Dynamic scaling near the Kasteleyn transition in spin ice: critical relaxation of monopoles and strings following a field quench

Abstract

We study dynamics in classical spin ice following a magnetic field quench to close to the Kasteleyn transition, using Monte Carlo simulations and dynamic scaling theory to characterize the relaxation of the magnetization and the density of magnetic monopoles. We have previously argued that this dynamics can be described in terms of seeding and growth of strings of flipped spins, and our results here demonstrate that a solvable stochastic model based on independent strings correctly describes the relaxation as well as the distribution of string lengths within the critical scaling regime near the transition. We also show how generalized scaling forms capture the behavior over a broader range of monopole densities and provide a clear understanding of the breakdown of the scaling picture further from the critical point.

Figures (26)

• Figure 1: Schematic phase diagram describing the Kasteleyn transition of spin ice in an applied magnetic field $h$ along the $[100]$ crystal direction. In equilibrium, there is a crossover at temperature $T = T_{\text{K}} \propto h$ between a spin-polarized regime and a string liquid (illustrated in top-left and bottom-right insets, respectively). This becomes a sharp transition (black dotted line) when the energy $\Delta$ of a magnetic monopole (see main text) is much larger than $T$. We consider a sudden field quench (solid orange arrow) from $h=+\infty$ to the critical scaling regime near the transition.
• Figure 2: Monte Carlo results for (a) magnetization density ($1 - \sigma$) and (b) monopole density ($\rho_1$) in equilibrium as a function of reduced temperature $\theta = (T - T_{\text{K}})/T_{\text{K}}$ for several values of the field $h$ after the quench. The system contains $N_{\text{s}}=128000$ spins and error bars are smaller than symbols.
• Figure 3: Monte Carlo results for (a) magnetization and (b) monopole density at equilibrium as a function of reduced temperature $\theta$ for several system sizes $N_{\text{s}}$. In each case, the magnetic field after the quench is $h=\qty{0.21}{\kelvin}$ corresponding to Kasteleyn temperature $T_{\text{K}} = \qty{0.606}{\kelvin}$. Finite-size effects are prominent only for the smallest $N_{\text{s}}$ shown. The solid gray lines connecting the points is a guide for the eye. Error bars are smaller than symbols.
• Figure 4: Illustration of string creation and growth. (a) In the starting configuration, all spins point upwards. (b) Flipping any spin (circled) creates a pair of monopoles on the adjacent tetrahedra, which can be interpreted as a string of length $\ell = 1$. Flipping this spin back upward would cause the two monopoles to annihilate and the string to disappear. (c) The monopoles can be separated by flipping other spins downwards. In this case, a second spin (circled) has been flipped, increasing the string length to $\ell=2$. At this point, flipping any of the four spins labeled with $+$ would extend the string to length $\ell = 3$, while flipping the down spin adjacent to either monopole would shrink the string to length $\ell = 1$.
• Figure 5: Graph representing the dynamical model for a single string with rate matrix $W_{\ell\ell'}$. Vertices (circles) denote strings of length $\ell$, with $\ell=0$ representing a string that has shrunk to zero length and hence disappeared. Edges (arrows) are transitions with associated rates for growth $r_+$, contraction $r_-$, and annihilation $r_0$. To describe the evolution of the string population, we also include string creation, which occurs with rate $r_*$.