Many-Body Localization in Periodically Driven Systems

We consider disordered many-body systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a many-body localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and (ii) a delocalized phase, in which eigenstates have volume-law entanglement and obey the ETH. The MBL phase exhibits logarithmic in time growth of entanglement entropy when the system is initially prepared in a product state, which distinguishes it from the delocalized phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion, which naturally explains the spectral and dynamical properties of this phase. Numerical data, obtained by exact diagonalization and time-evolving block decimation methods, suggest a direct transition between the two phases.

Figure 1

Disorder-averaged level statistics parameter $⟨r⟩$ as a function of the “kick” strength $T_1$. At small values of $T_1$, $⟨r⟩\approx 0.386$, indicating Poisson statistics of quasienergy levels (no level repulsion). At larger $T_1$ the system undergoes a transition into a delocalized phase with $⟨r⟩\approx 0.53$, consistent with the COE [27]. Data are for system sizes $L=10,12,14$, and averaging is performed over 1000 disorder realizations.

Figure 2

Averaged entanglement entropy $⟨S⟩$ and its fluctuations $⟨\mathrm{\Delta }S⟩$ (inset) as a function of $T_1$. The scaling of entropy and its fluctuations with system size $L$ are consistent with the existence of a MBL and a delocalized phase for small and large $T_1$, respectively.

Figure 3

Dynamical properties: Decay of magnetization at a given site $I=1$ for the Néel initial configuration. Inset: Long-time magnetization remains nonzero in the MBL phase as the system size is increased. In the delocalized phase, magnetization decays to zero at long times. Averaging was performed over 6000 disorder realizations.

Figure 4

Disorder-averaged entanglement entropy following a quantum quench, for the Néel initial state. Data for system sizes $L=12,14$ were obtained by ED, for $L=16,18$ using Krylov subspace projection, and $L=24,30$ using TEBD. Averaging is performed over 6000 disorder realizations.