Spontaneous symmetry breaking of (1+1)-dimensionalφ4theory in light-front field theory. II

We discuss spontaneous symmetry breaking of (1+1)-dimensional ${\phi }^4$ theory in light-front field theory using a Tamm-Dancoff truncation. We show that, even though light-front field theory has a simple vacuum state which is an eigenstate of the full Hamiltonian, the field can develop a nonzero vacuum expectation value. This occurs because the zero mode of the field must satisfy an operator-valued constraint equation. In the context of (1+1)-dimensional ${\phi }^4$ theory we present solutions to the constraint equation using a Tamm-Dancoff truncation to a finite number of particles and modes. We study the behavior of the zero mode as a function of coupling and Fock space truncation. The zero mode introduces new interactions into the Hamiltonian which breaks the $Z_2$ symmetry of the theory when the coupling is stronger than the critical coupling. We investigate the energy spectrum in the symmetric and broken phases, show that the theory does not break down in the vicinity of the critical coupling, and discuss the connection to perturbation theory. Finally, we study the spectrum of the field $\phi$ and show that, in the broken phase, the field is localized away from $\phi =0\mathrm{}$ as one would expect from equal-time calculations. We explicitly show that tunneling occurs.