Spin transport of weakly disordered Heisenberg chain at infinite temperature

We study the disordered Heisenberg spin chain, which exhibits many-body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as $C\sim t^{-\beta }$, whereas the conductivity exhibits a low-frequency power law $\sigma \sim {\omega }^{\alpha }$. The exponents depict subdiffusive behavior $\beta <1/2,\alpha >0$ at all finite disorders and convergence to the scaling result $\alpha +2\beta =1$ at large disorders.

Figure 1

Average exponents for the disordered Heisenberg model, evaluated by the continued fractions VER method. $h$ is the disorder strength. The local spin correlations decay in time as $t^{-\beta }$ ($\beta$ in blue circles). The low-frequency conductivity rises as ${\omega }^{\alpha }$ ($\alpha$ in red circles). Comparison is made with exact diagonalization results on 22 sites ${\beta }_{\mathrm{ED}}$ (black triangles). Error bars are given by the least-squares fit (see Sec. 4b). The sum $\alpha +2\beta$ converges to unity at higher disorder—as expected by scaling [Eq. (6)].

Figure 2

The local spin excess decay exponent $\beta$ of the disordered Heisenberg model as extracted from ED. The blue squares represent system size $N=18$, the green triangles represent $N=20$, and the red circles represent $N=22$. One observes a subdiffusive exponent well before the MBL transition, which occurs in the data at $h/J\sim 3.5$. Despite the relatively large system sizes, it is difficult to extract exponents for $h/J<0.4$. We utilize 1000 disorder realizations for $N=18$, 300 for $N=20$, and 44 for $N=22$. The inset shows the extraction of $\beta$ for $N=22$ at various disorder strengths.

Figure 3

The VER calculation of the local spin correlations. The blue circles are the calculated recurrents ${\mathrm{\Delta }}_n^2$ for a particular disorder realization of strength $h$. The red circles are variational recurrents ${\tilde{\mathrm{\Delta }}}_n^2$. The green solid lines are the extrapolated higher-order recurrents within the VER scheme (see the text). The vertical displacement is artificial, where ${\mathrm{\Delta }}_0=0$. Even-odd alternation of the recurrents reflect the asymptotic low-frequency power law of the correlation function.

Figure 4

ED calculation of $T\sigma \left(\omega \right)$ for $N=14$. Different disorder strengths (red diamonds) $h/J=0.5$, (blue circles) $h/J=0.7$, and (green squares) $h/J=1.0$ are shown. The structure of the functions resembles a Gaussian with a positive low-frequency power law.

Figure 5

Local spin correlation function with disorder strength $h=0.5$. VER scheme (see the text) results with an increasing number of recurrents $n_{\max }$. The inset: extraction of exponent $\beta$.

Figure 6

ac conductivity of a disordered Heisenberg chain of $N=14$ sites. The plot represents a single realization of disorder with $h=0.5$. Exact diagonalization of Eq. (3) (blue circles) is compared to the VER calculations (solid lines) for different values of $n_{\max }$. Good agreement is reached for $n_{\max }=13$. $\alpha$ represents the low-frequency power law.

Figure 7

$\beta$ (blue circles) of the clean Heisenberg Hamiltonian as extracted from varying the number of recurrents used in the VER scheme $n_{\max }$. The error bars are the uncertainty intervals of the least-squares procedure.

Figure 8

Probability distribution of $\beta$ for different disorder strengths. We witness a flow of weight towards $\beta \to 0$ as disorder is increased. An error estimate can be extracted from the width of the distribution.