Scaling and Universality at Noise-Affected Non-Equilibrium Spin Correlation Functions

TL;DR

The work investigates universal scaling and dynamical universality of nonequilibrium spin correlations in the XY chain under uncorrelated noise added to a linear ramp drive. By mapping to free fermions and exploiting Pfaffian representations and Toeplitz determinant formalism, it reveals that the noise lowers the boundary sweep velocity for singular spin correlations, with $v_c(\xi)=v_c(0)-b_\gamma\xi^2$, and introduces a noise-induced highly oscillatory regime marked by a finite window of maximally mixed modes where $p_k=1/2$. A secondary scale $v_m(\xi)=c_\gamma\xi^2$ delineates a multi-critical region, and both scales exhibit universal collapse under rescaling $\xi\to \xi/\sqrt{\gamma}$ and appropriate rescaling of $v$, implying a universal dynamical structure controlled by noise. The analysis also connects the singular behavior of spin correlations to Fisher–Hartwig zeros of the generating functions, with the noiseless case showing simple zeros ($\alpha_z=1/2$) and a merged double zero at the critical point ($\alpha_z=1$), while noise requires numerical extraction of exponents. Overall, the results demonstrate that stochastic driving can generate robust universal features and new dynamical phases, linking to dynamical quantum phase transitions and showing noise as a constructive ingredient in nonequilibrium quantum critical phenomena.

Abstract

We investigate scaling and universality in nonequilibrium spin correlation functions in the presence of uncorrelated noise. In the absence of noise, spin correlation functions exhibit a crossover from monotonic decay at fast sweep velocities to oscillatory behavior at slow sweeps. We show that, under a stochastically driven field, the critical sweep velocity at which the spin correlation functions undergo an abrupt change decreases with increasing noise strength and scales linearly with the square of the noise intensity. Remarkably, when the noise intensity and sweep velocity are comparable, the excitation probability becomes locked to pk = 1/2 over a finite momentum window, signaling the emergence of noise-induced maximally mixed modes. This gives rise to a highly oscillatory region in the dynamical phase diagram, whose threshold sweep velocity increases with noise and likewise exhibits quadratic scaling with the noise strength. Finally, we identify a universal scaling function under which all boundary sweep-velocity curves collapse onto a single universal curve.

Figures (2)

• Figure 1: (a) Schematic illustration of a linear ramp quench (red), with superposed noise fluctuations (gray). Here, $h(t)$ denotes the magnetic field, with $h_i$ and $h_f$ its initial and final values, and $t_i$ and $t_f = 0$ the corresponding times. Panels (b)–(d) show the probabilities $p_k$ of finding a fermionic mode with momentum $k$ in the upper band after a ramp across two quantum critical points at $h_c = \pm 1$ ($h_i = -100$, $h_f = 100$), for different noise amplitudes $\xi$ and sweep velocities: (b) $v = 0.5$, (c) $v = 1$, and (d) $v = 3$.
• Figure 2: (a) Dynamical phase diagram of the $XY$ model in the $\xi$--$v$ plane for different values of the anisotropy $\gamma$, following a noisy quench across two quantum critical points at $h_c = -1$ and $h_c = 1$ ($h_i = -100$, $h_f = 100$). The critical sweep velocity $v_c$ separates regions in which spin correlation functions exhibit oscillatory and monotonic decay behavior. The oscillatory regime further splits into two distinct regions: a highly oscillatory region associated with multi-critical modes (MCMs) and an oscillatory region characterized by two critical modes (TCMs), distinguished by the multi-critical-mode velocity $v_m(\xi)$. (b) The critical sweep velocity $v_c(\xi)$ and the multi-critical-mode velocity $v_m(\xi)$ are shown to scale linearly with the square of the noise strength, $\xi^2$. Inset: scaling of the slopes of $v_c(\xi)$ and $v_m(\xi)$, denoted by $b_\gamma$ and $c_\gamma$, respectively, as functions of the anisotropy $\gamma$. (c) Scaling invariance of the critical sweep velocity in the presence of noise, $v_c(\xi)$, and of $v_m(\xi)$ (inset). Data for different values of the anisotropy collapse onto a single universal curve.