Superdiffusion in spin chains

Abstract

This review summarizes recent advances in our understanding of anomalous transport in spin chains, viewed through the lens of integrability. Numerical advances, based on tensor-network methods, have shown that transport in many canonical integrable spin chains-most famously the Heisenberg model-is anomalous. Concurrently, the framework of generalized hydrodynamics has been extended to explain some of the mechanisms underlying anomalous transport. We present what is currently understood about these mechanisms, and discuss how they resemble (and differ from) the mechanisms for anomalous transport in other contexts. We also briefly review potential transport anomalies in systems where integrability is an emergent or approximate property. We survey instances of anomalous transport and dynamics that remain to be understood.

Fragmentation and emergent integrable transport in the weakly tilted Ising chain

“…Using Generalized Hydrodynamics (GHD) [39,40] (see also Refs. [41][42][43][44][45][46]) we analytically tackle transport within the isolated magnon sector. Since the Alcaraz-Bariev (AB) model is a close relative of the XXZ spin chain, it inherits its rich phenomenology: transport greatly depends on the interactions and can exhibit sharp jumps [47].…”

Quantum versus Classical Dynamics in Spin Models: Chains, Ladders, and Square Lattices

“…On the one hand, in the case of the quantum chain, this superdiffusive behavior is by now well established at the isotropic point (see Ref. 19 and references therein). On the other hand, in the case of the classical chain, the nature of spin transport at the isotropic point has been quite controversial [33][34][35][36][37][38][39] .…”

“…Moreover, plotted in a double-logarithmic representation [Fig. 1 (b)], we find that the hydrodynamic power-law tail C (M) (t) ∝ t −α at intermediate times is well described by α ≈ 2/3, which suggests superdiffusive transport within the Kardar-Parisi-Zhang (KPZ) universality class 18,19,57,59,62,63 (for more details see Sec. IV A 1 below).…”

“…From a theoretical point of view, quantum spin systems routinely serve as test beds to study concepts such as the eigenstate thermalization hypothesis [8][9][10][11][12] or the phenomenon of many-body localization 13,14 . Moreover, in the case of one-dimensional chain geometries, the integrability of certain spin models, accompanied by the existence of an extensive set of (quasi-)local conserved charges [15][16][17] , paves the way to obtain analytical insights, e.g., regarding their transport and relaxation behavior in the thermodynamic limit [18][19][20][21] . At the same time, the development of sophisticated numerical techniques [22][23][24] has significantly advanced our understanding of out-ofequilibrium processes in quantum spin models.…”

Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos

“…When a system is showing spatiotemporal chaos (STC), i.e., chaos characterized by spatial and temporal disorder, perturbations given to trajectories typically grow exponentially and fluctuate. Pioneering studies by Pikovsky, Kurths, and Politi [10,11] revealed that, for a wide range of systems, this growing perturbation belongs to the Kardar-Parisi-Zhang (KPZ) universality class [12], which is one of the most wide-ranging universality classes for various stochastic processes (see reviews [13][14][15][16][17][18][19]), including growing interfaces, directed polymers, stochastic particle transport, nonlinear fluctuating hydrodynamics [18,20], and most recently, to quantum spin chains [21][22][23][24], to name but a few.…”