From dissipationless to normal diffusion in the easy-axis Heisenberg spin chain

The anomalous spin diffusion of the integrable easy-axis Heisenberg chain originates in the ballistic transport of symmetry sectors with nonzero magnetization. Ballistic transport is replaced by normal dissipative transport in all magnetization sectors upon introducing the integrability-breaking perturbations, including external driving. Such behavior implies that the diffusion constant obtained for the integrable model is relevant for the spread of spin excitations but not for the spin conductivity. We present numerical results for closed systems and driven open systems, indicating that the diffusion constant shows a discontinuous variation as the function of perturbation strength.

Figure 1

High-$T$ dynamical diffusion $\mathcal{D}\left(\omega \right)$ evaluated for the XXZ chain at magnetization $m=0$ and fixed $\mathrm{\Delta }=1.5$, using MCLM on the $L=32$ system and various staggered fields $\delta h$. The inset shows the integrated quantity $I\left(\omega \right)={\int }_0^{\omega }d{\omega }^{\prime }\mathcal{D}\left({\omega }^{\prime }\right)$ with the extrapolation to limit $L\to \infty$ for the unperturbed system.

Figure 2

Canonical DC diffusion constant ${\mathcal{D}}_C\left(m\right)$ for $\mathrm{\Delta }=1.5$ as calculated for different magnetizations $m$ with MCLM in a system of $L=32$ sites for different staggered fields $\delta h>0$. Results are fitted with parabolic dependence Eq. (5) with fixed $\zeta =2.1$.

Figure 3

Diffusion constant $\mathcal{D}$ vs $1/L$ obtained for fixed $\mathrm{\Delta }=1.5$ from finite-$L$ results by three different approaches: Canonical $m=0$ sector MCLM results ${\mathcal{D}}_0$, GC average over $m$ sectors ${\mathcal{D}}_{GC}$, and open-system steady-state ${\mathcal{D}}_{ss}$ for two staggered fields: (a) $\delta h=0.2$, and (b) $\delta h=0.4$. Displayed is also unperturbed value ${\mathcal{D}}_I=0.55$. (c) Diffusion constant ${\mathcal{D}}_{ss}$ vs $\delta h$ as obtained for sizes $L=20,40,60$ within the steady-state open-system approach. Shown are also values obtained via linear $1/L\to 0$ scaling.

Figure 4

Time-dependent diffusion ${\mathcal{D}}_q\left(t\right)$ as extracted from corresponding spin-correlation functions $S(q,t)$ (shown in logarithmic scale as the inset) evaluated for $L=28$ and smallest finite $q=2\pi /L$ for staggered fields $\delta h=0\text{–}0.5$.

Figure 5

Dynamical diffusion $\mathcal{D}\left(\omega \right)$ for the XXZ spin chain perturbed with NNN interaction ${\mathrm{\Delta }}_2=0.4$ as obtained with MCLM for the canonical $m=0$ system with different sizes $L=16\text{–}32$.

Figure 6

Results for diffusion at various strengths of NNN interactions ${\mathrm{\Delta }}_2=0\text{–}0.8$ for XXZ chain with $\mathrm{\Delta }=1.5$. (a) High-$T$ dynamical $\mathcal{D}\left(\omega \right)$ as calculated with MCLM on chain of $L=32$. (b) Diffusion constant ${\mathcal{D}}_C\left(m\right)$ vs magnetization $m$. Results are fitted with parabola with fixed $\zeta =9.0$. (c) Time-dependent ${\mathcal{D}}_q\left(t\right)$ as extracted from spin correlations $S(q,t)$ with smallest finite $q=2\pi /L$ evaluated for the $L=28$ system (the inset shows the corresponding $S(q,t)$ on the logarithmic scale).

Figure 7

Results for diffusion for staggered-exchange perturbation $\delta J=0-0.8$ for the XXZ chain with $\mathrm{\Delta }=1.5$. (a) High-$T$ dynamical $\mathcal{D}\left(\omega \right)$ as calculated with MCLM on the chain of $L=32$. (b) Diffusion constant ${\mathcal{D}}_C\left(m\right)$ vs magnetization $m$ with results fitted to the parabola with $\zeta =2.1$. (c) Time-dependent ${\mathcal{D}}_q\left(t\right)$ as extracted from $S(q=2\pi /L,t)$ evaluated for the $L=28$ system [the inset shows corresponding $S(q,t)$ on the logarithmic scale].

Figure 8

Diffusion constant $\mathcal{D}$ vs $1/L$ obtained for fixed $\mathrm{\Delta }=1.5$ from finite-$L$ results by three different approaches: Canonical $m=0$ sector MCLM results ${\mathcal{D}}_0$, GC average over $m$ sectors ${\mathcal{D}}_{GC}$, and open-system steady-state ${\mathcal{D}}_{ss}$ for two staggered exchanges: (a) $\delta J=0.2$, and (b) $\delta J=0.4$. Displayed is also unperturbed value ${\mathcal{D}}_I=0.55$.

Figure 9

Time-dependent diffusion ${\mathcal{D}}_F\left(t\right)$ as extracted from the time evolution of the spin-density profile with the smallest finite $q=2\pi /L$ for the system of $L=28$ sites with $\mathrm{\Delta }=1.5$ for various driving fields $F=0,0.2,0.3$.