The supersymmetric method in random matrix theory and applications to QCD

Abstract

Abstract.The supersymmetric method is a powerful method for the nonperturbative evaluation of quenched averages in disordered systems. Among others, this method has been applied to the statistical theory of S-matrix fluctuations, the theory of universal conductance fluctuations and the microscopic spectral density of the QCD Dirac operator.We start this series of lectures with a general review of Random Matrix Theory and the statistical theory of spectra. An elementary introduction of the supersymmetric method in Random Matrix Theory is given in the second and third lecture. We will show that a Random Matrix Theory can be rewritten as an integral over a supermanifold. This integral will be worked out in detail for the Gaussian Unitary Ensemble that describes level correlations in systems with broken time reversal invariance. We especially emphasize the role of symmetries.As a second example of the application of the supersymemtric method we discuss the calculation of the microscopic spectral density of the QCD Dirac operator. This is the eigenvalue density near zero on the scale of the average level spacing which is known to be given by chiral Random Matrix Theory. Also in this case we use symmetry considerations to rewrite the generating function for the resolvent as an integral over a supermanifold.The main topic of the second last lecture is the recent developments on the relation between the supersymmetric partition function and integrable hierarchies (in our case the Toda lattice hierarchy). We will show that this relation is an efficient way to calculate superintegrals. Several examples that were given in previous lectures will be worked out by means of this new method. Finally, we will discuss the quenched QCD Dirac spectrum at nonzero chemical potential. Because of the nonhermiticity of the Dirac operator the usual supersymmetric method has not been successful in this case. However, we will show that the supersymmetric partition function can be evaluated by means of the replica limit of the Toda lattice equation.

Random matrix model forQCD3staggered fermions

“…The width of fluctuations of hermitian degrees of freedom decreases with the system size as 1/ √ N and the aspect ratio τ of the elliptic ensemble (19) approaches minus one as τ = −1 + 4 α 2 /N . The evolution of the shape with α can be found using for instance the supersymmetric method [16]. Having done that one can try to relate the parameter α to β in QCD 3 .…”

Untitled

“…Much of the RMT machinery -such as the saddle-point method, orthogonal polynomials, the Efetov's supersymmetric technique, or the diagrammatic expansion and the free random variables calculus (see for example [23,24,25,26,27,28,29,30,31,32,33,34,35,36] for reviews; these last two techniques are used in this paper) -has been developed in the Hermitian context, and tailored for real spectra. The necessity of dealing with complex spectra demands enhancements of these methods.…”

“…, L, with the cyclic convention 0 = L, where the normalized traces (22) have been used. Results (27), (28) mean that the four blocks of the matrix…”

Bosonic partition functions at nonzero (imaginary) chemical potential

“…In particular, it has been shown that the SO(2,1) symmetry of a hyperbolic spin chain is spontaneously broken also in one and two dimensions. In essence, the reason is that a partition function with a noncompact symmetry can only be defined if this symmetry is spontaneously broken to its compact subgroup SO (2). In a conformal invariant theory the spectral density of the Dirac operator also scales as ρ(λ) ∼ V λ α and this scenario might reconcile conformal behavior with universal random matrix statistics [15][16][17].…”

“…It has been well established that lattice QCD Dirac spectra fluctuate according to the corresponding random matrix theory in the microscopic domain (see [1][2][3]). Because this agreement is based on the spontaneous breaking of the flavor symmetry, one would expect that, as a consequence of the Coleman-Mermin-Wagner theorem, the agreement with Random Matrix Theory in two dimensions is structurally different from the agreement found in four dimensions.…”