Many-Body Localized Evolution

• Many-body localized evolution is the study of quantum dynamics in interacting, disordered systems characterized by a breakdown of thermalization and logarithmic entanglement growth.
• Real-space dynamical renormalization reveals emergent quasi-local integrals of motion that constrain the evolution and lead to nonergodic, nonthermal steady states.
• Experimental observations in cold-atom and quasiperiodic platforms, including subthermal entropy and suppressed transport, validate the theoretical framework of MBL.

Many-body localized (MBL) evolution refers to the quantum dynamical processes occurring in disordered, interacting many-body systems that fail to thermalize. In contrast to noninteracting Anderson insulators, MBL systems display an intricate interplay between disorder-induced localization and interaction-induced complexity, manifesting in distinctive features such as unbounded, slow entanglement growth, robust memory of initial conditions, and nonthermal steady states. This phenomenon traverses several conceptual domains: quantum information (entanglement dynamics), statistical mechanics (breakdown of thermalization), dynamical systems (emergent integrability), and computational complexity (tractability of simulation).

1. Model Systems and Microscopic Mechanisms

The foundational microscopic framework for MBL evolution is the disordered XXZ spin chain (or equivalently, a chain of interacting spinless fermions), governed by the Hamiltonian: $$H = J_\perp \sum_{i} (S_i^x S_{i+1}^x + S_i^y S_{i+1}^y) + \sum_i h_i S_i^z + J_z \sum_i S_i^z S_{i+1}^z$$ where $J_\perp$ is the transverse (XY) coupling, $J_z$ is the longitudinal (interaction) coupling, and $h_i$ are random fields drawn from a bounded interval. In the non-interacting limit ($J_z=0$), the system is an Anderson insulator with all single-particle excitations exponentially localized.

Crucially, even infinitesimal interactions ($J_z \neq 0$) render the many-body dynamics qualitatively distinct from noninteracting localization: instead of a strict freeze-out of correlations, interactions induce slow, unbounded (“glassy”) entanglement growth while particle transport remains strongly suppressed (Bardarson et al., 2012). The dynamical evolution after a quantum quench—starting, for example, from a simple product state—is thus the central probe of MBL physics.

2. Entanglement Growth: Logarithmic Evolution and Singular Perturbativity

A haLLMark of MBL dynamics is the unbounded, logarithmic-in-time growth of entanglement entropy in an infinite system. For a subsystem $A$ of the chain, the von Neumann entropy

$S(t) = -\operatorname{Tr}(\rho_A \log \rho_A)$

exhibits a two-stage evolution following a quench:

• An initial rapid (ballistic) increase over a timescale set by the single-particle localization length, independent of $J_z$.
• For $J_z=0$, $S(t)$ saturates at a finite (area-law) value. For $J_z \neq 0$, $S(t) \sim \log t$ continues without bound in the thermodynamic limit (Bardarson et al., 2012, Vosk et al., 2012).

This logarithmic growth reflects that interactions are a singular perturbation: the $J_z \rightarrow 0$ limit is fundamentally distinct from $J_z=0$. Even arbitrarily weak interactions promote decorrelation of local integrals of motion across exponentially long timescales, producing slow but unlimited entanglement propagation. This is in contrast to the typical expectation in localized systems and is reminiscent of classical glassy relaxations (Sinai diffusion, aging) where rearrangements proceed logarithmically with time.

For many-body systems with long-range interactions decaying as $1/|r|^p$, entanglement growth becomes a power law, $S(t) \sim t^{1/(p+1)}$, rather than logarithmic (Pino, 2014).

3. Dynamical Renormalization, Integrals of Motion, and Broken Thermalization

A central insight from real-space dynamical renormalization group (RG) approaches is that MBL can be identified with a dynamical infinite-randomness fixed point (Vosk et al., 2012). Strong local disorder separates energy scales such that rapidly oscillating pairs (strong bonds) are decimated, and effective interactions among residual degrees of freedom are renormalized to broader distributions at longer RG times. The process yields an infinite set of emergent, quasi-local integrals of motion (IOMs) or "l-bits," which become asymptotically exact as disorder strengthens.

MBL eigenstates are thus well-approximated as product states of l-bits, and the time evolution is governed by dephasing interactions between them. The fixed-point Hamiltonian has the form: $$H = \sum_i h_i \tau_i^z + \sum_{i<j} J_{ij} \tau_i^z \tau_j^z + \ldots$$ with rapidly decaying (exponential in distance) couplings $J_{ij}$ (Huse et al., 2014, Imbrie et al., 2016). These l-bits prevent the complete redistribution of energy and quantum correlations that would be enforced by the eigenstate thermalization hypothesis (ETH). Time evolution leads to relaxation within a restricted subspace defined by the conserved values of l-bits, resulting in nonergodic (nonthermal) steady states even in the presence of strong interactions.

In finite chains, entanglement entropy saturates to a volume-law value, yet the per-site entropy remains far below the infinite-temperature thermal equilibrium (Bardarson et al., 2012). The system "thermalizes" only within a constrained phase space, best described as a generalized Gibbs ensemble (GGE) conditioned on the emergent conservation laws (Vosk et al., 2012).

4. Observables, Experimental Signatures, and Glassy Dynamics

Experimentally, MBL can be distinguished from Anderson insulators and ergodic phases by a combination of entanglement diagnostics and transport measurements:

• Local entropy and nonthermal steady states: In cold-atom platforms with engineered disorder, local measurements detect large but subthermal entropy consistent with MBL predictions—indicative of local memory retention and lack of global ergodicity (Bardarson et al., 2012, Lukin et al., 2018).
• Slow dynamical relaxation: Observables such as number fluctuations and density imbalances relax extremely slowly (double-logarithmic in time or not at all), signaling transport blockade (Vosk et al., 2012).
• Configurational entanglement: Beyond number entropy, configurationally nonlocal entanglement builds up logarithmically with time, marking the difference between many-body and single-particle localization (Lukin et al., 2018).
• Glassy analogies: The logarithmic entanglement growth and slow rearrangement of degrees of freedom in MBL are directly analogous to relaxation in classical glasses (e.g., Sinai diffusion), with a distributed spectrum of local relaxation times and persistent nonthermal features (Bardarson et al., 2012).

5. Extensions: Quasiperiodic Systems, Mobility, and Rare Regions

MBL phenomenology extends to systems with quasiperiodic potentials, where localization can occur in the absence of traditional random disorder (Iyer et al., 2012). In such systems, the MBL transition separates a phase where local observables retain initial-state memory and transport is suppressed, from an ergodic phase with thermalizing behavior. Interactions can induce or destroy localization even where single-particle eigenstates are already extended or localized—yielding rich dynamical phase diagrams and nontrivial scaling behaviors.

In quasiperiodic cases, the lack of spatially uncorrelated disorder suppresses the formation of large Griffiths regions, modifying the mechanism and nature of localization transitions as compared to random systems (Iyer et al., 2012, Sierant et al., 11 Mar 2024).

6. Mathematical Formulations and Universal Features

Key formulas governing MBL evolution include:

• Entanglement entropy: $S(t) \sim \log t$ (interacting, short-range); $S(t) \sim t^{1/(p+1)}$ (long-range) (Bardarson et al., 2012, Pino, 2014).
• RG flow of couplings: Probabilities for logarithmic variables $\zeta$, $\beta$ evolve under flow equations reflecting the infinite randomness structure (Vosk et al., 2012).
• Particle number fluctuations: $\langle (\delta N)^2 \rangle \sim \log \log t$, supporting persistent localization in particle transport (Vosk et al., 2012).
• Generalized Gibbs ensemble: Steady states maximize entropy under constraints set by the emergent integrals of motion.

Universal features discovered across analytical, numerical, and experimental studies include: delayed onset of logarithmic entanglement (set by the inverse interaction strength); double-peaked entanglement entropy statistics (with peaks at 0 and $\ln 2$ due to localized spin pairs); and volume-law saturation of entanglement in finite systems, with inherent subthermal values signaling nonergodicity.

7. Broader Implications and Open Directions

The development of MBL evolution theory has yielded conceptually robust diagnostics of nonergodicity and localization in interacting systems, unified via the emergence of quasi-local integrals of motion and the accompanying unbounded, slow entanglement growth. Open questions persist regarding the ultimate stability of the MBL phase in the thermodynamic limit, the role of rare thermal regions (Griffiths effects), and the detailed structure of the phase transition to ergodicity.

MBL dynamics, especially as manifested through entanglement growth and blocked relaxation, have become pivotal in exploring fundamental questions at the intersection of quantum statistical mechanics, information theory, and nonequilibrium physics. Ongoing theoretical and experimental research continues to clarify the regimes of stability, the impact of dimensionality and potential landscapes, and the connections to glassy dynamics and quantum computational complexity.