Many-body-localization transition: sensitivity to twisted boundary conditions

Source abstract

Abstract For disordered interacting quantum systems, the sensitivity of the spectrum to twisted boundary conditions depending on an infinitesimal angle ϕ can be used to analyze the many-body-localization transition. The sensitivity of the energy levels E n ( ϕ ) is measured by the level curvature K n = E n ′ ′ ( 0 ) , or more precisely by the Thouless dimensionless curvature k n = K n / Δ n , where Δ n is the level spacing that decays exponentially with the size L of the system. For instance Δ n ∝ 2 − L in the middle of the spectrum of quantum spin chains of L spins, while the Drude weight D n = L K n studied recently by Filippone et al (arxiv: 1606.07291v1 ) involves a different rescaling. The sensitivity of the eigenstates | ψ n ( ϕ ) > is characterized by the susceptibility χ n = − F n ′ ′ ( 0 ) of the fidelity F n = | < ψ n ( 0 ) | ψ n ( ϕ ) > | . Both observables are distributed with probability distributions displaying power-law tails P β ( k ) ≃ A β | k | − ( 2 + β ) and Q ( χ ) ≃ B β χ − 3 + β 2 , where β is the level repulsion index taking the values β G O E = 1 in the ergodic phase and β l o c = 0 in the localized phase. The amplitudes A β and B β of these two heavy tails are given by some moments of the off-diagonal matrix element of the local current operator between two nearby energy levels, whose probability distribution has been proposed as a criterion for the many-body-localization transition by Serbyn et al (2015 Phys. Rev . X 5 041047).