Eigenvalue distributions of Wilson loops
Robert Lohmayer · University of Regensburg Publication Server (University of Regensburg) · 2010
Durhuus and Olesen discovered in 1981 that the infinite-N limit of the eigenvalue density of Wilson loops in SU(N) pure gauge theory in two Euclidean dimensions undergoes a phase transition at a critical size of the loop, where a gap in the eigenvalue density closes. A similar behavior occurs also in higher dimensions and the transition seems to have universal properties. In the first part of this thesis, we focus on the distribution of the eigenvalues of the unitary Wilson loop matrix in the two-dimensional case at arbitrary finite N. To characterize the distribution of the eigenvalues, we introduce three density functions (the „symmetric“, the „antisymmetric“, and the „true“ eigenvalue density) which differ at finite N but possess the same infinite-N limit, exhibiting the Durhuus-Olesen phase transition. These densities are related to the average of the characteristic polynomial, the average of the inverse of the characteristic polynomial, and the average of the ratio of characteristic polynomials at different arguments. Using expansions of determinants and inverse determinants in characters of totally symmetric or totally antisymmetric representations of SU(N), the densities at finite N can be expressed in terms of simple sums involving only dimensions and quadratic Casimir invariants of certain irreducible representations of SU(N), allowing for a numerical computation of the densities at arbitrary N to any desired accuracy. We find that the true eigenvalue density, adding N oscillations to the monotonic symmetric density, is in some sense intermediate between the symmetric and the antisymmetric density, which in turn is given by a sum of N delta peaks located at the zeros of the average of the characteristic polynomial. Furthermore, we show that the dependence on N can be made explicit by deriving integral representations for the resolvents associated to the three eigenvalue densities. Using saddle-point approximations, we confirm that all three densities reduce to the Durhuus-Olesen result in the infinite-N limit. In the second part, we study an exponential form of the multiplicative random complex matrix model introduced by Gudowska-Nowak et al. Varying a parameter which can be identified with the area of the Wilson loop in the unitary case, the region of non-vanishing eigenvalue density of the N-dimensional complex product matrix undergoes a topological change at a transition point in the infinite-N limit. For the complex model, eigenvalues are no longer confined to the unit circle, they spread out on a fattened arc in the complex plane. At the transition point, the domain of non-zero surface eigenvalue density becomes multiply connected (below the transition point, it is simply connected). We study the transition by a detailed analysis of the average of the modulus square of the characteristic polynomial. Supported by results of high-statistics numerical simulations, we observe that the boundary of the domain of non-zero eigenvalue density follows from the stability properties of a trivial saddle point of a multi-dimensional integral representation. Furthermore, the basic complex matrix model is generalized by introducing extra parameters in the probability distributions of the individual matrix factors allowing for a smooth interpolation between the original model and the two extreme cases where the factors in the product are Hermitian or unitary. Although the shape of the domain of non-vanishing infinite-N eigenvalue density is modified, the generalized model always leads to a transition in the topology of this domain. This transition can be viewed as a natural generalization of the Durhuus-Olesen transition occurring in the unitary case. In the last part of this thesis, we present a numerical study of the entanglement entropy which is obtained by tracing out the degrees of freedom residing inside an imaginary sphere for a free massless scalar field in four-dimensional Euclidean spacetime. Since existing analytical calculations of subleading terms to the area law rely on some non-trivial assumptions (e.g., the replica trick), we have determined the next order correction, a logarithmic term which might be universal, by numerical means. Using the regularization introduced by Srednicki, we find numerically that the coefficient of the logarithm is -1/90 to 0.2 percent accuracy. This is in agreement with an existing analytical result.