Equations of motion for noncompact symplectic dynamical symmetry

Abstract

When the high dimension of the quantum Hilbert space creates a serious challenge for conventional numerical diagonalisation of the Hamiltonian matrix, the equations of motion (EOM) method may solve efficiently the eigenvalue problem in some instances. EOM assumes the existence of a spectrum generating Lie algebra (SGA) for which the Hamiltonian operator is a polynomial (typically a quadratic) in the algebra’s basis operators. The method computes values of low-energy excitation energies and corresponding algebra matrix elements by solving a coupled set of commutation relations. To test EOM’s effectiveness, the paper applies it to SU(1, 1) Sp(1, ), whose unitary discrete series representation spaces are infinite dimensional. The paper investigates two (1, 1) Hamiltonians. The first Hamiltonian has a phase transition that challenges brute force diagonalisation. The second Hamiltonian is the one-dimensional quartic potential. For the quartic case, the algebraic mean field Hamiltonian, an approximation to the SGA Hamiltonian, provides a qualitative solution and a useful starting point for an EOM computation. EOM yields fast and accurate results on a laptop computer for both Hamiltonians.