Spin diffusion in a perturbed isotropic Heisenberg spin chain

The isotropic Heisenberg chain represents a particular case of an integrable many-body system exhibiting superdiffusive spin transport at finite temperatures. Here, we show that this model has distinct properties also at finite magnetization $m\ne 0$, even upon introducing the SU(2) invariant perturbations. Specifically, we observe nonmonotonic dependence of the diffusion constant ${\mathcal{D}}_0\left(\mathrm{\Delta }\right)$ on the spin anisotropy $\mathrm{\Delta }$, with a pronounced maximum at $\mathrm{\Delta }=1$. The latter dependence remains true also in the zero magnetization sector, with superdiffusion at $\mathrm{\Delta }=1$ that is remarkably stable against isotropic perturbation (at least in finite-size systems), consistent with recent experiments with cold atoms.

Figure 1

Diffusion constant ${\mathcal{D}}_0$ vs anisotropy $\mathrm{\Delta }$ at fixed magnetization density $m=1/4$ for $H_{\mathrm{I}}^{\prime }$ perturbation at strength $g=0.3$, as obtained for different system sizes via ED ($L=24$) and MCLM ($L=28\text{–}36$). (b) ${\mathcal{D}}_0$ vs $\mathrm{\Delta }$ for isotropic (full lines) and anisotropic (dashed lines) $g=0.15\text{–}0.3$, obtained via MCLM for $L=36$. (c), (d) Integrated optical conductivity $I\left(\omega \right)$ for (c) $\mathrm{\Delta }=1$ and (d) $\mathrm{\Delta }=1.1$ as the function of the rescaled frequencies by the square of the perturbation strength $\omega /g^2$. Dashed (solid) line depicts $L=20$ ($L=36$) ED (MCLM) data.

Figure 2

Diffusion constant ${\mathcal{D}}_0$ vs anisotropy $\mathrm{\Delta }$, as calculated via MCLM in ensembles with magnetization densities $m=[0,6]/28$ and $g=0.2$ for $H_{\mathrm{I}}^{\prime }$ (main panel) and $H_{\mathrm{II}}^{\prime }$ (left inset) perturbation, respectively. Right inset: Comparison of canonical $m=0$ and grand canonical results.

Figure 3

(a) Scaling of NESS diffusion constant vs $L$ for $H_{\mathrm{I}}^{\prime }$ perturbation at various perturbation strengths $g=0.15\text{–}0.4$, including the unperturbed result $g=0$, fitted with the power laws ${\mathcal{D}}_0\propto L^{\zeta }$ with different $\zeta =\left[0.24\text{–}0.5\right]$. In the inset, fitted $\zeta$ for different $g$ are shown for $\mathrm{\Delta }=1.0$ and $\mathrm{\Delta }=0.9$. (b) Diffusion constant ${\mathcal{D}}_0$ vs $\mathrm{\Delta }$ for $H_{\mathrm{I}}^{\prime }$ perturbation of different strength $g=0.15\text{–}0.4$, obtained via NESS method for sizes $L=30,60$.

Figure 4

(a) Normalized spin stiffness $D^{*}$ vs magnetization density $m$ obtained from (i) extrapolated $L\to \infty$ ED data [39] (solid black points), (ii) tDMRG result Ref. [43] (open black points), and (iii) GHD exact calculation (orange line). Simple linear relation $D^{*}=2\left|m\right|$ (green solid line) at large magnetization and $D^{*}\left(m\right)=6.19m^{2}ln(1/|m\left|\right)$ (green dashed line) at small magnetization is also presented.