Cheaper access to universal fluctuations in integrable spin chains from boundary effects

Abstract

Observing super-diffusive fluctuations from Kardar-Parisi-Zhang (KPZ) universality in isotropic integrable spin chains is usually challenging as it requires a fairly large number of spins in interaction. We demonstrate in this paper, in the context of classical spins, that accounting for boundary effects lowers the bar, down to a few dozen spins in some cases. Additionally, boundaries control the relaxation to stationarity, which leads to many new universal scaling functions to explore, both in periodic spin chains and for open chains with magnetization imposed by reservoirs at the ends.

Figures (4)

• Figure 1: Graphical representation of the dynamics of the KPLL model with open boundaries. Spins at various sites $i$ and times $T$ are coupled according to the dashed green links.
• Figure 2: (a) Stationary magnetization profile $C_{1}(x)$ plotted as a function of the position $x=i/L$. The green dots represent results from numerical simulations of the KPLL model with $L=1023$ spins, $T=10^{5}$ time steps, averaged over $10^{7}$ realizations, with boundary parameters $s_{a}=1$, $s_{b}=-0.5$. The solid black line in the middle is the exact KPZ result (\ref{['C1[c1-c1] open']}) normalized accordingly, with boundary densities $\sigma_{a}\approx1.33$, $\sigma_{b}\approx-0.63$ adjusted from $\langle S_{1}^{\rm z}(T)\rangle$, $\langle S_{L}^{\rm z}(T)\rangle$. The solid red and blue lines below and above correspond to keeping a single KPZ mode. (b) Difference between the KPLL simulations and the exact result (statistical average $\pm$ one standard deviation) for the stationary height profile (integral of $C_{1}(x)$ with respect to $x$), with $L=63,255,1023$ spins from top to bottom, roughly compatible with finite size corrections vanishing as $L^{-1/2}$. (c) Variance of the total magnetization $L^{1/2}\operatorname{Var}(M)$ with $L=511$ spins plotted as a function of $\arg\mathsf{s}$ for $\mathsf{s}=s_{a}+\mathrm{i} s_{b}$. (d) $m_{a},m_{b}$ from (\ref{['eq boundaries']}) for the values of $s_{a},s_{b}$ in (c).
• Figure 3: Stationary two-point correlation function $C_{2}(x,t)$ plotted as a function of the position $x$ for a few values of time $t$, for periodic (a) and open (b) spin chains. The solid line is the KPZ prediction, for times $t\approx0.020, 0.037, 0.074$ in (a) and times $t\approx0.0056, 0.0118, 0.0228$ in (b), in order of decreasing amplitude. The dots correspond to numerical simulations of the KPLL model with $L=32$ spins, $T=30, 50, 90$ time steps and averaging over $10^{5}$ samples in (a), and with $L=511$ spins, $T=1000, 2000, 4000$ time steps (in addition to $2\times10^{5}$ preliminary time steps used to reach the stationary state) and averaging over $10^{7}$ samples in (b). In (b), the boundary magnetizations $s_{a}=1$, $s_{b}=-1$ defined in (\ref{['Sab[sab]']}) correspond to KPZ boundary densities $\sigma_{a}=-\sigma_{b}\approx2.4$.
• Figure 4: First Fourier coefficients $a_{1}(t),\ldots,a_{5}(t)$ (from top to bottom) of the stationary two-point function $C_{2}(x,t)\simeq c_{2}(x,t)$ with periodic boundaries, plotted as a function of time. The solid lines are the exact results for the $a_{p}(t)$, and the dashed lines the short time approximation $\hat{c}_{2}^{\rm line}(2\pi k/\ell)$. The symbols represent the data from numerical simulations of the KPLL model (Ishimori chain) averaged over $10^{6}$ ($10^{5}$) samples. Inset: zoom around the first zero of $a_{2}(t)$.