Nonequilibrium resistive phase transition by exciting Goldstone modes

hile studies of nonequilibrium steady states in electronic systems under DC electric fields have shown that the threshold field for the metal–insulator transition is lower than that predicted by conventional Landau–Zener theory, it still exceeds the values observed in some charge-density-wave materials. In this work, we investigate the nonequilibrium behavior under DC E-field of the phase mode, or Goldstone mode, by receiving energy from accelerated electrons and thus destroying the spontaneous symmetry breaking. Using the Keldysh Green's function formalism, we numerically confirm that the Goldstone mode persists in the nonequilibrium steady state, with its dynamics primarily governed by electron–hole pair excitations. The effective temperature of the Goldstone bosons shows a sharp increase beyond a threshold field, similar to the electronic effective temperature, while the bosonic threshold field is several times smaller than that of the electrons, suggesting that collective phase dynamics can further reduce the transition field compared with single-electron processes.