---
title: Many-Body Localization Crossover
url: https://www.emergentmind.com/topics/many-body-localization-crossover
type: topic
---

# Many-Body Localization Crossover

Many-body localization crossover denotes the disorder-, quasiperiodicity-, geometry-, symmetry-, or drive-controlled evolution between an ergodic regime, in which eigenstates and dynamics are ETH-like, and a nonergodic regime, in which thermalization is strongly suppressed or absent. In finite systems this evolution is often robustly visible, yet many studies emphasize that the strongest supported statement is a crossover, or a transition-compatible crossover, rather than a definitive thermodynamic phase transition. The modern literature therefore treats the crossover as both a physical phenomenon in its own right and a window onto the unresolved status of asymptotic many-body localization (MBL) [2604.12464].

## 1. Defining the crossover and its principal diagnostics

At the broadest level, the crossover is identified by concurrent changes in spectral statistics, eigenstate structure, memory of initial conditions, entanglement growth, and transport. In the ergodic regime, level statistics are Wigner-Dyson-like, eigenstates exhibit thermal entanglement, and local observables relax to thermal values. In the localized regime, spectra become Poissonian, eigenstates obey area-law entanglement in the standard LIOM picture, local memory persists, and transport is suppressed; the review literature presents this as the canonical ETH-to-MBL phenomenology while emphasizing that finite systems show smooth disorder evolution and substantial drift of putative critical points [2604.12464].

A standard spectral diagnostic is the consecutive- or adjacent-gap ratio
\[
r_n=\frac{\min(\delta_n,\delta_{n-1})}{\max(\delta_n,\delta_{n-1})},
\qquad \delta_n=E_{n+1}-E_n,
\]
or its spacing-ratio equivalent. Benchmark averages are
\[
\langle r\rangle_{\rm GOE}=4-2\sqrt{3}\approx 0.5359,\qquad
\langle r\rangle_{\rm P}=2\ln 2-1\approx 0.3863,
\]
with the GUE value relevant when time-reversal symmetry is broken [2211.04816]. The same logic extends to Floquet systems, where quasienergy spacing ratios distinguish COE- or GOE-like ergodicity from Poisson-like localization [2408.02590].

Entanglement-based diagnostics sharpen this picture. The full entanglement spectrum, rather than entropy alone, can reveal an intermediate finite-size quantum-critical regime: for fixed subsystem size, the high-entanglement-energy peak of the entanglement density of states is essentially independent of total system size in thermal and deep MBL-like regimes, but drifts with system size in the intermediate regime, providing a criterion for finite-size criticality [1705.00631]. Dynamical observables play an analogous role: imbalance or autocorrelation probes memory retention, while logarithmic entanglement growth is a hallmark of LIOM-like localized dynamics [2604.12464]. In disorder-free Stark systems, long-time magic and entanglement likewise track an ETH-to-constrained-localized crossover, showing that nonergodicity can be diagnosed by observables beyond transport and level statistics [2512.16859].

## 2. Spectral-statistical organization of the crossover

A central development is the recognition that the spectral flow from Wigner-Dyson to Poisson is not generically a one-parameter interpolation. In the disordered XXZ chain, a Brownian-motion analysis tied to local-operator matrix elements yields a two-stage scenario. First, fractal enhancement of matrix elements near the delocalized side produces an effective power-law interaction between levels,
\[
U(s_i-s_j)\sim \frac{1}{|s_i-s_j|^\gamma},
\]
leading to a plasma-model regime with spacing distribution
\[
P(s)\propto s^\beta e^{-h_\gamma s^{2-\gamma}}
\]
and sublinear number variance \(\mathrm{var}\,N\propto N^\gamma\). Only at a second stage does the interaction become effectively local, with statistics crossing into a semi-Poisson universality class characterized by level repulsion at small spacing but exponential tails and finite compressibility [1508.07293].

Small-system exact diagonalization studies in different geometries arrived at a related phenomenology from a more phenomenological direction. The plasma-model form
\[
P(s,\beta,\gamma)=C_1 s^\beta e^{-C_2 s^{2-\gamma}}
\]
approximately describes the crossover in disordered Heisenberg clusters, with \(\beta\) controlling short-range level repulsion and \(\gamma\) the tail or long-range decorrelation. The observed evolution is explicitly two-stage: near the localized side \(\gamma=1\) remains pinned while \(\beta\) increases, and only after \(\beta\) reaches \(1\) does \(\gamma\) begin to flow toward the GOE limit \(\gamma=0\) [1805.07383]. This separates short-range and long-range spectral correlations in a way consistent with the two-stage microscopic account.

The spectral viewpoint becomes more delicate in symmetry-constrained problems. In the SU(2)-invariant exchange-disordered isotropic Heisenberg chain, exact internal multiplet degeneracies must be factored out, so level statistics are computed from multiplet energies rather than raw \(S^z\)-resolved eigenvalues. There the average gap ratio alone is inconclusive, but the sample-to-sample variance
\[
(\Delta_s r)^2 \approx \frac{1}{Q}\sum_{\alpha=1}^Q\bigl(\langle r\rangle_\alpha-\langle r\rangle\bigr)^2
\]
shows a maximum near the crossover and, for spin-\(\tfrac12\), increases with system size at strong disorder, distinguishing the strong-disorder regime from both the ergodic phase and conventional Poisson MBL [2211.04816].

## 3. One-dimensional disordered chains: resonances, bubbles, entanglement structure, and transport

The numerically accessible crossover in standard one-dimensional random chains is strongly shaped by finite-size effects. A constructive resonance theory models short chains by starting from an l-bit description and allowing a local perturbation to generate sparse many-body resonances. In that framework the resonance probability decays as
\[
q(r)=\frac{e^{-r/\xi}}{\lambda}
\]
in Floquet systems, with a large resonance length \(\lambda\), and the accessible exact-diagonalization regime satisfies \(L\lesssim 2\sqrt{\lambda}\). The model reproduces an apparent correlation-length exponent \(\nu=1\), exponential growth of Thouless time with disorder, linear drift of the apparent critical disorder with system size, scale-free resonances, apparent \(1/\omega\) spectral behavior, and subthermal entanglement entropy of small subsystems, while explicitly declining to describe the true asymptotic transition once resonances overlap strongly [2012.14393].

Real-space dynamical diagnostics support a compatible picture. In the random-field Heisenberg chain, neural-network detection of local ergodic bubbles from two-site correlators shows that on the MBL side the average bubble size grows logarithmically in time,
\[
\bar n(t)=\alpha\log t + C,
\]
with \(\alpha(W)\approx C e^{-\beta W}\), \(\beta=0.55\pm0.05\). Deep in the localized regime, the bubble-size distribution is exponential, \(p(n)\sim e^{-n\gamma}\); near the crossover it becomes broad and power-law-like, \(p(n)\sim n^{-\beta}\), and in random systems acquires a thermal peak near \(n\simeq L\), supporting an avalanche-mediated delocalization scenario [2106.01811].

Entanglement-structure analyses arrive at the same crossover from a complementary direction. In the disordered XXZ chain, recursive cluster decompositions based on normalized mutual information reveal a progression from global thermal entanglement in the ergodic phase to weak local correlations in the localized phase, with a critical regime composed of hierarchical contiguous clusters: small strongly entangled clusters weakly entangled together into larger ones. Near the transition, the cluster-size distribution is approximately power-law, \(P(n)\propto n^{-\alpha}\) with \(\alpha\) close to \(2\), whereas deeper in the localized phase it crosses to stretched exponential and then simple exponential form [1811.01925].

Basis-sensitive Fock-space diagnostics reinforce the existence of a pre-MBL crossover. In the standard interacting spinless-fermion model at half filling, the participation ratio in the Fock basis of natural orbitals exhibits a sharp finite-size crossover around \(W\approx 2.3\), well below the commonly cited \(W_c\approx 3.6\). The variance of \(\log_{10}\mathrm{PR}\) peaks there and sharpens with system size, suggesting a structural change in eigenstates closely tied to the onset of subdiffusive behavior rather than to full MBL itself [1712.06892].

Transport diagnostics show the same gradual but pronounced evolution. In the anisotropic Heisenberg chain at high temperature, the dynamical conductivity is fitted at low frequency by
\[
\sigma(\omega)\sim \sigma_0+\zeta |\omega|^\alpha.
\]
For \(\Delta=0.5,1\) and \(2\le W\le 8\), the d.c. conductivity varies approximately exponentially with disorder, \(\sigma_0\propto e^{-cW}\), indicating a crossover into an MBL regime rather than a sharply resolved transport transition. Over much of this range the fitted exponent remains close to \(\alpha\sim 1\), departing appreciably only for \(W>6\), while interacting systems display strongly correlated frequency-bin fluctuations absent in the Anderson limit [1603.01526].

## 4. Symmetry, geometry, topology, and long-range structure

The crossover depends sharply on symmetry content. In exchange-disordered isotropic Heisenberg chains,
\[
H_{\rm ex}=\sum_{i=1}^N J_i\,\vec S_i\cdot \vec S_{i+1},\qquad J_i=J+b_i,\qquad b_i\in[-b,b],
\]
global SU(2) symmetry prevents a conventional LIOM construction. Exact diagonalization finds a regime that is nonergodic but not fully localized: \(\langle r\rangle\) moves away from GOE beyond \(b_{\rm erg}\approx 0.5J\), a crossover region appears near \(b_{\rm cr}\gtrsim 1.0J\), yet even at infinite disorder the statistics do not become Poissonian. The decisive evidence is the strong-disorder scaling of \(\Delta_s r\), which remains broad and, for spin-\(\tfrac12\), increases with system size, motivating the description “imperfect” or “incompletely localized” MBL [2211.04816].

Lattice connectivity shifts the crossover in a more conventional way. In 16-site disordered Heisenberg clusters, the disorder scale associated with localization grows roughly with coordination number: about \(W\approx 3.7\) in a chain with \(z=2\), \(W\approx 8\) in a square lattice with \(z=4\), and \(W\approx 12\) in a triangular lattice with \(z=6\). The ladder lies between chain and square, and increasing transverse coupling in an anisotropic \(4\times 4\) geometry smoothly shifts the crossover from 1D toward 2D values [1805.07383]. This supports the interpretation that more resonant channels make localization harder.

Long-range couplings and topological single-particle structure can obstruct standard MBL even more strongly. In random antiferromagnetic XXZ chains with power-law exchange \(J_{ij}=J|r_i-r_j|^{-\alpha}\), strong-disorder RG finds a crossover in the full distribution of the lowest excitation gap as \(\alpha\) decreases, with evidence for a critical form near \(\alpha_c\approx 1.6\): large \(\alpha\) is localized/random-singlet-like, smaller \(\alpha\) becomes Wigner-like and extended [1408.2068]. By contrast, in a single disordered Landau level, finite-size exact diagonalization shows an apparently clear MBL-like crossover from GUE to Poisson, but the midpoint disorder drifts as
\[
W_c\propto L^{4-5},
\]
suggesting that the critical disorder scales to infinity and that true MBL is absent because a sub-thermodynamic number of effectively extended one-body states thermalizes the many-body spectrum [1610.08931].

## 5. Disorder-free, Floquet, and experimentally implemented crossovers

A major expansion of the subject concerns disorder-free and Floquet systems, where localization-like behavior emerges without quenched randomness. In a tilted transverse-field Ising chain with open boundaries,
\[
H = J \sum_{i=0}^{L-1} Z_i Z_{i+1} + h \sum_{i=0}^{L} X_i + F \sum_{i=0}^{L} i\, Z_i ,
\]
the Stark field \(F\) produces a finite-size ETH-to-SMBL crossover. Long-time stabilizer Rényi entropy \(M_2\) and half-chain entanglement show crossings and collapses at effective scales \(F_c\approx 0.19\) and \(F_c\approx 0.23\), respectively, while a Schrieffer–Wolff analysis yields factorially suppressed effective couplings \(J_{\rm eff}(r)\sim J_0 (h/F)^{r-1}/(r-1)!\). The strong-tilt regime is not frozen in a Clifford-trivial sense: magic remains nonzero and grows slowly to a plateau, with pronounced initial-state selectivity [2512.16859].

Periodic driving generates a related family of crossovers. In the driven Lipkin-Meshkov-Glick model,
\[
\hat H(t)=\hat H_0+\big[h\cos(\omega t)+h_0\big]\hat H_1,
\]
dynamical many-body localization occurs only near freezing conditions
\[
J_0\!\left(\frac{4h}{\omega}\right)=0.
\]
At low frequency the Floquet-state inverse participation ratio in the \(\hat S^x\) basis scales as \(1/N\), indicating Floquet-ETH-like heating, whereas along the high-frequency freezing manifold the IPR rises toward unity. The authors therefore present a drive-controlled crossover from thermalizing Floquet dynamics to frozen, localized Floquet states, while emphasizing that the characteristic crossover frequency drifts upward with system size and appears to diverge as \(N\to\infty\) [2308.03622].

A closely related clean Floquet mechanism appears in two-mode interacting bosons mapped to the kicked top. There the localization occurs in Fock space rather than real space. The long-time occupation profile in the imbalance basis is exponential,
\[
|\psi(n_1)|^2\sim \exp\!\left[-\frac{|n_1-N/2|}{\xi}\right],
\]
with localization length
\[
\xi \approx \frac{D}{\hbar_{\mathrm{eff}}^2}
      = \frac{\sin^2(a)}{4\hbar_{\mathrm{eff}}^2}
      = \frac{N^2\sin^2(a)}{16},
\qquad \hbar_{\mathrm{eff}}=\frac{2}{N}.
\]
The crossover is controlled by \(\xi\sim L\) in the finite Fock-space dimension \(L=N+1\), and the quasienergy statistics evolve from GOE-like to Poissonian as the Floquet eigenstates localize [2604.09224].

Flat-band systems provide a disorder-free crossover between distinct localization mechanisms. In the Creutz ladder at the flat-band point \(t_0=t_1\), compact Wannier states localized on four physical sites generate a disorder-free nonergodic regime. As disorder \(\mu\) is increased, the many-body level statistics evolve non-monotonically,
\[
\text{Poisson}\rightarrow \text{GUE-like}\rightarrow \text{Poisson},
\]
and the inverse participation ratio shows a minimum around \(\mu_c\sim 6\), supporting a crossover from flat-band-induced localization to conventional disorder-induced localization [1904.03463].

Recent quantum-processor experiments have moved this crossover physics into regimes beyond classical simulation. A 60-qubit disordered kicked Ising model on an IBM Heron device uses late-time out-of-time-ordered correlators to locate a Floquet MBL crossover around \(W_c\sim 0.18\), distinguishing a weak-disorder quantum-chaotic regime from a strong-disorder localized regime and validating the result by operator renormalization and zero-noise extrapolation [2510.01983]. A separate quasiperiodic Floquet Ising experiment on up to 129 qubits in 1D and 144 qubits in 2D reaches 5000 Floquet cycles, showing a smooth crossover in autocorrelations and logarithmic QFI growth \(F_Q(t)\simeq a+b\ln t\), with apparent onset scales \(W_*\sim 3.5\) in 1D and \(W_*\sim 4\) in 2D [2603.12675].

## 6. Interpretation, controversies, and current status

A persistent theme is that finite-size crossover phenomenology is robust, whereas the thermodynamic interpretation remains unsettled. The modern review literature emphasizes that level statistics, entanglement, imbalance, and other diagnostics clearly separate ETH-like and localized-like regimes in finite random XXZ chains, but apparent critical values drift strongly with system size. Simple gap-ratio scaling yields \(W_c\approx 3.7\) and \(\nu\approx 0.9\), yet Harris-bound violations, entanglement drifts, and late-time dynamical ambiguities make that value unreliable. More cautious extrapolations place the localization scale nearer \(W\sim 5\)–\(5.5\), while some resonance-based arguments suggest substantially larger disorder may be required [2604.12464].

The random-versus-quasiperiodic distinction sharpens this issue. In both kicked Ising and Heisenberg chains, the maximal slope of the crossover grows only linearly with \(L\) for random disorder but exponentially with \(L\) for quasiperiodic potentials,
\[
|\alpha_{\max}|\propto L \quad \text{(random)},\qquad
|\alpha_{\max}|\propto e^{\lambda L} \quad \text{(quasiperiodic)},
\]
with \(\lambda\approx 0.09\) for gap-ratio data and \(\lambda\approx 0.07\) for entanglement data in the quasiperiodic cases studied [2408.02590]. This strongly suggests that quasiperiodic systems approach non-analytic behavior much faster, plausibly because they lack Griffiths-type rare regions, but it does not by itself settle whether the thermodynamic limit contains a true MBL phase transition or merely a sharply sharpening crossover.

Several common misconceptions are therefore corrected by the current literature. Poisson-like spectral statistics in small systems are not sufficient evidence for stable thermodynamic MBL when delocalizing channels exist, as the Landau-level example shows [1610.08931]. Conversely, nonergodic or localization-like behavior need not flow to conventional LIOM MBL: SU(2)-symmetric chains show an incompletely localized regime incompatible with standard Poisson MBL [2211.04816], and disorder-free Stark or Floquet systems can display constrained localized dynamics without establishing a sharp thermodynamic MBL phase [2512.16859]. Finally, entanglement alone does not exhaust the relevant diagnostics: magic, bubble statistics, entanglement spectra, cluster hierarchies, and transport fluctuations each isolate different aspects of the crossover.

Taken together, these results support a precise but nontrivial picture. Many-body localization crossover is not a single universal curve between thermal and localized limits. It is a family of finite-size dynamical and spectral reorganizations whose form depends on resonance structure, symmetry class, lattice connectivity, quasiperiodicity, Floquet driving, and the presence or absence of protected delocalizing channels. What is robust across these settings is the existence of a broad regime in which ergodicity is strongly suppressed and conventional finite-size diagnostics change sharply. What remains open is when, and in which models, that finite-size crossover sharpens into a genuine thermodynamic many-body localized phase.

Source: https://www.emergentmind.com/topics/many-body-localization-crossover