Pram~.na, Vol. 15, No. 5, November 1980, pp. 501-505. ~) Printed in India.
Classical ~6-field theory in (1+ 1) dimensions 2. Proof of the existence of domain walls above the transition point
S N B E H E R A and A V I N A S H K H A R E * t Institute of Physics, Bhubaneswar 751 007, India
*Department of Theoretical Physics, The University, Manchester M13 9PL, U.K.
tPermanent address: Institute of Physics, A/105 Saheed Nagar, Bhubaneswar 751 007, India
MS received 27 August 1980
Abstract. The existence of a domain wall-like contribution to the free energy above the first order phase transition point is demonstrated for a system described by the ,~6-field theory in (1 + 1) dimensions.
Keywords. 96-field theory; domain walls: structural phase transition.
1. Introduction
In an earlier paper (Behera and Khare 1980a; hereafter referred to as I) the dynamics and t h e r m o d y n a m i c s o f the 4,e-field theory in (1-1-1) dimensions were extensively studied. T h e use o f the ffs-field theory as a model for first order structural phase tran- sitions (see I a n d the references therein) was also discussed. O f particular i m p o r t a n c e to the problem o f structural phase transitions is the existence o f d o m a i n wall-(kink) like solutions, which are responsible for the occurrence o f the central peak pheno- mena. It was shown that these domain wall solutions exist below the transition point (a < 9/8, notations are same as in I), the latter being determined by the parameters o f the fie-potential. However, the exact evaluation o f the free energy o f the system in this regime revealed that the tunnelling-like contribution expected for the d o m a i n wall free energy is absent i.e. identifying the d o m a i n wall free energy as the exact free energy minus the p h o n o n part, it was found to be large a n d p r o p o r t i o n a l to T ' instead o f exp ( - - c o n s t / T ) . In the concluding section o f I, it was conjectured that the presence o f local minima in the ~,n-potential, above the transition point, i.e.
9/8 < a < 3/2, will lead to the existence o f a tunnelling-like contribution to the free energy which can explain the experimentally observed central peak in ferroelectrics at temperatures a b o v e To. The purpose o f the present paper is to prove this conjecture.
The plan o f the rest o f the paper is as follows. In § 2 an upper b o u n d to the ground state energy eigenvalue o f the corresponding Schr6dinger equation (see I) for the eke-potential will be calculated for a 3> 9/8. This will then be used to calculate the free energy using equation (62) o f I, and the existence o f a domain wall-like contribution will be d e m o n s t r a t e d . The concluding § 3 is devoted to the discussions o f the results.
501
502 S N Behera a n d Avinash Khare
2. Calculation of free energy above the phase transition point
It was shown in I that the evaluation o f the free energy of the system reduces to the solution of an equivalent eigenvalue problem given by (some of the essential results of I are reproduced for the sake of completeness)
[~__# \3mC~ 1
In1 c9"
~ - ~ + v ( ~ ) v . ( ~ ) =]
, . ~ , (~), ( l )where the temperature-dependent effective m ~ s is
m * --: mfl~C~/la: fl = (kBT.)-I, (2)
and the potential is given by
v(~) - s ~" - [ A [ ~ + c ~ , c > 0. (s)
The free energy o f the system can be written in term,~ of the ground state eigen value
% as
F = N % - - N k B T In (2# m/fl) 1/~. (4)
In I, equation (1) was solved exactly for the case a ( ~ - - g B c / 2 i A [3)<9/8 when certain constraints on the coupling constants A, B, C are satisfied. We now show that for a :> 9/8 (i.e. above the phase transition point) even though the ground state energy cannot be calculated exactly, stringent upper and lower bounds on it can be obtained.
Let us first notice that if
B -- IAr" d- 3(C/2m*) 1/~, (5)
4C or equivalently
9 [ ( 9C V " ' l - '
a ~ - I - -
8 \2-m-VB-~I J > 918, (6}
then the Hamfltonian o f equation (1) can be written as
where A(~) - - - -
H = A * ( ~ ) A (~)
+~ .~/ ~ 2 ~ :
jI in ( 2~I~ I, ~.?~
e tAI4 c,:,4,
(2m*)I/~d~ + ( C ) I/z
--
. (8)rbr-field theory - - 2
Hence the ground state energy
>~ 1 {2Bi'/"[l__[
9CI1/'] ''z_i_Lln( 2~rl"i
~0 \m-~/ ~2m* B2I J z p ~flm C~o! , where the equality holds if and only if the solution ff (4) of the equation
A (4) ¢ (4) = 0,
is square integrable. The solution ~b (4) which we normalise by
~'4,=exp[-lal(~-~) 1 ' " 4 2 L 4 +
is however not square integrable and hence 1 (2B/112 [1
,o > ~ \ ~ - ~ ;
However, since ~ (¢k) obeys
,/,(4 = ± (I A tt2c)'~) = l,
503
(9)
( l l )
,1¢1I = o. (12)
l ~= -(141/2C)~/~
Hence following Herbst and Simon (1978, 1979) we can obtain a lower bound on c 0 i.e. we have the following stringent upper and lower bounds on it
0 < '~o- ~_(~,)1;" [1--t\2---~--~.-/~219C
1t~;'211/2 l in [ 2~F' t j 2,8 \3mC~! <~ N~
(13) In order to determine the constants N I and N 2, the trial function is chosen to be
~4) = I
IAI
2m*1,,,~
, ( 2 m , C ) , / 2 4 , _ f -IA I,
exp [ - - - - ~ - (--~?--) 4 -t- 4 ~ (~_~),/2],
<(I.~1/~'"
ll.a I'f"l"~ {lnt'f,"
I-'[ is' ' (14~
P--8
- ~--@/[ 9c V'l''j + ~ln 12~l~/.,,~mq,,
t A r - [ 2 m * ~ l / " ] (2m* C) 1/~" ~4 -b (10)
504
S N Behera and Avinash Khare
so that the boundary conditions are satisfied at ~ = ~: (I A
l/2c)x/~
and a is the variational parameter. On minimising the energy with the trial function (14) and after some complicated algebra one getsli'(2B/1/'~[,. 1 /" 9C \.,""11/2 3 2B[ 1 ( 9 C 11,"~t(2m*) 1/t
X exp [ - - [ ~-~---~' ( ~ - ~ ) 1/~ ] (15)
±,n(
or "o = Eo -- 213
\~SmC~]
(16)In evaluating equation (15), it has been assumed that C is small. On evaluating the free energy from equation (4) (using equation (2)) in the limit of T--> 0 yields;
1 1 (2B11/2[1 _ 1 ( 9C / 1'3
F = NkBT-Co 2,-ml 2\2---m~1 lokBT]
+ 3 (,1:,:(2m)'"2.[1_ ]""4 T]
~-~ \'~o I
C :/---~ m L \ 2 m B " l C okB
exp[ -(IAi2(2mll,2C°ll\-'8-C- \-C-, "7"11 kBT] --NkBTIn(2~rII\~'-C-"J (17a)
-=-- Fosc. + Ftunn" 4- O (Tin T) (17b)
3. Conclusion
It is clear from (17) that the free energy has a part corresponding to phonons and a part corresponding to tunnelling or domain walls. The domain wall contribution (Ftunn) has a structure similar to that obtained by Krumhansl and Schrieffer (1975) (KS) in the case of the ~4-field theory. The domain wall contribution vanishes either as T-r0 or as C-->0. This suggests that for a > 9/8, i.e. above the phase transition point, there exists a domain wall contribution to the free energy. However, such a contribution does not appear below the phase transition point (a<9/8) as was demonstrated by an exact calculation in § 4-2 o f I.
In order to compare the oscillatory part of the free energy (Fosc) as given by equation (17) with that of the phonons, one can estimate the later following KS.
In doing so one has to note that ~ = 0 corresponds to the absolute minimum of the potential, hence for small oscillations around 4 = 0 . Linearisation of the equation o f motion (equation (11) of I), yields the phonon dispersion to be
% ~ = c o ~ q~ + (2B/m), (18)
4,"-.field theory--2
505 which is the same as t h a t o f the if4 theory o f KS. Thus the p h o n o n free energy be- c o m e sl 1
Fos e ----
N k B T t-o'~" ~(2B/m) ~/2 q- ....
(19)which is the same as t h a t o f e q u a t i o n (17) for T-->0. On the o t h e r h a n d if one assumes small oscillations a r o u n d o n e o f the local m i n i m a
4, =-4-(l A I/2C)ll~,
then following the p r o c e d u r e o f I, the p h o n o n free energy can be written as ( T - > 0)F°sc = N kB T co(~--)I'/2[1--2\2~]3 ( 9C lllz '~-o kB T],
l ( 2 0 )which is m u c h larger t h a n the Fos c given by equation (17). This is physically reason- a b l e because it costs m o r e energy to m a k e the particles oscillate a b o u t the local m i n i m a , which are higher in energy than the absolute m i n i m u m o f the potential a t f f ---- 0.
Finally it is w o r t h pointing o u t that, proceeding as in § 2 it is easy to show that, f o r
9F t" 1
< 9t8;the H a m i l t o n i a n o f the system can be written as
(2B~1/211-~-(2~,C~.2)1/2} 1/2,
H : A * (4>) A (,¢,) - - ½
~--~/
where A (4,) _ _ + _ ~ / ~ 4,3,
(2 m*) 1/~- d ~ 2
"~/C
so t h a t
E0=--½\~-;~ (2Bt,," [l +tY-~'~/ J " ( 9c 7'T'
t h e r e b y the result obtained in I is reproduced.
R e f e r e n c e s
Behera S N and Khare A 1980a
Pramana
15 245 Herbst I W and Simon B 1978Phys. Lett.
B78 304 Herbst I W and Simon B 1979Phys. Lett. BSO
433Krumhansl J A and Schrieffer J R 1975
