Supersymmetry Method for Chiral Random Matrix Theory with Arbitrary Rotation Invariant Weights

Abstract

In the last 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.

References in corpus (7)

• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Superbosonization of invariant random matrix ensembles
• Arbitrary Rotation Invariant Random Matrix Ensembles and Supersymmetry
• On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices
• Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
• Integration of Grassmann variables over invariant functions on flat superspaces
• Norm-dependent Random Matrix Ensembles in External Field and Supersymmetry

Cited by in corpus (9)

• Exact Relation between Singular Value and Eigenvalue Statistics
• Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure
• Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances
• Dirac Spectrum of the Wilson Dirac Operator for QCD with Two Colors
• Superposition and higher-order spacing ratios in random matrix theory with application to complex systems
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Universal scaling of higher-order spacing ratios in Gaussian random matrices
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance
• Asymptotic Coincidence of the Statistics for Degenerate and Non-Degenerate Correlated Real Wishart Ensembles