Vacuum ground and excited states of the U(1) lattice gauge Hamiltonian.

Abstract

The method of correlated basis functions offers a promising new approach to Hamiltonian formulations of lattice gauge problems. In this contribution, ideas and techniques that have been successful in the variational theory of homogeneous and inhomogeneous boson quantum fluids are adapted and applied to the two-dimensional U(1) lattice gauge problem in the uncharged sector. The vacuum ground state is described by a variational trial function of Jastrow type, including single-plaquette and plaquette-pair factors. Euler-Lagrange equations for the optimization of this trial function are constructed and solved, in analogy with the paired-phonon analysis of quantum-fluid theory, to obtain results for the one-plaquette density and the plaquette-pair distribution function describing electromagnetic-field correlations. An associated set of optimized elementary excitations of Bijl-Feynman type emerges from the analysis. Combining the Feynman eigenvalue equation for the elementary excitations with the paired-lattice-photon equation, a numerically useful necessary condition is derived for stability of the trial vacuum ground state with respect to the optimized excitations. Numerical calculations are carried out with the aid of hypernetted-chain techniques analogous to those commonly applied for quantum fluids. Results for the vacuum ground-state energy, the pair distribution function and static structure function, the glueball mass, and the particle-hole potential are presented for a range of the coupling parameter $\ensuremath{\lambda}$ running from the strong- to the weak-coupling regime. These results are compared with previous analytical predictions and with data from numerical simulations. The local stability condition is found to be met for all $\ensuremath{\lambda}$, reaffirming the prediction that for $\ensuremath{\lambda}<\ensuremath{\infty}$ the glueball mass remains finite and a confinement-deconfinement transition does not occur. It is concluded that the optimized variational approach, carried to the Jastrow level, yields a quantitatively useful first description whose accuracy may be increased by practicable implementation of correlated-basis perturbation theory. The approach may be extended to finite temperature and to the charged, string sector of the U(1) model, as well as to other lattice gauge problems.