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Critical behaviour near lifshitz points
Journal of Magnetism and Magnetic Materials 9 (1978) 7-8 0 North-Holland Publishing Company
CRITICAL BEHAVIOUR NEAR LIFSHITZ POINTS W. SELKE Theoretische Physik, Universitiit,D-6600 Saarbriicken, Fed. Rep. Germany Received 1 March 1978
The phase diagram for the spherical model with third nearest neighbour interactions is calculated exhibiting a Lifshitz point of character three. Monte Carlo computations near the uniaxial Lifshitz point for a three dimensional lattice and a two dimensional order parameter are presented.
In the ordered phase one can distinguish four different regions for the wavevector dependent susceptibility ~(4) depending on the values of the nearest neighbour exchange integral Jr (>O; ferromagnetic), JZ (<0) and Js (XI), see fig 1. The point of intersection of these four regions (i2 = J, /J, = -5; j3 = J3 /J1 = k ; see ref. [3] ) is just the projection of the Lifshitz point of character 3 onto the jZ -ia-plane. For the uniaxial case, m = 1, this situation is displayed in fig. 2. A small part of the phase diagram (J3 = 0; the immediate neighbourhood of the LP) has been given before [ 141. For the isotropic case, m = 3, we did similar calculations [IS] ; there the LP and the LP of character 3 occur at zero temperature.
1. Introduction About two years ago, the Lifshitz point (LP) has been introduced as a new multicritical point denoting the triple point between the disordered, the uniformly ordered and modulated ordered phase [I]. Until now several aspects of the LP have been discussed including the notation of the more general “Lifshitz point of higher character” [2,3], the scaling properties of LP’s occurring at zero temperature [3,4], realizations of LPs in liquid crystals [ 5,6,7], in TTF-TCNQ [ 81 and magnets [ 1,9], high temperature series expansions [lo] as well as Monte Carlo calculations [ 111 for a simple model exhibiting a uniaxial LP, phase diagrams near LP’s [4,9,12] and the e-expansion .for the new exponent ok [ 131.
2. Spherical model .2 _.-
In this section we shall consider the spherical model with up to third nearest neighbour exchange interactions along m (m < 3) axes of a simple cubic lattice. The Fourier transform of the exchange integral is given by
, OK2 - _-’
5
j=l
COSS;
+I5i
taking the lattice constant
JJCOS(kJj))]
_Lnj; \ \T
: ,’
.6 / A!?&
/
‘cc
/’
&\
’
I,
,,,..(,
13 ,*’
J(4>= 2[Jl
-x:
xg
‘”
_I!4 9
,
Fig. 1. jz-j3-plane with different regions for the wavevectordependent susceptibility x(q) in the ordered phases (I,II-ferroIII,IV-helimagnetic). LP3 is the projection of the LP of charac ter 3.
to be one. 7
W. Selke / Critical behaviour near Lifshitz point9
display the data determing the critical exponent p of the order parameter, the magnetization. Taking into account only nearest neighbour interactions, J2 = 0, we get fl= 0.32 t 0.02; the critical temperature T, has been taken from the high temperature series expansion [ 171. Very close to the LP, see also ref. [lo], we made runs with 2000 Monte Carlo steps per spin discarding the first 250 steps to reach the equilibrium. Just as in the [sing case [ 1l] we observe a drastic decrease of the exponent of the order parameter for the planar model down to flLp = 0.20 + 0.02. The crossover effect near the LP is studied currently and the results will be published elsewhere [ 181.
Acknowledgements Fig. 2. Phase diagram for the spherical model. The transition temperature Tin units of 2.I,/kB for the LP of character 3 (LP$ is j”c = 1.066. P, F and H denote the para-, ferro- and helimagnetic phases.
I thank Profs. K. Binder and G. Meissner for their help. This work has been supported by the Deutsche Forschungsgemeinschaft SFB 130, Ferroelektrika.
3. Monte Carlo calculations References We did Monte Carlo calculations, see ref. [ 161, for the planar spin model (spin vectors with two components) on a simple cubic lattice with ferromagnetic interactions, Jr, between nearest neighbours and antiferromagnetic ones, JZ, between next nearest neighbours along one axis of the lattice (“R-S model”, see refs. [ 10 and 1 l]. The results for the analogous Ising case have been already given [ 1 I]. We used periodic boundary conditions for a 163 lattice. In fig. 3 we
AC/P -0.2 6_‘, --,
_.I
‘.-
-
.O a2-
0.1-
0.01
0.02
0.1
Q2 E
Fig. 3. Log-log-plot of the magnetization versus reduced temperature E = (Tc - 7)/T, for the planar R-S model with (ly, = 0 (0) and (2)J,/Jr = -0.26 (=). Tc is taken to be 2.204, resp. 1.81 in units of Jl/kg.
[l] R.M. Hornreich, [ 21 [3] [4] [S]
M. Luban and S. Shtrikman, Phys. Rev. Lett. 35 (1975) 1678. J.F. Nicoll, G.F. Tuthill, T.S. Chang and H.E. Stanley, Phys. Lett. A58 (1976) 1. W. Selke, Z. Physik B27 (1977) 81; Phys. Lett. A61 (1977) 443. A. Erzan and G. Stell, Phys. Rev. B16 (1977) 4146. J. Chen and T.C. Lubensky, Phys. Rev. Al4 (1976) 1202.
[6] C. Sigaud, F. Hardouin and M.F. Archard, Solid State Commun. 23 (1977) 35. [7] A. Michelson, Phys. Rev. Lett. 39 (1977) 464. [8] E. Abrahams and I.E. Dzyaloshinskii, Solid State Commun. 23 (1977) 883. [9] A. Michelson, Phys. Rev. B16 (1977) 585. [lo] S. Redner and H.E. Stanley, Phys. Rev. B16 (1977) 4901; J. Phys. Cl0 (1977) 4765. [ll] W. Selke, Z. Phys. B29 (1978) 133. [ 121 D. Mukamel and M. Luban, Phys. Rev., in print. [13] D. Mukamel, J. Phys. A10 (1977) 249. [ 141 R.M. Hornreich, M. Luban and S. Shtrikman, Physica A86 (1977) 465. [ 151 W. Selke, presented at Stat. Phys. 13, Haifa, Israel (August 1977). [ 161 K. Binder, in Phase transitions and critical phenomena, Vol. 5b, eds. C. Domb and M.S. Green (Academic Press, New York, 1976). [17] S. Singh and D. Jasnow, Phys. Rev. Bll (1975) 3445; R.G. Bowers and G.S. Joyce, Phys. Rev. Lett. 19 (1967) 630. [18] W. Selke, Solid State Commun. (1978) in print.
CRITICAL BEHAVIOUR NEAR LIFSHITZ POINTS W. SELKE Theoretische Physik, Universitiit,D-6600 Saarbriicken, Fed. Rep. Germany Received 1 March 1978
The phase diagram for the spherical model with third nearest neighbour interactions is calculated exhibiting a Lifshitz point of character three. Monte Carlo computations near the uniaxial Lifshitz point for a three dimensional lattice and a two dimensional order parameter are presented.
In the ordered phase one can distinguish four different regions for the wavevector dependent susceptibility ~(4) depending on the values of the nearest neighbour exchange integral Jr (>O; ferromagnetic), JZ (<0) and Js (XI), see fig 1. The point of intersection of these four regions (i2 = J, /J, = -5; j3 = J3 /J1 = k ; see ref. [3] ) is just the projection of the Lifshitz point of character 3 onto the jZ -ia-plane. For the uniaxial case, m = 1, this situation is displayed in fig. 2. A small part of the phase diagram (J3 = 0; the immediate neighbourhood of the LP) has been given before [ 141. For the isotropic case, m = 3, we did similar calculations [IS] ; there the LP and the LP of character 3 occur at zero temperature.
1. Introduction About two years ago, the Lifshitz point (LP) has been introduced as a new multicritical point denoting the triple point between the disordered, the uniformly ordered and modulated ordered phase [I]. Until now several aspects of the LP have been discussed including the notation of the more general “Lifshitz point of higher character” [2,3], the scaling properties of LP’s occurring at zero temperature [3,4], realizations of LPs in liquid crystals [ 5,6,7], in TTF-TCNQ [ 81 and magnets [ 1,9], high temperature series expansions [lo] as well as Monte Carlo calculations [ 111 for a simple model exhibiting a uniaxial LP, phase diagrams near LP’s [4,9,12] and the e-expansion .for the new exponent ok [ 131.
2. Spherical model .2 _.-
In this section we shall consider the spherical model with up to third nearest neighbour exchange interactions along m (m < 3) axes of a simple cubic lattice. The Fourier transform of the exchange integral is given by
, OK2 - _-’
5
j=l
COSS;
+I5i
taking the lattice constant
JJCOS(kJj))]
_Lnj; \ \T
: ,’
.6 / A!?&
/
‘cc
/’
&\
’
I,
,,,..(,
13 ,*’
J(4>= 2[Jl
-x:
xg
‘”
_I!4 9
,
Fig. 1. jz-j3-plane with different regions for the wavevectordependent susceptibility x(q) in the ordered phases (I,II-ferroIII,IV-helimagnetic). LP3 is the projection of the LP of charac ter 3.
to be one. 7
W. Selke / Critical behaviour near Lifshitz point9
display the data determing the critical exponent p of the order parameter, the magnetization. Taking into account only nearest neighbour interactions, J2 = 0, we get fl= 0.32 t 0.02; the critical temperature T, has been taken from the high temperature series expansion [ 171. Very close to the LP, see also ref. [lo], we made runs with 2000 Monte Carlo steps per spin discarding the first 250 steps to reach the equilibrium. Just as in the [sing case [ 1l] we observe a drastic decrease of the exponent of the order parameter for the planar model down to flLp = 0.20 + 0.02. The crossover effect near the LP is studied currently and the results will be published elsewhere [ 181.
Acknowledgements Fig. 2. Phase diagram for the spherical model. The transition temperature Tin units of 2.I,/kB for the LP of character 3 (LP$ is j”c = 1.066. P, F and H denote the para-, ferro- and helimagnetic phases.
I thank Profs. K. Binder and G. Meissner for their help. This work has been supported by the Deutsche Forschungsgemeinschaft SFB 130, Ferroelektrika.
3. Monte Carlo calculations References We did Monte Carlo calculations, see ref. [ 161, for the planar spin model (spin vectors with two components) on a simple cubic lattice with ferromagnetic interactions, Jr, between nearest neighbours and antiferromagnetic ones, JZ, between next nearest neighbours along one axis of the lattice (“R-S model”, see refs. [ 10 and 1 l]. The results for the analogous Ising case have been already given [ 1 I]. We used periodic boundary conditions for a 163 lattice. In fig. 3 we
AC/P -0.2 6_‘, --,
_.I
‘.-
-
.O a2-
0.1-
0.01
0.02
0.1
Q2 E
Fig. 3. Log-log-plot of the magnetization versus reduced temperature E = (Tc - 7)/T, for the planar R-S model with (ly, = 0 (0) and (2)J,/Jr = -0.26 (=). Tc is taken to be 2.204, resp. 1.81 in units of Jl/kg.
[l] R.M. Hornreich, [ 21 [3] [4] [S]
M. Luban and S. Shtrikman, Phys. Rev. Lett. 35 (1975) 1678. J.F. Nicoll, G.F. Tuthill, T.S. Chang and H.E. Stanley, Phys. Lett. A58 (1976) 1. W. Selke, Z. Physik B27 (1977) 81; Phys. Lett. A61 (1977) 443. A. Erzan and G. Stell, Phys. Rev. B16 (1977) 4146. J. Chen and T.C. Lubensky, Phys. Rev. Al4 (1976) 1202.
[6] C. Sigaud, F. Hardouin and M.F. Archard, Solid State Commun. 23 (1977) 35. [7] A. Michelson, Phys. Rev. Lett. 39 (1977) 464. [8] E. Abrahams and I.E. Dzyaloshinskii, Solid State Commun. 23 (1977) 883. [9] A. Michelson, Phys. Rev. B16 (1977) 585. [lo] S. Redner and H.E. Stanley, Phys. Rev. B16 (1977) 4901; J. Phys. Cl0 (1977) 4765. [ll] W. Selke, Z. Phys. B29 (1978) 133. [ 121 D. Mukamel and M. Luban, Phys. Rev., in print. [13] D. Mukamel, J. Phys. A10 (1977) 249. [ 141 R.M. Hornreich, M. Luban and S. Shtrikman, Physica A86 (1977) 465. [ 151 W. Selke, presented at Stat. Phys. 13, Haifa, Israel (August 1977). [ 161 K. Binder, in Phase transitions and critical phenomena, Vol. 5b, eds. C. Domb and M.S. Green (Academic Press, New York, 1976). [17] S. Singh and D. Jasnow, Phys. Rev. Bll (1975) 3445; R.G. Bowers and G.S. Joyce, Phys. Rev. Lett. 19 (1967) 630. [18] W. Selke, Solid State Commun. (1978) in print.