Disorder, Localization, and Glasses

Disorder introduces both random couplings and atypical regions that can dominate long-time behavior. This chapter connects interference and diffusive field theory to localization, interacting dynamics, and glassy order.

Enter this chapter

Begin with Quenched Disorder and Disorder Averaging to distinguish a quenched ensemble from averaging the disorder dynamically.

Helpful background. Quenched Disorder and Disorder Averaging and Probability Spaces, Random Variables, and Conditional Expectation develop the definitions used by the chapter’s opening pages.

Topics

1. Quenched Disorder and Disorder Averaging — Defines random couplings, quenched versus annealed averages, self-averaging, correlations, and rare samples.
2. Replica and Supersymmetry Methods for Disorder — Derives the normalization tricks and states the replica-limit and convergence qualifications.
3. Diffusive Modes and the Disorder Nonlinear Sigma Model — Obtains diffusons, cooperons, the $Q$ constraint, and the class-dependent long-wavelength action.
4. Weak Localization and Weak Antilocalization — Evaluates the coherent return correction and its magnetic, spin–orbit, dephasing, and dimensional cutoffs.
5. Interaction Corrections and Finkel’stein Scaling — Couples density-of-states and transport anomalies to the interacting diffusive RG.
6. Anderson Localization and Scaling Theory — Relates exponential localization, mobility edges, conductance flow, and finite-size critical fitting.
7. Symmetry Classes and Topological Localization — Connects the tenfold classification, sigma-model targets, topological terms, and protected delocalization.
8. Dirty Bosons and the Bose Glass — Distinguishes superfluid, Mott-insulating, and compressible Bose-glass diagnostics.
9. Griffiths Effects, Rare Regions, and Avalanches — Derives broad relaxation tails and tests whether an ergodic inclusion grows.
10. Many-Body Localization: Diagnostics and Evidence — Compares spectral, eigenstate, transport, and memory diagnostics with avalanche and finite-size limits.
11. Mean-Field Quantum Spin Glasses and Replica Symmetry Breaking — Develops the overlap order parameter, replicon instability, and mean-field scope.
12. Finite-Dimensional Quantum Glass Dynamics and Evidence — Evaluates aging, memory, susceptibilities, overlaps, and relaxation against competing slow mechanisms.

How the ideas fit together

Interference between multiple scattering paths changes transport, motivating a scale-dependent conductance and a long-wavelength theory of diffusive modes Abrahams et al. 1979, pp. 673–676. Symmetry and dimensionality determine which interference channels survive.

Interactions and rare regions add questions that a one-particle localization length does not answer. Long-lived memory can arise from several slow mechanisms, and an ergodic inclusion can destabilize a proposed many-body localized regime. A finite system’s slow relaxation is therefore insufficient to establish either a thermodynamic localized phase or glass order.

The comparison below separates the assumptions and tests needed for these different conclusions.

Disorder and glass claim test matrix

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Ensembles, scaling tests, and limits of disorder and glass claims.

Claim or regime	Ensemble and symmetry	Scaling variable or interaction effect	Rare-event diagnostic and size requirement	Competing explanation	Supported conclusion
Quenched thermodynamics	Fixed $V$ during equilibration; declared $P[V]$	$\overline{\ln Z}$ and cumulants, not $\ln\overline Z$	Distribution over samples; increase sample count and volume separately	Annealed disorder or nonstationary preparation	Exact only for the stated ensemble and order of limits
Replica or SUSY representation	Replica number $n$ or balanced boson–fermion fields; convergence prescription stated	$n\to0$ continuation or exact normalization cancellation	Compare moments and tails before analytic continuation	Wrong saddle, broken convergence contour, or nonunique continuation	A computational representation, not an independent physical phase
Diffusive nonlinear sigma model	Altland–Zirnbauer class fixed; $k_F\ell\gg1$	$q\ell$, $\lvert\omega\rvert\tau\ll1$; $Q^2=1$	Sample dimensions must exceed $\ell$ but remain within coherence window	Ballistic modes, additional soft channels, or strong disorder	Long-wavelength theory inside the declared class and cutoff
Weak localization or antilocalization	Orthogonal, unitary, or symplectic interference sector	$L_\phi/\ell$ and magnetic or spin–orbit crossover	Magnetoconductance shape over several coherent scales	Interaction correction, classical memory effect, or inhomogeneity	Perturbative coherent correction, not a phase diagnosis
Interaction correction	Diffusive fermions with specified singlet/triplet channels	Running resistance, $z$, and interaction amplitudes	Temperature and field scaling beyond one logarithmic window	Heating, phonons, granularity, or band-structure anomaly	Controlled only while conductance and retained couplings justify the expansion
Anderson transition	Noninteracting class and boundary conditions fixed	$\beta(g)=\mathrm d\ln g/\mathrm d\ln L$, $L/\xi$	Several sizes, disorder quantiles, and irrelevant-drift fit	Crossover, mobility-edge averaging, or contact resistance	Critical point only after stable scaling collapse and uncertainty analysis
Topological localization	Symmetry class plus bulk topological term or invariant	Mass, conductance, localization length, and boundary response	Bulk and boundary scaling with symmetry-breaking controls	Clean band transition or finite-size edge hybridization	Class- and dimension-specific delocalization statement
Bose glass	Quenched potential with interacting bosons and $U(1)$ symmetry	Vanishing stiffness, nonzero compressibility, no spectral gap	Size and trap scaling of $\rho_s$, $\kappa$, and gap proxies	Inhomogeneous superfluid puddles or Mott crossover	Compressible insulating phase only in the homogeneous thermodynamic limit
Griffiths dynamics	Spatial rare regions in a declared disorder ensemble	Tail exponent or dynamical exponent from full distributions	Rare-sample count grows with both volume and realization number	Broad but ordinary relaxation-rate distribution	Anomalous dynamics; does not alone imply localization or glass order
Thermal avalanche	Localized background plus an ergodic seed and local couplings	Inclusion entropy versus distance-dependent matrix element	Seed size and growth followed beyond the crossing scale	Long transient from a subcritical inclusion	Instability criterion under the stated localization-length convention
Many-body localization	Isolated interacting system with conserved quantities fixed	Level ratios, transport, entanglement, and local memory	Symmetry-resolved sizes, long times, bath-size drift, and rare seeds	Prethermalization, fragmentation, Stark localization, or finite-size crossover	No thermodynamic phase claim from accessible finite-size crossing alone
Mean-field quantum spin glass	Infinite-range or large-connectivity quenched couplings	Replicon stability and overlap distribution $P(q)$	Replica limit, imaginary-time discretization, and system-size convergence	Replica-symmetric solution or competing ordered saddle	Controlled mean-field RSB statement; finite-dimensional stability separate
Finite-dimensional quantum glass	Short-range model or material with declared preparation and bath	Aging, memory, nonlinear susceptibility, overlaps, and relaxation spectrum	Joint size–time scaling and protocol variation	Rare regions, domains, disorder pinning, MBL-like memory, or critical slowing	Model- and window-specific evidence through 10 August 2026

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Where to go next

Continue to Cold Atoms and Synthetic Matter to compare trap, interaction, temperature, and observation scales.

References

• Abrahams, E., Anderson, P. W., Licciardello, D. C., and Ramakrishnan, T. V. (1979). “Scaling theory of localization: Absence of quantum diffusion in two dimensions.” Physical Review Letters 42, 673–676. doi:10.1103/PhysRevLett.42.673.