Prethermal Many-Body Localization

• Prethermal MBL regime is a dynamical phase in disordered, interacting quantum systems that shows slow thermalization and retention of local observables.
• It is characterized by exponential decay of spatial correlations, logarithmic entanglement growth, and a plateau in local markers before eventual thermalization.
• Theoretical models and experimental probes using ultracold atoms and quantum simulators validate its unique non-ergodic behavior and potential for robust quantum memory.

A prethermal many-body localized (MBL) regime refers to a dynamical phase in which a closed interacting quantum system with disorder exhibits a robust non-ergodic window during which the system evolves as if it were many-body localized, before—on extremely long timescales—eventually thermalizing. This regime is characterized by locality-preserving dynamics, strongly suppressed transport, and the retention of local information, but lacks the strict asymptotic stability of a true MBL phase. The prethermal MBL scenario emerges across a range of models, dimensionalities, and experimental platforms, and is often controlled by subtle properties such as resonance proliferation, finite-size effects, energy resolution, or coupling to baths or measurements.

1. Defining Features and Theoretical Framework

The prethermal MBL regime arises in systems where conventional thermalization is hindered due to strong disorder and interactions, leading to extremely slow relaxation dynamics. In this regime, the eigenstates of the system do not satisfy the conventional eigenstate thermalization hypothesis (ETH); instead, local observables retain memory of the initial state for long times, and entanglement propagation is logarithmic or sub-thermal in growth (Pal et al., 2010, Iyer et al., 2012, Smith et al., 2015, Long et al., 2022). The essential diagnostic features include:

• Exponential decay of connected spatial correlations, $C_{n}^{zz}(i,j)\sim \exp(-|i-j|/\xi)$, where $\xi$ is a localization length (Pal et al., 2010).
• Persistence of local observables far from thermal values, with the difference in local magnetization between adjacent eigenstates remaining finite in the localized regime (Pal et al., 2010).
• Logarithmic or stretched exponential temporal growth of entanglement entropy after a quantum quench (Iyer et al., 2012, Long et al., 2022).
• Strongly suppressed thermalization, with equilibration times $\tau$ that grow exponentially with increasing disorder strength: $\log\tau=O(W)$ (Long et al., 2022).

Key mathematical constructs for the prethermal regime include the connected correlation function,

$$C_{n}^{zz}(i,j) = \langle n| S_i^z S_j^z |n\rangle - \langle n| S_i^z |n\rangle \langle n| S_j^z |n\rangle,$$

probability distributions of long-distance correlations, and stretched exponential forms for autocorrelators,

$$\langle Z(t) Z(0)\rangle \sim \exp\left[-(t/\tau)^\beta\right],$$

where $\beta$ is related to the microscopic statistics of many-body resonances (Long et al., 2022).

2. Mechanisms Underlying Prethermal MBL Behavior

Prethermal MBL phenomena are predicated on the slow proliferation of resonances and the associated separation of time scales for thermalization. The resonance model describes the system's dynamics as being controlled by a hierarchy of sparse many-body resonances. At intermediate times, the effective Hamiltonian is approximately diagonal in the local basis, with off-diagonal matrix elements (responsible for thermalization) distributed according to a power law:

$\rho_{\mathrm{dec}}(|H|) \sim |H|^{-2+\theta},$

where $\theta$ is a resonance exponent. The stretch exponent $\beta$ in the autocorrelation decay is directly related to the distribution of resonances via $\beta=-\theta$ (Long et al., 2022).

The dynamical phase diagram is thus revised from the previously assumed sharp transition between fully ergodic and fully MBL phases to one that features an extended prethermal MBL regime in which thermalization is not strictly arrested but is exponentially slow (Long et al., 2022). In this regime, finite chains may appear localized on any accessible timescale, although ultimate thermalization occurs asymptotically. The underlying features of prethermal MBL are supported by the existence of broad, log-normal distributions of observable fluctuations—an infinite-randomness phenomenology—near the MBL transition (Pal et al., 2010).

3. Correlational Diagnostics and Infinite-Randomness Scaling

The transition into and within the prethermal MBL regime is identified by examining full distributions of correlation functions, particularly for long-distance spin correlations and local magnetization differences:

• In the thermal regime, long-range spin correlations remain finite and approximately constant across the system.
• In the localized regime, these correlations decay exponentially with distance, and the local magnetizations exhibit strong sample-to-sample variability (Pal et al., 2010).

The paper (Pal et al., 2010) introduces scaled variables:

$$\phi = \log|C^{zz}_n(i, i+L/2)|,\qquad \eta = \phi / [\phi],$$

where $[\phi]$ is an average over disorder, eigenstates, and sites. The width

$\sigma = \sqrt{[\eta^2]-1}$

serves as a diagnostic: in both the thermal and localized phases, $\sigma$ decreases as the system size $L$ increases (self-averaging). However, near the MBL transition, $\sigma$ increases with $L$, indicating the emergence of infinite-randomness scaling and the predominance of rare regions and strong local fluctuations.

4. Eigenstate Evolution, Entanglement Structure, and Finite-Size Scaling

As the disorder strength $h$ increases, the system exhibits a continuous crossover from thermal to localized behavior:

• At low disorder (thermal/ergodic), eigenstates are locally indistinguishable, spatial correlations persist at large distances, and entanglement entropy follows a volume law.
• Near the critical region ($h_c \sim 3$–$4$ in the Heisenberg chain), correlation function distributions broaden, and their scaled width $\sigma$ increases with $L$.
• For strong disorder (localized), local observables differ significantly between adjacent eigenstates, correlations decay exponentially, and entanglement obeys an area law. In this phase, memory of initial conditions is retained over very long times, manifesting in persistent athermal configurations (Pal et al., 2010, Smith et al., 2015).

Prethermal MBL dynamics are thus characterized by extensive "plateaux" in observables: local autocorrelators, magnetization imbalance, and entanglement entropy remain near their initial values for timescales exponentially long in system size and disorder strength.

5. Implications for Nonequilibrium Dynamics and Experimental Probes

Prethermal MBL regimes have significant ramifications for the understanding and experimental paper of nonequilibrium quantum dynamics:

• In the thermal phase, rapid dephasing leads to equilibration, while in the localized regime, dephasing only erases off-diagonal matrix elements and local information is preserved in the diagonal ensemble for long times (Pal et al., 2010).
• Slow, often logarithmic, growth of entanglement entropy after a quench is observed (as in dynamical entanglement growth in quantum simulators) (Smith et al., 2015).
• Persistent memory of the initial state (magnetization or other local markers) is direct evidence for prethermal non-ergodicity (Smith et al., 2015).
• The short-range entangled structure of MBL eigenstates at high energy densities allows for the application of numerically efficient algorithms, such as DMRG, beyond their traditional ground-state applications (Pal et al., 2010).

These properties make prethermal MBL systems candidates for robust quantum information storage—quantum memories—where decoherence is suppressed by disorder-induced localization.

6. Connection to Infinite-Randomness Universality and Theoretical Approaches

The infinite-randomness fixed-point scenario, established through the scaling of correlation function distributions, connects the prethermal MBL regime to a universality class characterized by highly inhomogeneous, non-Gaussian fluctuations. This scenario is analogous to ground-state transitions in disordered systems, such as the random transverse-field Ising chain.

Moreover, strong disorder renormalization group (SDRG) methods and resonance-based models (successive Jacobi diagonalization) provide both an effective numerical framework and a conceptual foundation for understanding the slow thermalization and broad distribution of relaxation times in the prethermal MBL regime (Long et al., 2022, Pal et al., 2010). These models quantitatively predict stretched exponential decay of correlation functions, the scaling of relaxation times with disorder, and the statistics of resonant matrix elements.

7. Broader Outlook and Universality

The prethermal MBL regime is now understood to be a ubiquitous but ultimately transient phase in disordered many-body systems with finite size or at intermediate times. Its characteristic signatures—a plateau in local observables, slow entanglement growth, broad and non-Gaussian distribution of local fluctuations, and memory of initial states—form the experimental and theoretical basis for the diagnosis of localization phenomena away from ground state physics.

This phenomenology has utility in interpreting experiments in condensed matter systems, ultracold atomic gases, and quantum simulators, and motivates further research—including scaling analyses and strong disorder RG studies—to clarify the universality class of the MBL transition and the true fate of prethermal regimes in the thermodynamic limit.