Lax dynamics

TL;DR

This work introduces Lax dynamics as a framework to analyze perturbed Toda chains, where the time evolution of the Lax eigenvalues $\lambda_\alpha$ acts as a proxy for quasiparticle velocities and Toda actions under non-integrable perturbations. By deriving exact equations of motion for $\lambda_\alpha$ and drawing connections to the Pechukas-Yukawa gas, the authors show that the invariant measure of the perturbed dynamics closely matches the thermal density of states $\rho_{th}(\lambda)$ of the unperturbed Toda lattice, irrespective of the perturbation form. The results support a quasiparticle, Dyson-gas interpretation of the dynamics, with perturbations inducing elastic-like level rearrangements and scattering that drive thermalization toward the GGEs. This approach offers a practical route to model slow evolution of Toda actions in near-integrable systems and suggests broader applicability to other integrable discretizations and quantum-classical crossovers. The work thus provides a unifying lens for understanding thermalization and spectral statistics in perturbed integrable many-body systems.

Abstract

A novel approach is proposed to characterize the dynamics of perturbed many-body integrable systems. Focusing on the paradigmatic case of the Toda chain under non-integrable Hamiltonian perturbations, this study introduces a method based the time evolution of the Lax eigenvalues $λ_α$ as a proxy of the quasi-particles velocities and of the perturbed Toda actions. A set of exact equations of motion for the $λ_α$ is derived that closely resemble those for eigenenergies of a quantum problem (also known as the Pechukas-Yukawa gas). Numerical simulations suggest that the invariant measure of such dynamics is basically the thermal density of states of the Toda lattice, regardless of the form of the perturbation.

Figures (3)

• Figure 1: Simulations of Toda (upper panels) and Morse chains with $\varepsilon=0.1$ (lower panels); (a,d): Space-time evolution of the square modulus of a Lax eigenvector $|\psi_{\alpha,n}(t)|^2$$\alpha=166$ (b,e) quasiparticle velocities computed by Eq.(\ref{['qvel']}) (solid red line) and corresponiding Lax eigenvalue $\lambda_\alpha(t)$; (c,f) plots of $\lambda_\alpha(t)$ versus the time-averaged velocity $\langle v_\alpha \rangle$. For comparison, in the Morse case (f) the average eigenvalue is reported. In both cases, $N=200$ and initial conditions are sampled from a thermal GGE state of the Toda model with $\beta=1,\beta P=1$.
• Figure 2: Lax dynamics for the Morse chain, Eq. (\ref{['morse']}): time evolution of (a) a subset of eigenvalues $\lambda_\alpha(t)$ and (b) a couple of neighboring ones $\lambda_\alpha(t),\lambda_{\alpha+1}(t)$ illustrating the strong repulsion that yields almost elastic collisions ; (c,d): phase portraits ($\lambda_\alpha,\dot{\lambda}_\alpha$) for $\alpha=102,103$. The chain is initialized with random initial conditions sampled from the Toda thermal GGE state with $\beta=P=1$, $N=200$, $\varepsilon=0.1$.
• Figure 3: The DOS $\rho(\lambda)$ obtained from the Lax dynamics of the Morse (blue) and the Toda chain with alternating (staggered) coupling $\varepsilon_j = (-1)^j \varepsilon$ (orange lines), ($N=200$, $\varepsilon=0.1$), starting with initial thermal GGE initial conditions with three different $\beta P$, $\beta=1$. Magenta lines are the thermal DOS $\rho_{th}$ for the unperturbed Toda chain, obtained by sampling and diagonalizing the equilibrium Lax matrix. The dashed green line in (a) is the approximate Gaussian DOS expected predicted in the limit of small $\beta P$spohn2021hydrodynamic.