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World Scientific series in 20th century physics

Closed time path Green's functions and critical dynamics

Guangzhao Zhou, Zhao-bing SU, Bailin Hao, Lu Yu

Published May 1, 2009• 0 citations

Abstract

The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter fluctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGF's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function (free energy).

DOI: 10.1142/9789814280389_0081

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The set of coupled, self-consistent equations for the order parameter, the fermion field and the collective excitations developed previously in the framework of the closed time-path Green's functions (CTPGF) is applied to systems with four-fermion interaction. The Hartree, the Hartree-Pock (HP) and the random phase approximations (RPA) for the order parameter and its fluctuations are derived in a systematic way. The thermodynamical potential for the fermion system is calculated explicitly in these approximations. The formalism presented here can be applied to nuclear as well as other many fermion systems in both equilibrium and nonequilibrium states.

We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.

A bstract We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.

We study perturbative Wilsonian renormalization group (RG) for the scalar ϕ 4 theory at finite temperature to one loop order in the Schwinger-Keldysh closed-time-path (CTP) formalism. By explicitly integrating out the UV modes, we show how effective interactions coupling the two branches of the CTP contour arise. This provides a microscopic derivation of the type of effective CTP actions used in the literature. While the full effective theory has no critical points, we instead proceed to study critical properties of the RG flow in a reduced space of complex couplings that characterize certain time-ordered correlators and show the existence of two novel fixed points in d = 4 spacetime dimensions. We relate one to the Wilson-Fisher fixed point known to exist only in 4 − ε dimensions, and the other to the standard Gaussian fixed point.