Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains

TL;DR

Isotropic spin chains with continuous non-Abelian SU(2) symmetry pose a long-standing question about KPZ universality; this work tests whether KPZ scaling governs transport in both classical and quantum integrable spin chains. The authors combine the classical KPLL model and the quantum isotropic Heisenberg chain, and analyze multiple two-point quantities against exact KPZ results, notably the equal-time correlator $C_2(\ell,t)$, the height increment $h(x,t)$, and the temporal correlator $C_t(t_1,t_2)$. They find precise KPZ scaling for two-point quantities with no adjustable parameters: $C_2( obreak\ell,t) \simeq \frac{2\alpha t^{2/3}}{\xi(t)^2} f_{KPZ}(\ell/\xi(t))$ with $\xi(t) \sim t^{2/3}$ and $\mathrm{Var}[h] \sim \alpha t^{2/3}$, while higher-order cumulants such as kurtosis remain inconsistent with the Baik-Rains stationary state, indicating partial emergence of the KPZ class. Moreover, KPZ behavior persists in the presence of an energy current when $h(x,t)$ is measured in a comoving frame, but introducing local SU(2) symmetry breaking (e.g., a flat initial condition) triggers a crossover to diffusive scaling at a timescale $\mu^{-3}$, highlighting the boundary between KPZ-relevant and non-KPZ regimes.

Abstract

Integrable spin chains with a continuous non-Abelian symmetry, such as the one-dimensional isotropic Heisenberg model, show superdiffusive transport with little theoretical understanding. Although recent studies reported a surprising connection to the Kardar-Parisi-Zhang (KPZ) universality class in that case, this view was most recently questioned by discrepancies in full counting statistics. Here, by combining extensive numerical simulations of classical and quantum integrable isotropic spin chains with a framework developed by exact studies of the KPZ class, we characterize various two-point quantities that remain hitherto unexplored in spin chains, and find full agreement with KPZ scaling laws without adjustable parameters. This establishes the partial emergence of the KPZ class in integrable isotropic spin chains. Moreover, we reveal that the KPZ scaling laws are intact in the presence of an energy current, under the appropriate Galilean boost required by the propagation of spacetime correlation.

Figures (6)

• Figure 1: The two-point function and the magnetization transfer cumulants for the KPLL model. (a) Rescaled two-point function $\tilde{C}_2(\ell,t) = \frac{\xi(t)}{\Omega}C_2(\ell,t)$ against $\ell/\xi(t)$, compared with the Prähofer-Spohn solution $f_\mathrm{KPZ}(\cdot)$. (b) Correlation length $\xi(t)$. (c) Variance (main panel) and skewness (inset) of the magnetization transfer $h$. The dashed line in the inset indicates the skewness of the Baik-Rains distribution. (d) Ratio of $\alpha_1(t)$ [from Eq. (\ref{['main:eq:C2']})] and $\alpha_2(t)$ [from Eq. (\ref{['main:eq:var']})].
• Figure 2: Spatial (a) and temporal (b) correlation functions of the magnetization transfer for the KPLL model. (a) The rescaled spatial correlator $\tilde{C}_s(u) = C_s(\ell,t)/\alpha t^{2/3}$ (main panel) and its slope $\frac{d\tilde{C}_s}{du}(u)$ (left inset) against $u =\ell/\xi(t)$. The dashed lines show the curves for the KPZ class, obtained by TASEP simulations. The right inset compares the slope $\frac{d\tilde{C}_s}{du}(0)$ at $u=0$ with our exact result for the KPZ class, $\frac{d\tilde{C}_s}{du}(0)=-2$. (b) The rescaled temporal correlator $C_t(t_1,t_2)/C_t(t_2,t_2)$ against $t_1/t_2$, compared with the Ferrari-Spohn solution [Eq. (\ref{['main:eq:FS']})] for the KPZ class.
• Figure 3: Results for the quantum Heisenberg model, with system size $L=100$ and maximum bond dimension $\chi=1600$ (see End Matter for details). (a) Correlation length $\xi(t)$ and variance of the magnetization transfer, Var[$h$]. (b) Ratio $\alpha_2(t)/\alpha_1(t)$. See End Matter for the evaluation of the error bars. (c,d) Spatial (c) and temporal (d) correlation functions of the magnetization transfer.
• Figure 4: Results for KPLL with a finite energy current. (a) Two-point function $C_2(\ell,t)/t^{2/3}$ against $\ell/\xi(t)$ for different times, $t = 3000, 8000, 15000, 23000, 32000$ from left to right. Data smoothed by the locally weighted scatterplot smoothing method are displayed. Inset: total energy current $J_E(t)$. (b)(c) The location and the velocity of the peak of $C_2(\ell,t)$, $\ell_\mathrm{peak}(t)$ and $v_\mathrm{peak}$, respectively. The dashed line in (b) shows $\ell_\mathrm{peak}(t) = vt$ with $v = 0.5523$, wrapped by the periodic boundary. (d) Rescaled two-point function $\tilde{C}_2(\ell,t)$ centered at $\ell = \ell_\mathrm{peak}(t)$ (symbols, same colors as (a)), compared with the Prähofer-Spohn solution $f_\mathrm{KPZ}(\cdot)$ (dashed line). (e) Correlation length $\xi(t)$. The black dots are the data for the case without energy current, shown in Fig. \ref{['main:fig1']}(b). (f) Variance of the magnetization transfer $h(x,t)$, measured in the original and comoving frames (blue circles and red squares, respectively). (g)(h) Skewness and kurtosis of the magnetization transfer $h(x,t)$ in the comoving frame. The values for the Baik-Rains distribution are $0.359$ and $0.289$, respectively Prahofer.Spohn-PRL2000, which are far from the data.
• Figure 5: Results for KPLL with the flat initial condition. (a) Snapshots of the height $h_0(x,t) = h(x,t) + h_0(x,0)$ at $t=0, 2000, 4000, \cdots, 10000$ from bottom to top, for $\mu=1$. For visibility, every subsequent snapshot is shifted upward by $20$. (b) Variance of the magnetization transfer $h$ for different $\mu$. The dashed line is the KPZ growth law (\ref{['main:eq:var']}) with $\alpha$ determined from the equilibrium simulations [Fig. \ref{['main:fig1']}(d)]. The dashed-dotted line is a guide for the eyes showing Var$[h] \sim t^{1/2}$. Inset: rescaled variance Var$[h]/\alpha t^{2/3}$ against $\mu^3 t$, for $\mu = 0.1, 0.5, 1, 2$, from top to bottom at the leftmost part of the datasets. The bold solid line displays the behavior for KPZ interfaces Takeuchi-PRL2013 (with arbitrary horizontal shift), showing crossover from the Baik-Rains (BR) distribution (dotted line) to the characteristic distribution for flat interfaces, namely the GOE Tracy-Widom distribution (dashed line). Simulation parameters were $L = 40,000$ and $N=1,000$ for $\mu = 0.1, 0.5, 1$ and $L = 400,000$ and $N=33$ for $\mu = 2, 10$.