Many-body localization, thermalization, and entanglement

TL;DR

The paper surveys many-body localization (MBL) as a robust mechanism for ergodicity breaking in disordered, interacting quantum systems. It foregrounds an emergent quasi-local integrability via LIOMs, yielding area-law entanglement for highly excited eigenstates and a universal MBL Hamiltonian with exponentially decaying couplings. By combining entanglement theory, quantum quenches, and a suite of numerical and analytical methods, the authors explain the distinctive dynamical behavior, including logarithmic entanglement growth and non-thermal equilibration of local observables, and discuss the MBL delocalization transition through RG frameworks and Griffiths effects. The review also highlights experimental realizations across ultracold atoms, trapped ions, superconducting circuits, and solid-state systems, alongside Floquet-MBL, localization-protected orders, and ongoing challenges in higher dimensions and open-system dynamics.

Abstract

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 non-zero temperature, which do not thermalize and therefore cannot be described using statistical mechanics. In this Colloquium we review recent theoretical and experimental advances in studies of MBL systems, focusing on the new perspective provided by entanglement and non-equilibrium experimental probes such as quantum quenches. Theoretically, MBL systems exhibit a new kind of robust integrability: an extensive set of quasi-local integrals of motion emerges, which provides an intuitive explanation of the breakdown of thermalization. A description based on quasi-local 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. We outline 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. We review recent experiments with ultracold atoms, trapped ions, superconducting qubits, and quantum materials, in which different signatures of many-body localization have been observed. We conclude by listing outstanding challenges and promising future research directions.

Figures (12)

• Figure 1: In a quantum quench, interacting particles on a lattice are e.g. initially prepared in a state with non-uniform density. Following unitary quantum dynamics, the thermalizing system relaxes towards the state where all lattice sites are equally populated and the density profile is uniform (shown at the top). In contrast, the 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 non-interacting 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 Heisenberg spin chain (a) and spinless fermions in one dimension (b), which are used as a generic model for the MBL phase. Bottom panels show the phase diagram of the spin chain as a function of interaction (c) and disorder strength (d).
• Figure 4: Illustration of the area-law entanglement entropy in one and two spatial dimensions where only the shaded boundary regions $\propto\partial A$ contribute to the entanglement. In contrast, for systems with volume-law entanglement, extensively many degrees of freedom $\propto {\rm vol}(A)$ 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 quasi-local unitary transformations. (b,c) The same quasi-local unitary transformation can be used to obtain the quasi-local operators $\hat{\tau}^z$ and $\hat{\tilde{n}}_i$ which commute with the Hamiltonian.