How many-body chaos emerges in the presence of quasiparticles

TL;DR

This work addresses how chaos emerges from near-integrable, low-temperature dynamics by identifying a cascade of lightcones seeded by weakly interacting harmonic modes, diagnosed with a classical decorrelator in a classical Heisenberg chain. It introduces a defect ensemble to mimic low-temperature excitations and uses RK4 simulations plus a mode-coupling framework to reveal four dynamical regimes: an initial free regime, an intermediate scarred regime with secondary lightcones, an avalanche-like onset of chaos, and a fully chaotic late-time regime. The key contributions include a quantitative link between defect-induced secondary lightcones and the onset of chaos, a fat-tailed decorrelator distribution signaling rare hotspots, and a velocity-dependent Lyapunov analysis showing how chaos characteristics depend on defect density. The findings illuminate a universal, mechanism-based path from near-integrability to chaos, suggesting kinetic-theory descriptions may apply to both classical and quantum many-body systems at low temperatures and potentially guiding quantum simulations of chaotic dynamics.

Abstract

Many-body chaos is a default property of many-body systems; at the same time, near-integrable behaviour due to weakly interacting quasiparticles is ubiquitous throughout condensed matter at low temperature. There must therefore be a, possibly generic, crossover between these very different regimes. Here, we develop a theory encapsulating the notion of a cascade of lightcones seeded by sequences of scattering of weakly interacting harmonic modes as witnessed by a suitably defined chaos diagnostic (classical decorrelator) that measures the spatiotemporal profile of many-body chaos. Our numerics deals with the concrete case of a classical Heisenberg chain, for either sign of the interaction, at low temperatures where the short-time dynamics are well captured in terms of non-interacting spin waves. To model low-temperature dynamics, we use ensembles of initial states with randomly embedded point defects in an otherwise ordered background, which provides a controlled setting for studying the scattering events. The decorrelator exhibits a short-time integrable regime followed by an intermediate `scarred' regime of the cascade of lightcones in progress; these then overlap, leading to an avalanche of scattering events which finally yields the standard long-time signature of many-body chaos.

Figures (15)

• Figure 1: Schematic diagram of the different regimes leading to the onset of many-body chaos, as measured by the decorrelator (Eq. \ref{['eq:DecorrMain']}), in classical unfrustrated spin systems. The initial state can be viewed as a (quasi) long-ranged ordered state interspersed by disordered 'defect' regions; this constitutes a proxy for a low-temperature initial state where the disordered regions can be thought of as mimicking the excitations on top of a ground state. The quasiparticles corresponding to the excitations, denoted by the red circles, scatter off the primary lightcone of spatiotemporal chaos, seeding secondary lightcones inside the primary one. At intermediate times, this leads to a scarred regime, following which the secondary lightcones proliferate and further scatter off each other, leading to fully-developed chaos at late times.
• Figure 2: The decorrelator, ${\cal D}(i,t)$, shown as a heatmap (with logarithmic colour-scales) in space-time for a single initial configuration drawn from the defect ensemble. The scarred regime, indicated by the streaky patterns inside the lightcone, at intermediate times arising out of the secondary lightcones, is clearly visible along with their proliferation at late times, leading to chaos. The left and right panels correspond to the FM and AFM cases, respectively. For these plots $\rho_d=0.05$, $\theta_M=\pi$, $\phi_M=2\pi$ and $\varepsilon=10^{-4}$ with $L=2048$.
• Figure 3: The participation function $g(t)$ (defined in Eq. \ref{['eq:gt_main']}) plotted for different defect densities $\rho_d = 0.05,\ldots,0.50$ for ferromagnetic (upper panel) and antiferromagnetic (lower panel) couplings. The defect-free behaviour for the respective cases is shown in a cyan dashed line. For each defect density, we can distinguish three different regimes. Within the early times (up to ${O}(1)$), $g(t)$ coincides with the defect-free behaviour $g_0(t)$ (dashed cyan line). The onset of chaos in the intermediate time regime, as denoted by the plateau and the avalanche decay in $g(t)$, happens at the same time for FM and AFM cases, and therefore is independent of the sign of the interaction between the spins. Thereafter, the chaotic regime is indicated by $g(t) \sim 1/t$ as denoted by the black dashed line. In the insets, we show the scaling of the deviation $\Delta_g(t) = |g(t) - g_0(t)|$ from the free behaviour with defect density $\rho_d$. For these plots, $\theta_M = \pi$, $\phi_M = 2\pi$, system size $N = 2048$ and the results are averaged over 10240 configurations.
• Figure 4: Early-time free behaviour of the decorrelator for FM. Left: On-site time evolution of the decorrelation function at the $i=0$ site for different defect densities. The cyan dashed line denotes the defect-free case given by the Bessel function solution. The finite defect curves follow the free behaviour up to the time $t_\mathrm{free}$. Right: The distribution of time of free propagation for the decorrelator obtained numerically for $N=1024$ with $10240$ configurations. The black dashed lines denote the analytical results obtained from $t_{\rm free}=i_0/(2 v_B)$ and using the distribution of $i_0$ in Eq. \ref{['eq_probtfree']}. Given that the lowest probability we can attain is $\mathcal{O}(10^{-4})$ due to a finite number of samples, the numerical calculation of $P({\cal D})$ has a sharp cutoff at this value near the right edge.
• Figure 5: Seeding of a secondary lightcone due to a single defect illustrated for the FM case. The left panel shows the difference between the decorrelators $\Delta{\cal D}(i,t)\equiv |{\cal D}_0(i,t)-{\cal D}(i,t)|$ (rescaled by $\varepsilon^2$ for visibility) where ${\cal D}_0(i,t)$ and ${\cal D}(i,t)$ are the decorrelators in the absence of any defects and the presence of a single defect at site $i_d=200$ respectively. The red and grey solid lines denote the boundaries of the primary decorrelator lightcone and the defect dynamics lightcone, respectively. The three dashed horizontal lines denote three different time slices (at $t=20,50,80$) at which ${\cal D}(i,t)$ and the defect profile, $1-S_i^z(t)$ (multiplied by 10 to put it on visible scales) are shown in the right panels, in red and grey respectively. Note that at $t=50$ (right middle panel), the two profiles meet, and that is precisely when the secondary lightcone is seeded at site $i=i_d/2$ and $t_\ast=i_d/(2v_B)$ manifested in $\Delta{\cal D}(i,t)$ taking on a finite value for $t>t_\ast$. Results are for $\theta_d=\pi/10$ and $\phi_d=0$.