Spin Subdiffusion in the Disordered Hubbard Chain

We derive and study an effective spin model that explains the anomalous spin dynamics in the one-dimensional Hubbard model with strong potential disorder. Assuming that charges are localized, we show that spins are delocalized and their subdiffusive transport originates from a singular random distribution of spin exchange interactions. The exponent relevant for the subdiffusion is determined by the Anderson localization length and the density of the electrons. Although the analytical derivations are valid for low particle density, numerical results for the full model reveal a qualitative agreement up to half filling.

Figure 1

Two electrons on the disordered Hubbard chain. (a),(b) $⟨S_i^z⟩(t)$ and $⟨n_i⟩(t)$ for a single initial state and single realization of disorder. (c),(d) Frequency ${\omega }_2$ of spin oscillation obtained directly from the Hubbard model [see panel (a)] compared with Eq. (4). The distance between electrons is fixed at $d=4$ (c) and $d=6$ (d).

Figure 2

Points in (a),(b) show $\tilde{J}{f}_{\tilde{J}}(\tilde{J})$, generated directly from Eq. (4) for $N=2$ electrons at the average distance $d$, whereby results have been fitted to Eq. (8) by adjusting a single $\lambda$ (for all $d$). (c),(d) Local spin correlation function [Eq. (9)] for the effective model with various numbers of spins $\tilde{N}$. The results for $t\in [10,50]$ with the largest $\tilde{N}$ are fitted by $S_L(t)\propto t^{-\alpha }$ shown as the dashed line.

Figure 3

(a) Dynamical exponent $\alpha$ vs $\tilde{\lambda }$ obtained for the squeezed model. (b),(c) Spin-spin correlation function obtained for the Hubbard model ($L=18$, $N=6$) and compared with $t^{-\alpha }$ (dashed line), where $\alpha =\lambda /(\lambda +d)$. (d) $\alpha$ in the Hubbard model ($U=2$, $W=8$, various $N$ and $L$). $\lambda$ in (b)–(d) is obtained from fits in Fig. 2.