Spin transport in a long-range-interacting spin chain

We numerically study spin transport and spin-density profiles after a local quench in a clean one-dimensional spin chain with long-range interactions, decaying as a power law, $r^{-\alpha }$ with distance. We find two distinct regimes of transport: for $\alpha <1/2$, spin excitations relax instantaneously in the thermodynamic limit, and for $\alpha >1/2$, spin transport combines both diffusive and superdiffusive features. We show that while for $\alpha >3/2$ the spin diffusion coefficient is finite, transport in the system is never strictly diffusive, contrary to corresponding classical systems.

Figure 1

A cartoon describing the nature of transport in one-dimensional interacting systems, with an interaction decreasing as $r^{-\alpha }$ with the distance. For $0<\alpha <1$ the energy of the system is superextensive, resulting in the failure of conventional thermodynamics. For $0<\alpha <1/2$, dynamics corresponds to dynamics of the infinite-range $\left(\alpha \to 0\right)$ mean-field model in the limit of $L\to \infty$. For $\alpha >1/2$ transport combines diffusive and superdiffusive features, with a finite diffusion coefficient for $\alpha >3/2$ and $〈x^{2q}〉\left(t\right)\sim t^q$ for $q<\alpha -1/2$.

Figure 2

Upper panels: Spin excitation profiles as a function of time for two representative $\alpha$. The dashed black lines correspond to results obtained in the $\alpha \to \infty$. Darker tones represent longer times. Lower panels: Logarithmic derivative of the spin excitation profiles. The dashed black lines are guides to the eye for $2\alpha$. $L=201, \chi =256$.

Figure 3

Left panel: Spin excitation profiles at $t=2.0$ and various $\alpha$. Darker tones represent larger $\alpha$. The black dots represent a Gaussian fit. Right panel: Logarithmic derivative of the spin excitation profiles. The dashed black lines are guides to the eye for $2\alpha$. $L=201, \chi =256$.

Figure 4

The power-law exponent, $\gamma$, of the power-law tail in the spin excitation profiles obtained by averaging over the logarithmic derivative in Fig. 3 in different spatial regions. The yellow (light) line is the exponent $\gamma$ computed for the noninteracting model in Eq. (7). The dashed black line corresponds to $\gamma =2\alpha$.

Figure 5

Left panels: Time-dependent diffusion constant $D\left(t\right)$ for $\alpha =1.3$ and 3 and three different system sizes, $L=101(\chi =512),201$ and $301(\chi =256)$. Middle and right panels: Short-time relaxation of the central spin, $C_0\left(t\right)$ for $\alpha =0.3$ and 0.7 versus time (middle) and rescaled time (right) using the square root of the generalized harmonic numbers, $\sqrt{H_{L/2}^{\left(2\alpha \right)}}$ (see text) for $L=301,601$, and 1201 and $\chi =128$.

Figure 6

Convergence of the spin excitation profile with respect to bond dimension, $\chi$, at $t=2.0, L=201$, and $dt=0.005$.

Figure 7

Convergence of the time-dependent diffusion constant $D\left(t\right)$ with respect to bond dimension for $\alpha =1.3$ and $\alpha =3$ for $L=101$ and $dt=0.1$.

Figure 8

Convergence of the relaxation of the central spin $C_0\left(t\right)$ with respect to bond dimension for $\alpha =0.3$ and $\alpha =0.7\left(L=1201\right)$.

Figure 9

Tensor network diagram for Eq. (B2). Each tensor in the network is labeled with the physical site it represents. The upper MPO corresponds to the untranslated operator ${\widehat{S}}_{L/2}^z\left(-\frac{t}{2}\right)$, while the lower MPO is its translated version ${\mathit{T}}_3{\widehat{S}}_{L/2}^z\left(\frac{t}{2}\right)$.

Figure 10

Relative deviation between the spin excitation profiles obtained with and without the approximation described in the text. Data shown are for $t=2.0,\alpha =2.0,dt=0.1,\chi =128,L=201.$