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

Progress of Theoretical Physics, Sep 1977

The phase transition between commensurate and incommensurate phases in one-dimensional electron-phonon system is considered in the mean field approximation, taking account of the effect of all the harmonics of the charge density wave. We can show that the problem reduces to that of the G-L equation with a periodic boundary condition. The equation can be solved exactly at T = 0K and T ∼TC in the case of nearly 1/3-filled band. A part of the phase diagram in the µ-T plane is given. We also discuss the case of nearly half-filled band briefly.

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

Progress of Theoretical Physics Commensurate-Incommensurate Charge-Density Wave Phase Transition in One-Dimensional Electron-Phonon System Tetsuo 0 HMI 0 Hikaru YAMAMOTO 0 0 Department of Physics, Kyoto University , K·yoto 606 The phase transition between commensurate and incommensurate phases in one-dimensional electron-phonon system is considered in the mean field approximation, taking account of the effect of all the harmonics of the charge density wave. We can show that the problem reduces to that of the G-L equation with a periodic boundary condition. The equation can be solved exactly at T=OK and T~Tc in the case of nearly 1/3-filled band. A part of the phase diagram in the 11-T plane is given. We also discuss the case of nearly half-filled band briefly. Model Hamiltonian + (N(0)/4A) ~ [LfnQ[ 2, n § 3. Limiting situation of Q-'>2n'/3c Fig. 2. 21-th order graph of f2. Fig. 3. 2l-th order graph of f2 using .:1 (¢). -',l, ('"+') = L'\...'i L1 C3n+J)Q ei<sn+Jl¢' n (3 ·1) + l_(LJ3+LJ*3) +.l [.::112] 4Afp 2A and at T~Tc to fourth-order of Ll: (3 ·2) (3. 3) Fig.4. Umklapp graph. Fig.5. Four kinds of Green's functions. 3m2 -1 § 4. Expansion of fJ with respect to q --{J·Q--_- "L...".J {JQ aLi(¢) n !JL1(3n+!)Q (4 ·1) (4· 2) Fig. 6. Graphs of !J to O(q'). Fig. 7. Graph of !J to calculate the tt- and q-dependence of !J at T·--Ta. /}.2N(O) ~ err/3d¢[_!_ I(Y_.Jj8pt _ _!_ ~~j.JI2/8¢_L_J. 27C Jo 2 I.J 12 12 I.J 14 6 I.JI 2 8¢ 2 8¢ 2il.fp Tr'JTc § 5. The phase diagram (5 ·1) J G~=N(O)3- 12rr/3 d¢ [I!2:(_3(}-\;2+ ILJ 3 cos3(}. 2rr o 2 3¢, 2AfF (3(}) 2 -_-2ka2 - a ,, a cos 3(}, 30 and the limiting form 1s (5. 4) (5. 5) (5 · 6a) (5. 6b) where and a differential equation similar to (5 · 3) -VFq -_[-3 12rr/3 d ¢(f)-f)) 2] -1 2/1 2n o 8¢ 82() b (vFq) 2 -8¢+2 3ciL11 sin 3fJ =0 (5 ·10) (5 ·11) (5 ·12) Approaching the Terline, q vanishes like and the singularity of the specific heat C is where (iii) Phase diagram Employing the result at T=OK and T ~Te, we can draw the phase diagram illus­ trated in Fig. 8. Suppose Tis lovv·erecl from T>Te(/.L) with a constant /1 as shown in Fig. 8 vvith the arrow line. First the phase transition betvveen the normal and the in­ commensurate phase occurs at Te(f-1). The sinusoidal charge density wave near Te (!J.) continuously changes into the soliton wave near Tcr (!J.), at \vhich the transition between IC- and C-phases takes place. If T is lowered further, the transition occurs again from C- to IC-phase. Here ·we hav·e used the fact that the energy of the soliton phase is smaller than that of C-phase, i.e., the _Te-Ter= (!!___) 4 b3 ·fl4. Te 2 c2 C= N(O) c IAI 3/ (Tcrt), (5 ·13) (5 -14) (5 ·15) (5 ·16) (5 ·17) Minimizing G with respect to Ll, we get § 6. Nearly half-filled band (6·1) T=O (vFq=2fJ.) The free energy G is written as j (¢) = { L/ 0 ( -rr/2<¢<rr/2) -L/ 0 (-rr<¢<-rr/2, rr/2<¢<rr). TrvTc where D = _:!._ 1 4! (4n:T) 4 Re rjPl (_:!._ + i !J ) 2 2n:T Acknowledgements Appendix --Calculation of !2 of 0 (p2) - ­ From Figs. 6(e) and 6(f), we obtain (6 ·5) (A·1) over n, m 1 and m 2 in (A ·1), we have J*(¢2+¢)e1 ~' G0- 1 (+) X (2i-~-1)1JC¢1) (-2i_Q_+1)1JC¢2). a¢1 a¢2 at T=OK and obtain 2N(O) [ i 1 . aJ* + i J* a CIJI2) + 1 J* - 4 [JT2 -----w; 12 W a¢ 8 W Similarly we can get from (A· 2) 2 [ i 1 aJ* i J* 11 N(o) 4 TJf -----w;-_12 W a 2 1 J* 1 J* aJ aJ* a¢ CIJI) - 8 W - 2 !Jfa¢ -----w; 5 J* a CIJI2) + 1 J* a2 (IJ12)] 3 IJI4 a¢2 2 [1 J (aJ*) 2 1 1 a2J* 1 .::1* 2 a2J 11 N (O) 3 TJf -----w; - 3 [:1[2 a¢2 + 6 W a¢2 (A·3) (A·4) (A·5) (A·6) 1) L. B. Coleman , M. ]. Cohen , D. J. Sandman , F. G. Yamagishi , A. F. Garito and A. J. Heeger , Solid State Comm. 12 ( 1973 ), 1125 . M. C. Leung , Solid State Comm . 15 ( 1974 ), 879 ; Phys. Rev . Bll ( 1975 ), 4272 . Y. Ono , J. Phys. Soc. Japan 41 ( 1976 ), 817 . A. Kotani , J. Phys. Soc. Japan 42 ( 1977 ), 408 , 416 . W. L. McMillan , Phys. Rev. B14 ( 1976 ), 1496 . M. J. Rice , A. R. Bishop , J. A. Krumhansl and S. E. Trullinger , Phys. Rev. Letters 36 ( 1976 ), 432 . T. Ohmi and H. Yamamoto , Prog. The or . Phys . 57 ( 1977 ), 688 .


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Tetsuo Ohmi, Hikaru Yamamoto. Commensurate-Incommensurate Charge-Density Wave Phase Transition in One-Dimensional Electron-Phonon System, Progress of Theoretical Physics, 1977, 743-754, DOI: 10.1143/PTP.58.743