Colloquium: Many-body localization, thermalization, and entanglement

Thermalizing quantum systems are conventionally described by statistical mechanics at equilibrium. However, not all systems fall into this category, with many-body localization providing a generic mechanism for thermalization to fail in strongly disordered systems. Many-body localized (MBL) systems remain perfect insulators at nonzero temperature, which do not thermalize and therefore cannot be described using statistical mechanics. This Colloquium reviews recent theoretical and experimental advances in studies of MBL systems, focusing on the new perspective provided by entanglement and nonequilibrium experimental probes such as quantum quenches. Theoretically, MBL systems exhibit a new kind of robust integrability: an extensive set of quasilocal integrals of motion emerges, which provides an intuitive explanation of the breakdown of thermalization. A description based on quasilocal integrals of motion is used to predict dynamical properties of MBL systems, such as the spreading of quantum entanglement, the behavior of local observables, and the response to external dissipative processes. Furthermore, MBL systems can exhibit eigenstate transitions and quantum orders forbidden in thermodynamic equilibrium. An outline is given of the current theoretical understanding of the quantum-to-classical transition between many-body localized and ergodic phases and anomalous transport in the vicinity of that transition. Experimentally, synthetic quantum systems, which are well isolated from an external thermal reservoir, provide natural platforms for realizing the MBL phase. Recent experiments with ultracold atoms, trapped ions, superconducting qubits, and quantum materials, in which different signatures of many-body localization have been observed, are reviewed. This Colloquium concludes by listing outstanding challenges and promising future research directions.

Figure 1

In a quantum quench, interacting particles on a lattice are, e.g., initially prepared in a state with nonuniform density. Following unitary quantum dynamics, the thermalizing system relaxes toward the state where all lattice sites are equally populated and the density profile is uniform (shown at the top). In contrast, a nonthermalizing many-body localized system retains the memory of initial state even at infinite time (bottom).

Figure 2

(a) In a clean crystal, eigenstates are Bloch waves, which extend throughout the sample. (b) The essence of Anderson localization of noninteracting particles is that for sufficiently strong disorder there is a vanishing probability for a particle to make a resonant transition from one site to another one spatially separated from it. This leads to eigenstates which are localized in some region of space, decaying exponentially away from it. (c) Adding interactions to an Anderson localized system. To first order, the effect of interaction is to induce hopping of pairs of particles between the single-particle localized orbitals. One may ask if the localized phase, with vanishing particle and thermal conductivities, is robust to this process.

Figure 3

Sketch of the (a) Heisenberg spin chain and (b) spinless fermions in one dimension, which are used as generic models for the MBL phase. Bottom panels show the phase diagram of the spin chain as a function of (c) interaction and (d) disorder strength.

Figure 4

Illustration of the area-law entanglement entropy in one and two spatial dimensions where only the shaded boundary regions that include $\propto \mathrm{v}\mathrm{o}\mathrm{l}(\partial A)$ degrees of freedom contribute to the entanglement. In contrast, for systems with volume-law entanglement, extensively many degrees of freedom $\propto \mathrm{vol}(A)$ are entangled with the exterior region.

Figure 5

(a) Rotation of the product states into the exact many-body eigenstates can be achieved by a sequence of quasilocal unitary transformations. (b), (c) The same quasilocal unitary transformation can be used to obtain the quasilocal operators ${\widehat{\tau }}^z$ and ${\widehat{\tilde{n}}}_i$ which commute with the Hamiltonian.

Figure 6

Mechanism of dynamics in the MBL phase: (a) The central LIOM precesses with time, and interactions with other LIOMs lead to the dependence of its precession frequency on states of neighboring spins. This process is responsible for logarithmic growth of entanglement shown in (b) and also for the relaxation of fluctuations of $⟨{\widehat{\tau }}^x(t)⟩$ illustrated in (c).

Figure 7

(a) RG description starts with a collection of blocks each characterized by the hybridization rate and level spacing. Within one RG step two blocks with the strongest coupling rate are merged together resulting in a block with new effective parameters. (b) The inverse dynamical exponent $\alpha$ vanishes in the MBL phase and interpolates continuously between 0 and $1/2$ in the thermal phase.

Figure 8

Nonthermalizing out-of-equilibrium evolution of an initial density wave in the presence of a quasiperiodic detuning potential in the interacting Aubry-André model [see Eq. (17)]. Time traces of the imbalance $I$ for various strengths of the detuning potential $\mathrm{\Delta }$. Points are experimental measurements, averaged over six different phases $\varphi$ of the quasiperiodic detuning lattice. Lines denote DMRG simulations that take into account the trapping potential and the averaging over neighboring tubes, which are present in the experiment. From [134].

Figure 9

Probing many-body localization in two dimensions. (a) Almost arbitrary disorder potentials of light are projected onto an ultracold bosonic atom cloud. The subsequent quantum evolution of an initial nonequilibrium state can then be tracked in the experiment. (b) In the experiment an initial domain wall of a bosonic Mott insulator is prepared (“half circle” in images). Even for long evolution times of $\simeq 250$ tunneling times, the system fails to thermalize, indicated by the remnant domain wall still visible in the experiment. In contrast, a thermalized state would not carry any information about the initial state of the system. From [33].

Figure 10

Level spacing statistics obtained in a system of two interacting photons on nine lattice sites and a local disorder potential using many-body spectroscopy. For weak disorder the system exhibits a distribution resembling the one of a Wigner-Dyson Gaussian orthogonal ensemble, whereas for stronger disorder strengths a change toward the one of a Poissonian distribution is observed. Deviations at very small level spacings $r$ are attributed to the finite size and small number of particles in the system. From [130].

Figure 11

Schematic phase diagram of an open MBL system: Coupling an MBL system to a bath destroys the signatures of the system on a time scale that depends on the coupling strength. Signatures might, e.g., be a persistent imbalance in the MBL phase or a power-law decay of the imbalance in the thermalizing phase. The yellow and blue shaded regions indicate the regimes where those quantities are still accessible. In the white regime of strong couplings, the bath dominates the dynamics of the system. The ideal phases and an actual transition exist only at $\gamma =0$. From [95].

Figure 12

Engineered disorder pattern. Different techniques allow one to engineer disorder potentials in highly controllable ways. It is, for example, feasible to include nondisordered regions in the disorder pattern, where the system can locally thermalize. The stability of the overall MBL upon different densities and sizes of such thermalizing inclusions could thereby be investigated. This would enable one to probe the stability of MBL in fundamentally new ways.