Many body localization and thermalization in quantum statistical mechanics

TL;DR

The paper surveys how isolated quantum many-body systems can either thermalize or fail to thermalize, focusing on the Eigenstate Thermalization Hypothesis (ETH) and the emergence of many-body localization (MBL). It introduces the eigenstate perspective, including the l-bit framework, and explains how MBL supports non-ergodic dynamics, logarithmic entanglement growth, and localization-protected order at high energy densities. It also discusses open-system probes via local spectra, the possibility of localization in translationally invariant settings, and the challenging MBL phase transition, highlighting the potential for quantum memory applications and rich non-equilibrium physics. Overall, the work provides a comprehensive theoretical foundation for understanding non-thermal eigenstate physics and outlines key open questions and directions for experiments and theory.

Abstract

We review some recent developments in the statistical mechanics of isolated quantum systems. We provide a brief introduction to quantum thermalization, paying particular attention to the `Eigenstate Thermalization Hypothesis' (ETH), and the resulting `single-eigenstate statistical mechanics'. We then focus on a class of systems which fail to quantum thermalize and whose eigenstates violate the ETH: These are the many-body Anderson localized systems; their long-time properties are not captured by the conventional ensembles of quantum statistical mechanics. These systems can locally remember forever information about their local initial conditions, and are thus of interest for possibilities of storing quantum information. We discuss key features of many-body localization (MBL), and review a phenomenology of the MBL phase. Single-eigenstate statistical mechanics within the MBL phase reveals dynamically-stable ordered phases, and phase transitions among them, that are invisible to equilibrium statistical mechanics and can occur at high energy and low spatial dimensionality where equilibrium ordering is forbidden.

Figures (6)

• Figure 1: The system of interest can be represented as a set of spins on a lattice (a). Each spin has a two dimensional state space, which can be represented on a Bloch sphere (b). The many body pure state space for the full system consists of the outer product of the pure state spaces of each spin, as illustrated in (b).
• Figure 2: (a) Conventional quantum statistical mechanics assumes that the system of interest is coupled to a reservoir (or bath), with which it can exchange energy and particles. (b) Here we are interested in the statistical mechanics of a closed quantum system undergoing unitary time evolution. There is no external reservoir. (c) It can be useful to partition the closed quantum system into a subsystem (A) and 'everything else' (B). If the system quantum thermalizes, then the region (B) is able to act as a bath for the subsystem (A).
• Figure 3: An illustration of the dynamical behavior of the l-bits. The l-bits are a set of spins, whose $z$ component does not change, but which precess about the $z$ axis at a rate determined by the effective interactions with all other l-bits. This picture can be used to understand e.g. the logarithmic spreading of entanglement in the fully many body localized phase.
• Figure 4: One can consider an almost perfectly isolated quantum system, i.e. a quantum system that is coupled to an external bath with a weak but non-zero coupling $g$ that couples (weakly) to all of the degrees of freedom in the system.
• Figure 5: Figure showing the spectrum of a local spin flip operator $\sigma^x_i$ of a system governed by Hamiltonian (\ref{['eq: model2']}) coupled to a thermalizing spin chain (a bath) with a coupling $g$. Figure taken from Ref. spectralnumerics. The spectra are obtained by exact diagonalization, for a system and bath that consist of eight spins each. The spectra are averaged over spatial position in the system, and also over all many body eigenstates (i.e. the spectra are evaluated in an 'infinite temperature' Gibbs mixed state). Here $w$ controls the disorder strength in the system. For small $w$ (low disorder), the system is in its thermalizing phase (top panel), and the local spectrum is smooth and continuous. For large $w$ (lower three panels), the system is in its localized phase if isolated. For the coupled finite-size system and bath, the eigenstates become effectively thermal above a coupling $g \approx 0.15$. The second panel shows the local spectrum in the regime where the eigenstates are non-thermal - the local spectrum is highly inhomogenous, and contains a hierarchy of gaps, including the soft gap at zero frequency. The third and fourth panels show the local spectrum at coupling where the eigenstates of the coupled system and bath are thermal. We see that although spectral line broadening does smooth out the local spectrum, the local spectrum retains signatures of proximity to localization.