Rigorous bounds in structural phase transitions

Solid State Communications, Vol. 21, pp. 7 1 - 7 3 , 1977

Pergamon Press

Printed In Great Britain

RIGOROUS BOUNDS IN STRUCTURAL PHASE TRANSITIONS A Bunde and L. Lam InstItUt fur Theoretische Phystk, Unlversltat des Saarlandes, D66 Saarbrucken, FRG

(Received 1 September 1976 by P H Dedertchs) Exact inequalities for the local mean-square of displacement as well as its fluctuation are derived for a quantum anharmonIc phonon model Ragorous bounds for the transition temperature, the melting temperature and the kinetic energy are then obtained. IN THE STUDY of the static and dynamic properties 1 and, In particular, the critical central peak problem 2 of structural phase transitions the model Hamdtonlan of anharmonlc phonons,

1 X

H = 2-M ~,c,

X x?.+£ E 4n

2 ~,
[X~a, Pjt3] = tguSo¢, e t c , to obtain

(SXkcfiX-kc~)(e-A -- jk + 2B (x~) + B-n ~ (X~'r))>~ (3) where Xka IS the Fourier transform defined by

x,'.

1

Xka = N -l/z ~ X~a exp (tkR,) l

2 E s,,x,.x,~

(1)

and

t,j,O~

Jk =- ~. Juexp[tk(R, --R1)],

IS frequently used Pm and X,c~are the ath components (a = 1, , n) of momentum and displacement, respectively, of particles with mass M at the lattice site R, A, B and J~ (with J . = 0 and Jo = J~,) are model parameters which are assumed to be positive and finite e = 1 or -- 1 correspolIds to the so-called dIsplacive and o r d e r disorder type transitions respectively Unfortunately, except for n = ,~ case, this model cannot be solved exactly even with the use of classical mechanics On the other hand, quantum effects have been shown 1 to be important for certain range of the model parameters in regard to the transition temperature. In the following, we first derive quantum mechanically some exact inequalities concerning the local meansquare displacement (X~a) and the related fluctuation ((5X,~)2), with 8X,(~ - X ~ - - ( X ~ ) , for the model HamfltonIan (1) Both these quantities (In the classical regame) have been computed numerically wlth the molecular-dynamics technique a'4 for low dimensions (d = 1 or 2) Moreover, the mean-square displacement fluctuation is directly related to the line width in paramagnetic resonance experiments 5 Secondly, from these lnequahties, rigorous bounds on the transition temperature To, the melting temperature (with the use of the Dndemann criterion ~) and the kanetic energy (P2,~)/2Mare obtained In the Bogohubov lnequahty 7 (k B = 1),

½({D,D+})([[C,H],C+])>>-TI([C,D+])I2

1

etc Similarly, putting D = 6 X ~ and C = P~c~ in (2) we have another inequahty

((6X'a)2)(6"A + -~ (X~c~)+ B E"r (x~7)) ~

(4)

Alternatively, (4) may be derived 8 directly from (3) and thus gwes a weaker bound (see Fig 1). The stabdlty condition (Pro) = I([H, P,a ]) = 0 leads to

Jo- cA (X,c~) = 1 ~. (X~.yX,c~) B

n

(5)

"r

Using the inequality

In ~, (X'2"rX'~)>~ln (~ X'~) (x'c~)

(6)

which is valid i for all values of the dlmensionahty d for n = 1,2 and n = 0% and assuming a positive order parameter a (I e (Xt~) > 0) we combine (5) and (6) to get

Jo --eA >/1 ~ (XL) B

n

(7)

3,

It is mlportant to note that while (3) and (4) hold for arbitrary values of d and n and at all temperature T, (7) is established only in the ordered region for n = 1,2 and for arbitrary d From (7), the usual stability condition

(2)

let D = 5Xk,x and C = P~,~ and use commutators,

Jo - - ~ 71

1>0

(8)

72

RIGOROUS BOUNDS IN STRUCTURAL PHASE TRANSITIONS

(×z)

Vol. 21, No. 1

(al

60

f~"

J¢~A

t

ToIA21B1 E:-I 40

2O

3B

T~

~

r=

T2

2

Fig 1 Sketch of exact bounds for (X 2) as function of T(n = 1) To and T~ are both upper bounds for Tc Tm is an upper bound for the melting temperature Curves (a) and (b) are given respectively by equations (14) and (13) m text For n -~ 0% the upper bound becomes an Identity m the ordered region.

Fig. 2 Upper bound for transmon temperature for the displacwe (e = -- 1) and the order-disorder type (e = 1) transitions m a d = 3 and n = 1 simple cubic lattice with nearest-neighbor couphng

u 2 >~( r - e)/3

follows Because ([[C, H I , C+])>~ 0, by (3),

2B(x2~)+B ~ (X~)>~ Jk --cA n

tl

(9)

,y

for all k It is easy to obtain from (3) and (4) two more mequahtles,

(Xk'~X-kc')(eA--jk +2B(x~c~)+Bn ~v (x~7)) (X~)(eA + 2B (X~a) + B ~ (X~.r)) >~T n

n

7

/

n=1

For convenience, we introduce the following dimensionless and positive quantmes,

0 -- T(B/A 2) r -

u2 >~ [--e + (l +120)~/2]/6

(13)

follows from (4'). Also, from (3'), diwding both sides by the posltxve factor in the bracket and summing over k we get a more stringent lower bound given lmphcltly by u 2 >~(0/N) ~

[3u 2

+e--rJk/Jo] -1

(14)

k

Bounds for (X~) as a function of T for d = 3 are sketched m Fig 1. From the remark after (7) it is clear that To and T~, defined respectwely as the intersection between (11) and, m turn, (14) and (13), are both upper bounds for To, or

Tc < To
u 2 - (X 2,>(B/A)

(12)

In the case of order-disorder type transmons (e = -- 1), the single particle potential m (1) is a symmetric double well and the distance between the two mmtma of the potential Is given by 2Xo, with Xo - (A/B) 1/2 Equation (12) then lmphes that (X 2 ) is at least one-third o f x o2 0rrespectlve of dtmenslon), which seems to be consistent with computer calculations 4 Next, a temperature dependent lower bound,

(4')

Note that (3) and (3') together with (9) yield a lower k-dependent bound for the Ume independent correlation funcUons. In sum, (3) and (7) are the two basic mequalrues from which the rest follows To be more specific, let us consider two particular cases

Case 1

t+ Jo/A

(15)

with T~ = ( r - - e ) ( 3 r - -

(10)

Jo/A

2e)(A2/B)

To = [r(r--e)/g(3rr~2e)l(A2fB)

(16)

(17)

From (7), in the ordered region, we have an upper bound for u 2 r - - e ~ u 2 >10 (11)

where

In contrast, we have more than one lower bounds In fact, putting k = 0 In (9), where Jo > 0 is the maximum value o f J k , it follows that

For a three dimensional stmple cubic lattice with nearestneighbor couphng, g(0) = 0,g(1) = 1 516 and gO,) Is a

1

Jo

e(y) = ~ ~ (go~y)-J~

(18)

Vol. 21, No. 1

RIGOROUS BOUNDS IN STRUCTURAL PHASE TRANSITIONS

monotonic increasing function m the region 0 ~y Consequently, To ~< To as expected. The general trend of (X,2 ) and, in particular, the lncreasmg of (X,2) with T greater than T c as shown In Fig 1 is observed in computer simulations 3 [although it must be pointed out that the work in reference 3 is carried out in a finite two dimensional system w~th a Hamlltonlan slightly different from ours in (1)] In Fig 2, To of (17), the upper bound for To, is plotted against Jo/A (=- r) for both dlsplaclve (e = 1) and order-disorder type (e = -- 1) transitions in a d = 3 simple cubic lattice with nearest-neighbor coupling At the dlsplaclve hmlt, r = e (e = 1), To = 0 The condition for phase transition to take place, Jo ~>A, m the case of e = 1 should be compared to the analogous result in the transverse Islng model 9 Again, we emphasize that the results m Fig 2, as well as the rest In this paper, are rigorous. From (13) or (14), (X~) rises indefinitely as T Increases (see Fig 1) resulting in lnstablhty x° at a certain temperature. According to Llndemann 6 a solid melts when ((SX,) z) = sL 2 (19) where L is the lattice constant and s is a certain factor less than umty. In the region T > T e where the meeting takes place, 8Xi = X~ and thus companng (19) to (14) we obtain an exact upper bound for the melting temperature, 2 / Jo ) (20)

Tm = sL Jo/g ~ 3 B s ~ + eA

73

by (7) and (9), we have a 2 = r-- e

(21)

in the ordered regmn In other words, the horizontal lower bound m Fig. 1 rises up to colncxde with the upper bound m this case Equations (13) and (14) remain vahd when u 2 is replaced by t~/3 and 0 by 0/3. Correspondlngly, (17) becomes To = [r(r--e)/g(1)](A2/B)

(22)

which is nothmg but the classical T~ t and gives similar curves as shown an Fig 2 Note that the result Tc ~< Tffl derived here for a particular case is actually vahd for any quantum system a An Important consequence of (22) is that for an infinite system with d = 1 or 2, since g(1) = o% there exists no phase transition at any finite temperature Finally, from the Inequality s

MT <~(P•a) <<-MTff(X)

(23)

where f(x tanh x) - (tanh x)/x and X=

(e.A + 2+B ( B x ] a n)

n ~ (X]~))/4T(P2a))

(24)

we can use (7) to obtain in the ordered regmn upper and lower bounds for the lonetlc energy (P~a)/2M. More detads and discussions of various approximate calculations in connection with our exact results will be presented elsewhere a

/

Case 2

n -~ oo Acknowledgements - We have benefited from discussions

Defining

r? -= ( B / A ) l l m

-1

n--,- n

y~

( X , . 2r

with Karl H. Michel and Gunther Melssner. This work is supported in part by SFB Ferroelektrika der Deutschen Forschungsgememschaft.

),

REFERENCES 1

SCHNEIDER T , BECK H & STOLL E , Phys Rev B13, 1123 (1976), and references therem

2

For reviews, see FEDER J., in Local Propertws at Phase Transitions (Edited by MOLLER K A & RIGAMONTI A ). Academic, New York (1975), SCHWABL F , m Anharmonw Lattwes, Structural Transttton and Meltmg (Edited by RISTE T.). Leiden, Noordhoff (1974)

3

SCHNEIDER T. & STOLL E ,Phys Rev B13, 1216 (1976)

4

AUBRY S, J Chem. Phys 62, 3217 (1975)

5

Mt3LLERA.,BERLINGERW,WESTCH.&HELLERP.,Phys

6

LINDEMANN F, Z Phys. 11,609 (1910)

7

BOGOLIUBOVNN.,Phys. Abhandlg. SU. 6,113(1962)

8

LAM L & BUNDE A (to be published)

9

BUNDE A & L A M L , P h y s Lett 58A, 2 3 4 ( 1 9 7 6 )

10

MEISSNER G ,Phys. Rev B1, 1822 (1970)

Rev. Lett 32, 160(1974)