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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Coupled cluster description of field theories: Procedures and their application to the vacuum sector in (1+1)-dimensional Φ4 field theories

C.S. Hsue, Hermann G. Kümmel, P. Ueberholz

Published Sep 15, 1985• 36 citations
Publisher

Abstract

General procedures for the application of the nonperturbative coupled cluster method (CCM) are described for one-component boson field theories. The basis set is optimized first by a Hartree-Bogolubov transformation. Then ``correlations'' are added via the CCM. This procedure is applied to two-dimensional ${\ensuremath{\Phi}}^{4}$ field theory. The results are very stable outside the critical region.

DOI
10.1103/physrevd.32.1435

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General procedures for the application of the nonperturbative coupled cluster method (CCM) are described for one-component boson field theories. The basis set is optimized first by a Hartree-Bogolubov transformation. Then ``correlations'' are added via the CCM. This procedure is applied to two-dimensional \ensuremathΦ4{\ensuremath{\Phi}}^{4}\ensuremathΦ4 field theory. The results are very stable outside the critical region.
The boson field theory with ${\mathrm{\ensuremath{\Phi}}}^{4}$ and ${\mathrm{\ensuremath{\Phi}}}^{6}$ coupling in (1+1) dimensions is investigated. Rigorous duality relations covering the whole range of coupling constants are derived. Vacuum, one-, and two-particle states are computed via the coupled cluster method in the four-particle approximations. Outside a critical region around second-order phase transitions the method works very well even in regions with first-order phase transitions and bound states.
An ab initio formulation of microscopic quantum many-body theory, namely, the coupled-cluster method (CCM), is applied to the lattice gauge models, Z(2)Z(2)Z(2) in 2+1 dimensions, and U(1) in 1+1 and 2+1 dimensions. Both mode-mode couplings and plaquette-plaquette correlations are considered. In particular, within the one-plaquette approximation for the U(1) model, the CCM is able to include mode couplings of arbitrarily high order. It therefore reproduces essentially the exact numerical results for the equivalent Mathieu problem. These include the ground-state, plaquette, and excitation energies as functions of the coupling constant. Two-plaquette equations are solved by local approximations for both the U(1) and Z(2)Z(2)Z(2) models, and the results are impressive. Detailed comparisons with other methods, particularly perturbation theory, are made and discussed, with emphasis on the nonperturbative nature of the problem.
The coupled-cluster method (CCM) is applied to anisotropic-quantum-antiferromagnetic models in one and two dimensions (1D and 2D). Several hierarchical approximation schemes of the CCM are defined, which are specialized for the spin-lattice models. Good results are obtained for both ground-state and excited-state energies. The spin-spin correlation functions and the staggered magnetizations are also calculated. We have found that the CCM gives a qualitatively correct description of the entire Heisenberg-Ising phase, including the critical point where a phase transition occurs in 1D. Some interesting differences between 1D and 2D models near the critical point are discussed.
The phenomenon of pairing in atomic and nuclear many-body systems gives rise to a great number of different physical properties of matter, from areas as seemingly diverse as the shape of stable nuclei to superconductivity in metals and superfluidity in neutron stars. With the experimental realisation of the long sought BCS-BEC crossover observed in trapped atomic gases - where it is possible to fine tune the s-wave scattering length a of a many-fermion system between a dilute, correlated BCS-like superfluid of Cooper pairs and a densely packed BEC of composite bosons - pairing problems in atomic physics have found renewed interest in recent years. Given the high precision techniques involved in producing these trapped gas condensates, we would like to employ a suitably accurate many-body method to study such systems, preferably one which goes beyond the simple mean-field picture.The Coupled Cluster Method (CCM) is a widely applied and highly successful ab initio method in the realm of quantum many-body physics and quantum chemistry, known to be capable of producing extremely accurate results for a wide variety of different many-body systems. It has not found many applications in pairing problems however, at least not in a general sense. Our aim, therefore, is to study various models of pairing using a variety of CCM techniques - we are interested in studying the generic features of pairing problems and in particular, we are especially interested in probing the collective modes of a system which exhibits the BCS-BEC crossover, in either the BCS or BEC limit. The CCM seems a rather good candidate for the job, given the high precision results it can produce.
The coupled cluster method (CCM) is one of the most successful and universally applicable techniques in quantum many-body theory. The intrinsic nonlinear and non-perturbative nature of the method is considered to be one of its advantages. We present here a combination of CCM with the Wilsonian renormalization group which leads to a powerful framework for construction of effective Hamiltonian field theories. As a toy example we obtain the two-loop renormalized ϕ4ϕ^{4}ϕ4 theory.