Suppressed energy transport in the strongly disordered Hubbard chain

We study the dynamics of the energy fluctuations in the Hubbard chain with strong potential disorder that preserves the SU(2) spin symmetry. Within the effective spin-only model that correctly captures relaxation in the situation of localized charges, we show that the decay of local energy correlations is suppressed and is at least marginally nonergodic, being qualitatively different from the subdiffusive dynamics and relaxation of local spins. The anomalous behavior can be traced back to the singular distribution of effective exchange couplings. Numerical results for the dynamical thermal conductivity within the full Hubbard model confirm that the energy (thermal) transport is closer to the charge dynamics, which appears to be also marginally localized in a wide parameter range.

Figure 1

Results from the Lanczos propagation method averaged over $N_s=2000$ realizations of disorder. (a) Spin density $C^S\left(t\right)$ and energy density $C^h\left(t\right)$ correlation functions for $\tilde{\lambda }=0.5$ and $\tilde{N}=18$. Dashed line shows analytical prediction for asymptotic dynamics, $C^S\left(t\right)\sim t^{-\tilde{\lambda }/(1+\tilde{\lambda })}$. (b) and (c) $C^h\left(t\right)$ for $\tilde{\lambda }=0.3$ and $\tilde{\lambda }=1.0$, respectively. (d) Finite-size scaling of $C^h\left(t_m\right)$ for the longest accessible time, $2U{t}_m={10}^3$.

Figure 2

Results from the ED method averaged over $N_s⪆{10}^4$ realizations of disorder. (a) The same as in Fig. 1 for $\tilde{\lambda }=0.5$ and $\tilde{\lambda }=1$ but for longer times. (b) Finite-size scaling of the energy stiffness $C_0^h$ vs $1/\tilde{N}$. (c) and (d) Diagonal matrix elements, $s_n=\langle n|{\vec{S}}_a·{\vec{S}}_{a+1}|n\rangle$ vs eigenenergy $E_n$ obtained for a bond with the largest $J_a$ for $\tilde{N}=14$ and two particular realizations of disorder corresponding to the same $\tilde{\lambda }=0.5$.

Figure 3

Normalized charge, spin, and thermal dynamical conductivities, as obtained via the numerical MCLM in different Hubbard chains with large disorder. (a) and (b) show the finite-size effects (comparison of results for $L=10–14$ systems, of ${\overline{\sigma }}^S\left(\omega \right)$ and ${\overline{\sigma }}^t\left(\omega \right)$, respectively, for fixed disorder $W=8$ and $U=2$. (c) and (d) show the comparison of ${\overline{\sigma }}^{c,S,t}\left(\omega \right)$ for two different parameter sets, obtained for the largest $L=14$.