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.

This page is a quantum-matter evidence bridge. It owns the path from a platform Hamiltonian and preparation protocol to a defensible experimental or numerical claim. The canonical Many-Body Localization Preview owns the random-field-chain benchmark, l-bit construction, logarithmic entanglement derivation, spectral diagnostics, avalanche mechanism, and detailed status ledger. Relaxation and Thermalization owns the operational definition of thermalization. Those derivations are not repeated here.

The central inference discipline is

$$\text{memory at finite }(L,t_{\max}) \not\Rightarrow \text{localization as }L,t\to\infty.$$

Finite-window localization is still valuable quantum matter. It can protect memory, expose interaction-driven dephasing, benchmark quantum simulators, and reveal microscopic routes to ergodicity. It should simply be named at the strength established by the data.

Status at a Glance

Claim	Current status	Required qualifier
Strong disorder produces long-lived nonthermal dynamics in finite systems	Established across several synthetic platforms	State $L$, $t_{\max}$, preparation, disorder, and isolation scales
Interactions distinguish MBL phenomenology from Anderson localization	Established in benchmark models and finite-window experiments	Include an interaction toggle or independently calibrated interaction
Quasilocal integrals of motion describe a fully localized regime	Standard effective phenomenology	Their existence is conditional, not a universal theorem
A restricted one-dimensional random spin chain can be localized	Rigorous under stated assumptions	Do not generalize the theorem to every disordered chain
Generic short-range random systems have a sharp one-dimensional MBL transition	Active and unsettled	Strong finite-size drift and resonances obstruct extrapolation
Generic random systems support stable MBL in dimensions above one	Strongly challenged by avalanche arguments	Finite-time two-dimensional localization is not an asymptotic proof
Quasiperiodic systems evade every instability	Unsettled	Absence of random rare regions removes one mechanism, not all resonances
A disordered solid with very large resistance is many-body localized	Not established by transport alone	Phonons, leads, electron heating, and hopping must be excluded

The table separates phenomenology, which is robust and experimentally accessible, from an asymptotic phase verdict, which remains model dependent.

Start with a System Ledger

A common benchmark is the random-field XXZ chain,

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

The canonical preview derives its fermion representation and diagnostics. For a quantum-matter claim, the essential task is to map every symbol to a controlled or measured scale:

Ledger item	Question
Interaction	What is $\Delta J$, and can it be switched off or varied?
Disorder	Are the $h_i$ random, quasiperiodic, correlated, or calibrated site by site?
Geometry	What are $L$, dimension, boundaries, transverse couplings, and trap gradients?
Interaction range	Are couplings nearest-neighbor, dipolar, power-law, or cavity mediated?
Preparation	Which energy-density window and symmetry sector does the initial state occupy?
Observation	Which local memories, correlations, entropies, or currents are measured?
Isolation	What are the loss, dephasing, heating, and background-bath rates?
Resolution	What are the longest reliable time and the smallest resolvable signal?

Useful dimensionless controls include

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

where $W$ characterizes the declared disorder distribution and $\Gamma_{\mathrm{env}}$ is an independently bounded environmental rate. A plateau observed only after $\Gamma_{\mathrm{env}}t$ becomes order unity cannot be interpreted with a closed-system Hamiltonian alone.

Localization with Interactions

The cleanest experiment begins from a density wave or Néel product state and tracks a normalized imbalance,

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

For an ergodic, number-conserving system with no persistent staggered order, $\mathcal I(t)$ should decay toward zero after finite-size and symmetry effects are controlled. A nonzero finite-time value demonstrates retained memory of the preparation.

That observation alone does not distinguish:

1. Anderson localization of noninteracting orbitals;
2. interaction-induced many-body localization;
3. slow but ultimately thermal dynamics;
4. kinetic constraints or Hilbert-space fragmentation;
5. a trap, detuning gradient, or disconnected region;
6. decoherence that suppresses the intended dynamics.

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.

The logic is comparative:

$$\begin{gathered} U=0: \quad \text{memory} + \text{rapid entanglement saturation}, \\ U\ne0: \quad \text{memory} + \text{continued dephasing-driven entanglement growth}. \end{gathered}$$

This comparison supports interacting localization phenomenology. It does not by itself settle the infinite-time fate.

Failure of Thermalization Is a Test

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.

Observation	What it supports	What it does not establish alone
Persistent imbalance or autocorrelation	Local memory over the measured window	Interaction-driven localization
Poisson-like level statistics	Lack of Wigner–Dyson repulsion in a symmetry sector	Localization rather than integrability or mixed sectors
Area-law excited eigenstates	Weak spatial entanglement in finite eigenstates	Dynamical stability at larger $L$
Slow or logarithmic entanglement growth	Interaction-induced dephasing compatible with l-bits	A universal l-bit construction
Suppressed transport	Small current or spreading over the window	Failure of all local observables to thermalize
Reconstructed quasilocal conserved operators	Approximate conserved structure in the studied system	Exact integrals in the thermodynamic limit

Random Matrix Theory in Quantum Matter owns level statistics and symmetry resolution. Eigenstate Thermalization Hypothesis owns the matrix-element ansatz that an MBL claim challenges.

Local Integrals and Entanglement

In the fully localized phenomenology, physical spins are related by a quasilocal unitary transformation to conserved pseudospins $\tau_i^z$. The effective Hamiltonian is diagonal in those operators,

$$H_{\mathrm{eff}} = \sum_i\widetilde h_i\tau_i^z + \sum_{i<j}J_{ij}\tau_i^z\tau_j^z + \sum_{i<j<k} J_{ijk}\tau_i^z\tau_j^z\tau_k^z + \cdots,$$

with typical couplings that decrease rapidly with separation. This compact expression explains two observations at once:

• the $\tau_i^z$ retain local memory because they commute with $H_{\mathrm{eff}}$;
• the interaction terms dephase superpositions and spread entanglement without requiring particle transport.

An experiment does not measure an abstract l-bit merely by seeing slow dynamics. Stronger evidence reconstructs an operator with spatially decaying weight, tests its conservation under the measured evolution, and checks robustness across initial states. Entanglement-based reconstruction and controlled local tomography can probe such approximate conserved structure in finite systems, but the inferred localization length and conservation time remain window-dependent quantities.

Platform Comparison

Platform	Principal strengths	Leading cautions
Ultracold fermions in optical lattices	Tunable interactions, quasiperiodic or speckle potentials, direct density imbalance, long coherence	Traps, tube coupling, finite particle number, loss, and limited entropy access
Quantum gas microscopes	Site-resolved occupations, correlations, number entropy, planted thermal regions	Small systems, preparation bias, finite imaging fidelity, often quasiperiodic rather than random disorder
Trapped-ion spin simulators	Programmable random fields and state-resolved readout	Power-law interactions, modest $L$, inhomogeneous couplings, and finite coherence
Superconducting-qubit arrays	Local control, rapid sampling, tomography, engineered disorder, Fock-space probes	Decoherence, calibration drift, gate or Floquet errors, and device connectivity
Solid-state electronic or spin systems	Large size and long laboratory time	Uncontrolled phonons, leads, nuclear spins, charge noise, heating, and indirect observables

The table explains why synthetic platforms dominate direct MBL tests. A solid can show interaction-modified hopping, electron-glass memory, or a disorder-driven metal–insulator transition without realizing an isolated many-body Hamiltonian. Anderson Insulators owns bath-assisted hopping, while Glasses and Spin Glasses owns aging and overlap order. Neither phenomenon should be relabeled as MBL from slow response alone.

Experiments and Simulations

Use controls as part of the result

A strong experimental program compares:

1. weak and strong disorder;
2. interacting and noninteracting dynamics;
3. random and quasiperiodic potentials when the platform permits;
4. several system sizes or connected subsystem sizes;
5. several initial states at matched conserved densities;
6. local memory, correlations, entanglement, and transport;
7. nominally closed evolution with calibrated loss or dephasing controls;
8. homogeneous samples with intentionally planted thermal inclusions.

The planted-inclusion protocol is especially informative because it tests a mechanism. If a locally thermal region grows and erases memory farther away as its size increases, the experiment probes avalanche propagation rather than merely fitting another crossover curve.

Match the method to the claim

Method	Reliable contribution	Main limitation
Exact diagonalization	Full spectra, eigenstates, gap ratios, and exact finite-time evolution	Exponentially small accessible $L$ and severe drift
Krylov time evolution	Larger finite systems at controlled times	Does not reach arbitrary time or settle asymptotic eigenstates
Matrix-product-state evolution	Long one-dimensional systems in weak-entanglement regimes	Cost grows with entanglement and can bias the accessible window
Tensor-network eigenstate methods	Candidate localized eigenstates and quasilocal operators	Convergence can fail near resonances or the crossover
Analog quantum simulation	Large, directly evolving systems and microscopic interventions	Hamiltonian certification and finite-time inference
Digital quantum simulation	Programmable disorder, observables, and circuit-level controls	Noise, Trotterization, mitigation bias, and limited depth

Agreement among methods in an overlapping $(L,t)$ window is much stronger than extrapolation from any one method. Finite-Size Scaling in Numerics gives the canonical covariance, irrelevant-variable, and crossing-drift workflow.

Stability Questions

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

An external bath destroys exact closed-system local integrals of motion at sufficiently long times. The useful question is often whether the intrinsic memory time is parametrically shorter or longer than the calibrated bath time:

$$\tau_{\mathrm{int}} \ll \Gamma_{\mathrm{env}}^{-1} \quad \text{or} \quad \tau_{\mathrm{int}} \gtrsim \Gamma_{\mathrm{env}}^{-1}.$$

Only the first ordering gives a broad window in which an isolated-system interpretation can be tested. Controlled bath coupling is nevertheless scientifically useful: it measures how memory disappears and can distinguish dephasing, particle loss, and thermalization 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 owns the single-particle concept; the canonical MBL preview explains why the interacting claim needs additional resonance and avalanche tests.

Current Research Boundary

The 2025 review by Sierant et al. treats the robust finite-size, finite-time MBL regime as established while emphasizing persistent drift toward ergodicity and the unresolved conditions for an asymptotic phase. That is the stable status adopted here.

A 2025 preprint, revised in March 2026, reports two-dimensional ultracold-atom measurements up to $24\times24$ sites. Its random-disorder crossover shifts with size in a direction consistent with avalanche instability, while the quasiperiodic data show no clear size drift over the measured window. The result is important and unusually large-scale, but it remains a finite-window preprint result: “consistent with” and “no clear drift” are not thermodynamic-limit verdicts.

This page should be reviewed whenever a major experiment establishes a new size/time scaling window or a rigorous result changes the model classes known to localize.

Evidence Workflow

1. Declare the Hamiltonian. Include disorder ensemble, interaction range, symmetries, boundaries, gradients, and drive protocol.
2. Calibrate isolation. Measure loss, heating, dephasing, and detector backaction independently.
3. Define the claim window. Report $L$, $t_{\max}$, energy density, disorder count, and signal floor.
4. Use an interaction control. Compare to an Anderson-localized or otherwise noninteracting baseline.
5. Use several diagnostics. Combine memory, transport, entanglement or correlations, and symmetry-resolved spectra where available.
6. Scale the window. Increase size and time separately; do not hide crossing drift in a one-parameter collapse.
7. Probe a mechanism. Plant a thermal inclusion, vary transverse coupling, or tune a controlled bath.
8. State the strongest justified noun. Choose “slow dynamics,” “MBL-compatible regime,” or “asymptotic MBL phase” according to the evidence.

Common Mistakes

• Calling every long-lived imbalance MBL.
• Omitting the $U=0$ Anderson-localized control.
• Treating logarithmic-looking growth over less than a decade as an exponent measurement.
• Mixing symmetry sectors before computing level statistics.
• Averaging a heavy-tailed observable without reporting medians or distributions.
• Inferring a critical disorder from one crossing of small systems.
• Treating quasiperiodic modulation as statistically random disorder.
• Ignoring trap gradients or Stark localization.
• Using resistance or variable-range hopping in an open solid as direct evidence for MBL.
• Interpreting decoherence-suppressed dynamics as successful isolation.
• Calling a prethermal or fragmented plateau an infinite-time localized phase.
• Claiming a mobility edge without energy-window and avalanche audits.

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

The experiment establishes finite-window local memory in the implemented disordered circuit:

$$\mathcal I(t) \ne 0 \qquad \text{for} \qquad \frac{Jt}{\hbar}\le80.$$

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. “MBL-compatible finite-system memory” is defensible; “an MBL phase” is not.

2. Interaction-driven dephasing

Two localized experiments show the same persistent density imbalance. In the first, entanglement saturates rapidly when interactions are switched off. In the second, weak interactions are restored and entanglement continues to increase while particle transport remains suppressed. Why is the comparison stronger than either observation alone?

Solution

Persistent density memory occurs in both Anderson and many-body localization. The interaction toggle isolates the additional process: localized degrees of freedom dephase one another through interaction terms even when they do not transport particles. Continued entanglement growth with suppressed transport is therefore evidence for interacting localization phenomenology rather than one-particle localization alone.

The result remains finite-window evidence. Very slow transport or a later thermal avalanche could still appear beyond the measured scale.

3. Check the isolation window

An experiment has $\Gamma_{\mathrm{env}}=0.4\ \mathrm{s}^{-1}$ and records dynamics to $t_{\max}=0.25\ \mathrm{s}$. Compute $\Gamma_{\mathrm{env}}t_{\max}$ and interpret the result.

Solution

The dimensionless environmental exposure is

$$\Gamma_{\mathrm{env}}t_{\max} = (0.4\ \mathrm{s}^{-1})(0.25\ \mathrm{s}) = 0.10.$$

Environmental effects are not guaranteed to be negligible, but the calibrated rate leaves a plausible closed-system window because the expected exposure is well below unity. The claim should propagate uncertainty in $\Gamma_{\mathrm{env}}$, verify that the rate applies to the prepared state, and repeat the analysis at shorter times. A plateau appearing only near $t\sim\Gamma_{\mathrm{env}}^{-1}$ would be much harder to interpret intrinsically.

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.

5. Random versus quasiperiodic disorder

Why does the absence of Griffiths rare regions in a quasiperiodic potential strengthen, but not complete, a stability argument?

Solution

Random disorder samples contain statistically rare weak-disorder regions that can act as thermal seeds. A deterministic quasiperiodic potential lacks that particular rare-region distribution, so eliminating it removes one important avalanche source.

It does not eliminate every instability. Deterministic many-body resonances, transverse coupling, long-range interactions, boundary baths, finite-frequency heating, or other inhomogeneities can still thermalize the system. Stability must therefore be demonstrated for the actual quasiperiodic Hamiltonian and limits, not inferred solely from the absence of Griffiths regions.

6. Audit a solid-state claim

A disordered film has a resistance that rises rapidly at low temperature, logarithmic conductance relaxation after a gate quench, and memory of previous gate voltages. List six reasons this is not yet evidence for MBL.

Solution

At least the following are unresolved:

1. the film is coupled to phonons, leads, electromagnetic modes, and the measurement circuit;
2. rising resistance may be ordinary activation or variable-range hopping;
3. electron-glass aging can produce slow relaxation without isolated-system MBL;
4. contacts and electron heating can imitate nonlinear localization signatures;
5. no interaction toggle separates Anderson from many-body physics;
6. transport does not test failure of thermalization for several local observables;
7. the prepared energy density and closed-system Hamiltonian are undefined;
8. no size/time extrapolation or entanglement-compatible diagnostic is supplied.

The observations support glassy localized transport in an open disordered solid. They do not establish an MBL phase.

Connections

• Many-Body Localization Preview is the canonical home for the benchmark model, l-bits, entanglement growth, avalanches, finite-size analysis, and the detailed frontier ledger.
• Anderson Localization owns the noninteracting interference baseline and direct spatial or transport diagnostics.
• Disorder in Quantum Matter fixes disorder ensembles, averaging order, elastic lifetimes, and sample-specific observables.
• Anderson Insulators distinguishes closed-system MBL from phonon-assisted hopping and Coulomb-gap transport.
• Glasses and Spin Glasses distinguishes aging, overlap order, and slow collective freezing from eigenstate nonthermalization.
• Relaxation and Thermalization owns operational equilibration and ensemble-identification tests.
• Quantum Thermalization gives the platform-facing positive test for local equilibration and constrained thermal values.
• Quantum Quenches owns preparation and unitary-evolution conventions.
• Eigenstate Thermalization Hypothesis develops the matrix-element structure whose failure motivates MBL.
• Random Matrix Theory in Quantum Matter owns symmetry-resolved level repulsion and Poisson benchmarks.
• Finite-Size Scaling in Numerics supplies crossing-drift, covariance, correction-to-scaling, and uncertainty controls.
• Open Quantum Systems owns reduced dynamics and environmental coupling beyond the isolated model.

References

1. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metal–Insulator Transition in a Weakly Interacting Many-Electron System with Localized Single-Particle States,” Annals of Physics 321, 1126–1205 (2006), doi:10.1016/j.aop.2005.11.014. Develops the interacting localized-electron framework.
2. V. Oganesyan and D. A. Huse, “Localization of Interacting Fermions at High Temperature,” Physical Review B 75, 155111 (2007), doi:10.1103/PhysRevB.75.155111. Establishes early finite-size spectral diagnostics.
3. J. H. Bardarson, F. Pollmann, and J. E. Moore, “Unbounded Growth of Entanglement in Models of Many-Body Localization,” Physical Review Letters 109, 017202 (2012), doi:10.1103/PhysRevLett.109.017202. Demonstrates interaction-enabled entanglement growth without ordinary transport.
4. M. Serbyn, Z. Papić, and D. A. Abanin, “Local Conservation Laws and the Structure of the Many-Body Localized States,” Physical Review Letters 111, 127201 (2013), doi:10.1103/PhysRevLett.111.127201. Introduces quasilocal conserved-operator structure.
5. D. A. Huse, R. Nandkishore, and V. Oganesyan, “Phenomenology of Fully Many-Body-Localized Systems,” Physical Review B 90, 174202 (2014), doi:10.1103/PhysRevB.90.174202. Develops the l-bit effective Hamiltonian and its consequences.
6. J. Z. Imbrie, “On Many-Body Localization for Quantum Spin Chains,” Journal of Statistical Physics 163, 998–1048 (2016), doi:10.1007/s10955-016-1508-x. Gives a restricted rigorous one-dimensional construction.
7. 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.
8. T. Thiery, F. Huveneers, M. Müller, and W. De Roeck, “Many-Body Delocalization as a Quantum Avalanche,” Physical Review Letters 121, 140601 (2018), doi:10.1103/PhysRevLett.121.140601. Develops avalanche-driven delocalization.
9. 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.
10. 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.
11. 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.
12. J.-y. Choi et al., “Exploring the Many-Body Localization Transition in Two Dimensions,” Science 352, 1547–1552 (2016), doi:10.1126/science.aaf8834. Reports finite-time two-dimensional localization-compatible dynamics.
13. 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.
14. A. Lukin et al., “Probing Entanglement in a Many-Body-Localized System,” Science 364, 256–260 (2019), doi:10.1126/science.aau0818. Measures interaction-dependent entanglement growth in a quantum gas microscope.
15. 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.
16. M. Rispoli et al., “Quantum Critical Behaviour at the Many-Body Localization Transition,” Nature 573, 385–389 (2019), doi:10.1038/s41586-019-1527-2. Probes correlations and scaling in a finite quasiperiodic system.
17. 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.
18. B. Lu et al., “Measuring Out Quasi-Local Integrals of Motion from Entanglement,” Communications Physics 7, 17 (2024), doi:10.1038/s42005-023-01478-5. Develops finite-system reconstruction of approximate quasilocal conserved operators.
19. J. Hur et al., “Stability of Many-Body Localization in Two Dimensions,” arXiv:2508.20699, revised March 2026, arXiv:2508.20699. A preprint reporting size-resolved random and quasiperiodic two-dimensional ultracold-atom dynamics.

Further Reading

• 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.
• 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.