Complete Wiener-Hopf Solution of the X-Ray Edge Problem

We present a complete solution of the soft x-ray edge problem within a field-theoretic approach based on the Wiener-Hopf infinite-time technique. We derive for the first time within this approach critical asymptotics of all the relevant quantities for the x-ray problem as well as their nonuniversal prefactors. Thereby we obtain the most complete field-theoretic solution of the problem with a number of new experimentally relevant results. We make thorough comparison of the proposed Wiener-Hopf technique with other approaches based on finite-time methods. It is proven that the Fredholm, finite-time solution converges smoothly to the Wiener-Hopf one and that the latter is stable with respect to perturbations in the long-time limit. Further on we disclose a wide interval of intermediate times showing quasicritical behavior deviating from the Wiener-Hopf one. The quasicritical behavior of the core-hole Green function is derived exactly from the Wiener-Hopf solution and the quasicritical exponent is shown to match the result of Nozi\`eres and De Dominicis. The reasons for the quasicritical behavior and the way of a crossover to the infinite-time solution are expounded and the physical relevance of the Nozi\`eres and De Dominicis as well as of the Winer-Hopf results are discussed.Comment: 19 pages, RevTex, no figure

Two-particle renormalizations in many-fermion perturbation theory: Importance of the Ward identity

We analyze two-particle renormalizations within many-fermion perturbation expansion. We show that present diagrammatic theories suffer from lack of a direct diagrammatic control over the physical two-particle functions. To rectify this we introduce and prove a Ward identity enabling an explicit construction of the self-energy from a given two-particle irreducible vertex. Approximations constructed in this way are causal, obey conservation laws and offer an explicit diagrammatic control of singularities in dynamical two-particle functions.Comment: REVTeX4, 4 pages, 2 EPS figure

Replica trick with real replicas: A way to build in thermodynamic homogeneity

We use real replicas to investigate stability of thermodynamic homogeneity of the free energy of the Sherrington-Kirkpatrick (SK) model of spin glasses. Within the replica trick with the replica symmetric ansatz we show that the averaged free energy at low temperatures is not thermodynamically homogeneous. The demand of minimization of the inhomogeneity of thermodynamic potentials leads in a natural way to the hierarchical solution of the Parisi type. Conditions for the global thermodynamic homogeneity are derived and evaluated for the SK and $p$-spin infinite range models.Comment: 6 pages, presented at SPDSA2004 Hayama (Japan), to appear in Progr. Theor. Phy