Periodically driven (Floquet) interacting quantum systems generically absorb energy and heat to an infinite-temperature state, but strong disorder can produce many-body localization (MBL), which was shown experimentally and numerically (Bordia et al., Nature Physics 13, 460, 2017) to protect a non-ergodic driven phase in a one-dimensional disordered interacting fermion lattice: localized at high drive frequency, ergodic at low frequency, and — remarkably — apparently stable even at low frequencies and small amplitudes when the drive modulates all on-site disorder potentials synchronously. Whether this protection is asymptotic or only a long-lived prethermal regime, on what time scales and by what mechanisms delocalization or heating eventually sets in, whether the critical drive frequency stays finite or becomes extremely small, whether analogous Floquet-protected non-ergodic regimes exist in more than one spatial dimension, and how the frequency-resolved response behaves near the MBL transition itself, remain open.

Background

A closed interacting quantum system that is driven by a time-periodic (Floquet) Hamiltonian H(t)=H(t+T)H(t)=H(t+T) has no energy conservation, and generic interacting systems are expected to heat until they are described by an infinite-temperature (fully mixed) state, erasing any interesting order. One established exception is many-body localization (MBL): strong quenched disorder in an interacting system can produce eigenstates that violate the eigenstate thermalization hypothesis (the standard mechanism by which closed quantum systems thermalize) and an emergent set of quasi-local conserved quantities, so that transport and thermalization are arrested. Before 2017, theory (Ponte, Papić, Huveneers and Abanin, Phys. Rev. Lett. 114, 140401 (2015); Lazarides, Das and Moessner, Phys. Rev. Lett. 115, 030402 (2015); Abanin, De Roeck and Huveneers, Ann. Phys. 372, 1 (2016); von Keyserlingk, Khemani and Sondhi, Phys. Rev. B 94, 085112 (2016)) predicted that periodically driven disordered interacting chains should admit a genuinely localized Floquet phase in one dimension at sufficient disorder and drive frequency, separated from an ergodic, heated phase by a drive-induced transition. Bordia, Lüschen, Schneider, Knap and Bloch (Nature Physics 13, 460 (2017), preprint arXiv:1607.07868) tested this with ultracold fermionic potassium-40 atoms in a one-dimensional quasiperiodic (Aubry-André-type, i.e. with on-site energies modulated by a cosine incommensurate with the lattice spacing) optical lattice with repulsive interactions: they imprinted a charge-density-wave pattern (alternating occupation of even and odd lattice sites) and monitored its persistence through the imbalance I=(Neven−Nodd)/(Neven+Nodd)\mathcal{I}=(N_{\rm even}-N_{\rm odd})/(N_{\rm even}+N_{\rm odd}), modulating the lattice depth (i.e. the effective disorder strength, hopping and interactions) sinusoidally in time. They found a localized phase at high drive frequency ν\nu and delocalization below a critical frequency, consistent with numerics and with the level statistics of the effective Floquet Hamiltonian HFH_F defined by e−iHFTe^{-iH_FT}, whose eigenphases (quasienergies) are defined only modulo h/Th/T. Surprisingly, at low drive frequency the system remained localized as long as the drive amplitude was small: because the drive modulates all on-site energies synchronously (modulating the disorder itself), the localization length of the conserved quantities is modulated adiabatically and no percolating network of resonances forms, so the authors suggested a novel driven regime that 'would not delocalize for any frequency', supported by simulations to long but finite times. Two limitations of the experiment and numerics motivate the present question. First, all observations are restricted to finite times, finite system sizes and frequencies above a practical lower bound, and studying the extreme long-time and ultra-low-frequency limits remained, in the authors' words, challenging; whether the observed stability is asymptotic or prethermal — a long-lived transient that eventually gives way to heating — was left open, as were the time scales and mechanisms of eventual delocalization and the possibility that the critical drive frequency is pushed to extremely small values. Second, the experiment was one-dimensional, and the outlook explicitly proposed using periodic modulation to measure the frequency-resolved response of many-body systems, with the prospect of exploring the critical point of the MBL transition and the response in higher dimensions. The broader context makes these questions sharp: in clean driven systems, prethermalization theory (Else, Bauer and Nayak, Phys. Rev. X 7, 011026 (2017)) proves long-lived prethermal regimes with lifetimes nearly exponential in drive frequency, after which heating is still expected, and avalanche arguments (rare thermal inclusions that grow and thermalize their surroundings; De Roeck and Huveneers, Phys. Rev. B 95, 155129 (2017)) make even static MBL in random systems unstable for spatial dimension d>1d>1, while the thermodynamic-limit stability of MBL in one dimension remains actively debated.

Problem Statement

For a many-body localized system subjected to periodic driving — as realized with interacting fermions in a one-dimensional quasiperiodic optical lattice, where the drive synchronously modulates the on-site (disorder) potentials — determine the ultimate stability and scalability of driven many-body localization. Specifically: (1) Is the observed localization, in particular the low-frequency, small-amplitude regime in which the drive modulates the disorder strength itself and finite-time data suggested stability 'for any frequency', asymptotically stable, or a long-lived prethermal regime that eventually heats to an infinite-temperature state? Determine the possible time scales and microscopic mechanisms (e.g. many-body resonance proliferation, rare thermal inclusions or 'hot-spot' nucleation, avalanche processes) of eventual delocalization. (2) Determine the critical drive frequency below which the driven system delocalizes: does it remain finite in the thermodynamic limit, or can it be reduced to extremely small values, and what governs the extreme long-time and ultra-low-frequency limits that remain inaccessible to current experiments and numerics? (3) Determine whether an analogous disorder-protected, Floquet-induced non-ergodic regime exists in higher spatial dimensions, or whether heating is unavoidable for d≥2d\ge2. (4) Map the frequency-resolved response of the driven system near the many-body localization transition itself. The four parts above are posed jointly as one objective — establishing whether periodically driven many-body localization is a genuine asymptotic non-equilibrium phase and over what parameter range (frequency, amplitude, dimension) it can be used as a platform for Floquet-engineered phases of matter.

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