We study the nonequilibrium dynamics of random spin chains that remain integrable (i.e., solvable via Bethe ansatz): because of correlations in the disorder, these systems escape localization and feature ballistically spreading quasiparticles. We derive a generalized hydrodynamic theory for dynamics in such random integrable systems, including diffusive corrections due to disorder, and use it to study nonequilibrium energy and spin transport. We show that diffusive corrections to the ballistic propagation of quasiparticles can arise even in noninteracting settings, in sharp contrast to clean integrable systems. This implies that operator fronts broaden diffusively in random integrable systems. By tuning parameters in the disorder distribution, one can drive this model through an unusual phase transition, between a phase where all wave functions are delocalized and a phase in which low-energy wave functions are quasilocalized (in a sense we specify). Both phases have ballistic transport; however, in the quasilocalized phase, local autocorrelation functions decay with an anomalous power law, and the density of states diverges at low energy.
We study the nonequilibrium dynamics of random spin chains that remain integrable (i.e., solvable via Bethe ansatz): because of correlations in the disorder, these systems escape localization and feature ballistically spreading quasiparticles. We derive a generalized hydrodynamic theory for dynamics in such random integrable systems, including diffusive corrections due to disorder, and use it to study nonequilibrium energy and spin transport. We show that diffusive corrections to the ballistic propagation of quasiparticles can arise even in noninteracting settings, in sharp contrast to clean integrable systems. This implies that operator fronts broaden diffusively in random integrable systems. By tuning parameters in the disorder distribution, one can drive this model through an unusual phase transition, between a phase where all wave functions are delocalized and a phase in which low-energy wave functions are quasilocalized (in a sense we specify). Both phases have ballistic transport; however, in the quasilocalized phase, local autocorrelation functions decay with an anomalous power law, and the density of states diverges at low energy.
We address the hydrodynamics of operator spreading in interacting integrable lattice models. In these models, operators spread through the ballistic propagation of quasiparticles, with an operator front whose velocity is locally set by the fastest quasiparticle velocity. In interacting integrable systems, this velocity depends on the density of the other quasiparticles, so equilibrium density fluctuations cause the front to follow a biased random walk, and therefore to broaden diffusively. Ballistic front propagation and diffusive front broadening are also generically present in nonintegrable systems in one dimension; thus, although the mechanisms for operator spreading are distinct in the two cases, these coarse-grained measures of the operator front do not distinguish between the two cases. We present an expression for the front-broadening rate; we explicitly derive this for a particular integrable model (the ``Floquet-Fredrickson-Andersen'' model), and argue on kinetic grounds that it should apply generally. Our results elucidate the microscopic mechanism for diffusive corrections to ballistic transport in interacting integrable models.
High-temperature spin transport in integrable quantum spin chains exhibits a rich dynamical phase diagram, including ballistic, superdiffusive, and diffusive regimes. While integrability is known to survive in static and periodically driven systems, its fate in the complete absence of time-translation symmetry, particularly in interacting random quantum circuits, has remained unclear. Here we construct integrable random quantum circuits built from inhomogeneous XXZ R-matrices. Remarkably, integrability is preserved for arbitrary sequences of gate layers, ranging from quasiperiodic to fully random, thereby explicitly breaking both continuous and discrete time-translation symmetry. Using large-scale time-dependent density-matrix renormalization group simulations at infinite temperature and half filling, we map out the resulting spin-transport phase diagram and identify ballistic, superdiffusive, and diffusive regimes controlled by the spectral parameters of the R-matrices. The spatiotemporal structure of spin correlations within each regime depends sensitively on the inhomogeneity, exhibiting spatial asymmetry and sharp peak structures tied to near-degenerate quasiparticle velocities. To account for these findings, we develop a generalized hydrodynamics framework adapted to time-dependent integrable circuits, yielding Euler-scale predictions for correlation functions, Drude weights, and diffusion bounds. This approach identifies the quasiparticles governing transport and quantitatively captures both the scaling exponents and fine structures of the correlation profiles observed numerically. Our results demonstrate that exact Yang-Baxter integrability is compatible with stochastic quantum dynamics and establish generalized hydrodynamics as a predictive framework for transport in time-dependent integrable systems.
Many experimentally relevant quantum spin chains are approximately integrable, and support long-lived quasiparticle excitations. A canonical example of integrable model of quantum magnetism is the XXZ spin chain, for which energy spreads ballistically, but, surprisingly, spin transport can be diffusive or superdiffusive. We review the transport properties of this model using an intuitive quasiparticle picture that relies on the recently introduced framework of generalized hydrodynamics. We discuss how anomalous linear response properties emerge from hierarchies of quasiparticles both in integrable and near-integrable limits, with an emphasis on the role of hydrodynamic fluctuations. We also comment on recent developments including non-linear response, full-counting statistics and far-from-equilibrium transport. We provide an overview of recent numerical and experimental results on transport in XXZ spin chains.
The conventional wisdom suggests that transports of conserved quantities in nonintegrable quantum many-body systems at high temperatures are diffusive. Recent study suggests a potential counterexample of this paradigm. We independently complement this counterexample by directly uncovering anomalous infinite-temperature hydrodynamics in a spin-1/2 XXZ chain with power-law couplings. This model, classified as nonintegrable due to its Wigner-Dyson level-spacing statistics in the random matrix theory, exhibits a surprising superdiffusive-ballistic-superdiffusive transport transition by varying the power-law exponent of couplings for a fixed anisotropy. Our findings are verified by multiple observables, including the spin-spin autocorrelator, mean-square displacement, and spin conductivity. Interestingly, we further quantify the degree of quantum chaos using the Kullback-Leibler divergence between the entanglement entropy distributions of the model's eigenstates and a random state. Remarkably, an observed local maximum in the divergence near the transition boundary suggests a link between anomalous hydrodynamics and a suppression of quantum chaos. This work offers another deep understanding of emergent anomalous transport phenomena in a wider range of nonintegrable quantum many-body systems.
Anomalous transport in low dimensional spin chains is an intriguing topic that can offer key insights into the interplay of integrability and symmetry in many-body dynamics. Recent studies have shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved or broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality class. Similarly, energy transport can show ballistic or diffusive-like behavior. Although such behavior has been studied under equilibrium conditions, no results on nonequilibrium spin transport in classical spin chains has been reported so far. In this work, we investigate both spin and energy transport in classical spin chains (integrable and nonintegrable) when coupled to two reservoirs at two different temperatures/magnetizations. In both the integrable case and the broken-integrability (but spin-symmetry preserving) case, we report anomalous scaling of spin current with system size (J^{s}∝L^{-μ}), with an exponent value that, within error bars, is close to the KPZ universality class value μ≈2/3. On the other hand, it is noteworthy that the energy current remains ballistic (J^{e}∝L^{-η} with η≈0) in the purely integrable case while there is departure from ballistic behavior (η>0) when integrability is broken regardless of spin-symmetry. We also present results on other interesting observables in the nonequilibrium steady state such as the spatial profiles of magnetization and energy, and spin-spin correlations.
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.