Excited-state quantum phase transitions in constrained systems

Abstract

We extend the standard semiclassical theory of Excited-State Quantum Phase Transitions (ESQPTs), based on a classification of stationary points in the classical Hamiltonian, to constrained systems. We adopt the method of Lagrange multipliers to find all stationary points and their properties directly from the Hamiltonian constrained by an arbitrary number of integrals of motion, and demonstrate the procedure on an algebraic u(3) boson model with two independent constraints. We also elaborate the Holstein-Primakoff (HP) mapping, used to eliminate one degree of freedom in bosonic systems constrained by a conserved number of excitations, and address the fact that this mapping leads, in the classical limit, to a compact phase space with singular behaviour that conceals some stationary points at the phase space boundary. It is shown that the HP method reveals all ESQPTs only after constructing a complete atlas of different HP mappings.