Erratum: Nonequilibrium sum rules for the retarded self-energy of strongly correlated electrons [Phys. Rev. B77, 205102 (2008)]
Physical Review B
We have discovered a subtle error in our derivation of the nonequilibrium moments when the Hamiltonian in the Schrödinger representation has explicit time dependence. The problem arises from the fact that when we take the second or higher time derivatives of an operator in the Heisenberg picture, the operator inherits some additional time dependence from the explicit time dependence of the Schrödinger Hamiltonian and the way in which the operator is evolved as a function of time. This results
... ime. This results in terms proportional to derivatives of the Hamiltonian in the Schrödinger representation. We find one missing term in the expressions for the third momentum-dependent spectral moment for the retarded Green's function and for the first momentum-dependent retarded self-energy moment, 3 R ͑k , T͒ and C 1 R ͑k , T͒, in Eqs. ͑25͒ and ͑60͒ and three new terms in the expressions for the third momentum-dependent and local spectral moments for the lesser Green's function 3 Ͻ ͑k , T͒ and 3 Ͻ ͑T͒, in Eqs. ͑33͒ and ͑37͒. These terms come from the direct time derivative of the Hamiltonian in the Schrödinger picture in Eq. ͑13͒. More specifically, one cannot simply replace high order time derivatives of an operator O H in the Heisenberg picture i n d n O H / dT n ͑which has no explicit time dependence in the Schrödinger picture͒ by the multiple commutator operator L n O H = ͓¯͓͓O H , H H ͑T͔͒ , H H ͑T͔͒¯H H ͑T͔͒ ͓defined after Eq. ͑21͔͒, because the expression for the moments ͑beginning with n =2͒ will have additional terms proportional to time derivatives of the Hamiltonian ͑in the Schrödinger representation͒ with respect to T. Namely, the first derivative of an operator that has no time dependence in the Schrödinger picture can be substituted by a commutator with the Hamiltonian idO H / dT = ͓O H , H H ͑T͔͒ = L 1 O H . However, already the second derivative acquires an additional term: is the additional time dependence, with U͑T͒ the time evolution operator from the initial time to time T. These results can be easily generalized to the case of higher derivatives.