scholarly journals A bound on quantum chaos from Random Matrix Theory with Gaussian Unitary Ensemble

2019 ◽  
Vol 2019 (5) ◽  
Author(s):  
Sayantan Choudhury ◽  
Arkaprava Mukherjee
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Gaussian Unitary Ensemble ◽  
Unitary Ensemble

scholarly journals Replica approach in random matrix theory

2018 ◽  
Author(s):  
Thomas Guhr
Keyword(s):  
Partition Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Field Theories ◽  
Partition Functions ◽  
Gaussian Unitary Ensemble ◽  
Integrable Theory ◽  
Unitary Ensemble ◽  
Replica Approach

This article examines the replica method in random matrix theory (RMT), with particular emphasis on recently discovered integrability of zero-dimensional replica field theories. It first provides an overview of both fermionic and bosonic versions of the replica limit, along with its trickery, before discussing early heuristic treatments of zero-dimensional replica field theories, with the goal of advocating an exact approach to replicas. The latter is presented in two elaborations: by viewing the β = 2 replica partition function as the Toda lattice and by embedding the replica partition function into a more general theory of τ functions. The density of eigenvalues in the Gaussian Unitary Ensemble (GUE) and the saddle point approach to replica field theories are also considered. The article concludes by describing an integrable theory of replicas that offers an alternative way of treating replica partition functions.

scholarly journals Distribution of the Dirac modes in QCD

2018 ◽  
Vol 175 ◽  
pp. 04005
Author(s):  
M. Catillo ◽  
L. Ya. Glozman
Keyword(s):  
Dirac Operator ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
General Property ◽  
Matrix Theory ◽  
Spontaneous Breaking ◽  
Gaussian Unitary Ensemble ◽  
Zero Modes ◽  
Dynamical Simulations ◽  
Unitary Ensemble

It was established that distribution of the near-zero modes of the Dirac operator is consistent with the Chiral Random Matrix Theory (CRMT) and can be considered as a consequence of spontaneous breaking of chiral symmetry (SBCS) in QCD. The higherlying modes of the Dirac operator carry information about confinement physics and are not affected by SBCS. We study distributions of the near-zero and higher-lying modes of the overlap Dirac operator within NF = 2 dynamical simulations. We find that distributions of both near-zero and higher-lying modes are the same and follow the Gaussian Unitary Ensemble of Random Matrix Theory. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is related to some more general property of QCD in confinement regime.

scholarly journals Distribution law of the Dirac eigenmodes in QCD

2018 ◽  
Vol 33 (10) ◽  
pp. 1850054
Author(s):  
Marco Catillo ◽  
Leonid Ya. Glozman
Keyword(s):  
Dirac Operator ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Spontaneous Breaking ◽  
Distribution Law ◽  
Gaussian Unitary Ensemble ◽  
Zero Modes ◽  
Dynamical Simulations ◽  
Unitary Ensemble

The near-zero modes of the Dirac operator are connected to spontaneous breaking of chiral symmetry in QCD (SBCS) via the Banks–Casher relation. At the same time, the distribution of the near-zero modes is well described by the Random Matrix Theory (RMT) with the Gaussian Unitary Ensemble (GUE). Then, it has become a standard lore that a randomness, as observed through distributions of the near-zero modes of the Dirac operator, is a consequence of SBCS. The higher-lying modes of the Dirac operator are not affected by SBCS and are sensitive to confinement physics and related [Formula: see text] and [Formula: see text] symmetries. We study the distribution of the near-zero and higher-lying eigenmodes of the overlap Dirac operator within [Formula: see text] dynamical simulations. We find that both the distributions of the near-zero and higher-lying modes are perfectly described by GUE of RMT. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is linked to the confining chromo-electric field.

Introduction and guide to the handbook

2018 ◽  
Author(s):  
Gernot Akemann ◽  
Jinho Baik ◽  
Philippe Di Francesco
Keyword(s):  
Random Matrices ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
The Other ◽  
Random Operators ◽  
New Developments ◽  
Gaussian Unitary Ensemble ◽  
Mathematical Properties ◽  
Unitary Ensemble

This article discusses random matrix theory (RMT) in a nutshell — what it is about, what its main features are, and why it is so successful in applications. It first considers the simplest and maybe most frequently used standard example, the Gaussian Unitary Ensemble (GUE) of random matrices, before looking at several types of applications of RMT, focusing on random operators, counting devices, and RMT without matrices. It then provides a guide to the handbook, explaining how the other forty-two articles on mathematical properties and applications of random matrices are related and built one upon the other. It also lists some topics that are not covered in detail in the book and reviews recent new developments since the first edition of this handbook before concluding with a brief survey of the existing introductory literature.

scholarly journals Isospectral Twirling and Quantum Chaos

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1073
Author(s):  
Lorenzo Leone ◽  
Salvatore F. E. Oliviero ◽  
Alioscia Hamma
Keyword(s):  
Random Matrix ◽  
Finite Time ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Time Behavior ◽  
Unified Framework ◽  
Expectation Value ◽  
Gaussian Unitary Ensemble ◽  
Loschmidt Echo ◽  
Unitary Ensemble

We show that the most important measures of quantum chaos, such as frame potentials, scrambling, Loschmidt echo and out-of-time-order correlators (OTOCs), can be described by the unified framework of the isospectral twirling, namely the Haar average of a k-fold unitary channel. We show that such measures can then always be cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector stabilizer states does not possess chaotic features, unlike those Hamiltonians whose eigenvectors are taken from the Haar measure. As an example, OTOCs obtained with Clifford resources decay to higher values compared with universal resources. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior of probes to quantum chaos corresponding to chaotic spectra given by the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by Poisson distribution and the Gaussian Diagonal Ensemble (GDE).

scholarly journals UNIVERSAL CORRELATIONS IN RANDOM MATRICES: QUANTUM CHAOS, THE 1/r2 INTEGRABLE MODEL, AND QUANTUM GRAVITY

1996 ◽  
Vol 11 (15) ◽  
pp. 1201-1219 ◽  
Author(s):  
SANJAY JAIN
Keyword(s):  
Quantum Gravity ◽  
Random Matrices ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Correlation Functions ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Mathematical Formulation ◽  
Integrable Model ◽  
Moser System

Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-D integrable model with the 1/r2 interaction (the Calogero-Sutherland-Moser system) and 2-D quantum gravity. We review the connection of RMT with these areas. We also discuss the method of loop equations for determining correlation functions in RMT, and smoothed global eigenvalue correlators in the two-matrix model for Gaussian orthogonal, unitary and symplectic ensembles.

scholarly journals Quantum Chaos, Irreversible Classical Dynamics, and Random Matrix Theory

1996 ◽  
Vol 76 (21) ◽  
pp. 3947-3950 ◽  
Author(s):  
A. V. Andreev ◽  
O. Agam ◽  
B. D. Simons ◽  
B. L. Altshuler
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Classical Dynamics
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