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 $2 < α< \infty$) we observe long-lived $z=3/2$ superdiffusive spin transport and two-point correlators consistent with KPZ scaling functions, up to times $t \sim 10^3/J$. 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.
The emergence of irreversible classical hydrodynamics from charge-conserving, unitary quantum dynamics represents one of the central tenets of statistical physics. At high temperatures, conserved charges in a generic system are expected to equilibrate diffusively, resembling the motion of a random walk. Exceptions to diffusion typically rely on additional structure, such as subdiffusion in localized [1,2] and kinetically constrained [3,4] models or superdiffusion emerging from Levy flights [5,6] and nodal interactions [7]. Recent excitement has focused on a notable exception to diffusion: infinite-temperature spin transport in the integrable, nearest-neighbor (NN) quantum Heisenberg chain [8,9]. There, some but not all aspects of spin transport [9,10] are precisely governed by the Kardar-Parisi-Zhang universality class [11][12][13].Combined with extensive numerical explorations [14][15][16] and seminal experiments [17][18][19], this has led to the conjecture that local, integrable models with a continuous non-Abelian symmetry will generically exhibit KPZ-like superdiffusion with dynamical critical exponent z = 3/2 [15,20,21]. A counterexample to this conjecture is provided by the celebrated Haldane-Shastry (HS) model [22,23] -the prototypical long-range-interacting, integrable spin chain -where both energy and spin transport are ballistic (Fig. 1) [24,25]. This naturally raises the question: Do longer range interactions immediately preclude the observation of KPZlike dynamics? From an experimental perspective, this question is particularly relevant as isolated quantum simulators capable of probing such emergent hydrodynamics (e.g. Rydberg atom arrays, trapped atomic ions, po-lar molecules, etc.) often exhibit long-range interactions. This allows us to recast our original question: Does KPZlike transport ever control the physics of experiments away from fine-tuned, integrable fixed points?
In this Letter, we answer both questions by interpolating between the two aforementioned integrable “fixed points”, the NN Heisenberg spin chain and the 1/r 2interacting HS model (Fig. 1). We interpolate via two routes: (i) non-integrable, power-law-interacting 1/r α Heisenberg models [26,27] and (ii) the integrable Inozemtsev family of models [28]. For both routes, we investigate infinite temperature spin and energy transport, using tensor network algorithms to explore large systems (up to L ∼ 2000) and late times (up to t ∼ 1000/J).
Our main results are threefold. First, we demonstrate that power-law-interacting Heisenberg models, despite being non-integrable, defy diffusive expectations and exhibit KPZ-like spin dynamics up to the latest timescales numerically simulable. Although one expects such anomalous spin transport to ultimately cross over to diffusion, our results suggest that a multitude of experimental platforms realizing such long-range Heisenberg interactions are governed by KPZ-like hydrodynamics. Second, we provide a microscopic explanation for our observations above by demonstrating that many extended range Heisenberg models, including those with powerlaw interactions, can be viewed as a isotropic perturbation to an integrable Inozemtsev model; recent classical and quantum numerics has argued that similarly perturbated locally-interacting integrable models display exceptionally long crossovers to diffusion [29][30][31]. Sup-
Nearest-Neighbor and nearest-neighbor (NN) “fixed points” are integrable, and one can continuously tune between them either by the integrable Inozmentsev models parameterized by constant κ ≥ 0 or generic power-laws with decay α ≥ 2. The heat plots show the ballistic and superdiffuive “melting” of the spin domain wall (Eq. 2) for the HS and NN models, respectively. The dotted, white line indicates when the spin has deviated 1% from its initial value, with r ∼ t and r ∼ t 2/3 for HS and NN. (b) Interaction strengths for HS, Inozemtsev, and power-law-interacting models. (c) The linear magnetization profile in the HS model shows ballistic transport and collapses (inset) when rescaled by time t, indicating a dynamical critical exponent of zS = 1. Data is rescaled by the spinon speed vS = π/2, the speed at which the front propagates [25]. (d) Magnetization profile in Inozemtsev model with κ = 0.4 and collapsed data (left inset) display superdiffusive transport with zS = 3/2. For the Inozemtsev model, zS calculated from the polarization transfer is stable while that of the comparable non-integrable, next-nearest-neighbor model with coupling J2 ≈ 0.21 trends towards diffusion with z -1 S = 1/2 (right inset). (e) Energy profile for HS, κ = 1.0 Inozemtsev, and NN models, rescaled by time t show ballistic transport and dependence on κ. Profiles have been vertically shifted for visual clarity.
ported by a numerical investigation of scaling functions, we conjecture that spin transport in the Inozemtsev models, regardless of interaction range, falls into the same KPZ-like universality class as t
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