Broken Symmetry, Sum Rules, and Collective Modes in Many-Body Systems

An investigation is made of the relationship between long-wavelength, low-frequency normal modes and broken symmetry in nonrelativistic many-body systems. In particular, the relationship between broken symmetry as manifested through the so-called Goldstone pole and the normal-mode structure is examined. Through the study of various models, we show that the structure of the normal modes is correlated to the Goldstone pole either completely, partially, or not at all, according to the class of symmetry of the Hamiltonian of the system. For example, in the neutral superconductor, the Hamiltonian has such low symmetry that although the Anderson modes restore the symmetry of the ground state, they have no relationship at all to the Goldstone pole. It is observed that as the symmetry decreases, a dynamical sum rule takes the place of such a correlation and in all systems a sum rule gives the normal-mode frequency $\omega$. These sum rules also give the $\omega$ distribution of states through the Huang-Klein dispersion relation.