Kardar-Parisi-Zhang Physics in the Quantum Heisenberg Magnet

Equilibrium spatiotemporal correlation functions are central to understanding weak nonequilibrium physics. In certain local one-dimensional classical systems with three conservation laws they show universal features. Namely, fluctuations around ballistically propagating sound modes can be described by the celebrated Kardar-Parisi-Zhang (KPZ) universality class. Can such a universality class be found also in quantum systems? By unambiguously demonstrating that the KPZ scaling function describes magnetization dynamics in the $SU(2)$ symmetric Heisenberg spin chain we show, for the first time, that this is so. We achieve that by introducing new theoretical and numerical tools, and make a puzzling observation that the conservation of energy does not seem to matter for the KPZ physics.

Figure 1

Collapse of spin profiles for the continuous-time (top) and discrete-time (bottom) model in terms of the scaling parameter $\xi =x/t^{2/3}$ shown for several times. The continuous-time simulation was performed on a spin chain of length $L=400$ with bond dimension $\chi =400$ and polarization $\mu =0.0017$. The discrete-time simulation was performed with $L=7200$, $\chi =256$, and $\mu =0.0005$. The same parameters are used in other figures. In the discrete case there is an additional Floquet even-odd splitting whose size decays as $t^{-1/3}$ (the inset).

Figure 2

Scaling functions and numerical data: the left column corresponds to the continuous-time model while the right corresponds to the discrete-time model. We show data for the spin current density $⟨j{⟩}_{\mu }$ and the discrete spin derivative $\mathrm{\Delta }z$, defined as $\mathrm{\Delta }z=-(⟨s_r^z{⟩}_{\mu }-⟨s_{r-1}^z{⟩}_{\mu })$ in the continuous-time model and $\mathrm{\Delta }z=-\frac{1}{4}(⟨s_{r+1}^z{⟩}_{\mu }+⟨s_r^z{⟩}_{\mu }-⟨s_{r-1}^z{⟩}_{\mu }-⟨s_{r-2}^z{⟩}_{\mu })$ in the discrete-time model. All numerical data (yellow and red points) are appropriately scaled to the KPZ scaling functions, see Eqs. (10) and (11). The blue curves represent the KPZ scaling functions while the green ones are the best-fitting Gaussian profiles. We note that relatively long times are needed in order to observe the KPZ scaling, namely, $t⪆50$ for the continuous-time model and $t⪆600$ for the discrete-time model.

Figure 3

Plotting the ratio between the gradient of spin density and spin current density in scaled units we can observe that the numerical results for both models clearly do not obey Fick’s law. Instead, they are well described by the prediction from KPZ. Numerical data are plotted for maximum simulations times ($t=200$ for continuous and $t=3600$ for discrete time cases). The ratios are rescaled to 1 at $\phi =0$.

Figure 4

Dependence on bond dimension of the current profiles in a domain-wall state and for continuous ($t=200$) and discrete-time simulations ($t=3600$). Results are stable to increasing $\chi$ and converge to the KPZ scaling functions. We apply a moving average to the leftmost and rightmost 20% of the data so that it is easier to see the decreasing truncation error in the tails.