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Critical properties of the X-Y model near the lifshitz point
Solid State Communications, Vol. 27, pp. 1417—1419. © Pergamon Press Ltd. 1978. Printed in Great Britain.
0038—1098/78/0922—1417 $02.00/o
CRITICAL PROPERTIES OF THE X—Y MODEL NEAR THE LIFSHITZ POINT W. Selke* Universität des Saarlandes, Theoretische Physik, 66 Saarbruchen, West Germany (Received 23 June 1978 by B. Muhlschlegel) The Monte Carlo method is applied to a three-dimensional two-component spin model with nearest neighbour ferromagnetic interactions and next nearest neighbour antiferromagnetic interactions along one axis only (planar R—S model). The critical exponent of the order parameter is determined to be 0.20 ±0.02 very close to the Lifshitz point. The crossover effect is discussed. UFSHITZ POINTS (LP) [11 have attracted considerable theoretical efforts. Despite several theoretical suggestions for realizations of LPs in magnets [21, liquid crystals [3—51 and TTF—TCNQ [61 there are still only a few new experimental data [71showing just the phase diagram of a binary liquid crystal exhibiting a LP. Until now, precise measurements of the critical properties near a LP have not been published, Fortunately, there exists a simple model, the “R—S model” (see below), displaying a variety of LPs. Thus one can at least do easily computer experiments, i.e. Monte Carlo (MC) calculations [81 on systems belonging to the universality classes of different Lifshitz
even less accurate than the c-expansion for the interesting cases n = 1,2,3. The R—S model [13] is described by the Hamiltonian \ H = J(~~ $~Sj+ R ~ SiSj + S ~ sisi) (1)
points. Actually, there are only three types of LPs p05sibly realized in nature occuring at finite, non-zero tem~ peratures: the uniaxial [1] LPs with the order parameter of dimension (n) one, two or three. (Here we do not consider Ufshitz points of higher character [9, 101.) These three LPs are exhibited by the R—S model, and one may compute their universal properties like the critical exponents. We already did these computations for the Ising case (n = 1) [111 In this communication we shall extend the results to the planar (X—Y) model (n = 2) belonging to the same universality class as the liquid crystals [3—5,71and TFF—TCNQ [6] Certainly, the exponents can be calculated by other techniques, which show, however,serious deficiencies. The marginal dimension of a uniaxial LP is 4.5, so that the results of the c-expansion [1, 121 up to order e = 4.5 d for the realistic dimension d = 3 are not reliable in a quantitative sense. The high temperature series expansion (HTSE) requires a subtle analysis to avoid continuously varying exponents conflicting with universality [13] ; below the transition temperature the technique cannot be used at all. The 1/n-expansion [1] is expected to be * Present adress: Cornell University, Department of Chemistry, Baker Laboratory, Ithica, NY 14853,
ferromagnetic), JS = J2 (<0; antiferromagnetic) and n = 2[s~ = (sr, sfl] using the same notation as before [111 This model exhibits a transition line T~(J2/J1) between the paramagnetic and the ordered phase displaying a Lifshitz point at the point separating the ferromagnetic and helical region, see Fig. 1. According to the HTSE to order six [13, 141 the LP occurs at j2 ,j1 = 0.263 ±0.002; kB T/J1 = 1.804 ±0.002. At T = 0 the ferromagnetic ground state is the stable one for J2 /J1 ~ 0.25. The aim of the Monte Carlo calculations is to confirm the location of the LP and to study the crossover from the usual critical behaviour of a threedimensional ferromagnet with a two-component order parameter (J2 = 0) to the new critical behaviour near the LP. Therefore we did computations for J2/J1 = 0,—0.25 and 0.26 approaching the LP from the “ferromagnetic side” [11] The quantities recorded in the simulation were the energy, the specific heat, the root-mean-square magnetization (see also [151), and the structure factor. The calculations have been performed for lattices of 16 x 16 x 16 spins with periodic boundary conditions. We used 1000—1500 MC steps per spin forJ2 = 0 and 2000 steps per spin in the other cases. In order to reach
—
£,J
where the first two sums are over nearest neighbour pairs in the same and adjacent xy planes, respectively, and the third sum is over next-nearest neighbour spin pairs along the z-axis only. Sj denotes a n-component spin vector at lattice site i. In the following we shall take a simple cubic lattice and set R = 1 J J1 (>0;
,
.
.
—
U.S.A.
,
.
—
—
—
.
the equilibrium the first 250 steps per spin were always 1417
1418
CRITICAL PROPERTIES OF THE X—YMODEL
Vol. 27, No. 12
0,7~ Q26 0.3 0.7
5S
2.0
PARA ~~LP
1.8
FERRO 1.6 0
0.1
0.4
~
F I.
HELICAL Q25
[
0.25
~
~
~
e
Fig. 3. The root-mean-square 0.02 0.03 magnetization 0.1 0.2 ...JM~vs the
°~
Fig. 1. Transition temperatures ~ = 1~,/Jivs —J2/J1: the dashed line and the Lifshitz point (LP) are taken from the high temperature series expansion. The MC results are indicated by the error bars. The line separating the ferromagnetic (FERRO) and helical phases is a schematic one,
reduced temperature e = (T— T,~,)/T~ forj2/J1 —0.25 (‘v) and —0.26 (.).
=
HTSE [14, 17] the transition temperatures are 2.204, 1.825 and 1.807, respectively. The agreement is good; the differences can be explained partially by the finite size effect [11, 18] for the MC calculations. For determining the critical exponents (see below), we took as the “best estimates” T~/J1= 2.204, 1.83 and 1.81. In order to locate the Lifshitz point it is useful to consider the structure factor [11, 13]
1,0 ~ ‘S
x(q)
= ~r ((s~s~) + (s~s~’)) exp (iqz) where r = (x,y,z). By approaching the
(2) LP from the
0.5 ferromagnetic side the structure factor for a fixed value ofe=(T—T~)/7,should broaden [ll,13J.AttheLP 4 + constant). the shape of ~(q) for small values of q changes from a Lorentzian to aphase non-Lorentzian (~ 1 /(q If the ordered is a helical one, the maximum of ~(q)occurs at a non-zero value. Figure 2 shows the
—
0.26 0.25 0 1.0
•5 •
0.1 0
•.
•-
0.2
Fig. 2. The reduced structure factor ~(q)
=
q x(q)/x(0)
forf2/J1 = 0(.), —0.25 (1’) and —0.26(.) at = (T— T,~)/T~5 x 10~(T/J1 = 2.3 14, 1.92 and 1.9). discarded. The spins are classical two-component unit vectors described completely by generated one angle.by In changing the simulations new configurations were the angle by a random number within a certain range, so that about one half of the configurations was accepted for the averages [16] .
The transition temperatures T~(J2/J~)are identified with the position of the maxima of the specific heat as well as the turning points of the energy and the rootmean-square magnetization. From these data we estimated 7/J1 = 2.17 ±0.02 forJ2 = 0; 1.83 ±0.02 for J2 /.J~= —0.25 and 1.82 ±0.02 for 0.26 (see Fig. Here and in the following we set kB = 1. From the —
~,
broadening of x(q). The effect is of the same order by going from J 2/J1 = 0 to —0.25 and from —0.25 to —0.26. The drastic change near —0.26 agrees with the result of the HTSE [13] locating the LP at .121.11 = 0.263. The log—log plots for obtaining the critical e~po2 nent 13 of=the(162)_2 root-mean-square withM2 <(~~~,)2>) aremagnetization shown in Fig.(~‘M 3. The exponents are effective ones determined for 3 x 10-2 ~ e ~ 10’ ,which one has to distinguish from the asymptotic exponents [19]. For J 2/J1 = Owe get 13 = 0.32 ±0.02 in good agreement with the renormalization group calculation [20]. For J2 1.11 = —0.25 13 is lowered to 0.24 ±0.02. This is due to the closeness of the LP, as one sees by the even smaller value for J2 /J1 = —0.26, where f3 = 0.20 ±0.02. Since we cannot claim to be exactly at the LP the last value may still be slightly —
Vol. 27, No. 12
CRITICAL PROPERTIES OF THE X—Y MODEL
spoiled by the crossover effect, which in this case certainly would enlarge the “measured” exponent relative to the true LP exponent. The, general trend in the behaviour of 13 is the same as m the Ising case [11], where j3 decreases from 0.31 ±0.03 forJ2 = 0 to 0,21 ± 0.03 very close to the LP. In both cases (n = 1 and n
=
1419
2) the decrease of 13 is clearly large enough to be measurable in a real experiment. Acknowledgements I wish to thank Prof. K. Binder for providing me the computer program [15] and Prof. —
G. Meissner for the support. The work was supported in part by the SFB 130, Ferroelektrika.
REFERENCES 1.
HORNREICH R.M., LUBAN M.& SHTRIKMAN S.,Phys. Rev. Lett. 35, 1678 (1975).
2.
MICHELSON A.,Phys.Rev. B16, 585 (1977).
3. 4.
CHEN i.& LUBENSKYT.C.,Phys. Rev. A14, 1202 (1976). MICHELSON A.,Phys. Rev. Lett. 39,464(1977).
5.
BLINC R. & DE SA BARRETTO F.C. (to be published).
6.
ABRAHAMS E. & DZYALOSHINSKII I.E., Solid State Commun. 23, 883 (1977).
7.
SIGAUD G., HARDOUIN F. & ACHARD M.F.,Solid State Commun. 23,35 (1977).
8. 9.
BINDER K., Phase Transitions and Critical Phenomena, Vol. Sb. Academic Press, New York (1976). NICOLLJ.F.,TUTHILLG.F., CHANGTS.&STANLEYH.E.,Thys.Lett. A58, 1(1976).
A61, 443 (1977);J. Magn. Magn. Mat. (to be published).
10. 11.
SELKE W., Z. Phys. B27, 81(1977); Phys. Lett. SELKEW.,Z. Phys. B29, 133 (1978).
12.
MUKAMELD.,J.Thys. AlO, 249 (1977).
13.
REDNER S. & STANLEY H.E.,Phys. Rev. B16,(1977);J. Phys.
14.
REDNER S. & STANLEY H.E. (unpublished data).
15.
BINDERK.& LANDAU D.P.,Phys. Rev. B13, 1140(1976).
16. 17. 18.
BINDERK.&RAUCHH.,Z.Phys. 219,201 (1969). SINGH S. & JASNOW D., Phys. Rev, Bit, 3445(1975); PFEUTY P., JASNOW D. & FISHER M.E., Phys. Rev. BlO, 2088 (1974). LANDAU D.P.,Phys. Rev. B14, 255 (1976).
19. 20.
See, e.g. SINGH S. & JASNOW D.,Phys. Rev. B12, 493 (1975). LE GUILLOU J.C. & ZINN-JUSTIN J.,Phys. Rev. Lett. 39,95 (1977).
ClO, 4765 (1977).
NOTE ADDED IN PROOF GREST G.S. & SAK J.,Phys. Rev. B17, 3607 (1978), pointed out the possible occurrence of biaxial LP’s at finite temperatures for n = 1 and n = 2. Monte Carlo calculations, SELKE W. (unpublished), confirm this suggestion for n = 1.
0038—1098/78/0922—1417 $02.00/o
CRITICAL PROPERTIES OF THE X—Y MODEL NEAR THE LIFSHITZ POINT W. Selke* Universität des Saarlandes, Theoretische Physik, 66 Saarbruchen, West Germany (Received 23 June 1978 by B. Muhlschlegel) The Monte Carlo method is applied to a three-dimensional two-component spin model with nearest neighbour ferromagnetic interactions and next nearest neighbour antiferromagnetic interactions along one axis only (planar R—S model). The critical exponent of the order parameter is determined to be 0.20 ±0.02 very close to the Lifshitz point. The crossover effect is discussed. UFSHITZ POINTS (LP) [11 have attracted considerable theoretical efforts. Despite several theoretical suggestions for realizations of LPs in magnets [21, liquid crystals [3—51 and TTF—TCNQ [61 there are still only a few new experimental data [71showing just the phase diagram of a binary liquid crystal exhibiting a LP. Until now, precise measurements of the critical properties near a LP have not been published, Fortunately, there exists a simple model, the “R—S model” (see below), displaying a variety of LPs. Thus one can at least do easily computer experiments, i.e. Monte Carlo (MC) calculations [81 on systems belonging to the universality classes of different Lifshitz
even less accurate than the c-expansion for the interesting cases n = 1,2,3. The R—S model [13] is described by the Hamiltonian \ H = J(~~ $~Sj+ R ~ SiSj + S ~ sisi) (1)
points. Actually, there are only three types of LPs p05sibly realized in nature occuring at finite, non-zero tem~ peratures: the uniaxial [1] LPs with the order parameter of dimension (n) one, two or three. (Here we do not consider Ufshitz points of higher character [9, 101.) These three LPs are exhibited by the R—S model, and one may compute their universal properties like the critical exponents. We already did these computations for the Ising case (n = 1) [111 In this communication we shall extend the results to the planar (X—Y) model (n = 2) belonging to the same universality class as the liquid crystals [3—5,71and TFF—TCNQ [6] Certainly, the exponents can be calculated by other techniques, which show, however,serious deficiencies. The marginal dimension of a uniaxial LP is 4.5, so that the results of the c-expansion [1, 121 up to order e = 4.5 d for the realistic dimension d = 3 are not reliable in a quantitative sense. The high temperature series expansion (HTSE) requires a subtle analysis to avoid continuously varying exponents conflicting with universality [13] ; below the transition temperature the technique cannot be used at all. The 1/n-expansion [1] is expected to be * Present adress: Cornell University, Department of Chemistry, Baker Laboratory, Ithica, NY 14853,
ferromagnetic), JS = J2 (<0; antiferromagnetic) and n = 2[s~ = (sr, sfl] using the same notation as before [111 This model exhibits a transition line T~(J2/J1) between the paramagnetic and the ordered phase displaying a Lifshitz point at the point separating the ferromagnetic and helical region, see Fig. 1. According to the HTSE to order six [13, 141 the LP occurs at j2 ,j1 = 0.263 ±0.002; kB T/J1 = 1.804 ±0.002. At T = 0 the ferromagnetic ground state is the stable one for J2 /J1 ~ 0.25. The aim of the Monte Carlo calculations is to confirm the location of the LP and to study the crossover from the usual critical behaviour of a threedimensional ferromagnet with a two-component order parameter (J2 = 0) to the new critical behaviour near the LP. Therefore we did computations for J2/J1 = 0,—0.25 and 0.26 approaching the LP from the “ferromagnetic side” [11] The quantities recorded in the simulation were the energy, the specific heat, the root-mean-square magnetization (see also [151), and the structure factor. The calculations have been performed for lattices of 16 x 16 x 16 spins with periodic boundary conditions. We used 1000—1500 MC steps per spin forJ2 = 0 and 2000 steps per spin in the other cases. In order to reach
—
£,J
where the first two sums are over nearest neighbour pairs in the same and adjacent xy planes, respectively, and the third sum is over next-nearest neighbour spin pairs along the z-axis only. Sj denotes a n-component spin vector at lattice site i. In the following we shall take a simple cubic lattice and set R = 1 J J1 (>0;
,
.
.
—
U.S.A.
,
.
—
—
—
.
the equilibrium the first 250 steps per spin were always 1417
1418
CRITICAL PROPERTIES OF THE X—YMODEL
Vol. 27, No. 12
0,7~ Q26 0.3 0.7
5S
2.0
PARA ~~LP
1.8
FERRO 1.6 0
0.1
0.4
~
F I.
HELICAL Q25
[
0.25
~
~
~
e
Fig. 3. The root-mean-square 0.02 0.03 magnetization 0.1 0.2 ...JM~vs the
°~
Fig. 1. Transition temperatures ~ = 1~,/Jivs —J2/J1: the dashed line and the Lifshitz point (LP) are taken from the high temperature series expansion. The MC results are indicated by the error bars. The line separating the ferromagnetic (FERRO) and helical phases is a schematic one,
reduced temperature e = (T— T,~,)/T~ forj2/J1 —0.25 (‘v) and —0.26 (.).
=
HTSE [14, 17] the transition temperatures are 2.204, 1.825 and 1.807, respectively. The agreement is good; the differences can be explained partially by the finite size effect [11, 18] for the MC calculations. For determining the critical exponents (see below), we took as the “best estimates” T~/J1= 2.204, 1.83 and 1.81. In order to locate the Lifshitz point it is useful to consider the structure factor [11, 13]
1,0 ~ ‘S
x(q)
= ~r ((s~s~) + (s~s~’)) exp (iqz) where r = (x,y,z). By approaching the
(2) LP from the
0.5 ferromagnetic side the structure factor for a fixed value ofe=(T—T~)/7,should broaden [ll,13J.AttheLP 4 + constant). the shape of ~(q) for small values of q changes from a Lorentzian to aphase non-Lorentzian (~ 1 /(q If the ordered is a helical one, the maximum of ~(q)occurs at a non-zero value. Figure 2 shows the
—
0.26 0.25 0 1.0
•5 •
0.1 0
•.
•-
0.2
Fig. 2. The reduced structure factor ~(q)
=
q x(q)/x(0)
forf2/J1 = 0(.), —0.25 (1’) and —0.26(.) at = (T— T,~)/T~5 x 10~(T/J1 = 2.3 14, 1.92 and 1.9). discarded. The spins are classical two-component unit vectors described completely by generated one angle.by In changing the simulations new configurations were the angle by a random number within a certain range, so that about one half of the configurations was accepted for the averages [16] .
The transition temperatures T~(J2/J~)are identified with the position of the maxima of the specific heat as well as the turning points of the energy and the rootmean-square magnetization. From these data we estimated 7/J1 = 2.17 ±0.02 forJ2 = 0; 1.83 ±0.02 for J2 /.J~= —0.25 and 1.82 ±0.02 for 0.26 (see Fig. Here and in the following we set kB = 1. From the —
~,
broadening of x(q). The effect is of the same order by going from J 2/J1 = 0 to —0.25 and from —0.25 to —0.26. The drastic change near —0.26 agrees with the result of the HTSE [13] locating the LP at .121.11 = 0.263. The log—log plots for obtaining the critical e~po2 nent 13 of=the(162)_2 root-mean-square withM2 <(~~~,)2>) aremagnetization shown in Fig.(~‘M 3. The exponents are effective ones determined for 3 x 10-2 ~ e ~ 10’ ,which one has to distinguish from the asymptotic exponents [19]. For J 2/J1 = Owe get 13 = 0.32 ±0.02 in good agreement with the renormalization group calculation [20]. For J2 1.11 = —0.25 13 is lowered to 0.24 ±0.02. This is due to the closeness of the LP, as one sees by the even smaller value for J2 /J1 = —0.26, where f3 = 0.20 ±0.02. Since we cannot claim to be exactly at the LP the last value may still be slightly —
Vol. 27, No. 12
CRITICAL PROPERTIES OF THE X—Y MODEL
spoiled by the crossover effect, which in this case certainly would enlarge the “measured” exponent relative to the true LP exponent. The, general trend in the behaviour of 13 is the same as m the Ising case [11], where j3 decreases from 0.31 ±0.03 forJ2 = 0 to 0,21 ± 0.03 very close to the LP. In both cases (n = 1 and n
=
1419
2) the decrease of 13 is clearly large enough to be measurable in a real experiment. Acknowledgements I wish to thank Prof. K. Binder for providing me the computer program [15] and Prof. —
G. Meissner for the support. The work was supported in part by the SFB 130, Ferroelektrika.
REFERENCES 1.
HORNREICH R.M., LUBAN M.& SHTRIKMAN S.,Phys. Rev. Lett. 35, 1678 (1975).
2.
MICHELSON A.,Phys.Rev. B16, 585 (1977).
3. 4.
CHEN i.& LUBENSKYT.C.,Phys. Rev. A14, 1202 (1976). MICHELSON A.,Phys. Rev. Lett. 39,464(1977).
5.
BLINC R. & DE SA BARRETTO F.C. (to be published).
6.
ABRAHAMS E. & DZYALOSHINSKII I.E., Solid State Commun. 23, 883 (1977).
7.
SIGAUD G., HARDOUIN F. & ACHARD M.F.,Solid State Commun. 23,35 (1977).
8. 9.
BINDER K., Phase Transitions and Critical Phenomena, Vol. Sb. Academic Press, New York (1976). NICOLLJ.F.,TUTHILLG.F., CHANGTS.&STANLEYH.E.,Thys.Lett. A58, 1(1976).
A61, 443 (1977);J. Magn. Magn. Mat. (to be published).
10. 11.
SELKE W., Z. Phys. B27, 81(1977); Phys. Lett. SELKEW.,Z. Phys. B29, 133 (1978).
12.
MUKAMELD.,J.Thys. AlO, 249 (1977).
13.
REDNER S. & STANLEY H.E.,Phys. Rev. B16,(1977);J. Phys.
14.
REDNER S. & STANLEY H.E. (unpublished data).
15.
BINDERK.& LANDAU D.P.,Phys. Rev. B13, 1140(1976).
16. 17. 18.
BINDERK.&RAUCHH.,Z.Phys. 219,201 (1969). SINGH S. & JASNOW D., Phys. Rev, Bit, 3445(1975); PFEUTY P., JASNOW D. & FISHER M.E., Phys. Rev. BlO, 2088 (1974). LANDAU D.P.,Phys. Rev. B14, 255 (1976).
19. 20.
See, e.g. SINGH S. & JASNOW D.,Phys. Rev. B12, 493 (1975). LE GUILLOU J.C. & ZINN-JUSTIN J.,Phys. Rev. Lett. 39,95 (1977).
ClO, 4765 (1977).
NOTE ADDED IN PROOF GREST G.S. & SAK J.,Phys. Rev. B17, 3607 (1978), pointed out the possible occurrence of biaxial LP’s at finite temperatures for n = 1 and n = 2. Monte Carlo calculations, SELKE W. (unpublished), confirm this suggestion for n = 1.