Integrable spin chains with a continuous non-Abelian symmetry, such as the one-dimensional isotropic Heisenberg model, show superdiffusive transport with little theoretical understanding. Although recent studies reported a surprising connection to the Kardar-Parisi-Zhang (KPZ) universality class in that case, this view was most recently questioned by discrepancies in full counting statistics. Here, by combining extensive numerical simulations of classical and quantum integrable isotropic spin chains with a framework developed by exact studies of the KPZ class, we characterize various two-point quantities that remain hitherto unexplored in spin chains, and find full agreement with KPZ scaling laws without adjustable parameters. This establishes the partial emergence of the KPZ class in integrable isotropic spin chains. Moreover, we reveal that the KPZ scaling laws are intact in the presence of an energy current, under the appropriate Galilean boost required by the propagation of spacetime correlation.
The Heisenberg spin chain is a canonical integrable model. As such, it features stable ballistically propagating quasiparticles, but spin transport is subballistic at any nonzero temperature: An initially localized spin fluctuation spreads in time t to a width t2/3. This exponent as well as the functional form of the dynamical spin correlation function suggest that spin transport is in the Kardar–Parisi–Zhang (KPZ) universality class. However, the full counting statistics of magnetization is manifestly incompatible with KPZ scaling. A simple two-mode hydrodynamic description, derivable from microscopic principles, captures both the KPZ scaling of the correlation function and the coarse features of the full counting statistics, but remains to be numerically validated. These results generalize to any integrable spin chain invariant under a continuous nonabelian symmetry and are surprisingly robust against moderately strong integrability-breaking perturbations that respect the nonabelian symmetry.
Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the one-dimensional Heisenberg model were conjectured as to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we studied the probability distribution of the magnetization transferred across the chain's center, [Formula: see text]. The first two moments of [Formula: see text] show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments ruled out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide insights into universal behavior in quantum systems.
The large-scale dynamics of quantum integrable systems is often dominated by ballistic modes due to the existence of stable quasiparticles. We here consider as an archetypical example for such a system the spin-
21 XXX Heisenberg chain that features magnons and their bound states. An interesting question, which we here investigate numerically, arises with respect to the fate of ballistic modes at finite temperatures in the limit of zero magnetization
m=0. At a finite magnetization density
m, the spin-autocorrelation function $\mathrm{\ensuremath{\Pi}}(x,t)$ (at high temperatures) typically exhibits a trimodal behavior with left- and right-moving quasiparticle modes and a broad center peak with slower dynamics. The broadening of the fastest propagating modes exhibits a subdiffusive
t1/3 scaling at large magnetization densities $m\ensuremath{\rightarrow}\frac{1}{2}$, familiar from noninteracting models; it crosses over into a diffusive scaling
t1/2 upon decreasing the magnetization to smaller values. The behavior of the center peak appears to exhibit a crossover from transient superdiffusion to ballistic relaxation at long times. In the limit $m\ensuremath{\rightarrow}0$, the weight carried by the propagating peaks tends to zero; the residual dynamics is carried only by the central peak; it is sub-ballistic and characterized by a dynamical exponent
z close to the value
23 familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm that, employing elaborate finite-time extrapolations, that the spatial scaling of the correlator $\mathrm{\ensuremath{\Pi}}$ is in excellent agreement with KPZ-type behavior and analyze the corresponding corrections.
Isotropic integrable spin chains such as the Heisenberg model feature superdiffusive spin transport belonging to an as-yet-unidentified dynamical universality class closely related to that of Kardar, Parisi, and Zhang (KPZ). To determine whether these results extend to more generic one-dimensional models, particularly those realizable in quantum simulators, we investigate spin and energy transport in non-integrable, long-range Heisenberg models using state-of-the-art tensor network methods. Despite the lack of integrability and the asymptotic expectation of diffusion, for power-law models (with exponent
2<α<∞) we observe long-lived
z=3/2 superdiffusive spin transport and two-point correlators consistent with KPZ scaling functions, up to times
t∼103/J. We conjecture that this KPZ-like transport is due to the proximity of such power-law-interacting models to the integrable family of Inozemtsev models, which we show to also exhibit KPZ-like spin transport across all interaction ranges. Finally, we consider anisotropic spin models naturally realized in Rydberg atom arrays and ultracold polar molecules, demonstrating that a wide range of long-lived, non-diffusive transport can be observed in experimental settings.
Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the 1D Heisenberg model were conjectured to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we study the probability distribution,
P(M), of the magnetization transferred across the chain's center. The first two moments of
P(M) show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments rule out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide key insights into universal behavior in quantum systems.