We consider the system of Hamilton’s equations and the system of variation equations whose solutions determine the excitation spectrum of the model fermion system.
A single-component Fermi gas of polarized dipolar particles in a harmonic trap can undergo a mechanical collapse due to the attractive part of the dipole-dipole interaction. This phenomenon can be conveniently manipulated by the shape of the external trapping potential. We investigate the signatures of the instability by studying the spectrum of low-lying collective excitations of the system in the hydrodynamic regime. To this end, we employ a time-dependent variational method as well as exact numerical solutions of the hydrodynamic equations of the system.
The Hamiltonian of the massive Thirring model is explicitly diagonalized by formulating a Bethe ansatz for the eigenstates. The physical states are described by many-body wave functions representing the vacuum as a filled Fermi-Dirac sea and particle states as excitations built upon it. The spectrum of states is determined by imposing periodic boundary conditions on the wave functions. The energies are calculated by reducing the periodic boundary conditions to linear integral equations. For fermion-antifermion bound states the Dashen-Hasslacher-Neven spectrum is obtained. It is shown that the solution to the massive Thirring model can be understood as the critical limit of Baxter's solution of the eight-vertex model.