Slow diffusion and Thouless localization criterion in modulated spin chains

In recent years the ergodicity of disordered spin chains has been investigated via extensive numerical studies of the level statistics or the transport properties. However, a clear relationship between these results has yet to be established. We present the relation between the diffusion constant and the energy-level structure, which leads to the Thouless localization criterion. Together with the exponential-like dependence of the diffusion constant on the strength of quasiperiodic or random fields, the Thouless criterion explains the nearly linear drift with the system size of the crossover/transition to the nonergodic regime. Moreover, we show that the Heisenberg spin chain in the presence of the quasiperiodic fields can be well approached via a sequence of simple periodic systems, where diffusion remains finite even at large fields.

Figure 1

(a) Dynamical diffusivity $\mathcal{D}\left(\omega \right)$ obtained via MCLM for QP chain with different system sizes $L$, compared also with the case when $k$ is an irrational number. (b)–(c) Spin diffusion constant ${\mathcal{D}}_0$ vs. potential strength obtained from offdiagonal ME [Eq. (3)] and diagonal ME [Eq. (4)] via ED and compared to MCLM results for larger $L=27$ (periodic case with $P=3$) or $L=28$ (QP). (b) Results for $\mathrm{\Delta }=0.5$ where ED is carried out for $L=18$ (QP) or $L=21$ ($P=3$). (c) QP chain with $\mathrm{\Delta }=1$ and various $L$. For clarity, results for $L=16$ and $L=14$ are multiplied by factors 10 and ${10}^2$, respectively.

Figure 2

Level and current matrix-element criteria for the departure from the RMT universality vs. field strength $W$ for the QP case, calculated via ED for different system sizes $L=14-18$ and $\mathrm{\Delta }=1$. (a) the gap ratio $\overline{r}$, (b) $Q$ from diagonal ME, (c) off-diagonal/diagonal ME ratio $Y$, and (d) level sensitivity parameter $R$. Dashed lines denote GUE values.

Figure 3

dc diffusion ${\mathcal{D}}_0$ vs. field strength $W$, calculated with MCLM for $L=27$ at $\mathrm{\Delta }=0.5$ for a periodic $P=3$, and QP chains. The presented data depict five different choices of phase shift ${\varphi }_0$.

Figure 4

DC diffusion ${\mathcal{D}}_0$ vs. field strength $W$ as calculated with MCLM for $P=2$, $P=3$ and two QP approximants, and (a) $\mathrm{\Delta }=0.3$, (b) $\mathrm{\Delta }=0.5$, (c) and $\mathrm{\Delta }=1.0$. (d) depicts QP ($k=10/27$) case and different anisotropies $\mathrm{\Delta }=0.3,0.5,1.0$.

Figure 5

Integrated dynamical-diffusion spectra $I\left(\omega \right)$ calculated with MCLM for the spin chain with $\mathrm{\Delta }=0.5$ and various $W$ for (a) periodicity $P=2$ ($L=28$), (b) periodicity $P=3$ ($L=27$), and (c) QP approximant $k=10/27$ with $L=27$.

Figure 6

Diffusion constant ${\mathcal{D}}_0$ for $P=2$ with different anisotropies $\mathrm{\Delta }=0.3,0.5,1$, evaluated with ED on $L=16$ sites. At large $W\gg 1$ results confirm asymptotic scaling ${\mathcal{D}}_0\propto 1/W$.

Figure 7

Level and current matrix-element criteria for the departure from the GUE universality vs. field strength $W$ for the periodic field $P=3$ case, calculated via ED for $L=18,21$. (a) depicts the gap ratio $\overline{r}$, (b) $Q$ from the diagonal ME, (c) off-diagonal/diagonal ratio $Y$, and (d) level sensitivity parameter $R$. Dashed curves denote the GUE values.

Figure 8

Integrated level-distance distribution $F\left(x\right)$ for different $W$, as calculated via ED for: (a) the QP system with $\mathrm{\Delta }=1$ on $L=18$ sites, and (b) for $P=3$ system with $\mathrm{\Delta }=0.5$ on $L=21$ sites. Dashed line denotes the GUE dependence from Eq. (B3).