Many-Body Localization

Many-body localization, or MBL, denotes a proposed regime of strongly disordered, interacting, and effectively isolated quantum matter in which local memory persists and highly excited states fail to thermalize. The experimentally secure statement is narrower: many platforms exhibit an MBL regime over finite sizes and observation times. Whether that regime becomes a stable asymptotic phase depends on dimensionality, disorder structure, interaction range, rare resonances, and isolation from external baths.

The question here is how to interpret measurements on finite quantum systems: which information survives, whether interactions matter, and whether the apparatus changes the dynamics. Many-Body Localization Preview develops the benchmark models, local-integral description, entanglement growth and thermal-avalanche mechanism. Relaxation and Thermalization explains the thermal comparison. A finite observation can identify a long-lived nonthermal regime without determining its infinite-time fate.

Required background. Disorder in Quantum Matter supplies the disorder ensemble and its experimental controls. Anderson Localization supplies the noninteracting localization baseline. Many-Body Localization Preview supplies the interacting benchmark, local-integral and entanglement diagnostics, and rare-resonance qualifications. Relaxation and Thermalization supplies the distinction between local equilibration and thermal agreement under the appropriate conserved constraints.

Many-body localization in finite quantum materials

Strong disorder, long-lived memory and interacting dephasing provide a useful experimental regime (Abanin et al., 2019). Extrapolating it requires separate control of size, time, dimension, disorder structure and isolation. Random and quasiperiodic potentials need not have the same stability properties. For the evolving research assessment, see Many-body localization research.

The Hamiltonian and the observation window

A common benchmark is the random-field XXZ chain,

$$H = J\sum_{i=1}^{L-1} \left( S_i^xS_{i+1}^x + S_i^yS_{i+1}^y + \Delta S_i^zS_{i+1}^z \right) + \sum_{i=1}^{L}h_iS_i^z.$$

Use dimensionless spin operators $S_i^\alpha=\sigma_i^\alpha/2$, so $J$ and $h_i$ are energies and $\Delta$ is dimensionless. The canonical preview derives the fermion representation. In this chain, $\Delta=0$ is the free-fermion baseline; changing $J\Delta$ tests the effect of interactions.

Specify the field distribution, boundaries, trap gradients and coupling to neighboring chains. For uniform random fields, for example, $h_i\in[-W,W]$ defines $W$ as a half-width; a quoted disorder strength is otherwise ambiguous. Prepare states at matched conserved densities and compare measurements over the same time window.

Useful dimensionless controls include

$$\frac{W}{J}, \qquad \Delta, \qquad \frac{Jt_{\max}}{\hbar}, \qquad \Gamma_{\mathrm{env}}t_{\max}, \qquad L,$$

Here $\Gamma_{\mathrm{env}}$ must be a rate relevant to the observable, with its meaning stated. A rate per site is not the rate of any error anywhere in an $L$-site sample. A plateau reached after relevant environmental effects become appreciable requires an open-system analysis.

Density memory and interactions

A common experiment begins from a density wave or Néel product state and tracks a normalized imbalance,

$$\mathcal I(t) = \frac{N_{\mathrm e}(t)-N_{\mathrm o}(t)} {N_{\mathrm e}(t)+N_{\mathrm o}(t)}.$$

Here $N_{\mathrm e}$ and $N_{\mathrm o}$ are the measured even- and odd-site populations. A spatially uniform thermal state has zero imbalance. In one disordered sample, however, the thermal density can be inhomogeneous even without staggered order. The correct comparison is the independently predicted $\mathcal I_{\mathrm{ens}}$ for the same Hamiltonian, energy and particle number, not automatically zero. A symmetric disorder average can restore a zero baseline.

Also track the denominator: particle loss, sublattice-dependent detection and post-selection can change the normalized contrast. Retained information means a deviation from the appropriate thermal prediction or a distinguishable dependence on matched initial states, beyond these effects.

A plateau can arise from noninteracting localization, slow thermalization, kinetic constraints, a trap gradient or decoherence that suppresses motion. Interactions must therefore be tested separately.

An interaction toggle is therefore unusually informative. In the noninteracting localized baseline, local density memory can persist while bipartite entanglement typically saturates on a one-particle dephasing scale. With interactions, distant localized degrees of freedom can dephase one another without transporting charge, producing continuing entanglement growth. The canonical preview derives the logarithmic form expected from an exponentially decaying l-bit interaction.

For a Hubbard platform the interaction strength is often denoted $U$; for the chain above it is $J\Delta$. Compare the noninteracting memory and entanglement saturation with the interacting dynamics at matched preparation and disorder. Continuing entanglement growth with suppressed transport supports interaction-induced dephasing; it still leaves open whether transport resumes much later.

Comparing local observables with a thermal ensemble

Thermalization is not synonymous with decay of one contrast. For a local observable $O$, compare the late-time measurement with an ensemble fixed independently by the conserved quantities:

$$\delta_O(L,t) = \left| \langle O(t)\rangle - \langle O\rangle_{\mathrm{ens}} \right|.$$

A persuasive finite-window failure of thermalization requires several initial states in the same macroscopic energy and charge window to retain distinguishable local information, while calibration establishes that the difference is larger than statistical, systematic, and finite-size uncertainty.

Measurement	Interpretation and main limitation
Imbalance or autocorrelation	Memory relative to the thermal baseline; several mechanisms can preserve it.
Symmetry-resolved level statistics	A spectral correlation pattern; Poisson behavior also occurs in integrable models.
Excited-state entanglement	Spatial structure of finite eigenstates; does not alone settle larger-system stability.
Time-dependent entanglement or correlations	Can reveal interaction-induced dephasing; purity and measurement protocol matter.
Transport or spreading	Motion over the observed window; small current does not imply every local observable is nonthermal.
Reconstructed local conserved operators	Approximate conservation and spatial support over accessible sizes and times.

Random Matrix Theory in Quantum Matter explains spectral statistics, and Eigenstate Thermalization Hypothesis explains the eigenstate comparison.

What an entanglement measurement determines

In the fully localized effective description, conserved quasilocal pseudospins retain populations while interactions accumulate configuration-dependent phases. This permits entanglement to grow without efficient particle transport. The local-integral derivation explains the assumptions behind that description; a slow observable alone does not reconstruct a conserved operator.

Particle-number fluctuations probe only part of entanglement. For a globally pure state with fixed total particle number, the reduced state is block diagonal in subsystem particle number, and

$$S(\rho_A)=S_N+\sum_n p_n S(\rho_{A,n}), \qquad S_N=-\sum_n p_n\ln p_n.$$

Here $p_n$ is the probability of finding $n$ particles in $A$, and $\rho_{A,n}$ is its normalized state within that number sector. Thus $S_N$ can saturate while correlations within sectors continue to develop. Lukin et al. (2019) used particle fluctuations together with configuration correlations to probe these different contributions.

If the global state is mixed, subsystem entropy need not be entanglement entropy: classical preparation noise can produce the same number statistics. Purity, losses, measurement backaction and post-selection therefore affect what can be inferred. The exercise below gives a minimal counterexample.

Platforms and complementary controls

Platform	Useful capability and leading limitation
Optical-lattice gases	Tunable interactions and direct density readout; trapping, tube coupling and loss matter.
Quantum gas microscopes	Site-resolved correlations and controllable thermal regions; finite size and preparation constrain interpretation.
Trapped-ion spins	Programmable fields and local readout; power-law interactions differ from a nearest-neighbor chain.
Superconducting arrays	Local controls and tomography; decoherence and coherent calibration errors both affect dynamics.
Disordered solids	Large samples; phonons, leads and other environments complicate isolated-system interpretation.

Schreiber et al. (2015) provide an optical-lattice imbalance example; Smith et al. (2016) demonstrate a programmable ion platform. These different Hamiltonians should be compared through their measured observables and control parameters, not a shared disorder number.

Varying an interaction or coupling to a bath tests a mechanism directly. A deliberately prepared thermal region provides another intervention: its influence on progressively more distant sites tests whether it can absorb neighboring localized degrees of freedom. Léonard et al. (2023) implemented this idea. Its finite extent and growth time must still be resolved before drawing a thermodynamic conclusion.

Numerical methods supply overlapping checks. Exact diagonalization gives complete small-system spectra; Krylov methods reach longer chains at finite times; matrix-product methods extend the accessible sizes when entanglement stays manageable. Increasing bond dimension and comparing with exact evolution where possible exposes artificial suppression of entanglement. Agreement within a common size/time range is more informative than an uncontrolled extrapolation from one method.

An open solid can exhibit hopping transport or glassy aging without isolated-system MBL. Those mechanisms have their own observable predictions.

Limits on extrapolating the observed regime

Rare thermal regions

Random disorder permits atypically weak-disorder regions. If such a region has a sufficiently dense many-body spectrum, it can hybridize neighboring localized degrees of freedom, enlarge, and trigger a thermal avalanche. This is a mechanism for eventual delocalization that can remain invisible at accessible sizes.

Quasiperiodic potentials do not contain the same probabilistic rare regions. That distinction makes random-versus-quasiperiodic comparison valuable, but it does not prove quasiperiodic stability: deterministic resonances, dimensional coupling, and finite-frequency heating still require analysis.

Dimension and interaction range

Avalanche phase space generally becomes less favorable to localization as spatial dimension increases. Long-range interactions can also connect distant regions more strongly than an exponential l-bit model assumes. A result from a nearest-neighbor one-dimensional chain cannot be transferred unchanged to dipolar ions, cavity-mediated spins, or a two-dimensional lattice.

Environmental coupling

A generic bath coupling that does not commute with the l-bits can relax their populations and destroy localized memory at sufficiently long times. Pure dephasing need not do so: for an l-bit Hamiltonian diagonal in $\tau_i^z$ and Lindblad jumps $L_i=\sqrt{\gamma_i}\,\tau_i^z$, each $\tau_i^z$ remains conserved while off-diagonal coherences decay. The coupling operators, not only a bath rate, determine which memory survives. The Lindblad–GKSL Equation explains how these coupling operators determine reduced dynamics.

A finite-time test of nearly isolated dynamics needs a window after microscopic transients but before the relevant environmental processes dominate:

$$\tau_{\mathrm{micro}} \ll t_{\mathrm{obs}} \ll \Gamma_{\mathrm{env}}^{-1}.$$

The intrinsic memory time $\tau_{\mathrm{int}}$ can be much longer than the bath time and still permit such a finite observation window. Resolving an intrinsic eventual decay before the bath acts requires the stronger hierarchy $\tau_{\mathrm{int}}\ll\Gamma_{\mathrm{env}}^{-1}$. If it fails, finite-time memory can still be tested, but its intrinsic asymptotic fate remains unresolved. Controlled bath coupling is scientifically useful because it distinguishes dephasing, particle loss, and population-relaxation channels.

Energy-resolved claims

A many-body mobility edge would divide localized and thermal eigenstates by energy density. Finite spectra can display an apparent energy-dependent crossover even when rare thermal states destabilize the proposed edge at larger size. Mobility Edges explains the single-particle concept; the canonical MBL preview explains why the interacting claim needs additional resonance and avalanche tests.

Size-dependent drift is therefore a physical part of the result. Sierant et al. (2025) explain why naive single-parameter fits can fail in this regime. Track how conclusions change when size and time are increased separately, and use Finite-Size Scaling in Numerics to compare fit assumptions. The research discussion follows the open asymptotic questions; this page concentrates on what a finite measurement determines.

Exercises

1. Classify the claim

A $12$-site superconducting-qubit array retains a density imbalance through $Jt/\hbar=80$. The result is reproducible over $200$ disorder realizations, but no interaction toggle or system-size series is reported. What has been established?

Solution

If the contrast differs from the calibrated thermal prediction and detection artifacts are excluded, the experiment establishes finite-window local memory in the implemented disordered circuit:

$$\mathcal I(t) \ne \mathcal I_{\mathrm{ens}} \qquad \text{for} \qquad \frac{Jt}{\hbar}\le80.$$

It does not distinguish interacting localization from an Anderson-localized, constrained, disconnected, or calibration-limited mechanism because the interaction control is missing. It also cannot establish asymptotic stability because only one size is studied. The defensible result is finite-system memory compatible with MBL phenomenology; the asymptotic phase remains untested.

2. Number fluctuations without entanglement

Compare one particle shared between sites $A$ and $B$ in the pure state

$$|\psi\rangle=(|10\rangle+|01\rangle)/\sqrt2$$

with the incoherent mixture

$$\rho_{\mathrm{mix}}=\tfrac12|10\rangle\langle10| +\tfrac12|01\rangle\langle01|.$$

Do number measurements distinguish them? Which state is entangled across the two sites in the usual tensor-product description?

Solution

Both give probabilities $1/2$ for zero or one particle in $A$, so $S_N=\ln2$ and $\rho_A=I/2$ in each case. The pure state has subsystem entanglement entropy $\ln2$. The mixture is explicitly a convex sum of product states and is separable, despite the same subsystem entropy. Off-diagonal coherence distinguishes the two global states. Number statistics alone therefore do not certify entanglement for a mixed preparation. Restrictions from local particle-number conservation can further constrain which operations access the pure state’s entanglement.

3. Check the isolation window

An experiment has a Poisson error rate $\gamma=0.4\ \mathrm{s}^{-1}$ and records dynamics to $t_{\max}=0.25\ \mathrm{s}$. Find the probability of at least one error if $\gamma$ is the total rate. What if it is instead an independent rate at each of $L=12$ sites?

Solution

The exposure is $\gamma t_{\max}=0.10$. For a total Poisson rate,

$$P_{\geq1}=1-e^{-\gamma t_{\max}}\simeq0.0952.$$

If the same rate applies independently to each site, the total rate is $L\gamma$ and

$$P_{\geq1}^{\mathrm{sample}}=1-e^{-L\gamma t_{\max}}\simeq0.699.$$

Thus a small per-site exposure does not imply that the whole state is nearly error free. A local observable need not fail whenever any site experiences an error; its sensitivity depends on the coupling operator and dynamics. Isolation estimates must match the quantity being measured.

4. Read a drifting crossover

Adjacent-gap-ratio curves cross near disorder strengths

$$W_\times(12,14)=3.6, \qquad W_\times(14,16)=4.2, \qquad W_\times(16,18)=4.9.$$

What conclusion is justified?

Solution

The crossing drifts substantially toward stronger disorder as size increases. The data establish a finite-size crossover, not a size-independent critical point. A credible analysis should fit relevant and irrelevant scaling variables, vary the fitting window, inspect distributions and other observables, and test whether the drift saturates or continues.

Quoting $W_c=4.2$ from the middle pair would discard the dominant systematic trend.

References

• D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Colloquium: Many-Body Localization, Thermalization, and Entanglement,” Reviews of Modern Physics 91, 021001 (2019), doi:10.1103/RevModPhys.91.021001. Reviews theory, diagnostics, and experiments.
• W. De Roeck and F. Huveneers, “Stability and Instability towards Delocalization in Many-Body Localization Systems,” Physical Review B 95, 155129 (2017), doi:10.1103/PhysRevB.95.155129. Formulates thermal-inclusion stability constraints.
• J. Léonard et al., “Probing the Onset of Quantum Avalanches in a Many-Body Localized System,” Nature Physics 19, 481–485 (2023), doi:10.1038/s41567-022-01887-3. Plants thermal inclusions and tracks their influence.
• A. Lukin et al., “Probing Entanglement in a Many-Body-Localized System,” Science 364, 256–260 (2019), doi:10.1126/science.aau0818. Probes entanglement through number fluctuations and configuration correlations in a quantum gas microscope.
• A. Rubio-Abadal et al., “Many-Body Delocalization in the Presence of a Quantum Bath,” Physical Review X 9, 041014 (2019), doi:10.1103/PhysRevX.9.041014. Studies controlled environmental destruction of localization-compatible memory.
• M. Schreiber et al., “Observation of Many-Body Localization of Interacting Fermions in a Quasirandom Optical Lattice,” Science 349, 842–845 (2015), doi:10.1126/science.aaa7432. Establishes long-lived imbalance with interaction and disorder controls.
• P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, “Many-Body Localization in the Age of Classical Computing,” Reports on Progress in Physics 88, 026502 (2025), doi:10.1088/1361-6633/ad9756. Reviews finite-size drift, slow dynamics, numerical limits, and the present phase-status problem.
• J. Smith et al., “Many-Body Localization in a Quantum Simulator with Programmable Random Disorder,” Nature Physics 12, 907–911 (2016), doi:10.1038/nphys3783. Uses a trapped-ion spin simulator with programmable disorder.

Further Reading

• F. Alet and N. Laflorencie, “Many-Body Localization: An Introduction and Selected Topics,” Comptes Rendus Physique 19, 498–525 (2018), doi:10.1016/j.crhy.2018.03.003. A pedagogical review of diagnostics and phase structure.
• R. Nandkishore and D. A. Huse, “Many-Body Localization and Thermalization in Quantum Statistical Mechanics,” Annual Review of Condensed Matter Physics 6, 15–38 (2015), doi:10.1146/annurev-conmatphys-031214-014726. A concise conceptual introduction.