Factorization of correlation functions and the replica limit of the Toda lattice equation
Abstract
Exact microscopic spectral correlation functions are derived by means of the replica limit of the Toda lattice equation. We consider both Hermitian and nonHermitian theories in the WignerDyson universality class (class A) and in the chiral universality class (class AIII). In the Hermitian case we rederive twopoint correlation functions for class A and class AIII as well as several onepoint correlation functions in class AIII. In the nonHermitian case the average spectral density of nonHermitian complex random matrices in the weak nonHermiticity limit is obtained directly from the replica limit of the Toda lattice equation. In the case of class A, this result describes the spectral density of a disordered system in a constant imaginary vector potential (the HatanoNelson model) which is known from earlier work. New results are obtained for the average spectral density in the weak nonHermiticity limit of a quenched chiral random matrix model at nonzero chemical potential. These results apply to the ergodic or ∊ domain of the quenched QCD partition function at nonzero chemical potential. Our results have been checked against numerical results obtained from a large ensemble of random matrices. The spectral density obtained is different from the result derived by Akemann for a closely related model, which is given by the leading order asymptotic expansion of our result. In all cases, the replica limit of the Toda lattice equation explains the factorization of spectral one and twopoint functions into a product of a bosonic (noncompact integral) and a fermionic (compact integral) partition function. We conclude that the fermionic partition functions, the bosonic partition functions and the supersymmetric partition function are all part of a single integrable hierarchy. This is the reason that it is possible to obtain the supersymmetric partition function, and its derivatives, from the replica limit of the Toda lattice equation.
 Publication:

Nuclear Physics B
 Pub Date:
 April 2004
 DOI:
 10.1016/j.nuclphysb.2004.01.031
 arXiv:
 arXiv:hepth/0310271
 Bibcode:
 2004NuPhB.683..467S
 Keywords:

 High Energy Physics  Theory;
 Condensed Matter;
 High Energy Physics  Lattice
 EPrint:
 29 pages, 2 figures. Clarifying comments added in sec 3.2.3 and a few typos corrected. Version to appear in Nucl. Phys. B