Quantum-mechanical kink solution

A lattice model of a one-dimensional ${\phi }^4$ field theory exhibits a symmetry-broken ground state for $m^2$ less than some negative critical value. Using a truncated-basis set, we transform the ${\phi }^4$ theory into a one-dimensional spinless fermion problem which can be exactly solved. The excitation energy of a kink is calculated and compared with the results of a semiclassical solution. Near the onset of the broken symmetry we find that the kink energy varies as ${〈\phi 〉}^8$.