#### 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)
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