Mean field theory for U(n) dynamical groups

Abstract

Algebraic mean field theory (AMFT) is a many-body physics modeling tool which firstly, is a generalization of Hartree–Fock mean field theory, and secondly, an application of the orbit method from Lie representation theory. The AMFT ansatz is that the physical system enjoys a dynamical group, which may be either a strong or a weak dynamical Lie group G. When G is a strong dynamical group, the quantum states are, by definition, vectors in one irreducible unitary representation (irrep) space, and AMFT is equivalent to the Kirillov orbit method for deducing properties of a representation from a direct geometrical analysis of the associated integral co-adjoint orbit. AMFT can be the only tractable method for analyzing some complex many-body systems when the dimension of the irrep space of the strong dynamical group is very large or infinite. When G is a weak dynamical group, the quantum states are not vectors in one irrep space, but AMFT applies if the densities of the states lie on one non-integral co-adjoint orbit. The computational simplicity of AMFT is the same for both strong and weak dynamical groups. This paper formulates AMFT explicitly for unitary Lie algebras, and applies the general method to the Lipkin–Meshkov–Glick (2) model and the Elliott (3) model. When the energy in the (3) theory is a rotational scalar function, Marsden–Weinstein reduction simplifies AMFT dynamics to a two-dimensional phase space.

Equations of motion for noncompact symplectic dynamical symmetry

“…To find estimates for matrix elements of excited states spanning the small d×d EOM block, an approximation to the quartic Hamiltonian is made below in equation (60). The mean field Hamiltonian is derived from the energy functional in two steps: (1) take the differential of the energy considered as a function of ( ¯) w z z , , , and (2) replace the differentials by corresponding Lie algebra elements [29],…”

“…The algebraic mean field Hamiltonian, an element of the Lie algebra, is an approximation to the exact SGA Hamiltonian, an element of the enveloping algebra [29,32]. The purpose of this appendix is to review briefly the geometric symplectic structure which is the mathematical foundation of any AMFT.…”