Abstract
Quantum chaotic systems are often defined via the assertion that their spectral statistics coincides with, or is well approximated by, random matrix theory. In this paper we explain how the universal content of random matrix theory emerges as the consequence of a simple symmetry-breaking principle and its associated Goldstone modes. This allows us to write down an effective-field theory (EFT) description of quantum chaotic systems, which is able to control the level statistics up to an accuracy {O} \left(e^{-S} \right)O(e−S) with SS the entropy. We explain how the EFT description emerges from explicit ensembles, using the example of a matrix model with arbitrary invariant potential, but also when and how it applies to individual quantum systems, without reference to an ensemble. Within AdS/CFT this gives a general framework to express correlations between ``different universes’’ and we explicitly demonstrate the bulk realization of the EFT in minimal string theory where the Goldstone modes are bound states of strings stretching between bulk spectral branes. We discuss the construction of the EFT of quantum chaos also in higher dimensional field theories, as applicable for example for higher-dimensional AdS/CFT dual pairs.
Highlights
We explain how the effective-field theory (EFT) description emerges from explicit ensembles, using the example of a matrix model with arbitrary invariant potential, and when and how it applies to individual quantum systems, without reference to an ensemble
The systems are physically equivalent to random matrix models (which are likewise described by the action (2.17) as we will demonstrate in section 3): 5. Extracting the random-matrix physics: The partition sum describing a chaotic quantum system in the ergodic regime assumes the form
The main theme in this paper has been the universal nature of chaotic spectral correlations governed by a symmetry-based effective field theory
Summary
Important progress in lower-dimensional models of AdS/CFT duality has given renewed impetus to contemplate ensemble averages of boundary theories, which either entirely capture the bulk quantum gravity path integral, or at least agree with the latter in an averaged sense. We show that this symmetry breaking principle allows one to select a universal set of light modes – precisely the associated (pseudo–)Goldstone modes – which quantitatively express the RMT physics, both for ensembles as well as individual chaotic quantum systems, in terms of a simple effective (field) theory, well-known in the chaos community as the Efetov-Wegner sigma-model [1,2]. The essence of the sigma model approach is a reduction of theories defined in eS-dimensional Hilbert spaces to effective theories on much lower dimensional ‘flavor’ manifolds Both the realization of the flavor manifold and its exact dimensionality (which remains always of O(1) in terms of L) depend on the symmetries of the parent theory, and on the details of the correlation functions at hand, in a well-understood way.
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