Commensurate-Incommensurate Charge-Density Wave Phase Transition in One-Dimensional Electron-Phonon System. II

Commensurate-Incommensurate Charge-Density Wave Phase Transition in One-Dimensional Electron-Phonon System. II 0 0 Department of Physics, Kyoto University , Kyoto 606 The Commensurate-Incommensurate (C-IC) phase transition at finite temperature is considered in mean-field approximation taking account of higher harmonics. The phase diagram is given in the (!1, T) -plane, where tl represents a measure of the deviation from 1/3 filledness. It is shown that at proper values of fl successive phase transitions occur (a) from the normal (N) to the IC-phase, (b) from the IC- to the C-phase and (c) from the C- to the IC-phase as T is decreased to zero. The transition (b) is caused by the growth of the amplitude of the order parameter, whereas the transition (c) is due to the mcrease of the chemical potential. - Model Hamiltonian and formulation PnQ= 2:: a;+nQap, Q=2n/3c+q, p JF=JSJ(/1) -N(O) (/1-,U) 2/2+ (N(0)/2A.) 2:: JJ(3n+llQJ 2 , n .dfan,.+l)Q G + (P) G_(P- (3n2 + 1) Q) G_(p- (3n2 +1)Q+ (3n2 +1)Q- .. - (3n21 +1)Q) g(n1- n2 + .. - n2z), if 7l = 0' otherwise. + c.c. Jdrf;. where +~(vFq) 2 {Az(Zi, jiJj) (80) 2 +A4(Zi, lili) (~~J) 2 1f 2 ~ ~ +Aa(Zi, jiJj)cos 30() ]d, + Ii11 2 _ (Zi=JL) 2] 2A 2 2 Jo cosh f3!l +cosh (JEP (2. 4) Phase diagram at T~O Minimizing ilF with respect to q and (} () gives and a non-linear equation with k determined by 4 J:A2A~' (3 1) (3 2) (3 3) (3 5) eFITc =143.3 u. =0.1964) (f.10 1T, ,T0 1T,) Fig. 2. The behaviour of fdf, i1 and A, in the C-state. 4 VA2A3 (3 -6a) (36b) lJ. Yamamoto, I. 1Vakayama and T. Ohmi JE= (-_1_ Ll 02 4 .:1,/ )NCO) +/i.(O)pN(O), 2),fF (3 10) The \"alue (3 I) (3. 8) (3. 9) Phase diagram at T~Tc (41) II _3.0,o._o_ _ _,o,.s_q_1_2_f.J_1,.o 'F f.J=0.26 ( !0-2) Concluding remarks Acknowledgements Appendix G_(p- (3n2 +1)Q) = + (3n2+1)q + (3n2+1) 2q2 (ic.+71+P) 2 (ic.+71+P) 8 (A1) (A2) (A3) The summation over l is easily carried out to obtain 1s written as If we substitute Noting that 0 (q'): a a 1111 a " ' 3 } :.ILz= .::._.l 27C 0 S'"')T'' faLl* aLi 1 .::._. (- :-:,-:/_-- -::,~- I Al2- 1 iJA1uc_,_l-i-1(p.)G __ I+l(fJ- 2 ~1.;3 c) Z(~6+~) " . p,e Up U<p iJ (A-4) (A5) (AI) (A8) (A9) (AlO) (All) (Al2) (...truncated)