Localization of interacting fermions at high temperature

We suggest that if a localized phase at nonzero temperature $T>0$ exists for strongly disordered and weakly interacting electrons, as recently argued, it will also occur when both disorder and interactions are strong and $T$ is very high. We show that in this high-$T$ regime, the localization transition may be studied numerically through exact diagonalization of small systems. We obtain spectra for one-dimensional lattice models of interacting spinless fermions in a random potential. As expected, the spectral statistics of finite-size samples cross over from those of orthogonal random matrices in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. However, these data show deviations from simple one-parameter finite-size scaling: the apparent mobility edge “drifts” as the system’s size is increased. Based on spectral statistics alone, we have thus been unable to make a strong numerical case for the presence of a many-body localized phase at nonzero $T$.

Figure 1

(Color online) Disorder averaged probability distribution $P\left(r\right)$ for Poisson (solid line) and GOE distributed eigenvalues (Ref. 17) (dots) and for our interacting fermion model at randomness $W=3$ in diffusive regime [data in green, mostly overlapping with black (GOE) dots], $W=11$ in localized regime [data in red, mostly overlapping with black (Poisson) line], and $W=7$ in intermediate regime (data in blue) for length $L=16$.

Figure 2

(Color online) Size $L$ and disorder $W$ dependences of $r(W,L)$. The curves correspond to $L=8$ (diamonds), 10 (stars), 12 (squares), 14 (triangles), 16 (unadorned) from top to bottom for large $W$. Bottom: an enlargement of the crossing region to make the drift of the crossings more visible. Where not visible, the error bars are smaller than the points.