Stark Many-Body Localization

We consider spinless fermions on a finite one-dimensional lattice, interacting via nearest-neighbor repulsion and subject to a strong electric field. In the noninteracting case, due to Wannier-Stark localization, the single-particle wave functions are exponentially localized even though the model has no quenched disorder. We show that this system remains localized in the presence of interactions and exhibits physics analogous to models of conventional many-body localization (MBL). In particular, the entanglement entropy grows logarithmically with time after a quench, albeit with a slightly different functional form from the MBL case, and the level statistics of the many-body energy spectrum are Poissonian. We moreover predict that a quench experiment starting from a charge-density wave state would show results similar to those of Schreiber et al. [Science 349, 842 (2015)].

Figure 1

Schematic illustration of our model. On a finite lattice in one dimension, the single-particle states of spinless fermions, which can delocalize via hopping $J$, are exponentially localized via a strong electric field. For a purely uniform field (i.e., a linear electric potential), this is usually referred to as Wannier-Stark localization [26]. When the particles interact via nearest-neighbor interactions $V$, they share many properties with the well-studied MBL phenomenology.

Figure 2

Difference $\mathrm{\Delta }S$ between the bipartite entanglement in the interacting and noninteracting cases. The parameter $\alpha$ denotes the cumulative effect of field nonuniformity on the potential at the end site. For sufficiently small $\alpha$ and suitably large field $\gamma$, there is good qualitative agreement with our semianalytic calculation (magenta dashed line) and full analytic calculation [green line and Eq. (4)]. For larger $\alpha$, the entanglement growth becomes stronger than predicted due to the progressive delocalization of some of the single-particle orbitals as the position-dependent field gets weaker at the right-hand end of the chain (see right of inset). For the purely uniform field, there is an initial steep rise, which we attribute to additional degeneracies of the many-body spectrum. The numerical curves have been smoothed by convolution with a Gaussian, $w(n)=e^{-(n/\sigma {)}^2/2}$, with $\sigma =4$. Inset: Potential used in the respective main curves.

Figure 3

The many-body level statistics for the case of nonzero $\alpha$ in the half-filled sector. For all displayed values of $\gamma$ and $\alpha$, the probability distribution of the gap-ratio parameter, $r_n=\min ({\delta }_n/{\delta }_{n+1},{\delta }_{n+1}/{\delta }_n)$ [where ${\delta }_n$ is the gap between the $n$th and ($n-1$)th energy eigenvalues], agrees with the prediction for Poisson level statistics expected for integrable or localized models. The predicted average gap-ratio parameter in this case is $⟨r{⟩}_P=0.3863$. For comparison, we have included the prediction for Wigner-Dyson (WD) statistics.

Figure 4

The $L=16$ study of a hypothetical imbalance experiment, where an initial charge-density wave relaxes under unitary time evolution. The different curves represent different field strengths $\gamma$ with $\alpha =0.5$. Inset left: Visualization of the corresponding potential. Inset right: Average imbalance $\mathcal{I}$ at times $10^{13}\ge t/J\ge {10}^{11}$. The numerical curves have been smoothed by convolution with a Gaussian, $w(n)=e^{-(n/\sigma {)}^2/2}$, with $\sigma =4$.