Vortex states of the E_{u} model for Sr_{2}RuO_{4}

Based on the Ginzburg-Landau functional of E_{u} symmetry presented by Agterberg, vortex states of Sr_{2}RuO_{4} are studied in detail over $H_{c1}\lesssim H \leq H_{c2}$ by using the Landau-level expansion method. For the field in the basal plane, it is found that (i) the second superconducting transition should be present irrespective of the field direction; (ii) below this transition, a characteristic double-peak structure may develop in the magnetic-field distribution; (iii) a third transition may occur between two different vortex states. It is also found that, when the field is along the c axis, the square vortex lattice may deform through a second-order transition into a rectangular one as the field is lowered from H_{c2}. These predictions will be helpful in establishing the E_{u} model for Sr_{2}RuO_{4}.Comment: 4 pages, 5 figures, LaTeX; figure 5 replaced, corrected typo

Introduction to Nonequilibrium Statistical Mechanics with Quantum Field

In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".Comment: 78 pages, 14 figures, invited paper (free access) of Prog. Theor. Phys. http://ptp.ipap.jp/link?PTP/123/581