Anisotropic Landau-Lifshitz Model in Discrete Space-Time

TL;DR

This work constructs an integrable lattice model of classical spins in discrete space-time as a discrete-time analogue of the uniaxial lattice Landau–Lifshitz ferromagnet and provides an explicit, symplectic two-body propagator derived from a discrete zero-curvature condition. The model is realized as a brick-wall circuit built from two-body, canonical maps, with a Sklyanin Lax matrix written in Weyl variables that enables exact factorization and explicit propagation. Using this framework, the authors compute magnetization transport properties—specifically the spin Drude weight and diffusion constant—across easy-axis and easy-plane regimes, revealing ballistic transport away from zero magnetization in both cases, a divergence of the diffusion constant at the isotropic point signaling a crossover to spin superdiffusion, and KPZ-like scaling in the non-magnetized sector. The results highlight a robust discrete-integrable approach to nonequilibrium classical spin dynamics and connect to broader themes of generalized hydrodynamics and superdiffusive transport in integrable systems, with potential applicability to numerical studies of spin transport in discrete models.

Abstract

We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.

Figures (10)

• Figure 1: Elementary plaquette of the discrete light-cone lattice: classical spins ${\bf S}\in S^{2}$ (circles at the middle of the edges) are situated on the vertices of the discrete space-time lattice. Primed variables $\mathbf{S}^{\prime}_{1,2}$ pertain to spins $\mathbf{S}_{1,2}$ that are time-shifted forward by one unit of time by applying the propagator $\Phi$ (red square).
• Figure 2: Fabric of discrete space-time: the square space-time lattice comprising classical spin degrees of freedom $\mathbf{S}^{t}_{\ell}$ (circles) superimposed on a square light-cone lattice represented by a tilted checkerboard. In the middle of each plaquette, there is a two-body symplectic map $\Phi$ (red square) which maps the pairs of physical spin variables forward in time.
• Figure 3: Discrete Noether current, shown at an even site $2\ell$: the time-increment $t\to t+2$ of charge density, $(q^{t+2}_{2\ell}-q^{t}_{2\ell})/\tau$ equals the difference of two nearby current densities at consecutive times, $j^{2t+1}_{2\ell+1}-j^{2t}_{2\ell}$. Current densities $j^{t}_{\ell}$, depicted by pink arrows, can be compactly expressed as the difference of charge densities at consecutive times, $j^{t}_{\ell}=(q^{t+1}_{\ell}-q^{t}_{\ell})/\tau$.
• Figure 4: (a) Spin Drude weight $\mathcal{D}$ and (b) diffusion constant $D$ in the discrete Landau--Lifshitz model, cf. Eq. \ref{['eqn:st']}, in the easy-axis regime, depending on the chemical potential $\mu$ and the anisotropy $\varrho$. (a, inset) Drude weight (purple) and diffusion constant (olive green) as functions of $\mu$ at $\varrho = 1$ (vertical grey line). (b, inset) Divergence of $D$ at zero average magnetization, $\mu = 0$ (blue curve), in the isotropic limit $\varrho \rightarrow 0$, compared to $\sim 1/\varrho$ power-law scaling (black dashed line). Simulation parameters: $\tau = 1$, $L=2^{12}$, averaged over $10^{5}$ initial configurations.
• Figure 5: (a) Finite-time dynamical exponents $\alpha$ ($t \in [100, 300]$ dashed-dotted lines, $t \in [300,1000]$ solid lines) depending on the anisotropy $\varrho$ at four values of the chemical potentials $\mu$. Magnetized ensembles ($\mu > 0$) exhibit ballistic transport with exponent $\alpha = 1$. The sub-leading diffusive correction increases with $\varrho$ and overwhelms the ballistic contribution at finite times. In the non-magnetized sector at $\mu = 0$, the dynamical exponent is approximately $\alpha \approx 2/3$ (black circle) when approaching $\varrho \rightarrow 0$, consistent with KPZ-type superdiffusion at the isotropic point $\varrho = 0$. At large values of $\varrho$, the dynamical exponent approaches $\alpha \approx 1/2$ which is indicative of normal diffusion. (b) Rescaled diffusive stationary scaling profiles of the dynamical structure factors $t^{1/2}\,C(\ell,t)$ at $\mu = 0$ and different values of $\varrho$. Simulation parameters: $\tau = 1$, $L=2^{12}$, averaged over $10^{5}$ initial configurations.