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Critical properties of XY model on two-layer triangular lattice
Physica A 260 (1998) 131–149
Critical properties of XY model on two-layer triangular lattice Tsuyoshi Horiguchi ∗ , Yi Wang, Yasushi Honda Department of Computer and Mathematical Sciences, GSIS, Tohoku University 04, Sendai 980-8579, Japan Received 26 March 1998
Abstract We investigate phase transitions of the XY model on a two-layer triangular lattice when interactions are competing, by using Monte Carlo simulations. For the system with a ferromagnetic interaction on one of two layers and an antiferromagnetic interaction on the other layer, we nd two di erent types of critical phenomena with two phase-transition temperatures; (1) each of phase transitions due to chirality and (2) the Kosterlitz-Thouless-type phase transition and a chirality transition. We clarify the nature of these phase transitions by using nite-size scaling c 1998 Elsevier Science B.V. All rights reserved. analyses. PACS: 75.10.Hk; 05.50+q; 75.30.K; 75.40.c Keywords: XY model; KT transition; Chirality; Two-layer triangular lattice
1. Introduction A uniaxial antiferromagnet with an external magnetic eld H has been paid much attention for many years as for studies on a free surface of a Heisenberg antiferromagnet such as CsCl structure and a classical two-sublattice antiferromagnet such as MnF2 , for examples. It has been shown that there are several phases such as the antiferromagnetic phase with surface spins parallel to H , the antiferromagnetic phase with surface spins antiparallel to H , a surface spin op phase and a bulk spin op phase and so on. Thin lms composed of ferromagnetic planes, which are antiferromagnetically coupled to each other, have been investigated by using the same models. These models are treated as one-dimensional models after taking a mean- eld approximation for spins in the planes. The models are expressed in terms of one-dimensional XY models whose ground-state properties have been investigated [1 – 6]. Owing to the development of ∗
Corresponding author. Fax: +81 22 217 5851; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 2 9 1 - X
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epitaxy technology, it is now possible to make very thin magnetic lms. However, the theoretical investigation of the magnetic thin lms has been done mainly for the groundstate properties by means of phenomenological treatments or mean- eld approximations [7 – 12]. In this situation, we are interested in investigating of critical phenomena for an XY model on a two-layer lattice as one of the spin models related to the very thin magnetic lms. In a two-dimensional XY model, there is no long-range order due to the Mermin– Wagner theorem [13]. However, there is a phase transition, the so-called Kosterlitz– Thouless (KT) transition [14–16]. On the other hand, it was suggested by Miyashita and Shiba [17] that there are two di erent phase-transition temperatures, namely one for the chirality transition and the other for the KT transition in an antiferromagnetic XY model. Although there are still arguments going on whether both the transition temperatures are the same or not, we do not discuss this problem in the present paper; for examples, see Refs. [18–21]. New critical phenomena were discussed by Lee et al. and the domain walls are considered as elementary excitations [22]. Discussions for domain walls have been given recently by Denniston et al. [23]. Zhang et al. investigated and found that there are a spiral phase and other phases in some generalized XY models on a triangular lattice by using mean- eld approximations [24,25]. Thus, XY models are expected to show interesting critical phenomena on some lattices with structures di erent from those investigated so far. In these situations, we investigate the nature of phase transitions for the XY -model on a two-layer triangular lattice in the present paper. We consider the XY -model with a ferromagnetic intralayer interaction on one of two layers and an antiferromagnetic interaction on the other layer. We nd new critical phenomena due to the chirality; the chirality on a layer with ferromagnetic intralayer interaction starts to decrease at some temperature when the temperature decreases from a high-temperature side while the chirality on the layer with antiferromagnetic intralayer interaction does not show such change there. The speci c heat diverges with a positive value of critical index at this temperature, although we have the usual critical phenomena due to the chirality between the paramagnetic phase and the chirality phase. We describe our model in Section 2 and give phase diagrams of the ground state. In Section 3, we give the results obtained by Monte Carlo simulations and discuss the nature of phase transitions in terms of nite-size scaling analyses. The concluding remarks are given in Section 4.
2. XY model on two-layer triangular lattice We consider two triangular lattices 1 and 2 and the XY model on 1 + 2 whose Hamiltonian is given as follows: H = −I
X (i; j)
si · sj − J
X (i; j)
ti · tj − K
X i
si · ti ;
(1)
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where si and ti are XY spins on the triangular lattice 1 or 2 , respectively. We assume an exchange interaction between spins at nearest-neighbor lattice sites on 1 and that between spins at nearest-neighbor lattice sites on 2 and also an exchange interaction between spins at a site i on 1 and at its corresponding site i on 2 ; these interaction constants are expressed by I; J or K, respectively. The sum with (i; j) indicates a summation over the nearest-neighbor pairs of sites on 1 or 2 and the sum with i indicates a summation over the pairs of sites i on 1 and 2 . When we write si and ti as follows: si = (cos i ; sin i );
ti = (cos i ; sin i ) ;
(2)
Hamiltonian (1) is rewritten as H = −I
X (i; j)
cos(i − j ) − J
X
cos( i − j ) − K
X
(i; j)
cos(i − i ) :
(3)
i
We note that there is a relation Q(I; J; K) = Q(I; J; −K) for a thermodynamic quantity Q(I; J; K) and, hence, we assume K¿0 without loss of generality. The ground-state con guration is obtained by minimizing the energy, e, on a pair of triangles on 1 and 2 as shown in Fig. 1, e = −I [cos(1 − 2 ) + cos(2 − 3 ) + cos(3 − 1 )] −J [cos( 1 − 2 ) + cos( 2 − 3 ) + cos( 3 − 1 )] ˜ −K[cos( 1 − 1 ) + cos(2 − 2 ) + cos(3 − 3 )] ;
(4)
Fig. 1. A part of two-layer triangular lattice, 1 and 2 . The XY spins are denoted si on 1 and ti on 2 . The intralayer interaction constants are I on 1 and J on 2 and the interlayer infraction constant is K. For the energy (4), we choose K˜ = K=3.
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where 1 ; 2 ; 3 ; 1 ; 2 and 3 are assigned as shown in Fig. 1 and K˜ = K=3. By introducing new variables, 12 = 2 − 1 ;
31 = 1 − 3 ;
12 = 2 − 1 ;
31 = 1 − 3 ;
(5)
we have the following expression: e = −I [cos 12 + cos(12 + 31 ) + cos 31 ] −J [cos 12 + cos( 12 + 31 ) + cos 31 ] ˜ −K[cos( 1 − 1 ) + cos(1 − 1 + 12 − 12 ) + cos(1 − 1 − 31 + 31 )] : (6) By putting the derivative of e in terms of 1 or 1 to zero, we have sin(1 − 1 ) + sin(1 − 1 + 12 − 12 ) + sin(1 − 1 − 13 + 13 ) = 0 :
(7)
By putting the derivative of e in terms of 12 ; 12 ; 31 or 31 , we have the following equations, respectively, I [sin 12 + sin(12 + 31 )] + K˜ sin(1 − 1 + 12 − 12 ) = 0 ;
(8)
J [sin 12 + sin( 12 + 31 )] − K˜ sin(1 − 1 + 12 − 12 ) = 0 ;
(9)
I [sin 31 + sin(12 + 31 )] − K˜ sin(1 − 1 − 31 + 31 ) = 0
(10)
J [sin 31 + sin( 12 + 31 )] + K˜ sin(1 − 1 − 31 + 31 ) = 0 :
(11)
and
Now, we may express 1 by means of 1 ; 12 and 31 , and 12 and 31 by means of 1 ; 1 ; 12 and 31 as follows: 3I (sin 12 − sin 31 ) ; (12) 1 = 1 − arcsin K 3I {sin 12 + sin(12 + 31 )} (13) 12 = 1 − 1 + 12 − arcsin K and
31 = −1 + 1 + 31 + arcsin
3I {sin 31 + sin(12 + 31 )} : K
(14)
We set 1 = 0, without loss of generality. We vary the values of 12 and 31 from 0 to 2 and calculate the value of energy e. Then we nd spin con gurations which minimize energy e and construct the ground-state phase diagrams. In Figs. 2–4, we show the ground-state phase diagrams in the I=K − J=K plane, in the J=I − K=I plane with I ¿0 and in the J=|I | − K=|I | plane with I ¡0, respectively. In region F1, we have a ferromagnetic con guration of spins which are aligned in
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Fig. 2. Ground-state phase diagram in I=K − J=K plane. See text for details of each phase.
Fig. 3. Ground-state phase diagram in J=I − K=I plane for I ¿0. See text for details of each phase.
the same direction for all the lattice sites on both 1 and 2 . In region F2, we have a ferromagnetic con guration with i = , for example, for any site i on 1 and a ferromagnetic con guration with i = + for any site i on 2 . In region C1, we have a 120◦ structure with i = i for any site i on both 1 and 2 . In region C2, we have a 120◦ structure in 1 and also the 120◦ structure with i = i + in 2 . In region V1, the spin con guration on 2 is almost 120◦ structure and that in 1 is almost ferromagnetic for small I=K and large |J |=K in Fig. 2, for example; both structures are incomplete and the angle of each spin changes according to values of interaction parameters. In region V2, the spin con guration on 2 is almost ferromagnetic and that on 1 almost 120◦ structure for small J=K and large |I |=K in Fig. 2, for example; there is a similar situation to that in region V1.
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Fig. 4. Ground-state phase diagram in J=|I | − K=|I | plane for I ¡0. See text for details of each phase.
3. Results by Monte Carlo simulations In the present section, we present results obtained by standard Monte Carlo (MC) simulations and clarify the nature of phase transitions by investigating the results by means of the nite-size scaling method. We assume the periodic boundary conditions for each lattice, 1 and 2 , and we denote the number of total lattice sites on each triangular lattice as N , where N = L × L; |1 | = |2 | = N . Our MC simulations are performed for L = 12, 24, 36 and 48. Random spin con gurations have been taken as initial spin con gurations in our MC simulations. An MC average, hQi, for a quantity Q is calculated by hQi =
n X 1 Q(t) ; n − n0
(15)
t=n0 +1
where we choose n0 = 2:5 × 105 MC steps, n = 5 × 105 MC steps and also n0 = 5 × 105 MC steps, n = 106 MC steps, depending upon the system size and the value of temperature. We set the Boltzmann constant k to 1 in the present paper. We calculated the internal energy, the speci c heat and the chirality. The internal energy is given as E=
1 hH i ; 2N
(16)
where H is expressed as follows: H =−
X 1 XX [I cos(i − i+i ) + J cos( i − i+i )] − K cos(i − i ) : 2 i i i
(17)
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The notation i implies one of the six nearest-neighbor sites of a site i on each triangular lattice, 1 or 2 . The sum with i indicates a summation over the six nearest-neighbor sites of a site i. The speci c heat is obtained by C=
1 2 2 { H − hH i } : 2NT 2
(18)
The chirality is de ned at a smallest up-triangle, namely an elementary upward triangle, on 1 as 2 s = √ si × sj + sj × sk + sk × si · eˆz ijk 3 3 2 = √ [sin(j − i ) + sin(k − j ) + sin(i − k )] 3 3
(19)
and at a smallest up-triangle on 2 as 2 t = √ ti × tj + tj × tk + tk × ti · eˆz ijk 3 3 2 = √ [sin( j − i ) + sin( k − j ) + sin( i − k )] : 3 3
(20)
Here eˆz is the unit vector in the z-direction in spin space. A set of sites {i; j; k} indicates an allocation of sites counterclockwise on each up-triangle of the lattice 1 or 2 . The averaged chirality for the whole system is de ned by = 12 (s + t ) ;
(21)
where s =
1 X s ijk ; N
(22)
1 X t ijk ; N
(23)
(ijk)
t =
(ijk)
where the sum with (ijk) is taken over all of the elementary up-triangles on lattice 1 or 2 . We show in Figs. 5–8 the results obtained for the system with I = −1:0; K = 0:5 and J = −1:0; 0:2; 0:3; 0:5 and 1.0 for L = 48. In Fig. 5, the internal energy E=|I | is given as functions of T=|I |. We see that there are steep parts of the internal energy as functions of temperature at some temperature regions. These imply the possibility of phase transitions. We show the speci c heat for J = − 1:0; 0:2 and 0.3 in Fig. 6 and those for J = 0:5 and 1.0 in Fig. 7. From these gures, we see that there is a sharp peak of the speci c heat for J = −1:0, two sharp peaks for J = 0:2 and a sharp peak with a sluggish decrease in the low-temperature side for J = 0:3. On the other hand, there are a sharp peak and a broad peak for J = 0:5 and 1.0. The existence of two sharp peaks in the speci c heat for J = 0:2 is quite a new phenomena for the frustrated
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Fig. 5. Internal energy E=|I | as functions of T=|I | for I = −1:0 and K = 0:5. Pluses show the internal energy for J = 0:2, up-triangles for J = 0:3, squares for J = 0:5, pentagons for J = 1:0 and closed down-triangles for J = −1:0.
Fig. 6. Speci c heat C as functions of T=|I | for I = −1:0 and K = 0:5. Pluses show the speci c heat for J = 0:2, up-triangles for J = 0:3 and closed down-triangles for J = −1:0.
XY models on two-dimensional lattices, as far as we know. We will clarify this below. In Fig. 8, we show the chiralities, s on 1 and t on 2 , for J = 0:2; 0:3 and 0.5 and the total chirality for J = −1:0. The behaviors of chiralities, s and t , are the same for J = −1:0 as expected and, hence, we show the total chirality for J = −1:0 in Fig. 8. We note that di erent behaviors for s and t as functions of temperature are obtained for J = 0:2; 0:3; 0:5 and 1.0. We also note that t = 0 for J = 0:5 in Fig. 8. We did not show the results of the chirality for J = 1:0 in Fig. 8, since the values of the chirality for J = 1:0 and those for J = 0:5 are almost the same in the gure. We
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Fig. 7. Speci c heat C as functions of T=|I | for I = −1:0 and K = 0:5. Squares show the speci c heat for J = 0:5 and pentagons for J = 1:0.
Fig. 8. Chirality as functions of T=|I | for I = −1:0 and K = 0:5. Closed down-triangles show the total chirality for J = −1:0, pluses show that of chirality s and crosses the chirality t for J = 0:2, open up-triangles the chirality s and closed up-triangles the chirality t for J = 0:3, open pentagons the chirality s and closed pentagons the chirality t for J = 0:5.
will make nite-size scaling analyses for these data in order to clarify the nature of phase transitions. For J = −1:0, we show the speci c heat in Fig. 9 for L = 12; 24; 36 and 48. The peak height is getting higher when the linear system size, L, is getting larger. We found that the peak height is a linear function of ln L when we plotted the height values Ctop as a function of ln L, although we did not show it in the present paper [26]; Ctop denotes a height value of the speci c heat obtained in the present MC simulations. The nite-size scaling analysis is shown in Fig. 10 by using data for L = 24; 36, and
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Fig. 9. Size dependence of speci c heat for I = −1:0; J = −1:0 and K = 0:5. Pluses show the speci c heat for L = 12, up-triangles for L = 24, squares for L = 36 and closed down-triangles for L = 48.
Fig. 10. Finite-size scaling analysis for the speci c heat given in Fig. 9. The abscissa is (T=|I | − 0:7535)L1= where = 1:019 and the ordinate Cf(L) where f(L) = 1=(ln L)1=y with y = 0:765.
48; we have f(L) = 1=(ln L)1=y with y = 0:765, and hence we have = 0. We estimate Tc =|I | = 0:753(1). In this way, the speci c heat diverges logarithmically, and we obtain the critical index equal to 1.00(2). We show the chirality in Fig. 11, where the abscissa is − ln[(Tc − T )=|I |] and the ordinate − ln . We nd the critical index equal to 0.125(1). The critical temperature is also estimated as Tc =|I | = 0:753(1). In this way, there is a phase transition between the paramagnetic phase for T ¿Tc and the chirality phase for T ¡Tc ; the phase transition at Tc is due to the chirality. For J = 0:2, we show the speci c heat in Fig. 12 for L = 12; 24; 36 and 48. There are two peaks in Fig. 12; one is around T=|I | = 0:17 and the other around 0.435. These should give critical temperatures and, hence, we denote the critical temperature in the
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Fig. 11. Chirality for I = −1:0; J = −1:0 and K = 0:5. The abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:7533 and the ordinate y is − ln . Triangles show the result for chirality for L = 12, squares for L = 24, pentagons for L = 36 and closed up-triangles for L = 48. We have y = 0:125x − 0:02 as the straight line.
Fig. 12. Size dependence of speci c heat for I = −1:0; J = 0:2 and K = 0:5. Pluses show the result for L = 12, up-triangles for L = 24, squares for L = 36 and closed down-triangles for L = 48.
low temperature side as Tcl and that in the high temperature side as Tcu . The peak heights of the speci c heat near Tcu and Tcl are higher when the linear system size, L, is larger. We found that the behavior of height values Ctop of the speci c heat near Tcu is di erent from that near Tcu ; we have that ln Ctop ∼ ln L near Tcl but Ctop ∼ ln L near Tcu , although we did not show them in the present paper [26]. A nite-size scaling analysis for the speci c heat near Tcu is shown in Fig. 13 and that near Tcl in the inset of Fig. 13, where f(L) = 1=(ln L)1=y with y = 0:855 and g(L) = L− = . We estimate Tcl =|I | ∼ 0:173(3), Tcu =|I | ∼ 0:431(3) and ∼ 0:92(8) and ∼ 0:29(9) at Tcl and
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Fig. 13. Finite-size scaling analysis for the speci c heat given in Fig. 9. The abscissa is (T=|I | − 0:4316)L1= where = 1:098 and the ordinate is Cf(L) where f(L) = 1=(ln L)1=y with y = 0:855. In the inset, the abscissa is (T=|I | − 0:177)L1= with = 0:92 and the ordinate is Cg(L) where g(L) = L− = with = 0:29.
∼ 1:09(9) and = 0 at Tcu . The hyperscaling d = 2 − is satis ed within numerical errors. In Fig. 8, we give the behaviors of chiralities, s and t on both lattices, 1 and 2 ; we see the behavior of s and that of t are di erent. Both s and t are becoming non-zero at Tcu when the temperature decreases from the high-temperature side to Tcu and increase when the temperature decreases further. Now, we see that s continues to increase to 1 when temperature tends to 0, but t starts to decrease when temperature decreases from Tcl . This seems to be due to the competition between the interaction energy and the entropy. In Fig. 14, we show − ln as a function of − ln[(Tcu − T )=|I |] for (T . Tcu ) and nd that 1 ∼ 1:073 (Tcu − T ) (T . Tcu ) ; (24) |I | with = 0:125(1). We show − ln(0:2515−t ) as a function of − log[(Tcl −T )=|I |] (T . Tcl ) in the inset of Fig. 14 and then we nd that 1 (c) (Tcl − T ) t ∼ t − 0:249 (T . Tcl ) (25) |I | with t(c) = 0:2515 and = 0:253(3). This means that the chirality in 2 decreases from t(c) with index = 0:253. The divergent behavior of the speci c heat at Tcl is now clari ed to be due to the decreasing of the chirality in 2 . For J = 0:5, we have two peaks in the speci c heat in Fig. 7. We show their size dependence for L = 24; 36 and 48 in Fig. 15. The peaks at the high-temperature side do not show size-dependence and, hence, it is reasonable for us to think that the phase transition is a KT type. We will not discuss the details for the KT transition in this paper; we just estimate the KT transition temperature, TKT , by the peak-temperature and
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Fig. 14. Total chirality and chirality t for I = −1:0; J = 0:2 and K = 0:5. Triangles show the chirality for L = 12, squares for L = 24, pentagons for L = 36 and closed up-triangles for L = 48. The abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:4335 and the ordinate y is − ln . We have y = 0:125x − 0:071 as a straight line. In the inset, the abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:1734 and the ordinate y is − ln(0:2515 − t ). We have y = 0:253x + 1:395 as a straight line.
Fig. 15. Size dependence of speci c heat for I = −1:0; J = 0:5 and K = 0:5. Up-triangles show the speci c heat for L = 24, squares for L = 36 and closed down-triangles for L = 48.
the in ection-temperature of the speci c heat from the data by our MC simulations. The peaks at the low-temperature side do show size dependence and we nd that Ctop ∼ ln L near Tc [26]. A nite-size scaling analysis is shown in Fig. 16 near Tc and we obtained Tc =|I | = 0:420(1); = 0 and = 1:03(3). In Fig. 17, we show − ln s as a function of − ln[(Tc − T )=|I |]; we notice t = 0 in this case. We use Tc =|I | = 0:4205 and nd = 0:125; hence, the phase transition at Tc =|I | is due to the chirality. We
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Fig. 16. Finite-size scaling analysis for the speci c heat given in Fig. 15. The abscissa is (T=|I | − 0:4205)L1= with = 1:03 and the ordinate is Cf(L) where f(L) = 1=(ln L)1=y with y = 1:01.
Fig. 17. Chirality s for I = −1:0; J = 0:5 and K = 0:5. The abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:4205 and the ordinate y is − ln s . Triangles show the chirality for L = 12, squares for L = 24, pentagons for L = 36 and closed up-triangles for L = 48. We have y = 0:125x − 0:065 as a straight line.
made the same analysis for J = 1:0 and obtained the same conclusions for the nature of phase transitions for J = 0:5 and J = 1:0. Now in order to con rm the nature of phase transitions for the system with I = −1:0; J = 0:2 and K = 0:5, we made MC simulations for the system with I = −1:0; J = 0:2 and K = 1:0. We show the speci c heat for L = 48 in Fig. 18. The situation is similar to the case of the system with I = −1:0; J = 0:2 and K = 0:5; we have two sharp peaks in the speci c heat. The nite-size scaling analysis for the speci c heat near Tcu is shown in Fig. 19 and that near Tcl in the inset of Fig. 19. We estimate Tcl =|I | ∼ 0:176(1), Tcu =|I | ∼ 0:431(1) and ∼ 0:26(9), ∼ 0:885(20) at Tcl and = 0; = 0:92(9) at Tcu .
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Fig. 18. Speci c heat C as functions of T=|I | for I = −1:0; J = 0:2 and K = 1:0.
Fig. 19. Finite-size scaling analysis for the speci c heat given in Fig.18. The abscissa is (T=|I | − 0:4815)L1= where = 0:92 and the ordinate is Cf(L) where f(L) = 1=(ln L)1=y with y = 0:780. In the inset, the abscissa is (T=|I | − 0:1768)L1= with = 0:885 and the ordinate is Cg(L) where g(L) = L− = with = 0:26.
We show the behaviors of chirality s and t in Fig. 20. We see the behavior of s is di erent from that of t : their behaviors are similar to those for the system with I = −1:0; J = 0:2 and K = 0:5. We show nite-size scaling analysis for in Fig. 21 and that for t in set of Fig. 20. We nd that ∼
1 (Tcu − T ) |I |
(T . Tcu ) :
(26)
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Fig. 20. Chirality s and chirality t as functions of T=|I | for I = −1:0; J = 0:2 and K = 1:0. Open up-triangles show the chirality s and closed up-triangles the chirality t .
Fig. 21. Finite-size scaling analyses for the chirality in Fig. 20. The abscissa is (T=|I | − 0:4815)L1= where = 0:92 and the ordinate is L = with = 0:125. In the inset, the abscissa is (T=|I | − 0:1768)L1= with = 0:885 and the ordinate is −(t − 0:2734)L = .
with = 0:125(1), and 1 (c) (Tcl − T ) t − t ∼ − |I |
(T . Tcl )
(27)
with t(c) = 0:2734 and = 0:25(2). These results are consistent with that for the system with I = −1:0; J = 0:2 and K = 0:5. Finally, by using the analyses stated above for the system with I = −1:0; K = 0:5 and several values of J , we obtain a nite temperature phase diagram in Fig. 22 in the T=|I | − J plane for −1:56J 61:5. We nd the paramagnetic phase and the
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Fig. 22. Finite temperature phase diagram for I = −1:0 and K = 0:5. The abscissa is J and the ordinate is T=|I |. P denotes the paramagnetic phase, C the chirality phase and KT the Kosterlitz–Thouless phase.
chirality phase for J . 0:2; there is only one phase-transition temperature between these phases. For 0:2 . J . 0:3, we have the paramagnetic phase and two kinds of chirality phases. The chirality in the 2 shows small values according to the interaction energy for T ¡Tcl , but takes rather larger values for Tcl . T than those expected by the interaction energy; these larger values of t are due to e ect of the chirality, s , on 1 . We surmise that this phenomena is a kind of order-in-disorder; the values of chirality, t , are larger for higher temperatures than those for lower temperature in some temperature range. The speci c heat diverges at Tcl with ¿0. The critical phenomena at Tcl is due to the sharp change of the chirality in the 2 plane. We have two phasetransition temperatures, Tcu and Tcl for this case. At Tcu , there is a phase transition between the paramagnetic phase and the chirality phase; this is the usual chirality phase transition. For J & 0:3, there are two phase-transition temperatures, TKT and Tc . We have the paramagnetic phase for T ¿TKT and the KT phase for Tc ¡T ¡TKT and the chirality phase for T ¡Tc .
4. Concluding remarks In this paper, we have investigated critical phenomena for an XY model on a twolayer triangular lattice by using Monte Carlo simulations. We assume that the intralayer interaction in one of two triangular lattices is ferromagnetic and that in the other triangular lattice antiferromagnetic. The interlayer interaction is assumed to be ferromagnetic without loss of generality. We have made nite-size scaling analyses and clari ed the nature of the phase transitions that occurred in the present system. We have obtained a phase diagram in a plane spanned by one of the interaction parameters and temperature.
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We have found new critical phenomena for some range of interaction parameters; there is chirality even in the layer with the ferromagnetic intralayer interaction and its values increase sharply at a temperature Tcl when the temperature increases. We have the usual chirality transition at a temperature Tcu higher than Tcl ; there is the phase transition between the paramagnetic phase and the chirality phase at Tcu . Namely, we have two phase-transition temperatures, Tcu and Tcl . At the critical temperature Tcl we have found that the speci c heat diverges with the critical exponent = 0:28(9). We de ne a critical index, , which expresses the sharpness of increase of the chirality, and have estimated = 0:25(3). At the critical temperature Tcu , we have the chirality transition with = 0:125 and = 0. For other ranges of parameters, we have one phase-transition temperature, namely, the chirality transition in one case or two phasetransition temperatures, namely the Kosterlitz–Thouless (KT) transition at a higher temperature and the chirality transition at a lower temperature in the other case. Now, we make some comments. Although we have not investigated the KT phase transition in detail, we think the KT transition temperature is higher than the chirality transition temperature, if they both exist. The phase transition for an Ising model on a two-layer triangular lattice has been investigated recently in the present authors’ group [27,28]. For the system with a ferromagnetic intralayer interaction on one of the two layers and an antiferromagnetic interlayer interaction on the other layer, we have found that there are two phase-transition temperatures, Tcl and Tcu (Tcu ¿Tcl ), when the ground state has a six-fold degeneracy; the order parameter in the layer with the antiferromagnetic interaction has increased sharply at a temperature Tcl when temperature increases, while we have the usual phase transition between the ferromagnetic phase and the paramagnetic phase at a higher temperature Tcu . We expect that a similar phase transition will occur for other systems such as an XY model on a two-layer square lattice, on a three-layer triangular lattice and so on when there are frustrated layers and non-frustrated layers. These problems are left as future problems. Acknowledgements This work was partially supported by a Grand-In-Aid for Scienti c Research from the Ministry of Education, Science and Culture, Japan and also by the Computer Center of Tohoku University. References [1] [2] [3] [4] [5] [6] [7]
F.B. Anderson, H.B. Callen, Phys. Rev. 136 (1964) A1068. D.L. Mills, W.M. Saslow, Phys. Rev. 171 (1968) 488. D.L. Mills, Phys. Rev. Lett. 31 (1968) 18. F. Ke er, H. Chow, Phys. Rev. Lett. 31 (1973) 1061. F.C. Nortemann, R.L. Stemps, A.S. Carrico, R.E. Camley, Phys. Rev. B 46 (1992) 10 847. R.W. Wang, D.L. Mills, E.E. Fullerton, J.E. Mattson, S.D. Bader, Phys. Rev. Lett. 72 (1994) 920. R.W. Wang, D.L. Mills, Phys. Rev. B 50 (1994) 3931.
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[8] M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petro , P. Eitenne, G. Creuzet, A. Friederich, J. Chazelas, Phys. Rev. Lett. 61 (1988) 2472. [9] L. Trallori, P. Politi, A. Rettori, M.G. Pini, J. Villain, Phys. Rev. Lett. 72 (1994) 1925. [10] L. Trallori, P. Politi, A. Rettori, M.G. Pini, J. Villain, J. Phys. Condens. Matter 7 (1995) L451. [11] L. Trallori, P. Politi, A. Rettori, M.G. Pini, J. Phys. Condens. Matter 7 (1995) 7561. [12] M. Momma, H. Horiguchi (1998) Physica A 259 (1998) 105. [13] N.D. Mermin, H. Wagner, Phys. Rev. Lett. 22 (1966) 1133. [14] J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181. [15] V.L. Berezinskii, Sov. Phys. JETP 32 (1971) 493. [16] J.V. Jose, L.P. Kadano , S. Kirkpatrick, D.R. Nelson, Phys. Rev. B. 16 (1977) 1217. [17] S. Miyashita, H. Shiba, J. Phys. Soc. Japan 53 (1984) 1145. [18] B. Berge, H.T. Diep, A. Ghazali, P. Lallemand, Phys. Rev. B. 34 (1986) 3177. [19] G. Ramirez-Santiago, J.V. Jose, Phys. Rev. B 49 (1994) 9567. [20] T. Horiguchi, D. Loison, H.T. Diep, O. Nagai, Physica A 206 (1994) 508. [21] P. Olsson, Phys. Rev. Lett. 75 (1995) 2758. [22] D.H. Lee, J.D. Joannopoulos, J.W. Negele, D.P. Landau, Phys. Rev. Lett. 52 (1984) 433. [23] C. Denniston, C. Tang, Phys. Rev. Lett. 79 (1997) 451. [24] W.-M. Zhang, W.M. Saslow, M. Gabay, Phys. Rev. B 44 (1991) 5129. [25] W.-M. Zhang, W.M. Saslow, M. Gabay, M. Benakli, Phys. Rev. B 48 (1993) 10 204. [26] Y. Wang, Thesis, Tohoku Univeristy, 1997 (in Japanese). [27] N. Tsushima, Thesis, Tohoku University, 1998 (in Japanese). [28] N. Tsushima, T. Horiguchi, (1998) in preparation.
Critical properties of XY model on two-layer triangular lattice Tsuyoshi Horiguchi ∗ , Yi Wang, Yasushi Honda Department of Computer and Mathematical Sciences, GSIS, Tohoku University 04, Sendai 980-8579, Japan Received 26 March 1998
Abstract We investigate phase transitions of the XY model on a two-layer triangular lattice when interactions are competing, by using Monte Carlo simulations. For the system with a ferromagnetic interaction on one of two layers and an antiferromagnetic interaction on the other layer, we nd two di erent types of critical phenomena with two phase-transition temperatures; (1) each of phase transitions due to chirality and (2) the Kosterlitz-Thouless-type phase transition and a chirality transition. We clarify the nature of these phase transitions by using nite-size scaling c 1998 Elsevier Science B.V. All rights reserved. analyses. PACS: 75.10.Hk; 05.50+q; 75.30.K; 75.40.c Keywords: XY model; KT transition; Chirality; Two-layer triangular lattice
1. Introduction A uniaxial antiferromagnet with an external magnetic eld H has been paid much attention for many years as for studies on a free surface of a Heisenberg antiferromagnet such as CsCl structure and a classical two-sublattice antiferromagnet such as MnF2 , for examples. It has been shown that there are several phases such as the antiferromagnetic phase with surface spins parallel to H , the antiferromagnetic phase with surface spins antiparallel to H , a surface spin op phase and a bulk spin op phase and so on. Thin lms composed of ferromagnetic planes, which are antiferromagnetically coupled to each other, have been investigated by using the same models. These models are treated as one-dimensional models after taking a mean- eld approximation for spins in the planes. The models are expressed in terms of one-dimensional XY models whose ground-state properties have been investigated [1 – 6]. Owing to the development of ∗
Corresponding author. Fax: +81 22 217 5851; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 2 9 1 - X
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epitaxy technology, it is now possible to make very thin magnetic lms. However, the theoretical investigation of the magnetic thin lms has been done mainly for the groundstate properties by means of phenomenological treatments or mean- eld approximations [7 – 12]. In this situation, we are interested in investigating of critical phenomena for an XY model on a two-layer lattice as one of the spin models related to the very thin magnetic lms. In a two-dimensional XY model, there is no long-range order due to the Mermin– Wagner theorem [13]. However, there is a phase transition, the so-called Kosterlitz– Thouless (KT) transition [14–16]. On the other hand, it was suggested by Miyashita and Shiba [17] that there are two di erent phase-transition temperatures, namely one for the chirality transition and the other for the KT transition in an antiferromagnetic XY model. Although there are still arguments going on whether both the transition temperatures are the same or not, we do not discuss this problem in the present paper; for examples, see Refs. [18–21]. New critical phenomena were discussed by Lee et al. and the domain walls are considered as elementary excitations [22]. Discussions for domain walls have been given recently by Denniston et al. [23]. Zhang et al. investigated and found that there are a spiral phase and other phases in some generalized XY models on a triangular lattice by using mean- eld approximations [24,25]. Thus, XY models are expected to show interesting critical phenomena on some lattices with structures di erent from those investigated so far. In these situations, we investigate the nature of phase transitions for the XY -model on a two-layer triangular lattice in the present paper. We consider the XY -model with a ferromagnetic intralayer interaction on one of two layers and an antiferromagnetic interaction on the other layer. We nd new critical phenomena due to the chirality; the chirality on a layer with ferromagnetic intralayer interaction starts to decrease at some temperature when the temperature decreases from a high-temperature side while the chirality on the layer with antiferromagnetic intralayer interaction does not show such change there. The speci c heat diverges with a positive value of critical index at this temperature, although we have the usual critical phenomena due to the chirality between the paramagnetic phase and the chirality phase. We describe our model in Section 2 and give phase diagrams of the ground state. In Section 3, we give the results obtained by Monte Carlo simulations and discuss the nature of phase transitions in terms of nite-size scaling analyses. The concluding remarks are given in Section 4.
2. XY model on two-layer triangular lattice We consider two triangular lattices 1 and 2 and the XY model on 1 + 2 whose Hamiltonian is given as follows: H = −I
X (i; j)
si · sj − J
X (i; j)
ti · tj − K
X i
si · ti ;
(1)
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where si and ti are XY spins on the triangular lattice 1 or 2 , respectively. We assume an exchange interaction between spins at nearest-neighbor lattice sites on 1 and that between spins at nearest-neighbor lattice sites on 2 and also an exchange interaction between spins at a site i on 1 and at its corresponding site i on 2 ; these interaction constants are expressed by I; J or K, respectively. The sum with (i; j) indicates a summation over the nearest-neighbor pairs of sites on 1 or 2 and the sum with i indicates a summation over the pairs of sites i on 1 and 2 . When we write si and ti as follows: si = (cos i ; sin i );
ti = (cos i ; sin i ) ;
(2)
Hamiltonian (1) is rewritten as H = −I
X (i; j)
cos(i − j ) − J
X
cos( i − j ) − K
X
(i; j)
cos(i − i ) :
(3)
i
We note that there is a relation Q(I; J; K) = Q(I; J; −K) for a thermodynamic quantity Q(I; J; K) and, hence, we assume K¿0 without loss of generality. The ground-state con guration is obtained by minimizing the energy, e, on a pair of triangles on 1 and 2 as shown in Fig. 1, e = −I [cos(1 − 2 ) + cos(2 − 3 ) + cos(3 − 1 )] −J [cos( 1 − 2 ) + cos( 2 − 3 ) + cos( 3 − 1 )] ˜ −K[cos( 1 − 1 ) + cos(2 − 2 ) + cos(3 − 3 )] ;
(4)
Fig. 1. A part of two-layer triangular lattice, 1 and 2 . The XY spins are denoted si on 1 and ti on 2 . The intralayer interaction constants are I on 1 and J on 2 and the interlayer infraction constant is K. For the energy (4), we choose K˜ = K=3.
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where 1 ; 2 ; 3 ; 1 ; 2 and 3 are assigned as shown in Fig. 1 and K˜ = K=3. By introducing new variables, 12 = 2 − 1 ;
31 = 1 − 3 ;
12 = 2 − 1 ;
31 = 1 − 3 ;
(5)
we have the following expression: e = −I [cos 12 + cos(12 + 31 ) + cos 31 ] −J [cos 12 + cos( 12 + 31 ) + cos 31 ] ˜ −K[cos( 1 − 1 ) + cos(1 − 1 + 12 − 12 ) + cos(1 − 1 − 31 + 31 )] : (6) By putting the derivative of e in terms of 1 or 1 to zero, we have sin(1 − 1 ) + sin(1 − 1 + 12 − 12 ) + sin(1 − 1 − 13 + 13 ) = 0 :
(7)
By putting the derivative of e in terms of 12 ; 12 ; 31 or 31 , we have the following equations, respectively, I [sin 12 + sin(12 + 31 )] + K˜ sin(1 − 1 + 12 − 12 ) = 0 ;
(8)
J [sin 12 + sin( 12 + 31 )] − K˜ sin(1 − 1 + 12 − 12 ) = 0 ;
(9)
I [sin 31 + sin(12 + 31 )] − K˜ sin(1 − 1 − 31 + 31 ) = 0
(10)
J [sin 31 + sin( 12 + 31 )] + K˜ sin(1 − 1 − 31 + 31 ) = 0 :
(11)
and
Now, we may express 1 by means of 1 ; 12 and 31 , and 12 and 31 by means of 1 ; 1 ; 12 and 31 as follows: 3I (sin 12 − sin 31 ) ; (12) 1 = 1 − arcsin K 3I {sin 12 + sin(12 + 31 )} (13) 12 = 1 − 1 + 12 − arcsin K and
31 = −1 + 1 + 31 + arcsin
3I {sin 31 + sin(12 + 31 )} : K
(14)
We set 1 = 0, without loss of generality. We vary the values of 12 and 31 from 0 to 2 and calculate the value of energy e. Then we nd spin con gurations which minimize energy e and construct the ground-state phase diagrams. In Figs. 2–4, we show the ground-state phase diagrams in the I=K − J=K plane, in the J=I − K=I plane with I ¿0 and in the J=|I | − K=|I | plane with I ¡0, respectively. In region F1, we have a ferromagnetic con guration of spins which are aligned in
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Fig. 2. Ground-state phase diagram in I=K − J=K plane. See text for details of each phase.
Fig. 3. Ground-state phase diagram in J=I − K=I plane for I ¿0. See text for details of each phase.
the same direction for all the lattice sites on both 1 and 2 . In region F2, we have a ferromagnetic con guration with i = , for example, for any site i on 1 and a ferromagnetic con guration with i = + for any site i on 2 . In region C1, we have a 120◦ structure with i = i for any site i on both 1 and 2 . In region C2, we have a 120◦ structure in 1 and also the 120◦ structure with i = i + in 2 . In region V1, the spin con guration on 2 is almost 120◦ structure and that in 1 is almost ferromagnetic for small I=K and large |J |=K in Fig. 2, for example; both structures are incomplete and the angle of each spin changes according to values of interaction parameters. In region V2, the spin con guration on 2 is almost ferromagnetic and that on 1 almost 120◦ structure for small J=K and large |I |=K in Fig. 2, for example; there is a similar situation to that in region V1.
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Fig. 4. Ground-state phase diagram in J=|I | − K=|I | plane for I ¡0. See text for details of each phase.
3. Results by Monte Carlo simulations In the present section, we present results obtained by standard Monte Carlo (MC) simulations and clarify the nature of phase transitions by investigating the results by means of the nite-size scaling method. We assume the periodic boundary conditions for each lattice, 1 and 2 , and we denote the number of total lattice sites on each triangular lattice as N , where N = L × L; |1 | = |2 | = N . Our MC simulations are performed for L = 12, 24, 36 and 48. Random spin con gurations have been taken as initial spin con gurations in our MC simulations. An MC average, hQi, for a quantity Q is calculated by hQi =
n X 1 Q(t) ; n − n0
(15)
t=n0 +1
where we choose n0 = 2:5 × 105 MC steps, n = 5 × 105 MC steps and also n0 = 5 × 105 MC steps, n = 106 MC steps, depending upon the system size and the value of temperature. We set the Boltzmann constant k to 1 in the present paper. We calculated the internal energy, the speci c heat and the chirality. The internal energy is given as E=
1 hH i ; 2N
(16)
where H is expressed as follows: H =−
X 1 XX [I cos(i − i+i ) + J cos( i − i+i )] − K cos(i − i ) : 2 i i i
(17)
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The notation i implies one of the six nearest-neighbor sites of a site i on each triangular lattice, 1 or 2 . The sum with i indicates a summation over the six nearest-neighbor sites of a site i. The speci c heat is obtained by C=
1 2 2 { H − hH i } : 2NT 2
(18)
The chirality is de ned at a smallest up-triangle, namely an elementary upward triangle, on 1 as 2 s = √ si × sj + sj × sk + sk × si · eˆz ijk 3 3 2 = √ [sin(j − i ) + sin(k − j ) + sin(i − k )] 3 3
(19)
and at a smallest up-triangle on 2 as 2 t = √ ti × tj + tj × tk + tk × ti · eˆz ijk 3 3 2 = √ [sin( j − i ) + sin( k − j ) + sin( i − k )] : 3 3
(20)
Here eˆz is the unit vector in the z-direction in spin space. A set of sites {i; j; k} indicates an allocation of sites counterclockwise on each up-triangle of the lattice 1 or 2 . The averaged chirality for the whole system is de ned by = 12 (s + t ) ;
(21)
where s =
1 X s ijk ; N
(22)
1 X t ijk ; N
(23)
(ijk)
t =
(ijk)
where the sum with (ijk) is taken over all of the elementary up-triangles on lattice 1 or 2 . We show in Figs. 5–8 the results obtained for the system with I = −1:0; K = 0:5 and J = −1:0; 0:2; 0:3; 0:5 and 1.0 for L = 48. In Fig. 5, the internal energy E=|I | is given as functions of T=|I |. We see that there are steep parts of the internal energy as functions of temperature at some temperature regions. These imply the possibility of phase transitions. We show the speci c heat for J = − 1:0; 0:2 and 0.3 in Fig. 6 and those for J = 0:5 and 1.0 in Fig. 7. From these gures, we see that there is a sharp peak of the speci c heat for J = −1:0, two sharp peaks for J = 0:2 and a sharp peak with a sluggish decrease in the low-temperature side for J = 0:3. On the other hand, there are a sharp peak and a broad peak for J = 0:5 and 1.0. The existence of two sharp peaks in the speci c heat for J = 0:2 is quite a new phenomena for the frustrated
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Fig. 5. Internal energy E=|I | as functions of T=|I | for I = −1:0 and K = 0:5. Pluses show the internal energy for J = 0:2, up-triangles for J = 0:3, squares for J = 0:5, pentagons for J = 1:0 and closed down-triangles for J = −1:0.
Fig. 6. Speci c heat C as functions of T=|I | for I = −1:0 and K = 0:5. Pluses show the speci c heat for J = 0:2, up-triangles for J = 0:3 and closed down-triangles for J = −1:0.
XY models on two-dimensional lattices, as far as we know. We will clarify this below. In Fig. 8, we show the chiralities, s on 1 and t on 2 , for J = 0:2; 0:3 and 0.5 and the total chirality for J = −1:0. The behaviors of chiralities, s and t , are the same for J = −1:0 as expected and, hence, we show the total chirality for J = −1:0 in Fig. 8. We note that di erent behaviors for s and t as functions of temperature are obtained for J = 0:2; 0:3; 0:5 and 1.0. We also note that t = 0 for J = 0:5 in Fig. 8. We did not show the results of the chirality for J = 1:0 in Fig. 8, since the values of the chirality for J = 1:0 and those for J = 0:5 are almost the same in the gure. We
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Fig. 7. Speci c heat C as functions of T=|I | for I = −1:0 and K = 0:5. Squares show the speci c heat for J = 0:5 and pentagons for J = 1:0.
Fig. 8. Chirality as functions of T=|I | for I = −1:0 and K = 0:5. Closed down-triangles show the total chirality for J = −1:0, pluses show that of chirality s and crosses the chirality t for J = 0:2, open up-triangles the chirality s and closed up-triangles the chirality t for J = 0:3, open pentagons the chirality s and closed pentagons the chirality t for J = 0:5.
will make nite-size scaling analyses for these data in order to clarify the nature of phase transitions. For J = −1:0, we show the speci c heat in Fig. 9 for L = 12; 24; 36 and 48. The peak height is getting higher when the linear system size, L, is getting larger. We found that the peak height is a linear function of ln L when we plotted the height values Ctop as a function of ln L, although we did not show it in the present paper [26]; Ctop denotes a height value of the speci c heat obtained in the present MC simulations. The nite-size scaling analysis is shown in Fig. 10 by using data for L = 24; 36, and
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Fig. 9. Size dependence of speci c heat for I = −1:0; J = −1:0 and K = 0:5. Pluses show the speci c heat for L = 12, up-triangles for L = 24, squares for L = 36 and closed down-triangles for L = 48.
Fig. 10. Finite-size scaling analysis for the speci c heat given in Fig. 9. The abscissa is (T=|I | − 0:7535)L1= where = 1:019 and the ordinate Cf(L) where f(L) = 1=(ln L)1=y with y = 0:765.
48; we have f(L) = 1=(ln L)1=y with y = 0:765, and hence we have = 0. We estimate Tc =|I | = 0:753(1). In this way, the speci c heat diverges logarithmically, and we obtain the critical index equal to 1.00(2). We show the chirality in Fig. 11, where the abscissa is − ln[(Tc − T )=|I |] and the ordinate − ln . We nd the critical index equal to 0.125(1). The critical temperature is also estimated as Tc =|I | = 0:753(1). In this way, there is a phase transition between the paramagnetic phase for T ¿Tc and the chirality phase for T ¡Tc ; the phase transition at Tc is due to the chirality. For J = 0:2, we show the speci c heat in Fig. 12 for L = 12; 24; 36 and 48. There are two peaks in Fig. 12; one is around T=|I | = 0:17 and the other around 0.435. These should give critical temperatures and, hence, we denote the critical temperature in the
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Fig. 11. Chirality for I = −1:0; J = −1:0 and K = 0:5. The abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:7533 and the ordinate y is − ln . Triangles show the result for chirality for L = 12, squares for L = 24, pentagons for L = 36 and closed up-triangles for L = 48. We have y = 0:125x − 0:02 as the straight line.
Fig. 12. Size dependence of speci c heat for I = −1:0; J = 0:2 and K = 0:5. Pluses show the result for L = 12, up-triangles for L = 24, squares for L = 36 and closed down-triangles for L = 48.
low temperature side as Tcl and that in the high temperature side as Tcu . The peak heights of the speci c heat near Tcu and Tcl are higher when the linear system size, L, is larger. We found that the behavior of height values Ctop of the speci c heat near Tcu is di erent from that near Tcu ; we have that ln Ctop ∼ ln L near Tcl but Ctop ∼ ln L near Tcu , although we did not show them in the present paper [26]. A nite-size scaling analysis for the speci c heat near Tcu is shown in Fig. 13 and that near Tcl in the inset of Fig. 13, where f(L) = 1=(ln L)1=y with y = 0:855 and g(L) = L− = . We estimate Tcl =|I | ∼ 0:173(3), Tcu =|I | ∼ 0:431(3) and ∼ 0:92(8) and ∼ 0:29(9) at Tcl and
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Fig. 13. Finite-size scaling analysis for the speci c heat given in Fig. 9. The abscissa is (T=|I | − 0:4316)L1= where = 1:098 and the ordinate is Cf(L) where f(L) = 1=(ln L)1=y with y = 0:855. In the inset, the abscissa is (T=|I | − 0:177)L1= with = 0:92 and the ordinate is Cg(L) where g(L) = L− = with = 0:29.
∼ 1:09(9) and = 0 at Tcu . The hyperscaling d = 2 − is satis ed within numerical errors. In Fig. 8, we give the behaviors of chiralities, s and t on both lattices, 1 and 2 ; we see the behavior of s and that of t are di erent. Both s and t are becoming non-zero at Tcu when the temperature decreases from the high-temperature side to Tcu and increase when the temperature decreases further. Now, we see that s continues to increase to 1 when temperature tends to 0, but t starts to decrease when temperature decreases from Tcl . This seems to be due to the competition between the interaction energy and the entropy. In Fig. 14, we show − ln as a function of − ln[(Tcu − T )=|I |] for (T . Tcu ) and nd that 1 ∼ 1:073 (Tcu − T ) (T . Tcu ) ; (24) |I | with = 0:125(1). We show − ln(0:2515−t ) as a function of − log[(Tcl −T )=|I |] (T . Tcl ) in the inset of Fig. 14 and then we nd that 1 (c) (Tcl − T ) t ∼ t − 0:249 (T . Tcl ) (25) |I | with t(c) = 0:2515 and = 0:253(3). This means that the chirality in 2 decreases from t(c) with index = 0:253. The divergent behavior of the speci c heat at Tcl is now clari ed to be due to the decreasing of the chirality in 2 . For J = 0:5, we have two peaks in the speci c heat in Fig. 7. We show their size dependence for L = 24; 36 and 48 in Fig. 15. The peaks at the high-temperature side do not show size-dependence and, hence, it is reasonable for us to think that the phase transition is a KT type. We will not discuss the details for the KT transition in this paper; we just estimate the KT transition temperature, TKT , by the peak-temperature and
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Fig. 14. Total chirality and chirality t for I = −1:0; J = 0:2 and K = 0:5. Triangles show the chirality for L = 12, squares for L = 24, pentagons for L = 36 and closed up-triangles for L = 48. The abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:4335 and the ordinate y is − ln . We have y = 0:125x − 0:071 as a straight line. In the inset, the abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:1734 and the ordinate y is − ln(0:2515 − t ). We have y = 0:253x + 1:395 as a straight line.
Fig. 15. Size dependence of speci c heat for I = −1:0; J = 0:5 and K = 0:5. Up-triangles show the speci c heat for L = 24, squares for L = 36 and closed down-triangles for L = 48.
the in ection-temperature of the speci c heat from the data by our MC simulations. The peaks at the low-temperature side do show size dependence and we nd that Ctop ∼ ln L near Tc [26]. A nite-size scaling analysis is shown in Fig. 16 near Tc and we obtained Tc =|I | = 0:420(1); = 0 and = 1:03(3). In Fig. 17, we show − ln s as a function of − ln[(Tc − T )=|I |]; we notice t = 0 in this case. We use Tc =|I | = 0:4205 and nd = 0:125; hence, the phase transition at Tc =|I | is due to the chirality. We
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Fig. 16. Finite-size scaling analysis for the speci c heat given in Fig. 15. The abscissa is (T=|I | − 0:4205)L1= with = 1:03 and the ordinate is Cf(L) where f(L) = 1=(ln L)1=y with y = 1:01.
Fig. 17. Chirality s for I = −1:0; J = 0:5 and K = 0:5. The abscissa x is − ln[(Tc − T )=|I |] with Tc =|I | = 0:4205 and the ordinate y is − ln s . Triangles show the chirality for L = 12, squares for L = 24, pentagons for L = 36 and closed up-triangles for L = 48. We have y = 0:125x − 0:065 as a straight line.
made the same analysis for J = 1:0 and obtained the same conclusions for the nature of phase transitions for J = 0:5 and J = 1:0. Now in order to con rm the nature of phase transitions for the system with I = −1:0; J = 0:2 and K = 0:5, we made MC simulations for the system with I = −1:0; J = 0:2 and K = 1:0. We show the speci c heat for L = 48 in Fig. 18. The situation is similar to the case of the system with I = −1:0; J = 0:2 and K = 0:5; we have two sharp peaks in the speci c heat. The nite-size scaling analysis for the speci c heat near Tcu is shown in Fig. 19 and that near Tcl in the inset of Fig. 19. We estimate Tcl =|I | ∼ 0:176(1), Tcu =|I | ∼ 0:431(1) and ∼ 0:26(9), ∼ 0:885(20) at Tcl and = 0; = 0:92(9) at Tcu .
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Fig. 18. Speci c heat C as functions of T=|I | for I = −1:0; J = 0:2 and K = 1:0.
Fig. 19. Finite-size scaling analysis for the speci c heat given in Fig.18. The abscissa is (T=|I | − 0:4815)L1= where = 0:92 and the ordinate is Cf(L) where f(L) = 1=(ln L)1=y with y = 0:780. In the inset, the abscissa is (T=|I | − 0:1768)L1= with = 0:885 and the ordinate is Cg(L) where g(L) = L− = with = 0:26.
We show the behaviors of chirality s and t in Fig. 20. We see the behavior of s is di erent from that of t : their behaviors are similar to those for the system with I = −1:0; J = 0:2 and K = 0:5. We show nite-size scaling analysis for in Fig. 21 and that for t in set of Fig. 20. We nd that ∼
1 (Tcu − T ) |I |
(T . Tcu ) :
(26)
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Fig. 20. Chirality s and chirality t as functions of T=|I | for I = −1:0; J = 0:2 and K = 1:0. Open up-triangles show the chirality s and closed up-triangles the chirality t .
Fig. 21. Finite-size scaling analyses for the chirality in Fig. 20. The abscissa is (T=|I | − 0:4815)L1= where = 0:92 and the ordinate is L = with = 0:125. In the inset, the abscissa is (T=|I | − 0:1768)L1= with = 0:885 and the ordinate is −(t − 0:2734)L = .
with = 0:125(1), and 1 (c) (Tcl − T ) t − t ∼ − |I |
(T . Tcl )
(27)
with t(c) = 0:2734 and = 0:25(2). These results are consistent with that for the system with I = −1:0; J = 0:2 and K = 0:5. Finally, by using the analyses stated above for the system with I = −1:0; K = 0:5 and several values of J , we obtain a nite temperature phase diagram in Fig. 22 in the T=|I | − J plane for −1:56J 61:5. We nd the paramagnetic phase and the
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Fig. 22. Finite temperature phase diagram for I = −1:0 and K = 0:5. The abscissa is J and the ordinate is T=|I |. P denotes the paramagnetic phase, C the chirality phase and KT the Kosterlitz–Thouless phase.
chirality phase for J . 0:2; there is only one phase-transition temperature between these phases. For 0:2 . J . 0:3, we have the paramagnetic phase and two kinds of chirality phases. The chirality in the 2 shows small values according to the interaction energy for T ¡Tcl , but takes rather larger values for Tcl . T than those expected by the interaction energy; these larger values of t are due to e ect of the chirality, s , on 1 . We surmise that this phenomena is a kind of order-in-disorder; the values of chirality, t , are larger for higher temperatures than those for lower temperature in some temperature range. The speci c heat diverges at Tcl with ¿0. The critical phenomena at Tcl is due to the sharp change of the chirality in the 2 plane. We have two phasetransition temperatures, Tcu and Tcl for this case. At Tcu , there is a phase transition between the paramagnetic phase and the chirality phase; this is the usual chirality phase transition. For J & 0:3, there are two phase-transition temperatures, TKT and Tc . We have the paramagnetic phase for T ¿TKT and the KT phase for Tc ¡T ¡TKT and the chirality phase for T ¡Tc .
4. Concluding remarks In this paper, we have investigated critical phenomena for an XY model on a twolayer triangular lattice by using Monte Carlo simulations. We assume that the intralayer interaction in one of two triangular lattices is ferromagnetic and that in the other triangular lattice antiferromagnetic. The interlayer interaction is assumed to be ferromagnetic without loss of generality. We have made nite-size scaling analyses and clari ed the nature of the phase transitions that occurred in the present system. We have obtained a phase diagram in a plane spanned by one of the interaction parameters and temperature.
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We have found new critical phenomena for some range of interaction parameters; there is chirality even in the layer with the ferromagnetic intralayer interaction and its values increase sharply at a temperature Tcl when the temperature increases. We have the usual chirality transition at a temperature Tcu higher than Tcl ; there is the phase transition between the paramagnetic phase and the chirality phase at Tcu . Namely, we have two phase-transition temperatures, Tcu and Tcl . At the critical temperature Tcl we have found that the speci c heat diverges with the critical exponent = 0:28(9). We de ne a critical index, , which expresses the sharpness of increase of the chirality, and have estimated = 0:25(3). At the critical temperature Tcu , we have the chirality transition with = 0:125 and = 0. For other ranges of parameters, we have one phase-transition temperature, namely, the chirality transition in one case or two phasetransition temperatures, namely the Kosterlitz–Thouless (KT) transition at a higher temperature and the chirality transition at a lower temperature in the other case. Now, we make some comments. Although we have not investigated the KT phase transition in detail, we think the KT transition temperature is higher than the chirality transition temperature, if they both exist. The phase transition for an Ising model on a two-layer triangular lattice has been investigated recently in the present authors’ group [27,28]. For the system with a ferromagnetic intralayer interaction on one of the two layers and an antiferromagnetic interlayer interaction on the other layer, we have found that there are two phase-transition temperatures, Tcl and Tcu (Tcu ¿Tcl ), when the ground state has a six-fold degeneracy; the order parameter in the layer with the antiferromagnetic interaction has increased sharply at a temperature Tcl when temperature increases, while we have the usual phase transition between the ferromagnetic phase and the paramagnetic phase at a higher temperature Tcu . We expect that a similar phase transition will occur for other systems such as an XY model on a two-layer square lattice, on a three-layer triangular lattice and so on when there are frustrated layers and non-frustrated layers. These problems are left as future problems. Acknowledgements This work was partially supported by a Grand-In-Aid for Scienti c Research from the Ministry of Education, Science and Culture, Japan and also by the Computer Center of Tohoku University. References [1] [2] [3] [4] [5] [6] [7]
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