High-temperature spin dynamics in the Heisenberg chain: Magnon propagation and emerging Kardar-Parisi-Zhang scaling in the zero-magnetization limit

Abstract

The large-scale dynamics of quantum integrable systems is often dominated by ballistic modes due to the existence of stable quasi-particles. We here consider as an archetypical example for such a system the spin-1/2 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 Π(x, t) (at high temperatures) typically exhibits a trimodal behavior with left and right moving quasi-particle modes and a broad center peak with slower dynamics. The broadening of the fastest propagating modes exhibits a sub-diffusive t 1/3 scaling at large magnetization densities, m ≈ 1/2, familiar from non-interacting models; it crosses over into a diffusive scaling t 1/2 upon decreasing the magnetization to smaller values. The behavior of the center peak appears to exhibit a crossover from transient super-diffusion to ballistic relaxation at long times. In the limit m→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 3/2 familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm that, employing elaborate finite time extrapolations, that the spatial scaling of the correlator Π is in excellent agreement with KPZ-type behavior and analyze the corresponding corrections. :1908.11432v1 [cond-mat.stat-mech] arXiv

Finite-temperature transport in one-dimensional quantum lattice models

“…The same result has been derived via a GHD-based kinetic picture , and the time-dependent diffusion constant was predicted to scale as D (S) (t) ∝ t 1/3 . This was confirmed in another GHD study (Agrawal et al, 2020), yet without invoking the Kardar-Parisi-Zhang (KPZ) scaling mechanism (Kardar et al, 1986;Ljubotina et al, 2019aLjubotina et al, , 2017Spohn, 2020a;Weiner et al, 2020). This superdiffusive behavior is consistent with finitetime tDMRG data for D (S) (t) at ∆ = 1 [see Fig.…”

“…Moreover, numerical evidence for superdiffusive spin transport with a dynamical exponent of z = 3/2 at the Heisenberg point ∆ = 1 has been found in (Ljubotina et al, 2019a(Ljubotina et al, , 2017 and self-consistently explained within GHD (Bulchandani et al, 2020;De Nardis et al, 2019b, 2020b. This is the same exponent as in the Kardar-Parisi-Zhang universality class (Kardar et al, 1986) leading to the actively investigated question of whether this scenario is realized in the spin-1/2 Heisenberg chain and possibly other systems with SU(2)-symmetric exchange (De Nardis et al, 2019b;Dupont and Moore, 2020;Ljubotina et al, 2019aLjubotina et al, , 2017Spohn, 2020a;Weiner et al, 2020).…”

“…However, the origin and nature of this non-diffusive process is not fully understood yet. First attempts have been undertaken, and there is mounting support for the dynamical exponent of z = 3/2 (Ljubotina et al, 2017), consistent with KPZ-scaling (Ljubotina et al, 2019a) further corroborated and discussed in (Bulchandani, 2020;Bulchandani et al, 2019;De Nardis et al, 2020a;Spohn, 2020a;Weiner et al, 2020). Whether the KPZ-like scaling persists in other isotropic spin models with or without integrability is currently the object of intense scrutiny (see, e.g., (De Nardis et al, 2019b;Dupont and Moore, 2020)).…”

Diffusion from convection

“…Such situations were typically exempted from past NLFHD applications, where the non-degeneracy of velocities was usually assumed in order to justify the mode decoupling assumption. Such situations are, however, expected to be important in integrable chains, where super-diffusion can also be observed for the spin or charge degrees of freedom close to half-filling [56][57][58][59].…”

Integrable matrix models in discrete space-time

“…This work has been largely inspired by the recent discovery of superdiffusive magnetization transport in the isotropic Heisenberg spin-1/2 chain [66][67][68], subsequently scrutinized in a number of papers [41,42,69], collectively accumulating a convincing numerical evidence for the Kardar-Parisi-Zhang (KPZ) type universality [70] (see also [71,72] and [73][74][75]). It is remarkable that the same phenomenon is already visible at the classical level, namely in the integrable classical spin chains symmetric under global SO (3) rotations [24,76,77].…”