• Figure 1: (Top) Quasi-potential $U(m,s)$ with parameters $T=1$, $h=0$, $J=1/2$ and $K=1/2$. The two minima $(m^*,s^*)$, such that $U(m^*,s^*)=0$, are highlighted by the red squares. A few contour lines are shown in black. The red curve represents the solution of Eq. \ref{['eq:ssp']}. (Bottom) Potential barrier $\Delta U$ for $h=0$, separating the two local minima of $U(m,s)$, plotted as a function of $J$ and $K$, together with a few contour lines. The red point ($J=K=0.5$) corresponds to the parameter set of the top panel ($\Delta U \approx 0.018$), while the red square (where $\Delta U=0$) is the tricritical point $(J\!=\!-\frac{\log(3)}{4},K\!=\!\sqrt{3})$. The contour line $\Delta U=0$ denotes the second-order phase transition for $K < \sqrt{3}$ and the first-order one for $K >\sqrt{3}$.
• Figure 2: (Left) Time evolution of $\langle \vert m(t,N) \vert \rangle$, in log-log scale, at the critical point of the Curie-Weiss model, for decreasing values of $\epsilon=1/N$. We find that $z=2$ in an intermediate time range, before the magnetization reaches the equilibrium plateau proportional to $N^{-1/4}$ (see inset). (Right) Time evolution of $\langle \vert m(t,N) \vert \rangle$, in log-log scale, at the tricritical point of the Nagle-Kardar model [Eq. \ref{['eq:Hamiltonian']}]. Also for this model $z=2$, and the asymptotic magnetization fluctuates around $N^{-1/6}$. In all simulations, the average is performed over $1000$ independent trajectories, with $m(0)=1$.
• Figure 3: (Left) Average first passage time $\langle\tau\rangle$ versus $N$, for $h=0$, $J=0.5$, $K=0.5$ ($T=1$) in the Nagle-Kardar model. Zero external field entails bistability. We compare the Glauber (green squares) and Langevin (green dots) dynamics, obtaining a very good agreement. The plot is in semi-log scale to highlight the exponential growth with $N$, and the lines are drawn to guide the eyes. To depict the transitions (green arrows), we report in the upper-left inset the large deviation function $\psi(m)$ [Eq. \ref{['eq:LDVmagn']}]. Due to the symmetry of $\psi(m)$, the left-right passage time equals the right-left one. Using Eq. \ref{['eq:potentialU']}, the exponential growth rate is determined by the variation $\Delta U=0.018$, calculated between the stable and the saddle points in the $(m,s)$ plane. In the lower-right inset, we plot the time-dependence of the magnetization for $N=200$, using the Glauber dynamics. (Right) We repeat the same analysis with non-zero $h$ to study metastability, with $h=0.01$, $J=-0.28$, $K=1.86$. In this case, $\psi(m)$ is asymmetric (see the inset) and we need to compute two different scaling for the left-right (blue lines) and right-left (red lines) passage times. Here, the rates of exponential increase hold $\Delta U=0.011$ (blue) and $\Delta U=0.025$ (red), respectively.
• Figure 4: Comparison between the Glauber dynamics (points) and the coarse-grained dynamics (solid lines) given in Section III in the SM, for the parameters $N=7, T=1, h=0.1, J=0.5$ and $K=0.4$. Both positive and negative values of $M$ are considered. The points and lines sometimes overlap so perfectly that they are difficult to distinguish.
• Figure 5: Large deviation function $\psi(m)$ for several sets of parameters at $h=0$. Around $m=0$, $\psi(m) = O(m^2)$ in the disordered phase (dotted magenta), $\psi(m) = O(m^4)$ along the critical line (dash-dotted blue line), while $\psi(m) = O(m^6)$ at the tricritical point (TCP) (solid red line). We also plot the large deviation function for $J=0$, $K=1.25$, $h=0$, $\beta=1$, which belongs to the ferromagnetic phase (green dashed line).

Theorems & Definitions (9)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Proposition 8
• Proposition 9