Decay of density waves in coupled one-dimensional many-body-localized systems

This work analyzes the behavior of coupled disordered one-dimensional systems as modelled by identical fermionic Hubbard chains with the on-site potential disorder and coupling emerging through the interchain hopping $t^{\prime }$. The study is motivated by the experiment on fermionic cold atoms on a disordered lattice, where a decay rate of the quenched density wave was measured. We present a derivation of the decay rate $\mathrm{\Gamma }$ within perturbation theory and show that, even at large disorder along the chains, the interaction leads to finite $\mathrm{\Gamma }>0$, the mechanism being the interaction-induced coupling of in-chain localized and interchain extended single-fermion states. Explicit expressions for $\mathrm{\Gamma }$ are presented for a weak interaction $U<t,t^{\prime }$, but extended also to the regime $t>U>t^{\prime }$. It is shown that, in both regimes, $\mathrm{\Gamma }$ increases with the interchain hopping $t^{\prime }$, as well as decreases with increasing disorder.

Figure 1

Effective density of states $\tilde{\mathcal{D}}\left(\omega \right)$, emerging from the interchain hopping and entering the evaluation of the decay rate $\mathrm{\Gamma }$. Plotted is also the incoherent approximant ${\tilde{\mathcal{D}}}_I\left(\omega \right)$.

Figure 2

Decay rate $\mathrm{\Gamma }t/U^2$ vs interchain hopping $t^{\prime }/t$ for different disorders $W/t$ at fixed particle density $\overline{n}=1/2$.

Figure 3

Decay rate $\mathrm{\Gamma }t/U^2$ vs $t^{\prime }/t$ as calculated within the incoherent approximation for different $W/t$ at fixed $\overline{n}=1/2$.