Weak integrability breaking perturbations in classical integrable models on the lattice

Abstract

We show how to systematically construct weak integrability breaking perturbations (WIBs) for classical integrable models on the lattice. These perturbations, which allow quasi-conserved quantities, have mostly been explored in quantum systems, where they are expected to delay the onset of thermalization and diffusive transport to timescales far exceeding those predicted by Fermi's golden rule. However, accessing such long-time dynamics in quantum models is computationally challenging. Classical integrable lattice models offer a complementary setting for probing transport and long-time dynamics under WIBs. In this work, we specialize our general framework to construct several families of WIBs for the Ishimori model, the Toda chain, and the Harmonic Oscillator Chain (HOC). Such constructions can help quantify how WIBs contribute to anomalous transport and serve as a benchmark for thermalization studies in perturbed integrable models. An important example is the Fermi-Pasta-Ulam-Tsingou (FPUT) model: Starting from the HOC, we show that the cubic nonlinearity (the alpha-FPUT interaction) is a genuine WIB perturbation. Using the integrals of motion (IoMs) of the Toda lattice, we explicitly construct corrections to the entire hierarchy of the HOC IoMs, thereby obtaining an infinite tower of quasi-conserved quantities for the alpha-FPUT chain. We further identify the corresponding adiabatic gauge potential (AGP) as a nontrivial trilocal generator in real space, and show that, more generally, any cubic, translationally invariant, momentum-conserving perturbation of the HOC admits such a generator and is therefore a WIB. Together with our transport and AGP-variance studies, our results provide a unified classical framework for weak integrability breaking and for diagnosing anomalous thermalization and transport in nearly integrable Hamiltonian lattice systems.

Figures (11)

• Figure 1: Top: Illustration of a 1D chain of interacting particles (oscillators) with periodic boundary conditions. Bottom, left: Toda and HOC potentials. Bottom, right: $\alpha$-FPUT and $\beta$-FPUT potentials. We used Eq. (\ref{['eq:toda_pot3']}) with $\alpha = 0.5$ for the left plot and $\frac{1}{2} r^2 + \frac{\alpha}{3} r^3$ and $\frac{1}{2} r^2 + \frac{\alpha^2}{6} r^4$ for the right plot.
• Figure 2: Toda chain.Left, table: All perturbations studied with reference to their equation in the main text; time reversal $\Theta$ and inversion $I^{\text{Toda}}$ symmetries; and their observed AGP norm scaling. Middle, plot: Variance of the AGP averaged over 100 trajectories for the Toda chain under WIBs, and a few generic perturbations. The system size is $L=100$ with initial energy density $\epsilon=0.1$ and $\alpha=0.5$ (corresponds to setting $b=-1, a=-1$). After the initialization step, dynamics are simulated up to the final time $T=1.5\times10^6$ with a time step of $dt=0.04$ using the ABA864 symplectic integrator. For easier visualization, some of the perturbations have been rescaled by a constant ($L$-independent) prefactor. Right, plot: AGP norm as a function of system size $L$ for the WIBs studied. Generic perturbations are excluded from this plot as they do not reach saturation.
• Figure 3: HOC chain.Top left, plot: Fidelity susceptibility, $\chi$ as a function of time $t$, or the variance of AGP for cubic perturbations of the HOC at $L=100$ trajectories with initial energy density $\epsilon=0.1$, obtained by averaging over an ensemble of $N=100$ trajectories. Shaded bands indicate $\pm \Delta\chi(t)$, where $\Delta\chi(t)$ is the run-to-run variance of $\chi$ at time $t$ across all the runs. Top middle, plot: Scaling of the AGP norm (or fidelity susceptibility $\chi(T)$ at final time $T$) with system size $L$ for the WIB perturbations presented in the top right table. The exact fits, as well as theoretically predicted scalings, can be found in the Appendix \ref{['app:analytic_scalings']}. Top right, table: Perturbations studied (with references to equations in the main text), their time reversal ($\Theta$) and inversion ($I$) symmetries, and the observed AGP norm ($\chi^{\rm sat}(L)$) scaling. Bottom row: Everything is the same as in the top row, with the exception that perturbations in the bottom row are quartic perturbations to HOC.
• Figure 4: Ishimori chain.Left, table: All perturbations studied with reference to their equation in the main text; time reversal $\Theta$ and inversion $I$ symmetries; and their observed AGP norm scaling. Middle, plot: Variance of the AGP averaged over 100 trajectories for the Ishimori chain in the presence of WIB perturbations generated from boosted, bilocal, and extensive local generators obtained in Sec. \ref{['sec:ishimori']}, for system size $L=72$ with $\beta=1$. Right, plot: Scaling of the saturation of AGP variance with system size.
• Figure 5: Top: Rescaled dynamical exponent $z(t\lambda^2)$ for the Ishimori chain under a generic perturbation $V_g = \sum_j \vec{S}_j \cdot \vec{S}_{j+1}$ extracted from fluctuations in energy transfer across a central link. The curve collapse indicates that the crossover from ballistic to diffusive transport is consistent with the $t_\star \sim \lambda^{-2}$ timescale predicted by Fermi's Golden Rule. Bottom: The dynamical exponent curves for weak perturbation $V_{\rm ex}$ rescaled by $\lambda^2$ do not neatly collapse, indicating that the crossover timescale $t_\star$ associated with $V_{\rm ex}$ is anomalously long. Although the data are severely limited by a short numerically accessible timescale, the available data appear to be consistent with the predicted crossover timescale of $t_\star \sim \lambda^{-4}$ (inset). All data shown is run up to a final time of $t_f=920$ for a system of $L=4000$ sites and averaged over at least $5 \times 10^4$ independent realizations.