Abstract
Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are characterized by the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szeg"{o}’s limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of U(N)U(N) Chern-Simons theory on S^3S3, and the SFF of this ensemble is proportional to the HOMFLY invariant of (2n,2)(2n,2)-torus links with one component in the fundamental and one in the antifundamental representation. This is one example of a large class of ensembles with intermediate statistics arising from topological field and string theories. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.
Highlights
Random Matrix Theory in disordered and complex systems: brief overviewThe idea of Wigner [1] to describe complex physical systems by treating its Hamiltonian matrix as random has found since a wide variety of applications
Other classes correspond to ensembles of real symmetric matrices, with the probability measure being invariant under orthogonal transformations, or self-dual Hermitian matrices with probability distribution invariant under symplectic transformations, known as GOE and GSE, respectively [4]. Another notable generalization is the notion of circular Random matrix Ensemble (RME), where the eigenvalues are distributed across the complex unit circle instead of the real line
Quantum systems whose classical counterparts are somewhere in between ordered and chaotic have spectral statistics that exhibit a mixture of Wigner-Dyson and Poissonian features, which we will refer to as intermediate statistics
Summary
The idea of Wigner [1] to describe complex physical systems by treating its Hamiltonian matrix as random has found since a wide variety of applications. Other classes correspond to ensembles of real symmetric matrices, with the probability measure being invariant under orthogonal transformations, or self-dual Hermitian matrices with probability distribution invariant under symplectic transformations, known as GOE and GSE, respectively [4] Another notable generalization is the notion of circular Random matrix Ensemble (RME), where the eigenvalues are distributed across the complex unit circle instead of the real line. Deep inside a localized phase, the behavior of the system is nonergodic and the RMT level’s statistics follows a Poisson distribution, ppsq „ es This type of statistics is usually found in quantum integrable systems, where a sufficient number of conserved charges significantly constrains the dynamics
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