Many-body localization

arXiv:2604.12464 · quant-ph, cond-mat.dis-nn, cond-mat.quant-gas · Submitted 2026-04-14 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Many-body localization".

Mira: Here's the gist of the paper. The gist: This review presents an introductory overview of nonergodic dynamics in interacting many-body quantum systems, focusing primarily on many-body localization (MBL) and summarizing evidence for a crossover from ergodic to MBL regimes in finite systems using the XXZ model as an example.

Kai: A standard definition of “quantum chaos” is the behavior of quantum systems that are classically chaotic in the semiclassical limit, which traditionally involves few-body nonintegrable situations where such a limit is well controlled by, loosely speaking, taking all relevant quantum numbers to infinity For many-body physics the situation is somewhat different, as most models involve interacting fermions or spins that remain intrinsically quantum The dynamics of such systems are typically thermalizing when coupled to an environment, where a subsystem prepared in an arbitrary initial state and in contact with a large reservoir thermalizes and its long-time state can be described by an appropriate Gibbs ensemble By contrast, for a finite isolated system the whole evolution is unitary and time-reversible, so in principle the initial state can be recovered This apparent contradiction underlies the eigenstate thermalization hypothesis (ETH), which provides our current understanding of generic quantum many-body dynamics Dynamics that obey ETH are loosely termed ergodic, and their basic properties are discussed in Section 2.

Mira: Eigenstate Thermalization Hypothesis (ETH) The long-time behavior of an observable A (and the possibility of thermalization) is determined by the diagonal matrix elements Amm The standard formulation of ETH yields an ansatz for both diagonal and off-diagonal matrix elements of observables in the eigenbasis of Hˆ

6: In ETH, A(E¯) is the expectation value in the microcanonical ensemble while fA(E¯, ωmn) carries information on the nature of transport and the fine structure of response functions The long-time average of the observable Aˆ is independent of the specific values of the coefficients cm 2, and equals the average of Amm over the entire energy shell Em − ⟨E⟩ < δE, i.e., the microcanonical average.

Kai: ETH is thus believed to provide a sufficient criterion for the onset of thermalization.

Paper summary: Kai: So, we're looking at this paper now called "Many-body localization," and it’s diving into nonergodic dynamics in interacting many-body quantum systems. Basically, the whole thing focuses on many-body localization or MBL as a key phenomenon here.

Mira: Right, so the core idea seems to be summarizing what MBL is and looking at the evidence for a crossover from ergodic behavior to this localized regime in finite systems using something called the XXZ model as their main example.

Lev: If I'm hearing right, they're starting by setting up the usual picture of quantum chaos and thermalization, which usually involves systems coupled to an environment where they reach a Gibbs ensemble over long times.

Kai: Exactly, and then they introduce the idea that for a finite isolated system, everything is unitary and time-reversible so you can potentially get back to your initial state later on.

Mira: This leads them into the eigenstate thermalization hypothesis or ETH, which suggests that the long-time average of an observable is determined only by the diagonal matrix elements of the Hamiltonian, A mm, as they say in their text <ref:2604.12464#pg2>.

Lev: And they explain how ETH gives us a way to look at both those diagonal and off-diagonal elements, where the off-diagonal parts carry information about transport and the fine structure of response functions, which is important for understanding how systems evolve <ref:2604.12464#pg2>.

Kai: Then they define what many-body localization actually is as a property of at least a part of the excited spectrum in an interacting and disordered system that has several correlated characteristics.

Mira: Those characteristics include having a complete set of local integrals of motion, which means quantum integrability for each disorder realization <ref:2604.12464#pg3>.

Lev: And they mention statistical properties for the spectra corresponding to the Poisson ensemble, as well as area-law entangled eigenstates and a long-lasting memory of the initial nonstationary state in time dynamics <ref:2604.12464#pg3>.

Kai: They also point to a logarithmic growth of entanglement entropy when you quench from an initially low-entanglement state <ref:2604.12464#pg3>.

Paper summary: Mira: Then they bring in the disordered Heisenberg chain as the perfect example, showing a clear crossover from ergodic for small disorder W to localized regimes as seen through gap-ratio statistics <ref:2604.12464#pg3>.

Lev: They give some specific numbers for this crossover, pointing to a putative critical disorder amplitude Wc about three point seven and a correlation critical exponent nu equal to zero point nine in finite systems <ref:2604.12464#pg3>.

Kai: But then they add a caveat about the Harris bound suggesting that this finite-size scaling approach might be misleading because the drift of the disorder value, WT, at which differs from the GOE value, with system size is WT ∼ L <ref:2604.12464#pg3>.

Mira: That means extrapolating both borders to infinity for L gives an estimate for Wc about five point five, which suggests that the localization might actually persist in the thermodynamic limit <ref:2604.12464#pg3>.

Lev: And when you look at time dynamics, they see a slow decay in imbalance I(t) for large disorder values, sometimes described by a power-law decay I(t) about one/t beta <ref:2604.12464#pg3>.

Kai: They tie this back to the memory effect from those local integrals of motion, which is a big deal for how the system behaves over time in the MBL phase <ref:2604.12464#pg3>.

Mira: The entanglement entropy growth in that localized regime is area-law entangled eigenstates, which means it’s practically independent of the system size, or volume <ref:2604.12464#pg3>.

Lev: Moving to higher dimensions, they note that studies are scarcer but experiments show that in the presence of a strong disorder, the part of the two dimensional system that is initially occupied remains dominantly populated even at long times <ref:2604.12464#pg5>.

Kai: They also mention the Quantum Sun model as an example where they see many-body interacting systems convincing revealing MBL in the thermodynamic limit <ref:2604.12464#pg5>.

Mira: The authors note that while signatures of MBL are observed in small systems, no evidence for it in the thermodynamic limit seems to exist yet <ref:2604.12464#pg5>.

Lev: And they also mention a nonstandard MBL where positional disorder, which is randomizing the positions of interacting bodies instead of just diagonal disorder, can be realized <ref:2604.12464#pg5>.

Paper summary: Kai: So while they see signatures in small systems and some 2D models, the main point seems to be that we still lack convincing evidence for MBL in the thermodynamic limit of interacting systems <ref:2604.12464#pg5>.

Mira: It’s interesting because it shows that this crossover from ergodic to localized behavior is a real thing, even if we can't definitively say we've found the true MBL phase yet <ref:2604.12464#pg5>.

Lev: This whole discussion leads us into the conclusion section where they summarize the title and authors of this paper, "Many-body localization," and what it means for quantum computing.

Kai: So what does this all mean in simpler terms, focusing on the title and who wrote it? It’s about understanding how quantum systems behave when disorder is involved.

Mira: The implication is that we have a better idea of the transition point between a system behaving like a typical ergodic quantum system and one where localization effects take over <ref:2604.12464#pg5>.

Lev: For someone building real hardware, it suggests that if you can engineer the right level of disorder, you might be able to induce these localized states which could be useful for some kinds of computation <ref:2604.12464#pg5>.

Kai: So we're seeing a lot of evidence that this crossover from ergodic to localized behavior is happening in finite systems, and the paper shows the general applicability across different models like the XXZ model <ref:2604.12464#pg3>.

Mira: It really frames how we study these complex interacting quantum systems by looking at what happens when we push them into regimes where localization effects become dominant <ref:2604.12464#pg5>.

Lev: I think the main limitation they state is that they haven't found definitive evidence for MBL in the thermodynamic limit of interacting systems yet <ref:2604.12464#pg5>.

Kai: Exactly, so this paper gives us a roadmap of what's been observed and where the next big pieces of evidence need to come from.

Mira: We've seen how they discuss the crossover from ergodic to localized regimes in finite systems using the XXZ model as an example <ref:2604.12464#pg3>.

Lev: So, this paper provides a framework for understanding these nonergodic dynamics and helps us map out where we should look next when trying to understand many-body localization <ref:2604.12464#pg5>.

Conclusion: Kai: So, to wrap up this look at "Many-body localization," we're talking about how this paper summarizes nonergodic dynamics in interacting quantum systems, focusing on MBL and showing evidence for a crossover from ergodic behavior to the localized regime in finite systems.

Mira: Right, so what does that mean when you boil it down? It’s basically about understanding these weird quantum behaviors where a system stops behaving like a typical thermalizing thing and gets stuck in a localized state due to disorder.

Lev: For us building hardware, the implication is that if we can engineer the right level of disorder, we might be able to create these localized states which could be useful for some kinds of computation.

Kai: So the authors look at things like the XXZ model and show that this crossover happens in a few different ways, even though they haven't found solid proof for it happening in truly infinite systems yet.

Mira: They point out that while they see signatures in smaller systems, there hasn't been any real evidence for MBL in the thermodynamic limit of interacting systems.

Lev: That’s the sticking point for us right now, because running simulations or error correction on truly infinite lattices is impossible.

Kai: It means we’re still mapping out where we should look next when trying to understand these nonergodic dynamics and whether they are just finite-size effects or something deeper.

Uniwersytet Jagiellonski · Jagiellonian University

quant-ph, cond-mat.dis-nn, cond-mat.quant-gas

Submitted: 2026-04-14

Updated: 2026-10-22

Comments: revised version of the Chapter for the Quantum Chaos volume in 'Comprehensive Quantum Mechanics', to be published by Elsevier (Main editor: R.B. Mann; volume editors: S. Gnutzmann and K. {Ż}yczkowski)

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: The gist: This review presents an introductory overview of nonergodic dynamics in interacting many-body quantum systems, focusing primarily on many-body localization (MBL) and summarizing evidence

Key concepts

Eigenstate Thermalization Hypothesis (ETH): ETH suggests that the long-time average of an observable in a generic quantum system is determined by its diagonal matrix elements. This means that for systems obeying ETH, the system acts like it is in a microcanonical ensemble, providing a criterion for thermalization.
Many-Body Localization (MBL): MBL describes systems where disorder causes energy eigenstates to become localized. This localization is characterized by having local integrals of motion and exhibiting area-law entanglement, meaning the entanglement between parts of the system does not grow with system size.
Crossover from Ergodic to MBL: In finite systems, dynamics can switch between two regimes: ergodic (thermalizing) for small disorder and localized (MBL) for large disorder. This transition is studied using finite-size scaling and critical exponents to understand how the system's behavior changes with increasing disorder strength.

Terminology

Localization — 2× in this paper · explained in 2 episodes

Summary

The gist: This review presents an introductory overview of nonergodic dynamics in interacting many-body quantum systems, focusing primarily on many-body localization (MBL) and summarizing evidence for a crossover from ergodic to MBL regimes in finite systems using the XXZ model as an example.

Introduction to Quantum Chaos and Thermalization

A standard definition of “quantum chaos” is the behavior of quantum systems that are classically chaotic in the semiclassical limit, which traditionally involves few-body nonintegrable situations where such a limit is well controlled by, loosely speaking, taking all relevant quantum numbers to infinity For many-body physics the situation is somewhat different, as most models involve interacting fermions or spins that remain intrinsically quantum The dynamics of such systems are typically thermalizing when coupled to an environment, where a subsystem prepared in an arbitrary initial state and in contact with a large reservoir thermalizes and its long-time state can be described by an appropriate Gibbs ensemble By contrast, for a finite isolated system the whole evolution is unitary and time-reversible, so in principle the initial state can be recovered This apparent contradiction underlies the eigenstate thermalization hypothesis (ETH), which provides our current understanding of generic quantum many-body dynamics Dynamics that obey ETH are loosely termed ergodic, and their basic properties are discussed in Section 2.

Eigenstate Thermalization Hypothesis (ETH)

The long-time behavior of an observable A (and the possibility of thermalization) is determined by the diagonal matrix elements Amm The standard formulation of ETH yields an ansatz for both diagonal and off-diagonal matrix elements of observables in the eigenbasis of Hˆ [6] In ETH, A(E¯) is the expectation value in the microcanonical ensemble while fA(E¯, ωmn) carries information on the nature of transport and the fine structure of response functions The long-time average of the observable Aˆ is independent of the specific values of the coefficients cm 2, and equals the average of Amm over the entire energy shell Em − ⟨E⟩ < δE, i.e., the microcanonical average. ETH is thus believed to provide a sufficient criterion for the onset of thermalization.

Many-Body Localization (MBL) Properties

Many-body localization is a property of (at least) a part of the excited spectrum of a many-body interacting and disordered system characterized by several correlated characteristics These include the existence of a complete set of local integrals of motion (LIOMs), separate for each disorder realization, implying quantum integrability, statistical properties of spectra corresponding to the Poisson ensemble, area-law entangled eigenstates, a long-lasting memory of the initial nonstationary state in the time dynamics, and a logarithmic growth of the entanglement entropy in a quench from an initially low-entanglement state.

Spectral Measures and Crossover

The disordered Heisenberg chain is the paradigmatic case of studies, showing a clear crossover from ergodic (for small W) to localized regimes as revealed by gap-ratio statistics Standard MBL properties are observed in the disordered Heisenberg chain, such as the existence of LIOMs and statistical properties corresponding to the Poisson ensemble. The study of finite systems reveals a crossover from an extended, delocalized, ergodic regime to localized behavior manifestingly breaking the ETH. Finite-size scaling yields a putative critical disorder amplitude Wc ≈ 3.7 and a correlation critical exponent ν = 0.9. The breakdown of Harris’s bound suggests that the finite-size scaling approach may be misleading, as the drift of the disorder value WT, at which ¯r differs from the GOE value, with the system size is WT ∼ L. Extrapolating both borders to L → ∞ yields an estimate Wc ≈ 5.5.

Time Dynamics and Entanglement

The imbalance I(t) in experiments reveals, for large disorder values, a slow decay in time which, on some time intervals, may be considered a power-law decay I(t) ∼ 1/t β. In the MBL phase, the system exhibits some form of memory due to the existence of LIOMs. The entanglement entropy growth for an initially separable (or, more generally, weakly entangled) state grows logarithmically in time. In the localized regime, the eigenstates exhibit area-law entanglement, which is practically independent of the system size (i.e., volume).

Two Dimensional Models and Other Phenomena

In two (2D) or more dimensions, studies are scarcer, but experiments show that in the presence of a sufficiently strong disorder, the part of the 2D system that is initially occupied remains dominantly populated even at long times. The Quantum Sun model indicates that there exist many-body interacting systems that convincingly reveal MBL in the thermodynamic limit. Other examples include tilted chains, where strong localization may be observed due to Hilbert space shattering.

Nonstandard MBL and Conclusion

The exception to diagonal disorder is positional disorder, which can be realized by randomizing the positions of the interacting bodies The Quantum Sun model can reveal MBL in the thermodynamic limit, indicating that there exist many-body interacting systems that convincingly reveal MBL in the thermodynamic limit. While signatures of MBL are observed in small systems, no evidence for MBL in the thermodynamic limit seems to exist Real laboratory systems are finite, and for typical systems of a few tens of spins, MBL has been shown to exist. The application to facilitate quantum computing has also been suggested.

References

[1] R. Nandkishore and D.A. Huse, Many-Body Localization and Thermalization in Quantum Statistical Mechanics, Annual Review of Condensed Matter Physics 6 (2015) 15

[2] F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, Comptes Rendus Physique 19 (2018) 498

[3] D.A. Abanin, E. Altman, I. Bloch and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91 (2019) 021001

[4] S. Gopalakrishnan and S. Parameswaran, Dynamics and transport at the threshold of many-body localization, Physics Reports 862 (2020) 1

[5] 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 (2025) 026502

[6] M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, Journal of Physics A: Mathematical and General 32 (1999) 1163

[7] A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83 (2011) 863

[8] L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics 65 (2016) 239

[9] M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 EP

[10] F. Haake, Quantum Signatures of Chaos, Springer, Berlin (2010)

[11] V. Oganesyan and D.A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75 (2007) 155111

[12] Y.Y. Atas, E. Bogomolny, O. Giraud and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys.

Improvements for AI systems

A system that can predict whether a given many-body quantum state will thermalize based on its Hamiltonian structure could be improved by using criteria derived from eigenstate thermalization hypothesis (ETH). This system would be able to determine if the long-time average of an observable A hat follows the microcanonical average as described in Eq. 4, providing a criterion for ergodicity breaking.
An AI that simulates quantum dynamics on finite systems could be enhanced by incorporating methods to detect many-body localization (MBL) signatures like area-law entangled eigenstates and Poisson ensemble spectral statistics. This system would be able to distinguish between ergodic behavior and localized behavior by analyzing entanglement entropy growth and gap ratio statistics as a function of system size.
A quantum computing algorithm could be improved by using the concept of magic, quantified by stabilizer Renyi entropy (SRE), to guide optimization routines in many-body systems exhibiting MBL. This system would be able to quantify how far a state is from the set of stabilizer states, potentially enabling efficient simulation of nonstabilizer dynamics.
An AI designed for simulating interacting one-dimensional spin chains could be improved by incorporating knowledge of quasi-local integrals of motion (LIOMs). This system would be able to identify if a system possesses a complete set of LIOMs, which is a signature suggesting quantum integrability and vanishing transport in the MBL phase.
A model for simulating systems with positional disorder could be improved by incorporating the Quantum Sun model framework to explore the mobility edge. This AI could determine if an emergent mobility edge exists even in two dimensions or if localization occurs due to avalanche scenarios, providing insight into higher-dimensional localization phenomena.

Abstract

We present an introductory review of nonergodic dynamics in interacting many-body quantum systems, focusing on the phenomenon of many-body localization (MBL). We describe aspects of MBL and summarize the evidence for a crossover from the ergodic to the MBL regime in finite systems, using the paradigmatic XXZ model as an example. We then broaden the scope to other models to illustrate the generality of the phenomenon. We briefly touch on the largely unexplored relation between MBL and quantum computing.

Sources

Stability of many-body localization in two dimensions