Many-Body-Localization Transition : sensitivity to twisted boundary conditions

Abstract

For disordered interacting quantum systems, the sensitivity of the spectrum to twisted boundary conditions depending on an infinitesimal angle $\varphi$ can be used to analyze the Many-Body-Localization Transition. The sensitivity of the energy levels $E_n(\varphi )$ is measured by the level curvature $K_n=E_n"(0)$, or more precisely by the Thouless dimensionless curvature $k_n=K_n/{\Delta }_n$, where ${\Delta }_n$ is the level spacing that decays exponentially with the size $L$ of the system. For instance ${\Delta }_n\propto 2^{-L}$ 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 M. Filippone, P.W. Brouwer, J. Eisert and F. von Oppen [arxiv:1606.07291v1] involves a different rescaling. The sensitivity of the eigenstates $|{\psi }_n(\varphi )>$ is characterized by the susceptibility ${\chi }_n=-F_n"(0)$ of the fidelity $F_n=|<{\psi }_n(0)|{\psi }_n(\varphi )>|$. Both observables are distributed with probability distributions displaying power-law tails $P_{\beta }(k)\simeq A_{\beta }|k{|}^{-(2+\beta )}$ and $Q(\chi )\simeq B_{\beta }{\chi }^{-\frac{3+\beta }{2}}$, where $\beta$ is the level repulsion index taking the values ${\beta }^{GOE}=1$ in the ergodic phase and ${\beta }^{loc}=0$ in the localized phase. The amplitudes $A_{\beta }$ and $B_{\beta }$ 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 M. Serbyn, Z. Papic and D.A. Abanin [Phys. Rev. X 5, 041047 (2015)].

arXiv commentv2= revised version with many improvements , 11 pages

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• Holographic unitary renormalization group for correlated electrons -- I: a tensor network approach
• Quantum chaos and level dynamics
• Response of a quantum disordered spin system to a local periodic drive
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• Thouless Energy Across Many-Body Localization Transition in Floquet Systems
• Robustness and Independence of the Eigenstates with respect to the Boundary Conditions across a Delocalization-Localization Phase Transition