Transitions toward Quantum Chaos: With Supersymmetry from Poisson to Gauss

Guhr, Thomas

doi:10.1006/aphy.1996.0091

Cited by 75 publications

(132 citation statements)

References 33 publications

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“…We underline that the projection formula also holds if the source κ is chosen non-diagonal as it sometime happens in QCD [48] or if we add an external operator H 0 to the original random matrix H often consider in transition ensembles [49,50]. In both cases the integral (2.36) is slightly modified but the fundamental functional relation (2.37) still remains the same.…”

Section: Projection Formula For Dyson's Threefold Waymentioning

confidence: 94%

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

Kaymak¹,

Kieburg²,

Guhr³

2014

J. Phys. A: Math. Theor.

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Abstract. In the past few years, the supersymmetry method was generalized to real-symmetric, Hermitean, and Hermitean self-dual random matrices drawn from ensembles invariant under the orthogonal, unitary, and unitary symplectic group, respectively. We extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way. Thereby we generalize a projection formula providing a direct link and, hence, a 'short cut' between the probability density in ordinary space and the one in superspace. We emphasize that this point was one of the main problems and critiques of the supersymmetry method since only implicit dualities between ordinary and superspace were known before. As examples we apply this approach to the calculation of the supersymmetric analogue of a Lorentzian (Cauchy) ensemble and an ensemble with a quartic potential. Moreover we consider the partially quenched partition function of the three chiral Gaussian ensembles corresponding to four-dimensional continuum QCD. We identify a natural splitting of the chiral Lagrangian in its lowest order into a part of the physical mesons and a part associated to source terms generating the observables, e.g. the level density of the Dirac operator.

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Section: Projection Formula For Dyson's Threefold Waymentioning

confidence: 94%

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

Kaymak¹,

Kieburg²,

Guhr³

2014

J. Phys. A: Math. Theor.

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“…In the limit T → ∞ this model converges to Dyson's Brownian motion model, [9], with β = 2, and was considered in [13], see also [11]. We can also consider the same type of measure but on [0, ∞) and with appropriate boundary conditions at the origin.…”

Section: 1mentioning

confidence: 99%

Determinantal Processes with Number Variance Saturation

Johansson

2004

Commun. Math. Phys.

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Abstract. Consider Dyson's Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N → ∞. This limting determinantal proceess has the interesting feature that it shows number variance saturation. The variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity. Number variance saturation is also seen for example in the zeros of the Riemann ζ-function, [20], [2]. The process can also be constructed using nonintersecting paths and we consider several variants of this construction. One construction leads to a model which shows a transition from a non-universal behaviour with number variance saturation to a universal sine-kernel behaviour as we go up the line.

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“…In addition a position dependent potential may be present. Dyson's log-gas model [1,4,7] is of fundamental importance to the field of RMT and has been applied in various contexts [21][22][23][24][25]. Under this formalism the three invariant ensembles of random matrices, viz.…”

Section: Dyson Log Gasmentioning

confidence: 99%

Random matrix ensembles: Wang-Landau algorithm for spectral densities

Kumar

2013

EPL

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We propose a method based on the Wang-Landau algorithm to numerically generate the spectral densities of random matrix ensembles. The method employs Dyson's log-gas formalism for random matrix eigenvalues and also enables one to simultaneously investigate the thermodynamic properties. This approach is a powerful alternative to the conventionally used Monte Carlo simulations based on the Boltzmann sampling, and is ideally suited for investigating β-ensembles.

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Transitions toward Quantum Chaos: With Supersymmetry from Poisson to Gauss

Cited by 75 publications

References 33 publications

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

Determinantal Processes with Number Variance Saturation

Random matrix ensembles: Wang-Landau algorithm for spectral densities

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