24 April 2000




                                               Physics Letters A 269 Z         .
                                                                            2000 55­61
                                                                                                                    www.elsevier.nlrlocaterphysleta




            Phase transitions of classical XXZ model with easy-plane
                            anisotropy on a two-layer triangular lattice
                                               Yi Wang, Tsuyoshi Horiguchi )
Department of Computer and Mathematical Sciences, Graduate School of Information Sciences, Tohoku UniÕersity 04, Aramaki-aza-aoba,
                                                       Aoba-ku, Sendai 980-8579, Japan

                          Received 9 December 1999; received in revised form 13 March 2000; accepted 14 March 2000
                                                             Communicated by L.J. Sham


Abstract

       We investigate phase transitions of a classical XXZ model with easy-plane anisotropy, on a two-layer triangular lattice
with ferromagnetic and antiferromagnetic layers by using Monte Carlo simulations. It turns out that the chirality shows a
very steep increase as temperature increases in a temperature range; the value of a critical exponent for this change is
estimated. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 75.10.Hk; 05.50qq; 75.30K; 75.40.c
Keywords: Classical Heisenberg model; Frustration; Two-layer; Chirality



       Magnetic thin films, layered magnets or magnets                                 In the thermodynamic limit, there is no phase
on superlattices show interesting properties that are                               transition in the ferromagnetic classical Heisenberg
different from bulk properties w1­ x
                                             3 . We expect that                     model on two-dimensional lattices w x
                                                                                                                               16 . There is no
some systems on a lattice consisting of multi-layers                                long-range order but there exists the so-called
show quite interesting behaviors, which do not exist                                Kosterlitz­Thouless Z      .
                                                                                                           KT type transition in the anti-
in a system on a single layer nor in three-dimen-                                   ferromagnetic classical Heisenberg model on the tri-
sional bulk systems w4­1 x
                                   3 . Even in a simple mag-                        angular lattice w17,18x. As for the KT transition of
netic model such as an XY model, we may have a                                      the XY model Zthe plane rotator                    .
                                                                                                                                model , see for
completely different nature of phase transitions in                                 example Refs. w19­2 x
                                                                                                           1 . In the present Letter, we are
systems on a few layers from that in two-dimen-                                     interested in phase transitions in an anisotropic clas-
sional systems or from that in three-dimensional bulk                               sical Heisenberg model with an easy-plane anisotro-
systems w            x
              14,15 .                                                               py, namely a classical XXZ model with strong XY
                                                                                    anisotropy; we simply call this model an XXZ model
                                                                                    hereafter. For the XXZ model, it has been shown that
                                                                                    there exist the KT transition and also the chirality
  )                                                                                 transition, when the system is fully frustrated w22,23x
       Corresponding author. Tel.: q81-22-217-5842; fax: q81-22-                                                                                  .
217-5851.                                                                           This situation is similar to the XY model with full
       E-mail address: tsuyoshi@statp.is.tohoku.ac.jp ZT.             .
                                                             Horiguchi .            frustration w24­2 x
                                                                                                     6 . Hence we investigate phase tran-

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S0375-9601Z00.00223-1



56                                                  Y. Wang, T. HoriguchirPhysics Letters A 269 (                          )
                                                                                                                2000 55­61

sitions for an XXZ model on a two-layer triangular                                  ues of J in order to indicate the system. We have
lattice with a ferromagnetic layer and an antiferro-                                performed our MC simulations for Ls24,36 and 48.
magnetic layer.                                                                     We have used random spin configurations as initial
       We consider an XXZ model with an easy-plane                                  spin configurations in our MC simulations. A MC
anisotropy on a two-layer triangular lattice. Each of                               average, ²OO:, for a quantity OO is calculated by
two triangular lattices is denoted by L1 and L2,
respectively. We assume Heisenberg spins, s                                                                                          n
                                                                                                                1
                                                                    i and ti
for igL                                                                             ²OO:s                                            Ý OOZt.,                     Z3.
             1 and L2 , respectively, where N s < L <
                                                                        1 s                           nyn
< L <                                                                                                                 0 tsn q
                                                                                                                                     0    1
      2 ; N s L = L for each layer and L is the linear
size of L Z
               k k s 1,2.. Then the two-layer triangular                            where we choose n0s2.5=105 MC steps, ns5=
lattice is denoted by L: LsL1qL2 and hence
<                                                                                   105 MC steps, and also n0s5=105 MC steps and
     L<s2 N. The Hamiltonian is given as follows:                                   ns106 MC steps, depending upon the system size
                                                                                    and the values of temperature. We set the Boltzmann
HsyI Ý sxisxjqsyisyjqj szisz
                      Z                               j .                           constant k to 1 throughout the present Letter.
            Zi, j.                                                                         We have calculated the internal energy, the spe-
                                                                                    cific heat, the chirality and Binder parameters. The
         yJ Ý t xit xjqt yit yjqj t zit z
                       Z                              j .                           internal energy is given as
             Zi, j.
                                                                                                 1
         yK Ý Z sx                                                                  Es                     ² H :
                            i t x
                               i q s yi t y
                                         i q j s zi t zi . ,             Z1.                                          .                                           Z4.
                i                                                                           2 N
where j is an anisotropy parameter with 0Fj-1;                                      The specific heat is obtained by
we assume js0.5 in the present Letter. Here we                                                        1
have assumed that intralayer interactions exist only                                                                            2              2
                                                                                    Cs                          2  ² H : y ² H : 4 .                              Z5.
between spins at nearest-neighbor lattice sites on                                           2 NT
each of L1 and L2; those interaction constants are
denoted by I and J, respectively. The sum with Zi, j.                               The chirality is defined for a smallest up-triangle,
indicates a summation over nearest-neighbor pairs of                                namely an elementary upward triangle on each plane.
sites on L                                                                          For example, the chirality, k sijk, on L1 plane is
                     1 or L2 . We have also assumed that
interlayer exchange interactions exist only between                                 given as follows:
spins at a site i on L1 and at its corresponding site i                                                    2
on L                                                                                  s
          2 ; this interaction constant is denoted by K.                            kijks                            Zsi=sjqsj=skqsk=si. Pez,                     Z6.
The sum with i indicates a summation over pairs of                                                    '
                                                                                                 3 3
sites i on L1qL2. We assume the periodic bound-
ary conditions for each layer L                                                     where ez is the unit vector of z-component in the
                                                    1 or L2 .
       We notice that there is a relation:                                          spin space. A set of sites  i, j,k4 expresses three sites
                                                                                    labeled counterclockwise on an up-triangle of lattice
OOZ I, J,K . sOOZ I, J,yK .                                              Z2.        L . In a similar way, we define the chiralities k t
                                                                                      1                                                                            i jk
                                                                                    on lattice L2. The averaged chirality for the whole
                                                                                    system is defined by
for a thermodynamic quantity OOZ I, J, K .. Then we
assume that K)0 without loss of generality. In the                                          1
                                                                                    ks 2 Z ksqkt. ,                                                               Z7.
present Letter, we investigate the system with Is
y1,Ks1 and J)0; namely the exchange interac-                                        where
tions on the upper layer Z .
                                          Ul L1 is antiferromagnetic
and that on the lower layer Z .
                                               Ll L2 is ferromagnetic.                           1                                             1
We investigate the system by Monte Carlo Z                                  .
                                                                       MC                                                  s                                 t
                                                                                    kss                    Ý ²kijk ,: kts Ý ²kijk .:                              Z8.
simulations. Hereafter we just mention only the val-                                             N                                             N
                                                                                                      Zijk.                                         Zijk.



                                            Y. Wang, T. HoriguchirPhysics Letters A 269 (    )
                                                                                        2000 55­61                                         57

The sum with Zijk. in Eq. Z .
                                          8 is taken over all the           There are two sharp peaks for Jc1 -J-Jc2; al-
elementary up-triangles on the lattice L1 or L2. The                        though we do not show here, we find that these two
Binder parameter is defined as                                              peaks show size-dependencies for data by Ls24,36
                                                                            and 48. The peak near Ts0.4 is sharp as we see; we
                                2
                          2
                   ¦ZMs .;                                                  denote the peak temperature for this sharp peak by
Us Z M . s1y                         .
                                2                                Z9.        Tch and notice that the values of Tch do not depend
                           2
                    3² Ms :                                                 much on the value of J. The peak temperature for
                                                                            the other sharp peak of the specific heat is denoted
Here we have                                                                by Tcl. We have a sharp peak and a broad peak for
M                                                                           J)Jc2. We denote the peak temperature for this
  s s Ý s i ,                                                  Z10.         sharp peak by Tch and notice that the values of T
        igL                                                                                                                                ch
                                                                            do not depend much on the value of J for J)Jc2,
where si is either si or ti.                                                too. On the other hand, a broad peak at temperature
     The specific heat C is shown in Fig. 1 for Js                          higher than Tch shows no size dependence, and then
0.10, 0.18, 0.30 and 0.5 as functions of temperature.                       it is reasonable to think that the transition for this
We see two peaks in the specific heat for those                             broad peak is of the Kosterlitz­Thouless type. We
values of J except Js0.10. From the behaviors of                            denote this peak temperature by TKT. Although we
the specific heat, there are two critical values for J,                     need more comprehensive calculations in order to
namely Jc1 and Jc2; Jc1 is estimated as 0.17 and Jc2                        estimate TKT , for example, by calculating the helicity
as 0.35. We find only one sharp peak for J-Jc1.                             modulus, we estimate it here by the peak temperature
























Fig. 1. The specific heat as functions of temperature for Js0.1, 0.18, 0.3 and 0.5. In the inset, we show the chirality on the upper layer Z .
                                                                                                                                          Ul
and on the lower layer Z .
                        Ll for Js0.1,0.18,0.3 and 0.5.



58                              Y. Wang, T. HoriguchirPhysics Letters A 269 (            )
                                                                                      2000 55­61

because we are not interested in estimation of TKT in                  we have to estimate the critical temperature as pre-
the present Letter. In the inset of Fig. 1, we show                    cise as possible. For this purpose, we calculate the
behaviors of the chirality, ks and kt. We see clearly                  Binder parameters. Some results are shown in Figs. 3
that the chirality for Js0.18 increases steeply just                   and 4; we have estimated as Tch s0.3928 and Tcls
below T <
           r I <s0.1, as temperature increases. The                    0.0970, respectively, for Js0.18. We have found
chirality for Js0.30 shows a similar behavior as                       that the Binder parameters for each layer show more
that for Js0.18, although the increasing seems not                     clearly the behaviors of crossing than the Binder
so steep; this is due to the finite size effect. The                   parameter for the whole system. We make finite-size
chirality becomes zero at Tch as the temperature                       scaling analyses for the specific heat and the chiral-
increases.                                                             ity Zin the            .
                                                                                       inset at Tch s0.3928 in Fig. 5 and
      We estimate roughly the values of Tcl, Tch and                   Tcl s0.0970 in Fig. 6, respectively. From the finite-
TKT by using the system with Ls24 from the peak                        size scaling analyses for the specific heat and the
temperature of the specific heat. In Fig. 2, we give a                 chirality at Tch, we have found that the values of
phase diagram in the J versus temperature plane; we                    critical exponents, n,a and b are 1.00"0.05, Z
                                                                                                                            0 l .
                                                                                                                               n
denote the paramagnetic phase by P, the KT phase                       and 0.125"0.05, respectively. In order to investi-
by KT and the chiral phase by C. In the chiral phase,                  gate the behavior of the chirality at Tcl, we define the
we see a critical line between Jc -J-J                                 critical exponent b as follows:
                                    1              c         which
                                                        2
terminates at a low temperature; this is a critical end
point.                                                                        Zc.                   b
                                                                       kt;kt yaZTclyT .                  ZTQTcl. ,          Z11.
      In order to investigate the nature of the phase
transition at temperatures where the sharp peaks are                   where k Zc.
                                                                                t      is the maximum value of the chirality kt
observed in the specific heat between J                                on the ferromagnetic layer Z .
                                             c - J - J                                                    Ul and a is a constant.
                                              1                 c ,
                                                                 2
















































                                         Fig. 2. Phase diagram in J versus T <
                                                                              r I < plane.



                                       Y. Wang, T. HoriguchirPhysics Letters A 269 (        )
                                                                                       2000 55­61                                        59

























Fig. 3. Binder parameter for determination of critical temperature Tcl. In the inset, we show the whole behavior of the Binder parameter for
the lower layer.



























Fig. 4. Binder parameter for determination of critical temperature Tch. In the inset, we show the whole behavior of the Binder parameter for
the upper layer.



60                                       Y. Wang, T. HoriguchirPhysics Letters A 269 (          )
                                                                                            2000 55­61


























Fig. 5. Finite-size scaling analysis of the specific heat C for the system with Isy1,Ks1,js0.5 and Js0.18 near Tch. In the set, we
show a finite size scaling analysis of the chirality for the upper layer Z .
                                                                             Ul and that for the whole system Z       .
                                                                                                                   Tot .



























Fig. 6. Finite-size scaling analysis of the specific heat C for the system with Isy1,Ks1,js0.5 and Js0.18 near Tcl. In the set, we
show a finite size scaling analysis of the chirality for the upper layer.



                                  Y. Wang, T. HoriguchirPhysics Letters A 269 (                    )
                                                                                             2000 55­61                                                                   61

We have found that the values of n and b are                      effects as the strength of the coupling K varies. This
0.875"0.05 and 0.255"0.05, respectively. The er-                  is left as a future problem.
rors for the values of critical exponents are estimated
by the fitness for the finite-size scaling. We believe
that this critical behavior is a new finding for the              Acknowledgements
XXZ model. We found a similar behavior in the                        We thank Y. Fukui, K. Tanaka and Y. Honda for
two-layer XY model and in the three layer XY                      their helpful discussions. This work was partially
model w14,15x. In this way, we think that this quite              supported by Grand-In-Aid 08640472 for Scientific
new critical behavior occurs commonly in the XY                   Research from the Ministry of Education, Science,
model and also in the XXZ model of lattices consist-              Sports and Culture, Japan.
ing with layers when there is competing of ordering
among ferromagnetic planes and frustrated planes.
We have the chirality transitions similar to that in the          References
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