Experimental establishment of the many-body localization transition and the asymptotic stability of the localized phase in interacting disordered fermions

The 2015 observation of many-body localization (MBL) of interacting fermions in a one-dimensional quasirandom (Aubry-André) optical lattice showed that a prepared charge density wave persists above a critical disorder strength, but raised a connected set of unresolved questions that define the experimental MBL program: the mechanism of the slow decay of the imbalance on timescales of hundreds of tunneling times (extrinsic loss, heating, photon scattering, and inter-tube coupling versus intrinsic SU(2)-symmetry-related thermalization); the existence, location, and universality class of the MBL critical point, including whether quasiperiodic and truly random disorder share a universality class; site-resolved measurement of local relaxation and of imbalance fluctuations as a proxy for entanglement growth; extension to higher dimensions; and the critical dynamics under controlled coupling to an external bath, where the bath relaxation time would be the only relevant scale.

Background

A generic isolated interacting quantum system is expected to thermalize: its local observables relax to values fixed by a few conserved quantities, a property formalized in the eigenstate thermalization hypothesis (ETH). It was proposed in 2006 by Basko, Aleiner and Altshuler that sufficiently strong disorder can prevent thermalization even in the presence of interactions and at finite energy density, defining the many-body localized (MBL) phase. In an MBL phase the eigenstates possess an extensive set of quasi-local integrals of motion (emergent local constants of motion), the entanglement entropy of eigenstates obeys an area law (it scales with the boundary of a subsystem rather than its volume), and, following a global quench, the entanglement entropy of a subsystem grows only logarithmically in time, in contrast to the ballistic growth of thermal systems. Whether the MBL phase is truly stable in the thermodynamic limit, and what the transition separating it from the thermal phase looks like, remains among the central open problems of nonequilibrium quantum statistical mechanics.

For noninteracting particles, disorder localizes single-particle wavefunctions (Anderson localization). The quasiperiodic interacting model was introduced and studied numerically by Iyer, Oganesyan, Refael and Huse (2013). The 2015 experiment of Schreiber et al. used a bichromatic, incommensurate lattice potential — an Aubry-André quasirandom potential, which localizes single particles above a critical modulation strength — to induce disorder in an array of one-dimensional tubes filled with a two-spin-component gas of fermionic potassium-40 atoms. The two spin states realize a spin-$\frac{1}{2}$ Hubbard model (a lattice model of fermions with on-site repulsion and nearest-neighbor tunneling) with on-site interaction $U$ and tunneling $J$; because both spin components experience identical potentials and interactions, the Hamiltonian has a global $SU(2)$ spin-rotation symmetry. Loading a charge density wave (atoms on alternating sites) and quenching, they measured the imbalance $\mathcal{I}=(N_{\mathrm{even}}-N_{\mathrm{odd}})/(N_{\mathrm{even}}+N_{\mathrm{odd}})$, the memory of the initial density order, which serves as an order parameter for localization: $\mathcal{I}$ decays to zero in the ergodic regime and saturates at a nonzero value above a critical disorder strength whose dependence on $U$ agreed with numerical simulations. At strong repulsion, doubly occupied sites (doublons) introduce a second energy scale $J_D\approx 2J^2/|U|$ that competes with disorder and complicates the definition of a stationary state.

The experiment left a precisely stated set of open issues. First, even deep in the localized regime, $\mathcal{I}$ decays slowly on timescales of several hundred tunneling times; the authors attributed this to the imperfect closure of the system (background gas loss, technical heating, photon scattering, residual coupling between parallel one-dimensional tubes) but noted a possible intrinsic mechanism connected to the $SU(2)$ spin symmetry, which theory later showed to be incompatible with conventional MBL. Second, the quasirandom potential has deterministic, non-random statistics: it lacks the exponentially rare weakly disordered regions (Griffiths regions) that control subdiffusive dynamics and the avalanche instability of random disordered systems — the mechanism by which a locally thermal region can recruit neighboring sites and delocalize an entire chain — so the location and the universality class of the MBL transition may differ fundamentally between quasiperiodic and truly random (for example optical speckle) disorder. Third, the imbalance is a single global number; the entanglement growth that defines MBL, the full relaxation of local observables, and the distinction between quantum localization and classical glassy (slow but bath-insensitive) dynamics require single-site-resolved detection — a capability provided by quantum gas microscopes, which image individual lattice sites. Fourth, localization of interacting particles in two and three dimensions was untested, and theory is split between numerical evidence for two- and three-dimensional MBL signatures and avalanche-type arguments for instability at large sizes. Fifth, deliberately opening the system (enhanced photon scattering, or a Bose-Fermi mixture whose Bose condensate excitations act as a bath for the fermions) would, near criticality, make the bath relaxation time the only relevant timescale, enabling a systematic experimental study of the critical dynamics and a clean separation of MBL from insulators whose sluggishness does not depend on quantum coherence.

Problem Statement

Determine, experimentally and quantitatively, whether the nonergodic state observed for interacting fermions in one-dimensional disordered optical lattices is a genuine many-body localized phase with a well-defined transition, and characterize that phase, its transition, and its mechanisms of breakdown. Building on the charge-density-wave imbalance measurements of Schreiber et al. (2015) in the $SU(2)$-symmetric two-component Fermi-Hubbard system with an Aubry-André quasirandom potential, the problem comprises five jointly linked targets:

1. Long-time fate and decay mechanisms. Identify the relative contributions of extrinsic channels (background gas loss, technical heating, photon scattering, coupling between parallel tubes) and intrinsic many-body processes — in particular processes connected to the $SU(2)$ spin symmetry that may drive eventual thermalization — to the observed slow decay of the imbalance on timescales of several hundred tunneling times; and decide whether a genuinely stationary localized regime exists, including at very large $|U|$ and high doublon fractions where additional long timescales compete with heating and loss.

2. Critical point. Determine the existence, location, and universality class (critical behavior, critical exponents, scaling of relaxation times) of the many-body localization transition of the one-dimensional interacting fermion system, for quasiperiodic disorder and for truly random disorder such as optical speckle, and establish whether the two cases belong to the same or to distinct universality classes.

3. Site-resolved dynamics and entanglement. With single-site resolution, measure the full relaxation dynamics of local observables and correlations, and the temporal decay of the fluctuations of the imbalance around its stationary value, as an experimental proxy for the logarithmic entanglement growth characteristic of MBL; use these measurements to distinguish MBL from classical glassy dynamics.

4. Higher dimensions. Extend the localization study to two and three spatial dimensions and establish whether the localized phase of interacting fermions survives there in the thermodynamic limit.

5. Controlled opening to a bath. Couple the system in a controlled way to external baths — deliberately enhanced photon scattering, or a Bose-Fermi mixture whose bosonic excitations form a bath — and characterize the critical dynamics of the localization transition in this setting, where the bath relaxation time provides the only relevant timescale near criticality.

All five targets serve one overarching objective: to establish the MBL phase of interacting disordered fermions — its existence, its boundary, and its breakdown mechanisms — as an asymptotic property of the isolated disordered quantum system, rather than a transient or an extrinsically maintained effect. Whether the phase is an asymptotic property or a transient is settled by whether the apparent transition drifts with system size or time, or persists while the slow decay is quantitatively attributed to the identified channels.

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