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arXiv:2406.02782 · Journal of High Energy Physics

KPZ scaling from the Krylov space

A. Gorsky, Sergei Nechaev, Alexander Valov

Published Sep 5, 2024• 3 citations
arXivHTMLPDF

Abstract

A bstract Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling in late-time correlators and autocorrelators of certain interacting many-body systems, such as Heisenberg spin chains, has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis. We focus on the Heisenberg time scale, which approximately corresponds to the ramp–plateau transition for the Krylov complexity in systems with a large but finite number degrees of freedom. Two frameworks are under consideration: i) the system with growing Lanczos coefficients and an artificial cut-off, and ii) the system with the finite Hilbert space. In both cases via numerical analysis, we observe the transition from Gaussian to KPZ-like scaling, or equivalently, the diffusion-superdiffusion transition at the critical Euclidean time $$ {t}_E^{\ast } $$ t E ∗ = c cr K , for the Krylov chain of finite length K , and c cr = O (1). In particular, we find a scaling ~ K 1/3 for fluctuations in the one-point correlation function and a dynamical scaling ~ K −2/3 associated with the return probability (Loschmidt echo) corresponding to autocorrelators in physical space. In the first case, the transition is of the 3rd order and can be considered as an example of dynamical quantum phase transition (DQPT), while in the second, it is a crossover. For case ii), utilizing the relationship between the spectrum of tridiagonal matrices at the spectral edge and the spectrum of the stochastic Airy operator, we demonstrate analytically the origin of the KPZ scaling for the particular Krylov chain using the results of the probability theory. We argue that there is interesting outcome of our study for the double scaling limit of matrix models. For the case of topological gravity, the white noise term of order O (1/ N ) is identified, which should be taken into account in the controversial issue of ensemble averaging in 2D/1D holography.

DOI
10.1007/jhep09(2024)021

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A bstract Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling in late-time correlators and autocorrelators of certain interacting many-body systems, such as Heisenberg spin chains, has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis. We focus on the Heisenberg time scale, which approximately corresponds to the ramp–plateau transition for the Krylov complexity in systems with a large but finite number degrees of freedom. Two frameworks are under consideration: i) the system with growing Lanczos coefficients and an artificial cut-off, and ii) the system with the finite Hilbert space. In both cases via numerical analysis, we observe the transition from Gaussian to KPZ-like scaling, or equivalently, the diffusion-superdiffusion transition at the critical Euclidean time tE∗{t}_E^{\ast }tE∗​ t E ∗ = c cr K , for the Krylov chain of finite length K , and c cr = O (1). In particular, we find a scaling ~ K 1/3 for fluctuations in the one-point correlation function and a dynamical scaling ~ K −2/3 associated with the return probability (Loschmidt echo) corresponding to autocorrelators in physical space. In the first case, the transition is of the 3rd order and can be considered as an example of dynamical quantum phase transition (DQPT), while in the second, it is a crossover. For case ii), utilizing the relationship between the spectrum of tridiagonal matrices at the spectral edge and the spectrum of the stochastic Airy operator, we demonstrate analytically the origin of the KPZ scaling for the particular Krylov chain using the results of the probability theory. We argue that there is interesting outcome of our study for the double scaling limit of matrix models. For the case of topological gravity, the white noise term of order O (1/ N ) is identified, which should be taken into account in the controversial issue of ensemble averaging in 2D/1D holography.
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<∞2 < α< \infty2<α<∞) we observe long-lived z=3/2z=3/2z=3/2 superdiffusive spin transport and two-point correlators consistent with KPZ scaling functions, up to times t∼103/Jt \sim 10^3/Jt∼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.
Finite temperature spin transport in integrable isotropic spin chains is known to be superdiffusive, with dynamical spin correlations that are conjectured to fall into the Kardar-Parisi-Zhang (KPZ) universality class. However, integrable spin chains have time-reversal and parity symmetries that are absent from the KPZ (Kardar-Parisi-Zhang) or stochastic Burgers equation, which force higher-order spin fluctuations to deviate from standard KPZ predictions. We put forward a nonlinear fluctuating hydrodynamic theory consisting of two coupled stochastic modes: the local spin magnetization and its effective velocity. Our theory fully explains the emergence of anomalous spin dynamics in isotropic chains: it predicts KPZ scaling for the spin structure factor but with a symmetric, quasi-Gaussian, distribution of spin fluctuations. We substantiate our results using matrix-product states calculations.
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-12\frac{1}{2}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=0m=0m=0. At a finite magnetization density mmm, 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{t}^{1/3}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{t}^{1/2}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 zzz close to the value 32\frac{3}{2}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.
The PXP chain was recently shown to exhibit superdiffusive energy transport with Kardar-Parisi-Zhang-like scaling, z≈3/2z\approx3/2z≈3/2, joining a growing number of spin chains with this exponent. An understanding of how this anomalous hydrodynamics emerges from microscopics is, however, still lacking. In this work, we show that finite-temperature energy transport in this model provides a window into the emergence of superdiffusion. At finite temperature, the energy autocorrelation function exhibits a crossover from short-time coherent dynamics to long-time hydrodynamics. The short-time behavior is dominated by a single magnon band and can be understood analytically. In momentum space, this regime is characterized by spectral weight near q=πq=πq=π. The damping time τττ, which separates the short-time magnon-dominated behavior from the late-time hydrodynamics, grows rapidly upon cooling, consistent with an activated form τ(β)∼βeΔβτ(β)\sim βe^{Δβ}τ(β)∼βeΔβ with a gap scale set by the magnon band. At longer times, the spectral weight transfers to q=0q=0q=0 and the running decay exponent drifts toward the superdiffusive value z=3/2z=3/2z=3/2. Finite-temperature energy transport therefore provides a bridge between microscopic magnon physics and late-time superdiffusion in the PXP model.
Abstract In its original version, the KPZ equation models the dynamics of an interface bordering a stable phase against a metastable one. Over the past few years the corresponding two-dimensional field theory has been applied to models with different physics. Out of a wide choice, the spin–spin time correlations for the Heisenberg chain will be discussed at some length, also the equilibrium time correlations of the conserved fields for 1D fluids. An interesting recent theoretical advance is the construction of the scale-invariant asymptotic theory, the so-called KPZ fixed point.
We study the single-particle Green's function G(x,t)=⟨σx−(0)σ0+(t)⟩G(x,t)=\langle σ^-_x(0)σ^+_0(t)\rangleG(x,t)=⟨σx−​(0)σ0+​(t)⟩ in one-dimensional particle-number-conserving random unitary circuits coupled to an external bath. For fixed spacetime disorder, we argue that G(x,t)G(x,t)G(x,t) is governed, in both the strong- and weak-noise limits, by directed waves in a random medium. We find Kardar-Parisi-Zhang (KPZ) scaling in the wandering statistics of the normalized spatial distribution p(x,t)∝∣G(x,t)∣2p(x,t)\propto |G(x,t)|^2p(x,t)∝∣G(x,t)∣2 and in the associated free energy. In particular, its center ⟨x(t)⟩≡∑xx p(x,t)\langle x(t)\rangle\equiv\sum_x x\, p(x,t)⟨x(t)⟩≡∑x​xp(x,t) wanders on a length-scale O(t2/3)\mathcal{O}(t^{2/3})O(t2/3), while sample-to-sample fluctuations of −log⁡∑x∣G(x,t)∣2-\log\sum_x |G(x,t)|^2−log∑x​∣G(x,t)∣2 scale as t1/3t^{1/3}t1/3. At weak noise γ≪1γ\ll 1γ≪1, the crossover to the strong-disorder fixed point occurs at a parametrically long time O(γ−3/2)\mathcal{O}(γ^{-3/2})O(γ−3/2). These predictions are confirmed numerically using tensor-network simulations of the noisy operator dynamics in individual circuits at moderate noise, and of a phase-annealed proxy retaining hopping disorder at weak noise.