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Relaxational model near the Lifshitz point
Volume 69A, number 4
PHYSICS LETTERS
25 December 1978
RELAXATIONAL MODEL NEAR THE LIFSHITZ POINT R. FOLK Institut für Physik, Universitdt Linz, 4045 Linz, Austria
and W.SELKE’ Theoretische Physik, Universität des Saarlandes, 6600 Saarbrücken, Federal Republic of Germany Received 12 October 1978
The critical dynamics of a relaxational model near the Lifshitz point is studied by the nents z are calculated numerically for the uniaxial (m = 1) and biaxial (m 2) cases.
The Lifshitz point (LP) [1] is a new kind of multicritical point at which an uniform, e.g. ferromagnetic, phase transforms into a modulated, e.g. helical, phase. The static critical behaviour of such a phase transition has been studied extensively by means of the renormalization group method (RNG) by various authors [1—4].The dynamics near the Lifshitz point has only been considered in a paper by Huber [5] within the context of mode coupling theory determining the scaling behaviour of the critical mode and the scaling exponent z for multicomponent (n = 2, 3) order parameters (OP). In this letter the dynamics of a relaxational model, the model A [6,7], is studied by the RNG. This model is important in describing experiments for systems with a one-component OP (n = 1) [7] and in estimating the accuracy of Monte Carlo calculations [8]. Lifshitz points with an one-component OP haye been suggested for magnetic alloys [1,9] and liquid crystals [10]. For a three-dimensional system only LP’s with m = 1, 2 or 3 can be realized, where m characterizes the spatial anisotropy [1]. For n >2 the lower critical dimensionality, however, is = 2 + m/2, so that only the uniaxial LP (m = 1) occurs at a finite temperature. The situation for n ~ 2 is not clarified yet: Monte Carlo calculations for d = 3 show a finite transition temperature at least 1
Present address: NY 14850, USA.
Cornell University, Baker Laboratory, Ithaca,
expansion. The dynamical expo-
for n = 2 with m = 1 as well as n = 1 with m = 1 and m = 2, see ref. [11]. Thus the results presented below are useful mainly for the cases n = 1 with m = 1 and m = 2, although some of them will be shown to be valid for n> 1 too. The starting point for the following investigation of the critical dynamics is to assume that the time dependence of the one-component OP Sk(t) is governed by a simple TDGL model [6]. Thus Sk(t) obeys the equation of motion ~ ‘k’J) 1’u ?7L,öS_k~t,+ rk~t 1 with the Landau—Ginzburg—Wilson hamiltonian __
~
—
~( = —
—
,
1 {r + c
~ fd°~k (2ir)’
rddk
ddk
Uj
1
~<~(d)(~
+ 1
‘
+
2 + q2 + p4} SkS_k 1p
f
—3d
4~27T) k ~ 5 4’ k~
2 k4
Here the d-dimensional vector k is decomposed into the m- and (d m)-dimensional components p and q respectively. The relaxation constant I’ and the gaussian stochastic force rk(t) fulfill the Einstein relation. An equivalent description of the dynamics can be given within a lagrangian formalism [12]. We follow this —
formalism and proceed according to the procedure described in paper [13]. 255
Volume 69A, number 4
PHYSICS LETTERS
The scaling exponent z is then found to be z = (4— ii)/(1 X)
(3)
—
where A determines the w dependence of the correlation function at the Lifshitz point ,c’(k = 0, w) w~. By the Feynman graph (matching) technique A may be calculated from the contribution in the self energy Z(k, ~,)= ~stat(k) + EdY1~(k,w), which we split into a static and dynamic part. In second order the dynamical part of the self energy is given by
25 December 1978
into account the differences in the order parameter fluctuations, when k lies in the p or q subspace and ~ is the correlation length for the p fluctuation. In consequence the critical dispersion defines two different exponents, namely 1 W~(P,0,
~
w (0 q
~
c
4+ci ~i
—
0) = 0)
p
—
~
2+C12 ~~l2
~—
q
The coefficients c
(4)
14 and c12 are easily calculated from eqs. (3) and between u” and (7) in fl! second order in e. The connection 4 and flu2 may be taken for rn = 1 from refs. [2] or [3] leading to c =_l.4, c =—0.9, forml. (9a)
2 +p4 and the critical dimensionality
Form = 2 the situation in statics is controversial [15], since refs.ref. [2][2] and do not agree in their values for 77. From we[4] find
2
~dYn(k,w) = iw 96u
fd°’ck1da’ck2ddck3
2dc (2~)
X&~’c)(k1+k2+k3) r
/4
)j ~
x [u2(ki)u2(k2)u2(k3) (_ic~.+EU2(k~)
where (J2(k) = q d C = 4+ ~m. The exponent A can be extracted in corn-
c 1
pact form
2
=
—7,
4
2 fd’~cxF3(x)
(5)
X=96u*
c1
=
0.48
for m
2
=
,
(9b)
2
and from ref. [4] = —6, c 12 = 0.68 form = 2 (9c) These values should be compared with the Van Hove theory, where c14 = c~2= —1. Since the dimension (n) of the order parameter enters the calculations to second order of e only via the static exponents 77/4 and ~2 ,the values for the c’s, eqs. (9), are independent of n, to the order given above. In order to get a more realistic model for n> 1 one has to introduce, however, mode coupling terms in addition to the purely relaxational Preliminary calculations confirm the mode behaviour. coupling theory calculations of Huber [5] especially the appearance of two critical dimensions = 4 + m and dcq = 6 + m for p and q fluctuations respectively. For d < ~ one finds with z defined analogously to eq. (8): .
with the function F(x)
=
fddck(2~)_4cU~1(k)e1kx~2(k) .
(6)
In first order of e = d~—d the fixed point value u” is given by (n = 1) [2] u’~=-
K~K~’mi2[B(~rn, (8 m))} ~
—
1
2(F(~ rn))~). (Km = 2_(m_1)~—mI The integrations in eqs. (5) and (6) can be done numerically with the result X=0.9X103e2
forrnl,
(7a)
X’l.4Xl02e2
forrn2.
(7b)
1
r
4~fli4
z~[rn+~____(drn)+477l 4
Since the values of A are positive the general inequality Z >ZVanHove [14] is clearly fulfilled. Because of the anisotropy of the system in k-space the scaling law for the characteristic frequency w~ reads wc (p, q
~)= b~f(pb qbx, s/b).
The scaling exponent x 256
(4
—
77/4)1(2
(8) —
fl/2)
[1] takes
which differs from Huber’s [5] result by terms of 0(77). We thank Prof. M.E. Fisher for a critical reading. References [11 R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35 (1975) 1678.
:
Volume 69A, number 4 [2] [3] [4] [5] [6]
PHYSICS LETTERS
D. Mukamel, J. Phys. AlO (1977) L249. R.M. Hornreich and A.D. Bruce, J. Phys. All (1978) 595. J. Sak and G.S. Grest, Phys. Rev. B17 (1978) 3602. D.L. Huber, Phys. Lett. 55A (1976) 359. B.I. Halpering, P.C. Hohenber and S. Ma, Phys. Rev. Lett. 29 (1972) 1548. [7] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [8] K. Binder, in: Phase transitions and critical phenomena, Vol. Vb, eds. C. Domb and M.S. Green,(Academic, New York, 1976).
25 December 1978
[9] A. Michelson, Phys. Rev. B16 (1977) 577. [10] A. Michelson, Phys. Rev. Lett. 39 (1977) 464. [11] W. Selke, Z. Phys. B29 (1977) 133; Solid State Commun., to be published. [12] H.K. Janssen, Z. Phys. B23 (1976) 377. [131 R. Folk, H. Iro and F. Schwabl, Z. Phys. B27 (1977) 169. [14] K. Kawasaki, Phys. Rev. 148 (1966) 375. [15] J. Sak, private communication.
257
PHYSICS LETTERS
25 December 1978
RELAXATIONAL MODEL NEAR THE LIFSHITZ POINT R. FOLK Institut für Physik, Universitdt Linz, 4045 Linz, Austria
and W.SELKE’ Theoretische Physik, Universität des Saarlandes, 6600 Saarbrücken, Federal Republic of Germany Received 12 October 1978
The critical dynamics of a relaxational model near the Lifshitz point is studied by the nents z are calculated numerically for the uniaxial (m = 1) and biaxial (m 2) cases.
The Lifshitz point (LP) [1] is a new kind of multicritical point at which an uniform, e.g. ferromagnetic, phase transforms into a modulated, e.g. helical, phase. The static critical behaviour of such a phase transition has been studied extensively by means of the renormalization group method (RNG) by various authors [1—4].The dynamics near the Lifshitz point has only been considered in a paper by Huber [5] within the context of mode coupling theory determining the scaling behaviour of the critical mode and the scaling exponent z for multicomponent (n = 2, 3) order parameters (OP). In this letter the dynamics of a relaxational model, the model A [6,7], is studied by the RNG. This model is important in describing experiments for systems with a one-component OP (n = 1) [7] and in estimating the accuracy of Monte Carlo calculations [8]. Lifshitz points with an one-component OP haye been suggested for magnetic alloys [1,9] and liquid crystals [10]. For a three-dimensional system only LP’s with m = 1, 2 or 3 can be realized, where m characterizes the spatial anisotropy [1]. For n >2 the lower critical dimensionality, however, is = 2 + m/2, so that only the uniaxial LP (m = 1) occurs at a finite temperature. The situation for n ~ 2 is not clarified yet: Monte Carlo calculations for d = 3 show a finite transition temperature at least 1
Present address: NY 14850, USA.
Cornell University, Baker Laboratory, Ithaca,
expansion. The dynamical expo-
for n = 2 with m = 1 as well as n = 1 with m = 1 and m = 2, see ref. [11]. Thus the results presented below are useful mainly for the cases n = 1 with m = 1 and m = 2, although some of them will be shown to be valid for n> 1 too. The starting point for the following investigation of the critical dynamics is to assume that the time dependence of the one-component OP Sk(t) is governed by a simple TDGL model [6]. Thus Sk(t) obeys the equation of motion ~ ‘k’J) 1’u ?7L,öS_k~t,+ rk~t 1 with the Landau—Ginzburg—Wilson hamiltonian __
~
—
~( = —
—
,
1 {r + c
~ fd°~k (2ir)’
rddk
ddk
Uj
1
~<~(d)(~
+ 1
‘
+
2 + q2 + p4} SkS_k 1p
f
—3d
4~27T) k ~ 5 4’ k~
2 k4
Here the d-dimensional vector k is decomposed into the m- and (d m)-dimensional components p and q respectively. The relaxation constant I’ and the gaussian stochastic force rk(t) fulfill the Einstein relation. An equivalent description of the dynamics can be given within a lagrangian formalism [12]. We follow this —
formalism and proceed according to the procedure described in paper [13]. 255
Volume 69A, number 4
PHYSICS LETTERS
The scaling exponent z is then found to be z = (4— ii)/(1 X)
(3)
—
where A determines the w dependence of the correlation function at the Lifshitz point ,c’(k = 0, w) w~. By the Feynman graph (matching) technique A may be calculated from the contribution in the self energy Z(k, ~,)= ~stat(k) + EdY1~(k,w), which we split into a static and dynamic part. In second order the dynamical part of the self energy is given by
25 December 1978
into account the differences in the order parameter fluctuations, when k lies in the p or q subspace and ~ is the correlation length for the p fluctuation. In consequence the critical dispersion defines two different exponents, namely 1 W~(P,0,
~
w (0 q
~
c
4+ci ~i
—
0) = 0)
p
—
~
2+C12 ~~l2
~—
q
The coefficients c
(4)
14 and c12 are easily calculated from eqs. (3) and between u” and (7) in fl! second order in e. The connection 4 and flu2 may be taken for rn = 1 from refs. [2] or [3] leading to c =_l.4, c =—0.9, forml. (9a)
2 +p4 and the critical dimensionality
Form = 2 the situation in statics is controversial [15], since refs.ref. [2][2] and do not agree in their values for 77. From we[4] find
2
~dYn(k,w) = iw 96u
fd°’ck1da’ck2ddck3
2dc (2~)
X&~’c)(k1+k2+k3) r
/4
)j ~
x [u2(ki)u2(k2)u2(k3) (_ic~.+EU2(k~)
where (J2(k) = q d C = 4+ ~m. The exponent A can be extracted in corn-
c 1
pact form
2
=
—7,
4
2 fd’~cxF3(x)
(5)
X=96u*
c1
=
0.48
for m
2
=
,
(9b)
2
and from ref. [4] = —6, c 12 = 0.68 form = 2 (9c) These values should be compared with the Van Hove theory, where c14 = c~2= —1. Since the dimension (n) of the order parameter enters the calculations to second order of e only via the static exponents 77/4 and ~2 ,the values for the c’s, eqs. (9), are independent of n, to the order given above. In order to get a more realistic model for n> 1 one has to introduce, however, mode coupling terms in addition to the purely relaxational Preliminary calculations confirm the mode behaviour. coupling theory calculations of Huber [5] especially the appearance of two critical dimensions = 4 + m and dcq = 6 + m for p and q fluctuations respectively. For d < ~ one finds with z defined analogously to eq. (8): .
with the function F(x)
=
fddck(2~)_4cU~1(k)e1kx~2(k) .
(6)
In first order of e = d~—d the fixed point value u” is given by (n = 1) [2] u’~=-
K~K~’mi2[B(~rn, (8 m))} ~
—
1
2(F(~ rn))~). (Km = 2_(m_1)~—mI The integrations in eqs. (5) and (6) can be done numerically with the result X=0.9X103e2
forrnl,
(7a)
X’l.4Xl02e2
forrn2.
(7b)
1
r
4~fli4
z~[rn+~____(drn)+477l 4
Since the values of A are positive the general inequality Z >ZVanHove [14] is clearly fulfilled. Because of the anisotropy of the system in k-space the scaling law for the characteristic frequency w~ reads wc (p, q
~)= b~f(pb qbx, s/b).
The scaling exponent x 256
(4
—
77/4)1(2
(8) —
fl/2)
[1] takes
which differs from Huber’s [5] result by terms of 0(77). We thank Prof. M.E. Fisher for a critical reading. References [11 R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35 (1975) 1678.
:
Volume 69A, number 4 [2] [3] [4] [5] [6]
PHYSICS LETTERS
D. Mukamel, J. Phys. AlO (1977) L249. R.M. Hornreich and A.D. Bruce, J. Phys. All (1978) 595. J. Sak and G.S. Grest, Phys. Rev. B17 (1978) 3602. D.L. Huber, Phys. Lett. 55A (1976) 359. B.I. Halpering, P.C. Hohenber and S. Ma, Phys. Rev. Lett. 29 (1972) 1548. [7] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [8] K. Binder, in: Phase transitions and critical phenomena, Vol. Vb, eds. C. Domb and M.S. Green,(Academic, New York, 1976).
25 December 1978
[9] A. Michelson, Phys. Rev. B16 (1977) 577. [10] A. Michelson, Phys. Rev. Lett. 39 (1977) 464. [11] W. Selke, Z. Phys. B29 (1977) 133; Solid State Commun., to be published. [12] H.K. Janssen, Z. Phys. B23 (1976) 377. [131 R. Folk, H. Iro and F. Schwabl, Z. Phys. B27 (1977) 169. [14] K. Kawasaki, Phys. Rev. 148 (1966) 375. [15] J. Sak, private communication.
257