Conservation laws and chaos propagation in a non-reciprocal classical magnet

TL;DR

The paper analyzes a nonreciprocal generalization of the classical Heisenberg spin chain, showing that a hidden symplectic structure emerges under a transformed description for nearest-neighbour interactions, with conserved magnetization and energy in transformed variables. Chaos spreads ballistically across all studied models as measured by the decorrelator, but only in the antisymmetric case are the front velocities symmetric and a simple hydrodynamic picture recovered; longer-range couplings generally destroy the conservation laws. The work highlights how nonreciprocity can coexist with conserved densities in a transformed frame while also delineating the limits of thermodynamic limit and hydrodynamics when extending interactions beyond nearest neighbours. These findings advance the understanding of thermalization and information spreading in nonreciprocal classical many-body systems, and point to distinct transport and chaotic signatures across model variants with potential relevance to driven or active spin networks.

Abstract

We study a nonreciprocal generalization [EPL 60, 418 (2002)] of the classical Heisenberg spin chain, in which the exchange coupling is nonsymmetric, and show that it displays a ballistic spreading of chaos as measured by the decorrelator. We show that the interactions are reciprocal in terms of transformed variables, with conserved quantities that can be identified as magnetization and energy, with a Poisson-bracket algebra and Hamiltonian dynamics. For strictly antisymmetric couplings in the original model the conserved quantities diffuse, the decorrelator spreads symmetrically, and a simple hydrodynamic theory emerges. The general case in which the interaction has symmetric and antisymmetric parts presents complexities in the limit of large scales. Ballistic propagation of chaos survives the inclusion of interactions beyond nearest neighbours, but the conservation laws in general do not.

Figures (11)

• Figure 1: Two-point correlation functions of: (a) the staggered magnetization, $C_{N}(x,t)$ and (b) the pseudo-energy , $C_{\mathcal{E}}(x,t)$, with $x \in \{1,\dots, L\}$ for $L=512$, sampled over 5000 initial configurations as functions of $x$ for different values of $t$. The left inset of both panels shows the scaling collapse to a form $C(x,t) = t^{-1/2}f(x/t^{1/2})$ consistent with diffusion while the right inset shows a plot of $C(0,t)$ versus $t$ with a fit to $t^{-1/2}$.
• Figure 2: Two-point correlation functions: (a) $C_{\mathcal{M}}(x,t)$ and (b) $C_{\mathcal{E}}(x,t)$ for $\alpha=1.1$, with $x \in \{1,\dots, L\}$, $L=256, \text{ and} \Delta t = 0.002$, sampled over 5000 initial configurations as functions of $x$ for different values of $t$. There is no power law that scales $C(x,t)$ at different time steps since the log-log relation between the correlation peak $C(0,t)$ starts saturating, as seen in the right inset plot of $C^{-1}(0,t)$ vs $t$. Taking $C^{-1}(0,t)$ as the length scale, we plot in the left inset $C(x,t)/C(0,t)$ which shows a collapse.
• Figure 3: Two-point correlation functions of: (a) the conserved magnetization, $C_{\mathcal{M}}(x,t)$ for $\alpha=1.2$ and (b) the pseudo-energy , $C_{\mathcal{E}}(x,t)$, with $x \in \{1,\dots, L\}$ for $L=256$, sampled over 5000 initial configurations as functions of $x$ for different values of $t$.
• Figure 4: Site-anchored correlator $C^{\mathcal{M}}_n(x,t)$ for $n = L/8, 3L/16 (L=256)$ as defined in \ref{['eqn:corr_site_magplus']} for $\alpha=1.1$
• Figure 5: Site-anchored correlator $C^{\mathcal{E}}_n(x,t)$ for $n = L/4, L/2 (L=256)$ as defined in \ref{['eqn:corr_site_Ehyb']} for $\alpha=1.1$