Many-Body Localization

MBL is an ergodicity-breaking phase in disordered quantum systems. In the ETH (thermal) phase, entanglement grows as volume law S ~ L. In the MBL phase, disorder localizes eigenstates and entanglement follows an area law S ~ const, growing logarithmically in time.

W = 1.50
L = 12
t = 0
ETH PHASE — Ergodic, thermalized
ETH phase: S(t) ~ t (ballistic) → S ~ L (volume law at long times)
MBL phase: S(t) ~ log(t) (slow dephasing) → S ~ const (area law)
Transition at W_c ≈ 3.5 (Pal-Huse 2010) | Level spacing ratio: r → 0.53 (GOE) vs 0.39 (Poisson)