C T symmetry in nonlinear time-dependent electronic structure theories

Nonlinear time-dependent (NLTD) electronic structure theories with effective Hamiltonians depending on the time-dependent state can exhibit "complex excitation energy" solutions with exponential rather than oscillatory time-dependence. These instabilities greatly limit the scope of popular NLTD approaches, such as time-dependent Hartree-Fock and density functional theory, as well as some time-dependent correlated wavefunction and Green's function methods. We show that CT symmetry breaking is responsible for these instabilities, where C denotes unitary frequency or quasiparticle charge reversal and T denotes antiunitary time reversal. While CT symmetry is trivially conserved in Hermitian quantum mechanics, it can be broken in NLTD response theories, which are generally non-Hermitian. This non-Hermitian structure is a consequence of the nonanalytic dependence of the effective Hamiltonian on the time-dependent state encountered in most NLTD approaches. Analytic continuation of the underlying many-electron Hilbert space to a Krein space of twice the dimension admits a well-defined response theory. Stable solutions exhibit double "generalized Kramers" degeneracy of solutions and their CT reverses, which can be interpreted as particle-antiparticle pairs of quasiparticle excitations. In non-Hermitian NLTD response theory, real observables and an oscillatory time-evolution are guaranteed by CT symmetry. Electronic stability of the initial state is found to be sufficient, but not necessary, for the conservation of CT symmetry. The results are illustrated for an analytically solvable two-state model.