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arXiv:hep-ph/9512314 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Initial-condition problem for a chiral Gross-Neveu system

Paulo Laerte Natti, A. F. R. de Toledo Piza

Published Dec 15, 1996• 4 citations
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Abstract

A time-dependent projection technique is used to treat the initial-value problem for self-interacting fermionic fields. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type for the set of one-body dynamical variables. A nonperturbative mean-field expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Gaussian mean-field approximation, for a uniform system described by the chiral Gross-Neveu Hamiltonian. Standard stationary features of the model, such as dynamical mass generation due to chiral symmetry breaking and a phenomenon analogous to dimensional transmutation, are reobtained in this context. The mean-field time evolution of nonequilibrium initial states is discussed.

DOI
10.1103/physrevd.54.7867

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A framework to describe the real-time evolution of interacting fermion-scalar field models is set up. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type to the set of one-body dynamical variables. A time-dependent projection technique is used then to generate a nonperturbative mean-field expansion leading to a set of self-consistent equations of motion for these observable, where the lowest order corresponds to the Gaussian approximation. As an application, we consider an uniform system of relativistic spin-1/2 fermion field coupled, through a Yukawa term, to a scalar field in 3+1 dimensions, known as quantum scalar plasma. The renormalizability for the Gaussian mean-field equations, both static and dynamical, are examined and initial conditions discussed. We also investigate solutions for the gap equation and show that the energy density has a single minimum.
We study the quantum relaxation dynamics for a lattice version of the one-dimensional (1D) NNN-flavor Gross-Neveu (GN) model after a Hamiltonian parameter quench. Allowing for a system-reservoir coupling γγγ, we numerically describe the system dynamics through a time-dependent self-consistent Lindblad master equation. For a closed (γ=0γ=0γ=0) finite-size system subjected to an interaction parameter quench, the order parameter dynamics exhibits oscillations and revivals. In the thermodynamic limit, our results imply that the order parameter reaches its post-quench stationary value in accordance with the eigenstate thermalization hypothesis (ETH). However, time-dependent finite-momentum correlation matrix elements equilibrate only if γ>0γ>0γ>0. Our findings are consistent with the system being described by a pertinent Generalized Gibbs Ensemble (GGE) and, accordingly, highlight subtle yet important aspects of the post-quench relaxation dynamics of quantum many-body systems.
The self-energy functional theory (SFT) is generalized to describe the real-time dynamics of correlated lattice-fermion models far from thermal equilibrium. This is achieved by starting from a reformulation of the original equilibrium theory in terms of double-time Green's functions on the Keldysh-Matsubara contour. With the help of a generalized Luttinger-Ward functional, we construct a functional $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\Omega}}[\ensuremath{\Sigma}]$ which is stationary at the physical (nonequilibrium) self-energy $\ensuremath{\Sigma}$ and which yields the grand potential of the initial thermal state $\ensuremath{\Omega}$ at the physical point. Nonperturbative approximations can be defined by specifying a reference system that serves to generate trial self-energies. These self-energies are varied by varying the reference system's one-particle parameters on the Keldysh-Matsubara contour. In the case of thermal equilibrium, this approach reduces to the conventional SFT. Contrary to the equilibrium theory, however, ``unphysical'' variations, i.e., variations that are different on the upper and the lower branches of the Keldysh contour, must be considered to fix the time dependence of the optimal physical parameters via the variational principle. Functional derivatives in the nonequilibrium SFT Euler equation are carried out analytically to derive conditional equations for the variational parameters that are accessible to a numerical evaluation via a time-propagation scheme. Approximations constructed by means of the nonequilibrium SFT are shown to be inherently causal, internally consistent, and to respect macroscopic conservation laws resulting from gauge symmetries of the Hamiltonian. This comprises the nonequilibrium dynamical mean-field theory but also dynamical-impurity and variational-cluster approximations that are specified by reference systems with a finite number of degrees of freedom. In this way, nonperturbative and consistent approximations can be set up, the numerical evaluation of which is accessible to an exact-diagonalization approach.