Operator growth in disordered systems, a translation invariant approach*

Operator growth in disordered systems, a translation invariant approach

Recently, we proposed [1] a general framework to study operator growth and complexity, using the Lanczos algorithm (aka Krylov subspace or recursion method). Here, we apply the framework to disordered systems, especially random spin chains and many-body localization (MBL). An argument based on l-bits shows that in MBL phases, the Krylov-complexity [1] grows polynomially in time (as opposed to exponentially in chaotic systems). An equivalence between quenched disorder and static ancilla allows us to carry out numerical study to in a translation invariant systems, bypassing finite size effects and statistical fluctuations of disorder averaging. Under strong disorder, the Lanczos coefficients define a quasi-random Krylov chain with multiple bound states, which we identify as local integral of motions and long-lived oscilating modes. We propose to study the MBL transition as the divergence of localization length on the Krylov chain. Finally, we apply the disorder-free method to a mean-field random magnet realized in a recent cold-atom experiment [2].

[1] Phys. Rev. X 9, 041017 (2019).
[2] Phys. Rev. X 9, 041011 (2019).