Spatiotemporal dynamics of classical and quantum density profiles in low-dimensional spin systems

We provide a detailed comparison between the dynamics of high-temperature spatiotemporal correlation functions in quantum and classical spin models. In the quantum case, our large-scale numerics are based on the concept of quantum typicality, which exploits the fact that random pure quantum states can faithfully approximate ensemble averages, allowing the simulation of spin-$1/2$ systems with up to 40 lattice sites. Due to the exponentially growing Hilbert space, we find that for such system sizes even a single random state is sufficient to yield results with extremely low noise that is negligible for most practical purposes. In contrast, a classical analog of typicality is missing. In particular, we demonstrate that to obtain data with a similar level of noise in the classical case, extensive averaging over classical trajectories is required, no matter how large the system size. Focusing on (quasi-)one-dimensional spin chains and ladders, we find remarkably good agreement between quantum and classical dynamics. This applies not only to cases where both the quantum and classical models are nonintegrable but also to cases where the quantum spin-$1/2$ model is integrable and the corresponding classical $s\to \infty$ model is not. Our analysis is based on the comparison of space-time profiles of the spin and energy correlation functions, where the agreement is found to hold not only in the bulk but also in the tails of the resulting density distribution. The mean-squared displacement of the density profiles reflects the nature of emerging hydrodynamics and is found to exhibit similar scaling for quantum and classical models.

Figure 1

Exemplary plot of an initially peaked density profile that broadens over time by diffusion. The data shows quantum and classical magnetization dynamics in the XXZ spin chain with anisotropy $\mathrm{\Delta }=1.5$ and system size $L=36$. For a more detailed discussion of the results, see Sec. 4a below. Note that the legend in the lower right panel applies to all other panels.

Figure 2

Single-state trajectories of $C_{L/2}\left(t\right)$ (blue lines) for 300 random initial states in the XXZ chain with $\mathrm{\Delta }=1.5$ in the classical case (a) and the quantum case (b). Red and orange lines show the corresponding averages over $N={10}^3$ and $N={10}^9$ (only classical) trajectories. (c) Relative variance $R\left(t\right)$ of sample-to-sample fluctuations as obtained by Eq. (33) for $N={10}^3$ and different system sizes $L=2^4,2^5,\cdots ,2^{19}$ (classical) and $L=8,9,\cdots ,20$ (quantum). Dashed line indicates the value of $R\left(0\right)$ for the classical case which is essentially the second moment of the probability distribution for ${\varrho }_{L/2}^2\left(0\right)$ that arises from the random initial configuration of the state. Note that we consider the high-temperature limit $\beta \to 0$ and that $C_{L/2}\left(0\right)$ is set to 1.

Figure 3

(a1)–(c2) Profiles $C_r\left(t\right)$ of magnetization densities in the XXZ spin chain Eq. (3) with anisotropy $\mathrm{\Delta }=1.5$ and $L=36$, sampled over $N$ initial states. Solid lines are Gaussian fits (24) to the QM data. (d) Time-dependent spatial width $\mathrm{\Sigma }\left(t\right)$ as obtained by Eq. (22). Dashed and solid lines indicate scaling $t^{\alpha }$ for $\alpha =1$ and $1/2$. To illustrate the necessity for large sample sizes in the simulations of classical dynamics, additional CM data is shown for different sample sizes $N=2^l\times {10}^3$ with $l=0,1,\cdots ,20$ (grey scales). Note that we consider the high-temperature limit $\beta \to 0$ and that the legend in panel (d) applies to all other panels.

Figure 4

(a1)–(c2) Profiles $C_r\left(t\right)$ of magnetization densities in the XXZ spin chain Eq. (3) with anisotropy $\mathrm{\Delta }=1.0$ and $L=36$, sampled over $N$ initial states. Solid lines are Gaussian fits Eq. (24) to the QM data. Dashed lines indicate KPZ scaling functions [104]. Dotted lines indicate a function $\propto exp(-a|r-L/2{|}^3)$. (d) Time-dependent spatial width $\mathrm{\Sigma }\left(t\right)$ as obtained by Eq. (22). Dashed and solid lines indicate scaling $t^{\alpha }$ for $\alpha =1$ and $2/3$. Note that we consider the high-temperature limit $\beta \to 0$ and that the legend in panel (d) applies to all other panels.

Figure 5

(a1)–(c2) Profiles $C_r\left(t\right)$ of magnetization densities in the XXZ spin chain Eq. (3) with anisotropy $\mathrm{\Delta }=0.5$ and $L=36$, sampled over $N$ initial states. (d) Time-dependent spatial width $\mathrm{\Sigma }\left(t\right)$ as obtained by Eq. (22). Solid line indicates scaling $t^{\alpha }$ for $\alpha =1$. Note that we consider the high-temperature limit $\beta \to 0$ and that the legend in panel (d) applies to all other panels.

Figure 6

(a1)–(c2) Profiles $C_r\left(t\right)$ of magnetization densities in the XXX spin ladder Eq. (4) with $L=20$, sampled over $N$ initial states. Solid lines are Gaussian fits Eq. (24) to the QM data. (d) Time-dependent spatial width $\mathrm{\Sigma }\left(t\right)$ as obtained by Eq. (22). Dashed and solid lines indicate scaling $t^{\alpha }$ for $\alpha =1$ and $1/2$. Note that we consider the high-temperature limit $\beta \to 0$ and that the legend in panel (d) applies to all other panels.

Figure 7

(a1)–(c2) Profiles $C_r\left(t\right)$ of energy densities in the XXX spin ladder Eq. (4) with $L=20$, sampled over $N$ initial states. Solid lines are Gaussian fits Eq. (24) to the QM data. (d) Time-dependent spatial width $\mathrm{\Sigma }\left(t\right)$ as obtained by Eq. (22). Solid line indicates scaling $t^{\alpha }$ for $\alpha =1/2$. Note that we consider the high-temperature limit $\beta \to 0$ and that the legend in panel (d) applies to all other panels.

Figure 8

(a1)–(c2) Profiles $C_r\left(t\right)$ of charge densities in the Fermi-Hubbard chain Eq. (5) with $L=18$ and $U/t_{\mathrm{h}}=4$, sampled over $N$ initial states. Solid lines are Gaussian fits Eq. (24) to the QM data. (d) Time-dependent spatial width $\mathrm{\Sigma }\left(t\right)$ as obtained by Eq. (22). Dashed and solid lines indicate scaling $t^{\alpha }$ for $\alpha =1$ and $1/2$. Note that we consider the high-temperature limit $\beta \to 0$ and that the legend in panel (d) applies to all other panels.

Figure 9

Classical dynamics of $C_r\left(t\right)$ for $r=L/2$ and $r=L/2+100$ in a XXZ spin chain with anisotropy $\mathrm{\Delta }=0.5$ and large system size $L=2000$. Dashed lines indicate a Gaussian fit to the equal-site correlation function $C_{L/2}\left(t\right)$ [cf. Eq. (A1)]. Note that we consider the high-temperature limit $\beta \to 0$.