Spatiotemporal Crossover between Low- and High-Temperature Dynamical Regimes in the Quantum Heisenberg Magnet

The stranglehold of low temperatures on fascinating quantum phenomena in one-dimensional quantum magnets has been challenged recently by the discovery of anomalous spin transport at high temperatures. Whereas both regimes have been investigated separately, no study has attempted to reconcile them. For instance, the paradigmatic quantum Heisenberg spin-$1/2$ chain falls at low temperature within the Tomonaga-Luttinger liquid framework, while its high-temperature dynamics is superdiffusive and relates to the Kardar-Parisi-Zhang universality class in $1+1$ dimensions. This Letter aims at reconciling the two regimes. Building on large-scale matrix product state simulations, we find that they are connected by a temperature-dependent spatiotemporal crossover. As the temperature $T$ is reduced, we show that the onset of superdiffusion takes place at longer length and timescales $\propto 1/T$. This prediction has direct consequences for experiments including nuclear magnetic resonance: it is consistent with earlier measurements on the nearly ideal Heisenberg $S=1/2$ chain compound ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$, yet calls for new and dedicated experiments.

Figure 1

Log-scale intensity plot of the Euclidean norm of the spin-spin correlation (2) at $T=0.25$. Simulation obtained for $L=256$ with $\chi =1024$. The goal of this Letter is to determine and study the superdiffusive region delimited by the spatiotemporal crossover $t^{\star }$ of Eq. (3) versus the temperature (white circles and dashed white line). As the temperature is decreased, we find that the superdiffusive region is shifted vertically to longer and longer times by a factor $\propto 1/T$ and eventually disappears at exactly zero temperature.

Figure 2

Time dependence of the norm of the spin-spin correlation (2) at $x=0$ for various temperatures $T$. Simulations obtained for $L=256$ with $\chi =1024$. At long time, it displays an algebraic decay with time, according to Eq. (3). It is well fitted by the form $\mathrm{\Upsilon }(T)t^{-2/3}$ with $\mathrm{\Upsilon }(T)$, a temperature-dependent prefactor decreasing with the temperature reported in Fig. 3. The deviation from the genuine power law at long time is the result of the bond dimension being too small [49].

Figure 3

The data points are extracted from Fig. 2. (a) Temperature dependence of the crossover timescale $t^{\star }(x=0,T)$ beyond which the algebraic decay $\propto t^{-2/3}$ for superdiffusive hydrodynamics emerges, see Eq. (3). It shows a linear dependence with the inverse temperature (dashed line). (b) Temperature dependence of the prefactor $\mathrm{\Upsilon }(T)$ of the algebraic decay $\propto t^{-2/3}$ for superdiffusive hydrodynamics. At low temperatures $T\lesssim 1$, it follows a quadratic dependence $\propto T^2$ (dashed line).

Figure 4

(a) Time dependence of the norm of the spin-spin correlation (2) at $T=0.25$ for various distances $x$. Simulations obtained for $L=256$ with $\chi =1024$. The curves have been shifted vertically for visibility. At long time, it displays an algebraic decay with time, according to Eq. (3), well fitted by the form $\propto t^{-2/3}$. The deviation from the genuine power law at long time is the result of the bond dimension being too small [49]. (b) Spatial dependence of the crossover time $t^{\star }(x,T)$ beyond which the algebraic decay $\propto t^{-2/3}$ for superdiffusive hydrodynamics emerges, see Eq. (3). The dashed lines are fits of the form $A+B|x{|}^{3/2}$ with $A\equiv t^{\star }(0,T)$ and $B=0.17(3)$ found to be temperature independent [49].