Random Matrix Theories in Quantum Physics: Common Concepts

TL;DR

This review surveys a decade of rapid Random Matrix Theory (RMT) development, unifying classical Gaussian and circular ensembles with supersymmetry methods to address spectral fluctuations across chaotic, disordered, and many-body quantum systems. It highlights how universal laws emerge from symmetry principles rather than dynamics, linking RMT to localization, quantum chaos, and mesoscopic transport, including two-body embedded ensembles and crossover transitions between universality classes. By detailing spectral observables, parametric correlations, wave-function statistics, and scattering, the paper demonstrates RMT’s role as a comprehensive framework for a broad class of quantum systems, while also noting boundaries where nonuniversal effects or interactions become important. The work foregrounds a new statistical mechanics, where stochasticity and symmetry yield universal predictions applicable across nuclei, atoms, molecules, quantum dots, and mesoscopic devices—extending beyond pure quantum mechanics into wave phenomena and field theory contexts.

Abstract

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Figures (45)

• Figure 1: Nearest neighbor spacing distribution for the "Nuclear Data Ensemble" comprising 1726 spacings (histogram) versus $s=S/D$ with $D$ the mean level spacing and $S$ the actual spacing. For comparison, the RMT prediction labelled GOE and the result for a Poisson distribution are also shown as solid lines. Taken from Ref. Boh83.
• Figure 2: Total cross section versus neutron energy for scattering of neutrons on $^{238}$U. The resonances all have the same spin $1/2$ and positive parity. Taken from Ref. Garg64.
• Figure 3: Differential cross section at several lab angles versus proton c.m. energy (in MeV) for the reaction $^{26}$Mg$(p,p)^{26}$Mg leaving $^{26}$Mg in its second excited state. Taken from Ref. Hauss68.
• Figure 4: The nearest neighbor spacing distribution versus $s$ (defined as in Fig. \ref{['fig1']}) for the Sinai billiard. The histogram comprises about 1000 consecutive eigenvalues. Taken from Ref. Boh84b.
• Figure 5: Nearest neighbor spacing distribution versus $s$ (as in Fig. \ref{['fig1']}) for the hydrogen atom in a strong magnetic field. The levels are taken from a vicinity of the scaled binding energy $\tilde{E}$. Solid and dashed lines are fits, except for the bottom figure which represents the GOE. The transition Poisson $\rightarrow$ GOE is clearly visible. Taken from Ref. Win87b.