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Phase diagrams of a two-dimensional Heisenberg antiferromagnet with single-ion anisotropy
Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
Phase diagrams of a two-dimensional Heisenberg antiferromagnet with single-ion anisotropy B.V. Costa, A.S.T. Pires* Departamento de F!ısica ICEX, Universidade Federal de Minas Gerais Caixa Postal 702, 30123-970 Belo Horizonte, MG, Brazil Received 30 August 2002; received in revised form 4 December 2002
Abstract A two-dimensional Heisenberg antiferromagnet with a single-ion anisotropy is studied in the presence of a uniform magnetic field along the easy axis. The concept of effective field-dependent anisotropy is applied to the theoretical description of the phase diagrams. We perform calculations using the self-consistent harmonic approximation to treat the XY phase and the Green function technique for the Ising phase. Monte Carlo simulations on L L square lattices, with L ¼ 8; 16; 32 and 64; and with periodic boundary conditions were performed for the classical model. In order to test the validity of the effective anisotropy concept we compare our theoretical calculation with simulation data. We also calculate theoretically the phase diagram of the compound Rb2 MnF4 ; a quasi-two-dimensional spin 5=2 antiferromagnet, finding good agreement with experimental data. Our estimates for the transition temperature of the compounds K2 MnF4 and MnðHCOOÞ2 2H2 O; in zero magnetic field, are also in agreement with the experimental values. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Two-dimensional Heisenberg antiferromagnet; Single-ion anisotropy; Phase diagrams; Monte Carlo simulations
1. Introduction The interest in the properties of two-dimensional Heisenberg magnets has been greatly revived since the discovery of high-Tc superconductivity. Thus, the study of magnetism in two-dimensions has become interesting to both, theoreticians and experimentalists, and significant progress in the understanding of two-dimensional Heisenberg magnets has been achieved as a result of a symbiotic interplay between theory, simula*Corresponding author. E-mail addresses: bvc@fisica.ufmg.br antpires@fisica.ufmg.br (A.S.T. Pires).
(B.V.
Costa),
tion and experiments. It is now well known, that most quasi-two-dimensional magnetic materials exhibit some kind of anisotropy: it is rare that anisotropic properties arise from an anisotropy in the interaction, (which can be wholly isotropic), but, instead, it is caused by other sources, such as the presence of a crystal field that couples the spins along a certain direction in the crystal. The analysis of the anisotropic Heisenberg model is then quite interesting because, from the experimental point of view, the presence of some degree of anisotropy in the interaction is to be expected in nearly all cases, and because even a very small anisotropy can play an important role when the temperature goes to zero.
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-8853(02)01527-5
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
Although the quantum nature of magnetism cannot be forgotten, the study of classical models continues to be an important subject of research. Field theory yields results which agree closely with experiments on spin-1=2 Heisenberg systems but display strong deviations from the predicted behavior in systems with S > 1=2: In those systems, a broad crossover from quantum to classical behavior occurs at high temperatures [1]. From the theoretical point of view, not everything is settled for the easy axis model. For instance, a renormalization group calculation suggests the existence of a finite critical value of the anisotropy parameter, at which the critical temperature goes to zero [2]. However, numerical simulation and spin wave calculations do not confirm the existence of this critical anisotropy. In this paper we will be interested in the study of the phase diagram of the classical anisotropic Heisenberg antiferromagnet in two dimensions described by the following Hamiltonian: H¼J
X /i;jS
~j D ~i S S
X 2 X Siz gmB H Siz ; ð1Þ i
i
where the summation ð/i; jSÞ is over nearest neighbor pairs, J > 0 and the anisotropy term D establishes an easy axis (Ising-type anisotropy). In zero field the system undergoes a transition to a two-dimensional magnetically ordered structure. As it is well known when a magnetic field of sufficient strength is applied various interesting classes of phase transitions may be induced [3]. Here we consider the case of H applied along the z direction. The low-temperature, low field ordered state is an antiferromagnet (AF) which is separated from the paramagnetic state (P) by a line of second-order phase transitions. At sufficiently low temperature the system undergoes a first-order transition to a canted ‘‘spin-flop’’ state (SP) as the field is increased. The spin-flop phase behaves like a two-dimensional XY model and thus shows no long-range order. The state is separated from the P phase by a second-order phase transition line. The point at which AF and P phases simultaneously become critical is termed the ‘‘bicritical point’’. In two dimensions, if a simple bicritical point occurs, it would be at T ¼ 0 K:
317
The behavior of an anisotropic Heisenberg model similar to Hamiltonian (1) but with an P exchange anisotropy (D i Siz Sjz ) was first studied by Binder and Landau [4] using Monte Carlo computer simulations. The exchange anisotropic model, in zero magnetic field, has been recently studied by Cuccoli et al. [5] using the pure quantum-self-consistent harmonic approximation. Although the qualitative features of both models, at low temperatures with single-site anisotropy or exchange anisotropy, are expected to be the same, quantitative differences do occur. In a theoretical approach, the field dependence of Hamiltonian (1) can be taken into account as an effective field-dependent anisotropy Deff ðHÞ [3]. For a field H; applied along the easy-axis, smaller than the spin-flop field Hsf ; the effective anisotropy is given by H2 Deff ðHÞ ¼ D 1 2 ; ð2Þ Hsf where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4JD D2 Hsf ¼ 2S gmB
ð3Þ
for a square lattice. Clearly, while HoHsf ; the effective anisotropy Deff is an Ising-type anisotropy that vanishes at Hsf : On the other hand, as H increases to H > Hsf ; Deff becomes negative playing the role of a planar anisotropy which forces the spins to the lie in the XY -plane. In this XY -case, the planar behavior is enhanced as H is increased but this effective anisotropy approximation is valid for fields H52He ; where He is the exchange field given by He ¼ 4JS=gmB when the system is far from saturation. The critical field Hc ; where the system saturates at T ¼ 0; is given by Hc ¼ 2S
ð4J DÞ : gmB
ð4Þ
It is well known that in the classical limit, the thermodynamic properties of the antiferromagnet are the same as the ones for the ferromagnet. This allows us to do the theoretical calculations using a ferromagnet. Therefore, we may now treat the system by means of the effective
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
Hamiltonian X X 2 ~j Deff ðHÞ ~i S Siz Heff ¼ J S /i;jS
þ constant:
i
ð5Þ
A detailed derivation of this effective Hamiltonian can be found in Refs. [3,6]. Of course, the concept of an effective field-dependent anisotropy can be applied only to the classical limit (i.e. for large values of the spin). In fact for S ¼ 1=2 the single-ion anisotropy plays no role. In the usual studies of phase diagrams of 2D AF, experimental data have been qualitatively compared with simulations [1,3]. In this work, we will go one step further by comparing experimental data and simulations with theory. This paper is organized as follows: in Section 2, we present Monte Carlo simulations. In Section 3 we study the Ising phase using the self consistent renormalized spin-wave theory. The study of the XY phase using a self consistent harmonic approximation is presented in Section 4. Finally in Section 5 we analyse some previously reported experimental data for the compound Rb2 MnF4 :
2. Monte Carlo Monte Carlo calculations for the two-dimensional exchange anisotropic Heisenberg model were performed by Binder and Landau [4]. Those authors obtained the phase diagram for that model, but a plot of Tc as a function of the anisotropy was not presented. Also, at the time that the work was done, the possibility of a phase transition for the 2D isotropic Heisenberg model (as suggested by high-temperature series extrapolation) had not been completely ruled out. Later, Serena et al. [7], using Monte Carlo simulation with an improved algorithm, obtained reliable results for several thermodynamic properties of the same model. Our simulations of Hamiltonian (1) were carried out using the standard Metropolis algorithm. We have used lattices of size L L with L ¼ 8; 16; 32; 64 with periodic boundary conditions. In order to reach thermodynamic equilibrium, we
performed long runs of size 100 L L: In order to extract the critical temperature Tc in the Ising phase, the position of the maxima of the specific heat and magnetic susceptibility and, also, the fourth order Binder cumulant were analyzed. To get the Berezinskii–Kosterlitz–Thouless (BKT) temperature TBKT we analyzed the helicity modulus [8]. The procedure we have adopted is as follows: we fixed the anisotropy D while the magnetic field H was varied. For each pair of values, ðH; DÞ; we then obtained the specific heat. It is known that the specific heat maximum Cmax does not change with L in the BKT region but behaves as Cmax pln L in the Ising region. This distinct behavior of Cmax can guide us in deciding in which region, Ising or BKT we are. Then, the corresponding critical temperature is obtained as discussed above. Although the model has an Ising or a BKT transition for any value of D; it is very difficult to perform the simulation for low values of the anisotropy, because the critical temperature goes down monotonically with D; and close to T ¼ 0 the system suffers of a strong slowing down. For this reason, we have used the following values for the anisotropy parameter, D=J ¼ 0:25; 0:50; 1:0; 5:0 and 10:0: Our results are shown in Figs. 1, 2 and 3. For all simulations, we have adopted kB ¼ 1; J ¼ 1; S ¼ 1 , and H is in units of gmB : We must note that having an anisotropy parameter of 6 D=0.25 D=0.5 D=1.00
5 4 H
318
3 2 1 0
0
1
0.5 Tc
Fig. 1. Phase diagramm H=JS2 Tc for some anisotropy values as indicated in the insert. Error bars are smaller than the symbols when not indicated. Lines are guide to the eyes.
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
takes into account the effects of dynamic interactions between spin waves in the Hartee-Fock approximation. This theory was introduced long ago for the Heisenberg magnet with uniaxial anisotropy by Rastelli et al. [9] and here we reproduce their final result for the classical limit. The temperature dependent spin-wave spectrum for Hamiltonian (5) is given by
6 5
Field
4 3 2
Eq ðTÞ ¼ 4JSð1 gq Þ½1 bðTÞ þ ZðTÞ þ 2DS½1 2bðTÞ ;
1
where 0
319
0
0.5 Temperature
T bðTÞ ¼ 2 p
1
2
Fig. 2. Phase diagramm H=JS Tc for D=J ¼ 1:0: Circles and squares are simulation and theoretical results, respectively. The lines are guide to the eyes.
ZðTÞ ¼
T p2
Z
p
Z
0
Z
0 p
0
p
Z 0
p
ð6Þ
d~ q ; Eq ðTÞ
ð7Þ
gq d~ q ; Eq ðTÞ
ð8Þ
and 2
Ln (Tc/ D)
1
0
-1
-2
-2
-1
0
1
2
3
Ln (D) Fig. 3. Logarithm of Tc D as a function of ln D: Circles are simulation data and the line is the best fit.
the order of the exchange constant (D=JE1) is not an unrealistic case since we have D=J ¼ 1:58 [9] for the compound FeCl2 :
3. Ising region The self-consistently renormalized (SCR) spinwave theory extends the temperature range in which the spin-wave theory is valid because it
1 gq ¼ ðcos qx þ cos qy Þ: ð9Þ 2 Eqs. (6)–(8) constitute a set of self-consistent equations for Eq ðTÞ which were solved numerically by an iterative method. These equations have a solution up to a maximum temperature Tc which we consider to constitute the transition temperature. As can be easily seen, the SCR theory correctly predicts that the transition temperature is zero when D ¼ 0: In Fig. 2, we compare our theoretical calculation with Monte Carlo data for D=J ¼ 1:0: Here H is in units of gmB : As we can see, the overall agreement in the Ising region is reasonable. Then, when an uniform field, smaller than Hsf ; is applied, the Ising-like anisotropic model continues to provide an adequate description of the critical behavior. For the exchange anisotropic case with H ¼ 0 and when D-N (or J-0), the system behaves like the pure Ising model and Tc =D-1 [4]. However, for Hamiltonian (1), we know that there is no phase transition for J ¼ 0: It is therefore interesting to study the behavior of Tc as a function of D: In Fig. 3 we present our simulation data for lnðTc =DÞ as a function of lnðDÞ: The data obtained can be modelled using the following expression Tc ln ¼ a b lnðDÞ; ð10Þ D
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
320
with a ¼ 0:070ð12Þ and b ¼ 0:80ð10Þ: As we can see for large values of D we have Tc =DpDb quite different from the exchange anisotropic case [10]. Notice that we cannot obtain results using the SCR theory for the single-ion anisotropic Hamiltonian, for D-N; because this region is out of the validity range of the theory. However, for the exchange anisotropic case, the SCR theory gives good agreement with classical Monte Carlo simulation for any value of the anisotropy parameter [10].
4. XY region The two-dimensional classical model has a transition into a phase with algebraically decaying correlation functions and infinite susceptibility. The transition temperature, called the BKT temperature TBKT ; can be calculated using a selfconsistent harmonic approximation [11]. In order to properly treat the magnetic field we will first introduce the parametrization.
~n ¼ ð1Þn S sin½yn þ ð1Þn y0 cosf ; S n sin ½yn þ ð1Þn y0 sin fn ; cos½yn þ ð1Þn y0 ; ð11Þ where cos y0 ¼
H Hsf : Hc Hsf
ð12Þ
Inserting Eq. (12) into Hamiltonian (1) we find that keeping only terms up to second order the Hamiltonian becomes X 2 H J 2 X ¼ sin y0 fr frþa 2J ð yr y0 Þ 2 2 S 4 r;a r JX gmB H ðyr y0 Þðyrþa y0 Þ þ 2 r;a 2 X cos y0 ð yr y0 Þ 2 : ð13Þ
Hamiltonian
J X 2 x x y z sin y0 Sr Srþa þ Sry Srþa þ Srz Srþa H¼ 2 r;a X þ Deff ðSrz Þ2 ; ð15Þ r;a
with Deff the effective field-dependent anisotropy. Now, in order to use the SCHA we follow the same procedure used in Ref. [11] obtaining the quadratic form for the Hamiltonian J X H¼ rð1 gq Þfq fq 2 q Deff z z þ ð1 gq Þ þ 2 ð16Þ Sq Sq ; J where gq is given by Eq. (9) and the stiffness r is given by D
1=2
1=2 z r ¼ sin2 y0 1 ðSrz Þ2 1 ðSrþa Þ2 E ð17Þ cos ðfrþa fr Þ : The stiffness takes into account anharmonic terms neglected when we write the original Hamiltonian in the harmonic form. Following Ref. [11] we find that we can write Eq. (17) as t r ¼ sin2 y0 ½1 tIðHÞ exp ; ð18Þ r where t ¼ T=4J; and Z pZ p 1 d~ q IðHÞ ¼ 2 : 2p 0 0 1 gq þ 2Deff =J
For large fields, magnetic saturation effects become important and the phase line bends back towards the field axis intersecting it at H ¼ Hc : The stiffness goes abruptly to zero at a temperature T0 given by
1 T0 ðHÞ ¼ 4JS 2 sin2 y0 e þ sin2 y0 IðHÞ : ð20Þ For small values of Deff we find
r
T0 ¼
Redefining the spin component Snz by Snz ¼ S cosðyn y0 Þ;
ð14Þ
we see that, from the thermodynamical point of view, our original model is equivalent to an anisotropic ferromagnet described by the
ð19Þ
4JS2 : lnðpJ=4Deff Þ
ð21Þ
This equation is in agreement with detailed renormalization group calculation [12]. The stiffness calculated using the SCHA does not incorporate the effect of polarization by bound vortex
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
5. Rb2 MnF4 A material which is particularly close to classical two-dimensional models seems to be Rb2 MnF4 which has spin S ¼ 5=2 [16]. At zero field this compound is a weakly Ising antiferromagnet with J ¼ 7:31 K: The principal spin anisotropy is an uniaxial magnetic interaction with gmB HA ¼ 0:371 K evaluated at a temperature of T ¼ 4:2 K; along the z-axis perpendicular to the magnetic plane. Using the relation DS 2 ¼ gmB HA S; we find D ¼ 0:148 K: The magnetic ordering has been described by Birgeneau et al. [17]. In zero field, the system undergoes a transition to a 2D magnetically ordered structure at 38:4 K: The line of zero anisotropy was experimentally found to be
approximately given by [1] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hsf ¼ 28:09 þ 0:23T ;
ð22Þ
where T is the temperature in Kelvin, and H is given in Tesla. For a 2D system, described by Hamiltonian (1), we would have just a bicritical point at T ¼ 0: A line of zero anisotropy implies the existence of long range order for H ¼ Hsf : The comparison of our theoretical calculation with the experiments of Ref. [1] is shown in Figs. 4 and 5. A quantum SCHA for an easy-plane model with S ¼ 5=2 gives results similar to the ones using the classical approximation [18] and, therefore, in the XY region we are allowed to use the simpler classical approach presented in Section 4. The higher-field phase boundary can be fitted reasonably well by our theory. In Fig. 4 we use the value D ¼ 0:148 K; and in Fig. 5 we use a temperature dependent value obtained using Eq. (22). For H
7
6
5
4 Field (T)
pairs. This latter mechanism is responsible for a shift in the transition temperature [13]. At TBKT renormalization group analysis [14] shows that the stiffness should exhibit a universal jump given by 2TBKT =p: The BKT temperature for our model can then be determined by the crossing between the rðTÞ curve, calculated using Eq. (18) and the line g ¼ 2T=p: Our calculation for the XY region is also presented in Fig. 2. The two phase boundaries meet at T ¼ 0 for H ¼ Hsf ; with a horizontal tangent. At low temperatures, the two critical lines are so close to each other, that they cannot be distinguished from a single spin-flop line. In the XY region, an applied field is not really equivalent to an anisotropy: increasing the field would lead the spins to tilt out of the XY plane while a true anisotropy would force the spins to lie in the XY plane. The SCHA works reasonably well for the anisotropic easy-plane Heisenberg model [11–15], and the effect of the tilting out of the plane was taken into account by the sin2 y0 term in Eq. (18). However, even so, the quantitative agreement is good only for the regions HBHsf and HBHc ; for other values of H; the agreement is not so good. It seems, then, that the reduction of the spin in-plane component was not completely taken into account in our calculation and a more elaborated theory should be developed.
321
3
2
1
0
0
10
20 Temperature (K)
30
40
Fig. 4. Phase diagram for Rb2 MnF4 in an external magnetic field perpendicular to the magnetic planes. The experimental data (circles) are from Ref. [1]. Squares represent Green function calculation in the Ising region and SCHA in the XY region. Diamonds are the SCR calculation.
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
322
This technique has been used by Dalton and Wood [20] and Reinehr Figueiredo [21] to study the ferromagnet with exchange anisotropy. The retarded Green function for Heisenberg operators A and B is defined as
7
Field (T)
6
GA;B ðtÞ iyðtÞ/½AðtÞ; Bð0Þ S;
ð23Þ
in which yðtÞ is the step function equal to 1 for t > 0; 0 for to0: The equation of motion for the time-Fourier transformation of GAB ðtÞ given by Z 1 þN GAB ðtÞeiot dt; 0A; BT ¼ 2p N
5
is 4
o0A; BT ¼
3
0
10
20 Temperature (K)
30
40
Fig. 5. Phase diagram for Rb2 MnF4 in an external magnetic field perpendicular to the magnetic planes.The experimental data (circles) are from Ref. [1]. Squares are Green function calculations taking into account the interplanar coupling. Diamonds are Green function calculations in the Ising region and SCHA in the XY region, with a temperature dependent anisotropy.
up to 7 T, the effective anisotropy formulation is expected to work since, in this case, Hc E65 T: In the Ising phase, for a more precise comparison with the experimental data, we performed a quantum SCR calculation. We obtain the quantum SCR expression by replacing T=Eq by ðexpðEq =TÞ 1Þ1 in Eqs. (7) and (8) [9]. However, now, due to the exponential term, the numerical solution of the self consistent equations must be performed in a very carefull way since the accuracy decreases with decreasing temperature. In Fig. 4 we also show the theoretical calculation using the SCR technique. The results are not very different from the ones obtained using the Green function technique described below which is a more convenient approach and easier to calculate [19].
1 /½A; B S þ 0½AðtÞ; H ; Bð0ÞT; 2p ð24Þ
where H is the Hamiltonian of the system. For Hamiltonian (1) we find 1 o0Snþ ; Sm T ¼ /Snz Sdm;n p X þ þJ 0Snþd Snz ; Sm T d z þ T 0Sn Snþd ; Sm
z þ 0Snþd Sn ; Sm T T þ Deff 0Snþ Snz ; Sm
z þ þ 0Sn Sn ; Sm T :
ð25Þ
In order to solve Eq. (25) we use the random-phase approximation to decouple the higher order Green function, this is þ þ ; Sj T ¼ /Snz S0Sm ; Sj T; 0Snz Sm
ð26Þ
to obtain T o0Snþ ; Sm 1 ¼ /Snz Sdm;n þ 2J/Snz S pX
þ 0Snþd ; Sm T þ 0Snþ ; Sm T d T: þ 2Deff /Snz S0Snþ ; Sm
ð27Þ
We remark that in the limit J-0 (and H ¼ 0) we have /S z S-0 even if Da0; and the above
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
procedure does not work. Making use of the translational invariance of the system we may define the Fourier transforms X Gðq; oÞ ¼ 0Snþ ; Sm TeiqðnmÞ ; ð28Þ m;n
in terms of which Eq. (27) becomes Gðq; oÞ ¼
/Sz S 1 ; p o oðqÞ
ð29Þ
where oðqÞ ¼ 8/Sz S½Jð1 gq Þ þ Deff ;
ð30Þ
is the magnon energy spectrum and gq was defined in (9). Using the relation Z þN Gðo þ ieÞ Gðo ieÞ do; /ABS ¼ lim i e-0 expðboðqÞÞ 1 N ð31Þ we find the following expression for the magnetization: Z 1 S boðqÞ coth ¼ ð32Þ dq2 : /Sz S ð2pÞ2 2 Near the critical temperature, where /Sz S goes to zero, we can expand the argument in the integral and obtain Z 1 1 1 d2 q ¼ : ð33Þ 2 Tc 4S 2p Jð1 gq Þ þ Deff Performing the integral in the qy variable, we find Z p 1 1 1 dqx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ ð34Þ Tc 4JS 2p2 0 ð2 þ B cosqx Þ2 1 where B ¼ 2Deff =J: For B small we can solve the integral analytically and obtain p 1 1 ln ¼ : ð35Þ Tc 4JSp B Before interpreting the data for Rb2 MnF4 let us apply Eq. (34) to two other compounds [22]. For K2 MnF4 we have J ¼ 8:4 K; D ¼ 0:134 K which leads to Tc ¼ 42:73 K: The experimental value is 42:3 K: The second compound is MnðHCOOÞ2
323
2H2 O: We have J ¼ 0:70 K; D ¼ 0:0114 K: We find Tc ¼ 3:57 K; to be compared with the experimental value 3.68 K. Just for comparison, the quantum SCR gives Tc ¼ 42:19 K and Tc ¼ 4:09 K respectively. For Rb2 MnF4 ; if we take D temperature dependent, using for D the effective anisotropy defined in Eq. (2) with Hsf given by Eq. (22) and calculate Tc using Eq. (34) we obtain Tc ¼ 40:53 K: Otherwise using the temperature independent value for the anisotropy, D ¼ 0:148 K; we find Tc ¼ 38:73 K; to be compared with the experimental value 38:4 K: In the Ising region are seen deviations from the calculations. This can be attributed to the destruction of the pure planar anisotropy in the high field phase. The 2D XY phase is extremely sensitive to symmetry breaking interactions and to the interplanar coupling. Any of these effects move the bicritical point from T ¼ 0 to a non-zero temperature. The interplanar coupling is often so small that the observed transition is primarily induced by anisotropy. The experimental data are consistent with a BCP at a temperature near 30 K. Let us consider an orthorhombic anisotropy: this would correspond to Hamiltonian (1) with an additional small anisotropy term Dx ðSnx Þ2 : For HoHsf the effective anisotropy is of the Ising-type and the behavior is comparable to the uniaxial case. Let us take H ¼ Heff ; use the experimental value Tc ¼ 30 K and determine the value of the parameter Dx : For a positive Dx a Kosterlitz– Thouless transition should occur at the bicritical point Tbc ¼ 30 K if Dx ¼ 0:0212 K: Otherwise a negative term would lead to an order–disorder (Ising) transition with Dx ¼ 0:028 K: An interplanar coupling J 0 ; treated by the Green function method, would have to have the value a ¼ J 0 =J ¼ 0:01; to give Tbc ¼ 30 K; a value too high since it is expected that a should be around 106 : We remark that this last value of a leads to Tbc ¼ 12:5 K: The dashed line in Fig. 5 describes the result considering interplanar coupling with a ¼ 0:01: A similar curve would be obtained in the case of an orthorhombic anisotropy. Here we have used the values of J and D given in Ref. [1]. The experimental data of those authors differ slightly from the ones obtained by Cowley et al. [16]. Of course we could vary D and Dx to get a better fit to
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experimental data. However, considering the approximate nature of the theories we have used, and the precision of the experimental data we do not see a valid reason for doing so. We conclude that it is not yet completely clear what the mechanism is which is responsible for the bicritical point in Rb2 MnF4 : More detailed experiments are needed to elucidate this point.
Acknowledgements This work was partially supported by CNPq and FAPEMIG (Brazilian agencies). Numerical work was done in the LINUX parallel cluster at the ! Laboratorio de Simulac-ao * Departamento de F!ısica - UFMG.
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Phase diagrams of a two-dimensional Heisenberg antiferromagnet with single-ion anisotropy B.V. Costa, A.S.T. Pires* Departamento de F!ısica ICEX, Universidade Federal de Minas Gerais Caixa Postal 702, 30123-970 Belo Horizonte, MG, Brazil Received 30 August 2002; received in revised form 4 December 2002
Abstract A two-dimensional Heisenberg antiferromagnet with a single-ion anisotropy is studied in the presence of a uniform magnetic field along the easy axis. The concept of effective field-dependent anisotropy is applied to the theoretical description of the phase diagrams. We perform calculations using the self-consistent harmonic approximation to treat the XY phase and the Green function technique for the Ising phase. Monte Carlo simulations on L L square lattices, with L ¼ 8; 16; 32 and 64; and with periodic boundary conditions were performed for the classical model. In order to test the validity of the effective anisotropy concept we compare our theoretical calculation with simulation data. We also calculate theoretically the phase diagram of the compound Rb2 MnF4 ; a quasi-two-dimensional spin 5=2 antiferromagnet, finding good agreement with experimental data. Our estimates for the transition temperature of the compounds K2 MnF4 and MnðHCOOÞ2 2H2 O; in zero magnetic field, are also in agreement with the experimental values. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Two-dimensional Heisenberg antiferromagnet; Single-ion anisotropy; Phase diagrams; Monte Carlo simulations
1. Introduction The interest in the properties of two-dimensional Heisenberg magnets has been greatly revived since the discovery of high-Tc superconductivity. Thus, the study of magnetism in two-dimensions has become interesting to both, theoreticians and experimentalists, and significant progress in the understanding of two-dimensional Heisenberg magnets has been achieved as a result of a symbiotic interplay between theory, simula*Corresponding author. E-mail addresses: bvc@fisica.ufmg.br antpires@fisica.ufmg.br (A.S.T. Pires).
(B.V.
Costa),
tion and experiments. It is now well known, that most quasi-two-dimensional magnetic materials exhibit some kind of anisotropy: it is rare that anisotropic properties arise from an anisotropy in the interaction, (which can be wholly isotropic), but, instead, it is caused by other sources, such as the presence of a crystal field that couples the spins along a certain direction in the crystal. The analysis of the anisotropic Heisenberg model is then quite interesting because, from the experimental point of view, the presence of some degree of anisotropy in the interaction is to be expected in nearly all cases, and because even a very small anisotropy can play an important role when the temperature goes to zero.
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-8853(02)01527-5
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
Although the quantum nature of magnetism cannot be forgotten, the study of classical models continues to be an important subject of research. Field theory yields results which agree closely with experiments on spin-1=2 Heisenberg systems but display strong deviations from the predicted behavior in systems with S > 1=2: In those systems, a broad crossover from quantum to classical behavior occurs at high temperatures [1]. From the theoretical point of view, not everything is settled for the easy axis model. For instance, a renormalization group calculation suggests the existence of a finite critical value of the anisotropy parameter, at which the critical temperature goes to zero [2]. However, numerical simulation and spin wave calculations do not confirm the existence of this critical anisotropy. In this paper we will be interested in the study of the phase diagram of the classical anisotropic Heisenberg antiferromagnet in two dimensions described by the following Hamiltonian: H¼J
X /i;jS
~j D ~i S S
X 2 X Siz gmB H Siz ; ð1Þ i
i
where the summation ð/i; jSÞ is over nearest neighbor pairs, J > 0 and the anisotropy term D establishes an easy axis (Ising-type anisotropy). In zero field the system undergoes a transition to a two-dimensional magnetically ordered structure. As it is well known when a magnetic field of sufficient strength is applied various interesting classes of phase transitions may be induced [3]. Here we consider the case of H applied along the z direction. The low-temperature, low field ordered state is an antiferromagnet (AF) which is separated from the paramagnetic state (P) by a line of second-order phase transitions. At sufficiently low temperature the system undergoes a first-order transition to a canted ‘‘spin-flop’’ state (SP) as the field is increased. The spin-flop phase behaves like a two-dimensional XY model and thus shows no long-range order. The state is separated from the P phase by a second-order phase transition line. The point at which AF and P phases simultaneously become critical is termed the ‘‘bicritical point’’. In two dimensions, if a simple bicritical point occurs, it would be at T ¼ 0 K:
317
The behavior of an anisotropic Heisenberg model similar to Hamiltonian (1) but with an P exchange anisotropy (D i Siz Sjz ) was first studied by Binder and Landau [4] using Monte Carlo computer simulations. The exchange anisotropic model, in zero magnetic field, has been recently studied by Cuccoli et al. [5] using the pure quantum-self-consistent harmonic approximation. Although the qualitative features of both models, at low temperatures with single-site anisotropy or exchange anisotropy, are expected to be the same, quantitative differences do occur. In a theoretical approach, the field dependence of Hamiltonian (1) can be taken into account as an effective field-dependent anisotropy Deff ðHÞ [3]. For a field H; applied along the easy-axis, smaller than the spin-flop field Hsf ; the effective anisotropy is given by H2 Deff ðHÞ ¼ D 1 2 ; ð2Þ Hsf where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4JD D2 Hsf ¼ 2S gmB
ð3Þ
for a square lattice. Clearly, while HoHsf ; the effective anisotropy Deff is an Ising-type anisotropy that vanishes at Hsf : On the other hand, as H increases to H > Hsf ; Deff becomes negative playing the role of a planar anisotropy which forces the spins to the lie in the XY -plane. In this XY -case, the planar behavior is enhanced as H is increased but this effective anisotropy approximation is valid for fields H52He ; where He is the exchange field given by He ¼ 4JS=gmB when the system is far from saturation. The critical field Hc ; where the system saturates at T ¼ 0; is given by Hc ¼ 2S
ð4J DÞ : gmB
ð4Þ
It is well known that in the classical limit, the thermodynamic properties of the antiferromagnet are the same as the ones for the ferromagnet. This allows us to do the theoretical calculations using a ferromagnet. Therefore, we may now treat the system by means of the effective
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
Hamiltonian X X 2 ~j Deff ðHÞ ~i S Siz Heff ¼ J S /i;jS
þ constant:
i
ð5Þ
A detailed derivation of this effective Hamiltonian can be found in Refs. [3,6]. Of course, the concept of an effective field-dependent anisotropy can be applied only to the classical limit (i.e. for large values of the spin). In fact for S ¼ 1=2 the single-ion anisotropy plays no role. In the usual studies of phase diagrams of 2D AF, experimental data have been qualitatively compared with simulations [1,3]. In this work, we will go one step further by comparing experimental data and simulations with theory. This paper is organized as follows: in Section 2, we present Monte Carlo simulations. In Section 3 we study the Ising phase using the self consistent renormalized spin-wave theory. The study of the XY phase using a self consistent harmonic approximation is presented in Section 4. Finally in Section 5 we analyse some previously reported experimental data for the compound Rb2 MnF4 :
2. Monte Carlo Monte Carlo calculations for the two-dimensional exchange anisotropic Heisenberg model were performed by Binder and Landau [4]. Those authors obtained the phase diagram for that model, but a plot of Tc as a function of the anisotropy was not presented. Also, at the time that the work was done, the possibility of a phase transition for the 2D isotropic Heisenberg model (as suggested by high-temperature series extrapolation) had not been completely ruled out. Later, Serena et al. [7], using Monte Carlo simulation with an improved algorithm, obtained reliable results for several thermodynamic properties of the same model. Our simulations of Hamiltonian (1) were carried out using the standard Metropolis algorithm. We have used lattices of size L L with L ¼ 8; 16; 32; 64 with periodic boundary conditions. In order to reach thermodynamic equilibrium, we
performed long runs of size 100 L L: In order to extract the critical temperature Tc in the Ising phase, the position of the maxima of the specific heat and magnetic susceptibility and, also, the fourth order Binder cumulant were analyzed. To get the Berezinskii–Kosterlitz–Thouless (BKT) temperature TBKT we analyzed the helicity modulus [8]. The procedure we have adopted is as follows: we fixed the anisotropy D while the magnetic field H was varied. For each pair of values, ðH; DÞ; we then obtained the specific heat. It is known that the specific heat maximum Cmax does not change with L in the BKT region but behaves as Cmax pln L in the Ising region. This distinct behavior of Cmax can guide us in deciding in which region, Ising or BKT we are. Then, the corresponding critical temperature is obtained as discussed above. Although the model has an Ising or a BKT transition for any value of D; it is very difficult to perform the simulation for low values of the anisotropy, because the critical temperature goes down monotonically with D; and close to T ¼ 0 the system suffers of a strong slowing down. For this reason, we have used the following values for the anisotropy parameter, D=J ¼ 0:25; 0:50; 1:0; 5:0 and 10:0: Our results are shown in Figs. 1, 2 and 3. For all simulations, we have adopted kB ¼ 1; J ¼ 1; S ¼ 1 , and H is in units of gmB : We must note that having an anisotropy parameter of 6 D=0.25 D=0.5 D=1.00
5 4 H
318
3 2 1 0
0
1
0.5 Tc
Fig. 1. Phase diagramm H=JS2 Tc for some anisotropy values as indicated in the insert. Error bars are smaller than the symbols when not indicated. Lines are guide to the eyes.
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
takes into account the effects of dynamic interactions between spin waves in the Hartee-Fock approximation. This theory was introduced long ago for the Heisenberg magnet with uniaxial anisotropy by Rastelli et al. [9] and here we reproduce their final result for the classical limit. The temperature dependent spin-wave spectrum for Hamiltonian (5) is given by
6 5
Field
4 3 2
Eq ðTÞ ¼ 4JSð1 gq Þ½1 bðTÞ þ ZðTÞ þ 2DS½1 2bðTÞ ;
1
where 0
319
0
0.5 Temperature
T bðTÞ ¼ 2 p
1
2
Fig. 2. Phase diagramm H=JS Tc for D=J ¼ 1:0: Circles and squares are simulation and theoretical results, respectively. The lines are guide to the eyes.
ZðTÞ ¼
T p2
Z
p
Z
0
Z
0 p
0
p
Z 0
p
ð6Þ
d~ q ; Eq ðTÞ
ð7Þ
gq d~ q ; Eq ðTÞ
ð8Þ
and 2
Ln (Tc/ D)
1
0
-1
-2
-2
-1
0
1
2
3
Ln (D) Fig. 3. Logarithm of Tc D as a function of ln D: Circles are simulation data and the line is the best fit.
the order of the exchange constant (D=JE1) is not an unrealistic case since we have D=J ¼ 1:58 [9] for the compound FeCl2 :
3. Ising region The self-consistently renormalized (SCR) spinwave theory extends the temperature range in which the spin-wave theory is valid because it
1 gq ¼ ðcos qx þ cos qy Þ: ð9Þ 2 Eqs. (6)–(8) constitute a set of self-consistent equations for Eq ðTÞ which were solved numerically by an iterative method. These equations have a solution up to a maximum temperature Tc which we consider to constitute the transition temperature. As can be easily seen, the SCR theory correctly predicts that the transition temperature is zero when D ¼ 0: In Fig. 2, we compare our theoretical calculation with Monte Carlo data for D=J ¼ 1:0: Here H is in units of gmB : As we can see, the overall agreement in the Ising region is reasonable. Then, when an uniform field, smaller than Hsf ; is applied, the Ising-like anisotropic model continues to provide an adequate description of the critical behavior. For the exchange anisotropic case with H ¼ 0 and when D-N (or J-0), the system behaves like the pure Ising model and Tc =D-1 [4]. However, for Hamiltonian (1), we know that there is no phase transition for J ¼ 0: It is therefore interesting to study the behavior of Tc as a function of D: In Fig. 3 we present our simulation data for lnðTc =DÞ as a function of lnðDÞ: The data obtained can be modelled using the following expression Tc ln ¼ a b lnðDÞ; ð10Þ D
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320
with a ¼ 0:070ð12Þ and b ¼ 0:80ð10Þ: As we can see for large values of D we have Tc =DpDb quite different from the exchange anisotropic case [10]. Notice that we cannot obtain results using the SCR theory for the single-ion anisotropic Hamiltonian, for D-N; because this region is out of the validity range of the theory. However, for the exchange anisotropic case, the SCR theory gives good agreement with classical Monte Carlo simulation for any value of the anisotropy parameter [10].
4. XY region The two-dimensional classical model has a transition into a phase with algebraically decaying correlation functions and infinite susceptibility. The transition temperature, called the BKT temperature TBKT ; can be calculated using a selfconsistent harmonic approximation [11]. In order to properly treat the magnetic field we will first introduce the parametrization.
~n ¼ ð1Þn S sin½yn þ ð1Þn y0 cosf ; S n sin ½yn þ ð1Þn y0 sin fn ; cos½yn þ ð1Þn y0 ; ð11Þ where cos y0 ¼
H Hsf : Hc Hsf
ð12Þ
Inserting Eq. (12) into Hamiltonian (1) we find that keeping only terms up to second order the Hamiltonian becomes X 2 H J 2 X ¼ sin y0 fr frþa 2J ð yr y0 Þ 2 2 S 4 r;a r JX gmB H ðyr y0 Þðyrþa y0 Þ þ 2 r;a 2 X cos y0 ð yr y0 Þ 2 : ð13Þ
Hamiltonian
J X 2 x x y z sin y0 Sr Srþa þ Sry Srþa þ Srz Srþa H¼ 2 r;a X þ Deff ðSrz Þ2 ; ð15Þ r;a
with Deff the effective field-dependent anisotropy. Now, in order to use the SCHA we follow the same procedure used in Ref. [11] obtaining the quadratic form for the Hamiltonian J X H¼ rð1 gq Þfq fq 2 q Deff z z þ ð1 gq Þ þ 2 ð16Þ Sq Sq ; J where gq is given by Eq. (9) and the stiffness r is given by D
1=2
1=2 z r ¼ sin2 y0 1 ðSrz Þ2 1 ðSrþa Þ2 E ð17Þ cos ðfrþa fr Þ : The stiffness takes into account anharmonic terms neglected when we write the original Hamiltonian in the harmonic form. Following Ref. [11] we find that we can write Eq. (17) as t r ¼ sin2 y0 ½1 tIðHÞ exp ; ð18Þ r where t ¼ T=4J; and Z pZ p 1 d~ q IðHÞ ¼ 2 : 2p 0 0 1 gq þ 2Deff =J
For large fields, magnetic saturation effects become important and the phase line bends back towards the field axis intersecting it at H ¼ Hc : The stiffness goes abruptly to zero at a temperature T0 given by
1 T0 ðHÞ ¼ 4JS 2 sin2 y0 e þ sin2 y0 IðHÞ : ð20Þ For small values of Deff we find
r
T0 ¼
Redefining the spin component Snz by Snz ¼ S cosðyn y0 Þ;
ð14Þ
we see that, from the thermodynamical point of view, our original model is equivalent to an anisotropic ferromagnet described by the
ð19Þ
4JS2 : lnðpJ=4Deff Þ
ð21Þ
This equation is in agreement with detailed renormalization group calculation [12]. The stiffness calculated using the SCHA does not incorporate the effect of polarization by bound vortex
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
5. Rb2 MnF4 A material which is particularly close to classical two-dimensional models seems to be Rb2 MnF4 which has spin S ¼ 5=2 [16]. At zero field this compound is a weakly Ising antiferromagnet with J ¼ 7:31 K: The principal spin anisotropy is an uniaxial magnetic interaction with gmB HA ¼ 0:371 K evaluated at a temperature of T ¼ 4:2 K; along the z-axis perpendicular to the magnetic plane. Using the relation DS 2 ¼ gmB HA S; we find D ¼ 0:148 K: The magnetic ordering has been described by Birgeneau et al. [17]. In zero field, the system undergoes a transition to a 2D magnetically ordered structure at 38:4 K: The line of zero anisotropy was experimentally found to be
approximately given by [1] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hsf ¼ 28:09 þ 0:23T ;
ð22Þ
where T is the temperature in Kelvin, and H is given in Tesla. For a 2D system, described by Hamiltonian (1), we would have just a bicritical point at T ¼ 0: A line of zero anisotropy implies the existence of long range order for H ¼ Hsf : The comparison of our theoretical calculation with the experiments of Ref. [1] is shown in Figs. 4 and 5. A quantum SCHA for an easy-plane model with S ¼ 5=2 gives results similar to the ones using the classical approximation [18] and, therefore, in the XY region we are allowed to use the simpler classical approach presented in Section 4. The higher-field phase boundary can be fitted reasonably well by our theory. In Fig. 4 we use the value D ¼ 0:148 K; and in Fig. 5 we use a temperature dependent value obtained using Eq. (22). For H
7
6
5
4 Field (T)
pairs. This latter mechanism is responsible for a shift in the transition temperature [13]. At TBKT renormalization group analysis [14] shows that the stiffness should exhibit a universal jump given by 2TBKT =p: The BKT temperature for our model can then be determined by the crossing between the rðTÞ curve, calculated using Eq. (18) and the line g ¼ 2T=p: Our calculation for the XY region is also presented in Fig. 2. The two phase boundaries meet at T ¼ 0 for H ¼ Hsf ; with a horizontal tangent. At low temperatures, the two critical lines are so close to each other, that they cannot be distinguished from a single spin-flop line. In the XY region, an applied field is not really equivalent to an anisotropy: increasing the field would lead the spins to tilt out of the XY plane while a true anisotropy would force the spins to lie in the XY plane. The SCHA works reasonably well for the anisotropic easy-plane Heisenberg model [11–15], and the effect of the tilting out of the plane was taken into account by the sin2 y0 term in Eq. (18). However, even so, the quantitative agreement is good only for the regions HBHsf and HBHc ; for other values of H; the agreement is not so good. It seems, then, that the reduction of the spin in-plane component was not completely taken into account in our calculation and a more elaborated theory should be developed.
321
3
2
1
0
0
10
20 Temperature (K)
30
40
Fig. 4. Phase diagram for Rb2 MnF4 in an external magnetic field perpendicular to the magnetic planes. The experimental data (circles) are from Ref. [1]. Squares represent Green function calculation in the Ising region and SCHA in the XY region. Diamonds are the SCR calculation.
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
322
This technique has been used by Dalton and Wood [20] and Reinehr Figueiredo [21] to study the ferromagnet with exchange anisotropy. The retarded Green function for Heisenberg operators A and B is defined as
7
Field (T)
6
GA;B ðtÞ iyðtÞ/½AðtÞ; Bð0Þ S;
ð23Þ
in which yðtÞ is the step function equal to 1 for t > 0; 0 for to0: The equation of motion for the time-Fourier transformation of GAB ðtÞ given by Z 1 þN GAB ðtÞeiot dt; 0A; BT ¼ 2p N
5
is 4
o0A; BT ¼
3
0
10
20 Temperature (K)
30
40
Fig. 5. Phase diagram for Rb2 MnF4 in an external magnetic field perpendicular to the magnetic planes.The experimental data (circles) are from Ref. [1]. Squares are Green function calculations taking into account the interplanar coupling. Diamonds are Green function calculations in the Ising region and SCHA in the XY region, with a temperature dependent anisotropy.
up to 7 T, the effective anisotropy formulation is expected to work since, in this case, Hc E65 T: In the Ising phase, for a more precise comparison with the experimental data, we performed a quantum SCR calculation. We obtain the quantum SCR expression by replacing T=Eq by ðexpðEq =TÞ 1Þ1 in Eqs. (7) and (8) [9]. However, now, due to the exponential term, the numerical solution of the self consistent equations must be performed in a very carefull way since the accuracy decreases with decreasing temperature. In Fig. 4 we also show the theoretical calculation using the SCR technique. The results are not very different from the ones obtained using the Green function technique described below which is a more convenient approach and easier to calculate [19].
1 /½A; B S þ 0½AðtÞ; H ; Bð0ÞT; 2p ð24Þ
where H is the Hamiltonian of the system. For Hamiltonian (1) we find 1 o0Snþ ; Sm T ¼ /Snz Sdm;n p X þ þJ 0Snþd Snz ; Sm T d z þ T 0Sn Snþd ; Sm
z þ 0Snþd Sn ; Sm T T þ Deff 0Snþ Snz ; Sm
z þ þ 0Sn Sn ; Sm T :
ð25Þ
In order to solve Eq. (25) we use the random-phase approximation to decouple the higher order Green function, this is þ þ ; Sj T ¼ /Snz S0Sm ; Sj T; 0Snz Sm
ð26Þ
to obtain T o0Snþ ; Sm 1 ¼ /Snz Sdm;n þ 2J/Snz S pX
þ 0Snþd ; Sm T þ 0Snþ ; Sm T d T: þ 2Deff /Snz S0Snþ ; Sm
ð27Þ
We remark that in the limit J-0 (and H ¼ 0) we have /S z S-0 even if Da0; and the above
B.V. Costa, A.S.T. Pires / Journal of Magnetism and Magnetic Materials 262 (2003) 316–324
procedure does not work. Making use of the translational invariance of the system we may define the Fourier transforms X Gðq; oÞ ¼ 0Snþ ; Sm TeiqðnmÞ ; ð28Þ m;n
in terms of which Eq. (27) becomes Gðq; oÞ ¼
/Sz S 1 ; p o oðqÞ
ð29Þ
where oðqÞ ¼ 8/Sz S½Jð1 gq Þ þ Deff ;
ð30Þ
is the magnon energy spectrum and gq was defined in (9). Using the relation Z þN Gðo þ ieÞ Gðo ieÞ do; /ABS ¼ lim i e-0 expðboðqÞÞ 1 N ð31Þ we find the following expression for the magnetization: Z 1 S boðqÞ coth ¼ ð32Þ dq2 : /Sz S ð2pÞ2 2 Near the critical temperature, where /Sz S goes to zero, we can expand the argument in the integral and obtain Z 1 1 1 d2 q ¼ : ð33Þ 2 Tc 4S 2p Jð1 gq Þ þ Deff Performing the integral in the qy variable, we find Z p 1 1 1 dqx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ ð34Þ Tc 4JS 2p2 0 ð2 þ B cosqx Þ2 1 where B ¼ 2Deff =J: For B small we can solve the integral analytically and obtain p 1 1 ln ¼ : ð35Þ Tc 4JSp B Before interpreting the data for Rb2 MnF4 let us apply Eq. (34) to two other compounds [22]. For K2 MnF4 we have J ¼ 8:4 K; D ¼ 0:134 K which leads to Tc ¼ 42:73 K: The experimental value is 42:3 K: The second compound is MnðHCOOÞ2
323
2H2 O: We have J ¼ 0:70 K; D ¼ 0:0114 K: We find Tc ¼ 3:57 K; to be compared with the experimental value 3.68 K. Just for comparison, the quantum SCR gives Tc ¼ 42:19 K and Tc ¼ 4:09 K respectively. For Rb2 MnF4 ; if we take D temperature dependent, using for D the effective anisotropy defined in Eq. (2) with Hsf given by Eq. (22) and calculate Tc using Eq. (34) we obtain Tc ¼ 40:53 K: Otherwise using the temperature independent value for the anisotropy, D ¼ 0:148 K; we find Tc ¼ 38:73 K; to be compared with the experimental value 38:4 K: In the Ising region are seen deviations from the calculations. This can be attributed to the destruction of the pure planar anisotropy in the high field phase. The 2D XY phase is extremely sensitive to symmetry breaking interactions and to the interplanar coupling. Any of these effects move the bicritical point from T ¼ 0 to a non-zero temperature. The interplanar coupling is often so small that the observed transition is primarily induced by anisotropy. The experimental data are consistent with a BCP at a temperature near 30 K. Let us consider an orthorhombic anisotropy: this would correspond to Hamiltonian (1) with an additional small anisotropy term Dx ðSnx Þ2 : For HoHsf the effective anisotropy is of the Ising-type and the behavior is comparable to the uniaxial case. Let us take H ¼ Heff ; use the experimental value Tc ¼ 30 K and determine the value of the parameter Dx : For a positive Dx a Kosterlitz– Thouless transition should occur at the bicritical point Tbc ¼ 30 K if Dx ¼ 0:0212 K: Otherwise a negative term would lead to an order–disorder (Ising) transition with Dx ¼ 0:028 K: An interplanar coupling J 0 ; treated by the Green function method, would have to have the value a ¼ J 0 =J ¼ 0:01; to give Tbc ¼ 30 K; a value too high since it is expected that a should be around 106 : We remark that this last value of a leads to Tbc ¼ 12:5 K: The dashed line in Fig. 5 describes the result considering interplanar coupling with a ¼ 0:01: A similar curve would be obtained in the case of an orthorhombic anisotropy. Here we have used the values of J and D given in Ref. [1]. The experimental data of those authors differ slightly from the ones obtained by Cowley et al. [16]. Of course we could vary D and Dx to get a better fit to
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experimental data. However, considering the approximate nature of the theories we have used, and the precision of the experimental data we do not see a valid reason for doing so. We conclude that it is not yet completely clear what the mechanism is which is responsible for the bicritical point in Rb2 MnF4 : More detailed experiments are needed to elucidate this point.
Acknowledgements This work was partially supported by CNPq and FAPEMIG (Brazilian agencies). Numerical work was done in the LINUX parallel cluster at the ! Laboratorio de Simulac-ao * Departamento de F!ısica - UFMG.
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