On the Interaction of Electrons with Lattice Vibrations

The present paper treats the interaction of electrons with lattice vibrations, using the intermediate coupling method developed by Lee, Low and Pines. The energy of the system is calculated to the first approximation. The expression for the energy is equivalent to that obtained by Frohlich. The interaction energy between electrons which is given rise to by the virtual emission and absorption of phonons has so far been calculated by perturbation theory. However the perturbation calculation will not give a good approximation owing to the large value of the interaction parameter. Tomonaga1) introduced a variational method, so called "intermediate coupling method" in treating meson-nucleon problem. It is difficult to apply this method in its original form to the many-electron system. Recently Lee, Low and Pines2) have developed a variational technique, which is closely related to Tomonaga's method. They have calculated the self-energy of an electron in.a polar crystal and have obtained a better result than that obtained by the perturbation calculation. Their method is equivalent to a canonical transformation, which we .generalize and apply to the manyelectron system. Frohlich has carried out a renormalization calculation. He, has applied a canonical transformation to the l;Iamiltonian: this transformation leads to a renormalization of the phonon velocity and of the interaction parameter. He has thus removed the objection of Wentzel against the use of large interaction parameter. It is the object of the present paper to elucidate the relation between Frohlich's renormalization calculation and Lee-Low-Pines method. Introduction Resume of Lee-Low-Pines method We consider one electron interacting with phonons. Let the Hamiltonian be + 2J nws bw *bw +~ (rwbwe"wx + rw *bw*e-!u'X) , w w U is a unitary operator and is given by We find U- 1bw U =bw +fCw)e-tU'''', U-1bw*U=b", * + f* (w)etWX, = f2 +2J(r,./('w) +rw*f" (w 2m W + tz2{2J If(r,/) 12w}2 2m w .. Eq. (6) is minimized by setting +_1[S, [S, [So [S,HJJ]]}'o). 24 {3E - - - (3f(w) Manyelectron .system H=Ho+.fI;.+H2 , Ho=2J E"a,,*a,,+ 2J fi W s(bw+bw+~), w w 2 H,=t~ D w (bwi1,,*a,,_w- bw*ai-Wak) ' The renormalized velocity s will be determined below. We choose for our trial wave function The unitary operator U is given by U-IHU=Ho +.~ +H 2 -[5, Ho]- [5, H l ] + ~[5, [5, Ho]] + ... 2 = 2J E"ak*ak +~ n'lVs (bw*bw+~) + ~ bw{iD,,+ (E'H.- Ek +hws) ". k 2 'w,k -b"bw(ak-W.qak*aq-V-ak,q_va/at._w) f* (k, 'w) } 1 +-~ (E k_w - Ek+n'lVS) (aq-W *aqak*ak-wf(q, w) f* (k, '1.0) 2 w,Ic,q +aq*aq_wak_W*ad*(q, w)f(k, wi) - ~ 2J (E,,_w- E,,+n'lVs) {bwb/ (aq,laq-v*ak W-aq-",k-wak*aq)f(q,v)f* (k,w) 2 t.,'W.k.q + ~ kW(s' -S) (bw*bw+ 1/2) +......... . w lIS low as' possible. Using eq. (10), we see that to the first approximation We minimize (13) by setting We get which can be rewritten as _ 41'~(>: (liws) 2 nknk_ , 3nVt;; (E k- w- E,Y- (liws) 2 " W liws. (16) (...truncated)