Possible phase transition of anisotropic frustrated Heisenberg model at finite temperature

Abstract

The frustrated spin-1/2 J_1a–J_1b–J₂ antiferromagnet with anisotropy on the two-dimensional square lattice was investigated, where the parameters J_{1a} and J_{1b} represent the nearest neighbor exchanges and along the x and y directions, respectively. J₂ represents the next-nearest neighbor exchange. The anisotropy includes the spatial and exchange anisotropies. Using the double-time Green’s function method, the effects of the interplay of exchanges and anisotropy on the possible phase transition of the Néel state and stripe state were discussed. Our results indicated that, in the case of anisotropic parameter , the Néel and stripe states can exist and have the same critical temperature as long as J_2=J_{1b}/2. Under such parameters, a first-order phase transformation between the Néel and stripe states can occur below the critical point. For J_2\ne J_{1b}/2, our results indicate that the Néel and stripe states can also exist, while their critical temperatures differ. When J_2>J_{1b}/2, a first-order phase transformation between the two states may also occur. However, for , the Néel state is always more stable than the stripe state.

Graphical abstract

Keywords

frustrated Heisenberg model / quantum phase transition / magnetic anisotropy / antiferromagnetics

1 Introduction

In recent years, the study of frustrated spin model has become very active on two-dimensional (2D) square lattice with nearest-neighbor (NN) and next-nearest-neighbor (NNN) antiferromagnetic exchange interactions (known as J₁–J₂ model) [1-12]. The frustration is caused by the mutual competition between J₁ and J₂, which can trigger a rich phase diagram [1]. Therefore, in the past, this model had been widely investigated by means of various theoretical methods (see, e.g., [12], and references). It was well known that the ground state usually showed two possible antiferromagnetic states. One is called Néel state (NS), and the other is called stripe state (SS). The NS was characterized by an antiparallel alignment of NN spins with a corresponding magnetic wave vector Q_{\text{N}\text{S}}(\pi \mathrm{,}\pi ), see Fig. 1. The SS was twofold degenerated and the corresponding magnetic wave vectors were Q_{\text{S}\text{S}}^1(\pi \mathrm{,0}) and Q_{\text{S}\text{S}}^2(\mathrm{0,}\pi ), which were characterized by the fact that the NN spins take a parallel orientation in the vertical (horizontal) direction and an antiparallel orientation in the horizontal (vertical) direction, respectively, see Fig. 1. Investigations indicated that the system was an NS for \alpha =J_2/J_1\lesssim 0.38, and for \alpha \gtrsim 0.6 it turned to be an SS. When 0.38\lesssim \alpha \lesssim 0.6, it was a nonmagnetic quantum phase [1-12]. At zero temperature, the key question is to understand the nature of the non-magnetic phase in the intermediate J_2/J_1 regime [3, 13].

For finite temperature, there is no long-range order for isotropic 2D model [14]. Usually, an anisotropy is considered. This is because that an anisotropy, no matter how is faint, will cause a long-range order at finite temperature. Therefore, at finite temperature, the investigative emphasis is to discuss the effect of the anisotropy on the phase diagram. For example, Viana et al. studied the phase diagram of a exchange anisotropic J₁–J₂ model at finite temperature [2]. Their results indicated that between the paramagnetic and stripe phases the system underwent a first-order transition at low temperature and a second-order transition at high temperature. The investigation of Roscilde et al. showed that when an Ising type exchange anisotropy was induced, there would be Chandra–Coleman–Larkin transition and Berezinskii–Kosterlitz–Thouless transition [9]. These investigations indicated that the anisotropy played an important role.

A generalization of the J₁–J₂ Heisenberg antiferromagnetic model on a square lattice was introduced by Nersesyan and Tsvelik [15], the so-called J_1a–J_1b–J₂ model [16, 17]. It possesses more degrees of freedom to tune the fluctuation of system compared to the J₁–J₂ model, since the NN exchange interactions J_{1a} and J_{1b} along the x and y directions in this model are of different strengths. This feature leads naturally to an increased sensitivity of the underlying Hamiltonian to the presence of small perturbations.

Extensive band structure calculations for the vanadium phosphate compounds Pb₂VO(PO₄)₂, SrZnVO(PO₄)₂, BaZnVO(PO₄)₂, BaCdVO(PO₄)₂ had shown four different exchange couplings: J_{1a} and J_{1b} between the NN and J₂ and J_2^{\prime } between NNN [18]. For example, J_{1b}/J_{1a}\approx 0.7 and J_2^{\prime }/J_2\approx 0.4 were obtained for SrZnVO(PO₄)₂. A possible candidate for J_1a–J_1b–J₂ model might be the compound (NO)Cu(NO₃)₃ though band-structure calculations showed a uniform spin chain model with different types of anisotropy and weak interchain couplings [19, 20].

Our primary purpose of this work is to study the phase transition of the J_1a–J_1b–J₂ model at finite temperature by using the double-time Green’s function (DTGF) method. Our results indicate that at (where η is the anisotropic parameter.) both NS and SS can exist and have the same critical temperature as long as J_2=J_{1b}/2. When J_2\ne J_1/2, these two states may also exist, but their critical temperatures differ. The calculated free energies for these two cases show that a first-order phase transformation between NS and SS below critical points can occur. Which of the states is more stable, it will depend on the spatiality and exchange anisotropies of the system.

This paper is organized as follows. In Section 2, we introduce the theoretical model, and present the formulism of Green’s function to derive the self-consistent equation for evaluation of magnetizations. In Section 3, the numerical results are presented and discussed. Section 4 is our concluding remarks.

2 Model and method

The Hamiltonian of J_1a–J_1b–J₂ model can be written as

\begin{array}{}\text{(1)}& \begin{array}{ll}H=\hfill & J_{1a}{\displaystyle \sum _{\langle i\mathrm{,}j\rangle }}\left[\eta (S_i^x{S}_j^x+S_i^y{S}_j^y)+S_i^z{S}_j^z\right]\hfill \\ \hfill & +J_{1b}{\displaystyle \sum _{\langle i\mathrm{,}j\rangle }}\left[\eta (S_i^x{S}_j^x+S_i^y{S}_j^y)+S_i^z{S}_j^z\right]\hfill \\ \hfill & +J_2{\displaystyle \sum _{[i\mathrm{,}j]}}\left[\eta (S_i^x{S}_j^x+S_i^y{S}_j^y)+S_i^z{S}_j^z\right]\mathrm{,}\hfill \end{array}\end{array}

where S_i^x, S_i^y and S_i^z represent the three components of the spin-S operator for a spin at site i. The sums \langle i\mathrm{,}j\rangle and [i\mathrm{,}j] run over the NN and NNN lattice sites, respectively. η denotes the anisotropic parameter with . Spin quantum number is S=1/2 and the lattice is the 2D square lattice. We fix J_{1a}=1 and change other parameters in computation.

For the sake of convenience, we let Boltzmann constant k_B=1. The sublattice magnetization is defined as m=\langle S^z\rangle. where \langle S^z\rangle is the assembly statistical average of spin operator S^z.

In this paper, we will use the DTGF method to investigate the phase transition properties of the system. The DTGF method used in Refs. [21] and [22] is known as the standard method in the study of magnetic systems. In this method one obtains a nonlinear differential equation in which the higher-order Green’s functions are coupled with the lower-order ones. Each of the higher-order Green’s functions is again written down in the form of a nonlinear equation and so on. To obtain tractable solutions, decoupling procedures have been invoked to terminate the hierarchy of Green’s functions generated by the equations of motion. Many results for the thermodynamic properties have been obtained in a frame of the simplest decoupling, i.e., the random phase approximation decoupling [21, 22]. The decoupling provides a simple enough way for giving results in good agreement with other approaches and experiments in a wide range of temperatures and magnetic fields [23, 24].

The Green’s function (GF) is defined as [21, 22]

\begin{array}{}\text{(2)}& G_{ij}^{+-}=\langle \langle S_i^{+};{\text{e}}^{u{S}_j^z}{S}_j^{-}\rangle \rangle \mathrm{.}\end{array}

Here, u is a parameter [22]. After solving the GF by means of the method of equations of motion, u will be ultimately set as zero to give the expression of magnetization [22]. We derive the equation of motion of the GF by a standard procedure [21, 22]. In the course of derivation, the higher order GFs have to be decoupled. In this paper, we apply random phase approximation [21, 22] to decouple the higher order GFs in the equations of motion,

\begin{array}{}\text{(3)}& \langle \langle S_l^z{S}_i^{+};{\text{e}}^{u{S}_j^z}{S}_j^{-}\rangle \rangle =\langle S_l^z\rangle \langle \langle S_i^{+};{\text{e}}^{u{S}_j^z}{S}_j^{-}\rangle \rangle ;l\ne i\mathrm{.}\end{array}

After decoupling of the higher order GFs and standard procedure [21, 22], we obtain

\begin{array}{}\text{(4)}& \frac{2}{N}{\displaystyle \sum _k}\langle {\text{e}}^{u{S}_i^z}{S}_i^{-}{S}_i^{+}\rangle (k)=\theta (u){\varphi }_F\mathrm{,}F=\text{N}\text{S}\mathrm{,}\text{S}\text{S}\mathrm{,}\end{array}

where the summation of wave vector k runs over the first Brillouin zone. N is the number of lattice sites and

\begin{array}{}\text{(5)}& \theta (u)=\langle [S_i^{+}\mathrm{,}{\text{e}}^{u{S}_j^z}{S}_j^{-}]\rangle \mathrm{.}\end{array}

For u=0, \theta (u)=2m. Using Eqs. (4) and (5), we obtain

\begin{array}{}\text{(6)}& {\varphi }_F=\frac{2}{N}{\displaystyle \sum _k}\frac{E_{1F}}{2\sqrt{E_{1F}^2-E_{2F}^2}}\coth \frac{\sqrt{E_{1F}^2-E_{2F}^2}}{2T}-\frac{1}{2}\mathrm{.}\end{array}

Here for the NS,

\begin{array}{}\text{(7)}& \begin{array}{l}{E}_{\text{1}\text{N}\text{S}}=4J_{2}m(\eta \cos k_x\cos k_y-1)+2m(J_{1a}+J_{1b})\mathrm{,}\\ E_{\text{2}\text{N}\text{S}}=2\eta m(J_{1a}\cos k_x+J_{1b}\cos k_y)\mathrm{,}\end{array}\end{array}

and for the SS,

\begin{array}{}\text{(8)}& \begin{array}{l}{E}_{\text{1}\text{S}\text{S}}=2J_{1b}m(\eta \cos k_y-1)+2m(J_{1a}+2J_2)\mathrm{,}\\ E_{\text{2}\text{S}\text{S}}=2\eta m(J_{1a}\cos k_x+2J_2\cos k_x\cos k_y)\mathrm{.}\end{array}\end{array}

Using Eqs. (5), (6) and the relation \langle S_i^{-}{S}_i^{+}\rangle =S(S+1)-\langle S_i^z\rangle -\langle {(S_i^z)}^2\rangle, the sublattice magnetization is expressed by following formula [21, 22]

\begin{array}{}\text{(9)}& m_F=\frac{({\varphi }_F+1+S){\varphi }_F^{2S+1}-({\varphi }_F-S)({\varphi }_F+1{)}^{2S+1}}{{({\varphi }_F+1)}^{2S+1}-{\varphi }_F^{2S+1}}\mathrm{.}\end{array}

The Néel points of the two configurations are expressed by

\begin{array}{}\text{(10)}& \frac{k_B{T}_{\text{N}\mathrm{,}\text{N}\text{S}}}{S(S+1)}=\frac{m}{3}{\left(\frac{2}{N}{\displaystyle \sum _k}\frac{E_{\text{1}\text{N}\text{S}}}{E_{\text{1}\text{N}\text{S}}^2-E_{\text{2}\text{N}\text{S}}^2}\right)}^{-1}\mathrm{,}\end{array}

and

\begin{array}{}\text{(11)}& \frac{k_B{T}_{\text{N}\mathrm{,}\text{S}\text{S}}}{S(S+1)}=\frac{m}{3}{\left(\frac{2}{N}{\displaystyle \sum _k}\frac{E_{\text{1}\text{S}\text{S}}}{E_{\text{1}\text{S}\text{S}}^2-E_{\text{2}\text{S}\text{S}}^2}\right)}^{-1}\mathrm{,}\end{array}

respectively.

3 Results and discussion

At zero temperature, our numerical calculations show that the phase diagram of the system is a classical result based on the DTGF. This case will not be discussed here. In this paper, we focus on the properties at finite temperature. Therefore, when we say zero temperature, we actually mean that temperature is very close to zero, which is denoted by T=0^{+}.

3.1 Magnetic properties

In this section, the basic magnetic properties of the system under various parameters are presented and discussed.

Figure 2 plots the results of magnetization m as a function of temperature T at different parameter values. Figure 2(a) shows the case of \eta =0.5 and J_{1b}=0.8. It is seen that it is NS configuration at J_2\le 0.4. For J_2\ge 0.4, it is SS configuration. As J_2=0, it is an ordinary 2D antiferromagnetic model. There is no competition to cause frustration. As J₂ increases from zero, the competition between J_{1b}/J_{1a} and J_2/J_{1a} emerges. Since we have fixed J_{1a}=1 in computation, this competition is actually between J_{1b} and J₂. When the value of J_{1b} is fixed, the frustration becomes stronger with the increase of J₂ value. This leads to that T_{\text{N}\mathrm{,}\text{N}\text{S}} and m_{\text{N}\text{S}} decrease with the increase of J₂. At J_2=0.4, the frustration reaches maximum. Therefore, as J₂ rises from 0.4, the role of J₂ becomes more important and the frustration becomes comparatively weaker. It leads to that T_{\text{N}\mathrm{,}\text{S}\text{S}} and m_{\text{S}\text{S}} increase with increasing J₂ value.

Figure 2(a) indicates that the NS and SS have the same order-disorder transition point at J_2=0.4. It reflects that both configurations can coexist. Since the value of J₂ is exactly equal to J_{1b}/2, it is necessary to see whether at J_2=J_{1b}/2 this conclusion is true for any other η and J_{1b} values. We change J_{1b} values and fix \eta =0.5 at J_2=J_{1b}/2, and the results are shown in Fig. 2(b). It is seen that, as J_{1b} takes value from 0.1 to 1, the critical temperatures of the two states are always equal. It indicates that this conclusion holds for different J_{1b} values at J_2=J_{1b}/2. As J_{1b}=0 and J_2=0, both NS and SS recover an ordinary one-dimensional anisotropic Heisenberg antiferromagnetic model. In this case, only J_{1a} plays a role. When J_{1b} increases from 0, under the condition of J_2=J_{1b}/2, the role of J_{1a} will gradually become weaker, see Fig. 2(b). However, for J_{1b}=0.1, numerical results indicate that the magnetization curves of the two states are the same, see Fig. 2(c). It reflects that the role of J_{1a} is predominant for small J_{1b} values.

In Fig. 3, J_{1b}=0.8 is fixed and the η varies. It is seen that the two states have the same critical temperature at as long as J_2=J_{1b}/2, see Fig. 3(a). The magnetization curves of NS and SS are the same at \eta =0, see Fig. 3(b). This is because that both NS and SS recover an ordinary two-dimensional Ising model. In this case, there are longitudinal but no transverse magnetic correlations between spins. Therefore, as T\to 0, the sublattice magnetization becomes fully saturated, i.e., m(0^{+})=0.5. Figure 3(a) demonstrates two features as the η value decreases from large to small. One is that both m_{\text{N}\text{S}}(0^{+}) and m_{\text{S}\text{S}}(0^{+}) increase until full saturation. The other is that the relative difference between m_{\text{N}\text{S}}(0^{+}) and m_{\text{S}\text{S}}(0^{+}) decreases. Both features are ascribed to the weaker transverse quantum fluctuation for stronger anisotropy. In combination of Figs. 2(b) and 3(a), it is drawn that at the two states always have the same critical temperature as long as J_2=J_{1b}/2.

Figure 4 plots the critical temperature as a function of J₂ at different parameters. When J_{1b}=0 and J_2=0, as mentioned above, both NS and SS recover an ordinary one-dimensional exchange anisotropic Heisenberg antiferromagnetic model. As a consequence, they have the same critical temperature. For NS, as J_{1b} increases from 0, the competition between J_{1b} and J₂ becomes weaker, the frustration of the system also becomes weaker, which leads to rise of the T_{\text{N}\mathrm{,}\text{N}\text{S}}. SS is just a contrary case, i.e., the T_{\text{N}\mathrm{,}\text{S}\text{S}} decreases with increasing J_{1b}. Due to these two reasons, the cross point of the critical temperature curves of the two states shifts rightwards with increasing J_{1b}, see Fig. 4(a). This is actually caused by the competition between J_{1b} and J₂.

Figure 4(b) presents the enlargement of Fig. 4(a) when J₂ is in the vicinity of 0.4 for J_{1b}=0.8. In this panel, the two lines divide the figure into four regions. The upper region, marked by “P”, means that the system is in paramagnetic state. The left and right regions are that the system is in NS and SS configurations, respectively. The lower region can be the coexistence of NS and SS.

Figure 4(c) plots the critical temperature as a function of J₂ for different η values when J_{1b}=0.8. It is seen that the critical temperature increases with the decrease of η. The reason is that the anisotropy suppresses the fluctuation of the system. It leads to that the critical temperature rises. Therefore, the slopes of the critical temperatures of the two states become slower and slower with increasing η, see Fig. 4(c). It leads to the drop of the cross point of the critical temperature between NS and SS. Accordingly, the temperature corresponding to the cross point decreases with the increase of η. As the η value is close to 1, the crossing area disappears. One obtains from Fig. 4 that at J_{1b}>0 and the two states always have the same critical temperature as long as J_2=J_{1b}/2.

We have known from the discussion above that both configurations can coexist at J_2=J_{1b}/2 and J_2\ne J_{1b}/2 when . One may ask which one is more stable. In the following, we manage to answer this question. The two configurations are different from each other, and so are their entropies at a fixed temperature. Therefore, the internal energy cannot be used to determine which one is more stable at each temperature. Under the same volume and temperature, the state with lower free energy is more stable.

The free energy can be evaluated numerically by means of the internal energy via F(T)=E(0)-T{\displaystyle {\int }_0^T}\frac{E(T^{\prime })-E(0)}{{T^{\prime }}^2}\text{d}{T}^{\prime }, where E(T) represents the internal energy of the system, which is defined as the thermostatistical average of Hamiltonian, E=\frac{\langle H\rangle }{N} [22]. Computing internal energy involves the calculation of the transverse ({\displaystyle {\sum }_{i\mathrm{,}j}}\langle S_i^{+}{S}_j^{-}\rangle) and longitudinal ({\displaystyle {\sum }_{i\mathrm{,}j}}\langle S_i^z{S}_j^z\rangle) correlation functions. We do not present the lengthy derivation here. Please refer to Ref. [25].

3.2 Possible phase transition at J_2=J_{1b}/2

In this section, the free energies of NS and SS at J_2=J_{1b}/2 are presented at various parameters, respectively. Since the critical temperatures of the two states equal at J_2=J_{1b}/2, their critical temperatures are uniformly described by T_N.

Figure 5 plots the internal energy E(T) and free energy F(T) as a function of temperature for different J_{1b} values when \eta =0.9. The E(T) increases with temperature as it should be. The F(T) decreases monotonically with temperature. As J_{1b} increases from 0.001 to 0.038, the F(T) curves of NS and SS are identical, i.e., the difference between their free energies is negligible, see Figs. 5(a) and (b). It means that the system can be either the NS or SS, or coexistence of them. For convenience, we denote J_{1b}^c=0.038. As J_{1b} increases from J_{1b}^c, the free energy curves of the two states begin to separate, and , see Fig. 5(c). In this case, NS is more stable in the range of T\le T_{\text{N}}. Meanwhile, the free energy of SS with increasing temperature drops faster than of NS. Therefore, when J_{1b} further increases, the free energy curves of the two states cross at an intermediate temperature point, see Fig. 5(d). Therefore, below the temperature of this cross point, . It means that NS is more stable. Above the temperature of this cross point, F_{\text{N}\text{S}}(T)>F_{\text{S}\text{S}}(T), so that SS is more stable. At this cross point, there can occur a NS–SS phase transition. Note that, at the cross point, although the free energies of the NS and SS are the same, their internal energies differ, see Fig. 5(d). Subsequently, the specific heat at this cross point is discontinuous (not shown). Therefore, it is a first-order transition.

Figure 6 plots E(T) and F(T) as a function of temperature for different η values when J_{1b}=0.9. Similar to Fig. 5(b), for a fixed J_{1b}, there is an upper limit {\eta }^c. When 0\le \eta \le {\eta }^c, the difference of the free energy between NS and SS is negligible, i.e., F_{\text{N}\text{S}}(T)=F_{\text{S}\text{S}}(T), see Figs. 6(a) and (b). When η increases from {\eta }^c, , see Fig. 6(c). As η continues increasing, the free energy curves of the two states cross, see Fig. 6(d). Therefore, its results are similar to Fig. 5. However, it can be found from Figs. 5(d) and 6(d) that, when the values of η and/or J_{1b} increase, it can lead to that the free energy curves of SS with increasing temperature drops faster than of NS. Therefore, when η and/or J_{1b} takes a certain value, the F_{\text{N}\text{S}}(T) value may be greater than F_{\text{S}\text{S}}(T). To be explicit, Fig. 7 presents free energy as a function of temperature for J_{1b}=0.95 when \eta =0.9, 0.999. It is seen that the free energy curves of SS with increasing η drops faster than of NS. When \eta =0.999, F_{\text{N}\text{S}}(T)>F_{\text{S}\text{S}}(T), see Fig. 7(b). In this case, it means that SS is more stable.

In Figs. 5–7, we only calculate the free energies of the NS and SS as a function of temperature at fixed J_{1b} and η when J_2=J_{1b}/2, one may expect to obtain a number of transitions between NS and SS. Therefore, in Fig. 8, we present a comprehensive recognition of the effect of parameters J_{1b} and η on the free energies of the two states below the T_{\text{N}}. All possible relationships are presented in Fig. 8(a). There are four regions in Fig. 8(a). Some of them are very narrow. Then we show a enlargement of the region in Fig. 8(b).

In region I, the difference of the free energy between the two states is negligible. So it is denoted as F_{\text{N}\text{S}}=F_{\text{S}\text{S}}. The examples are the curves in Figs. 5(a, b) and Figs. 6(a, b). It indicates that in this region the system may be either the NS or SS, or coexistence of them.

In region II, the free energy of NS is always less than SS. The examples are the curves in Fig. 5(c) and Fig. 6(c). This region is denoted as . It shows that in region II NS is more stable.

In region III, the F(T) curves of the two states cross. The examples are curves in Figs. 5(d), 6(d), and 7(a). When the temperature closes to zero, . However, when the temperature approaches to the T_N, F_{\text{N}\text{S}}(T)>F_{\text{S}\text{S}}(T). Therefore, as temperature is near zero, the state of the system should be NS, and as temperature rises, it is possible to occur a first-order phase transformation from the NS to SS at the cross point below the T_{\text{N}}.

In region IV, the free energy of F_{\text{N}\text{S}} is always larger than F_{\text{S}\text{S}}. It shows that in region IV SS is more stable. The example is the curves in Fig. 7(b).

In Fig. 8, two points need to be stressed. One is that in region I, the free energies of NS and SS are very close to each other and one cannot tell which one is more stable. However, as J_{1b}(\eta ) increases from J_{1b}^c({\eta }^c), region I gradually transits to region II. The other is that for region IV, its boundary seems to be \eta =0.9999. This is because we only compute the case of \eta =0.9999, see Fig. 8(b). Actually, its boundary is \eta \to 1. For \eta =1, the system is a two-dimensional isotropic antiferromagnetic model. According to the well-known Mermin–Wagner theorem [14], there is no long-range order at finite temperature for 2D isotropic antiferromagnetic system. Therefore, this case shall be not discussed at finite temperature.

3.3 Possible phase transition at J_2\ne J_{1b}/2

When the J₂ value is apart from J_{1b}/2, the NS and SS can also coexist, as revealed by Fig. 4(b). In this case, the system should also be in the state with the lower free energy at any temperature. If the F(T) curves of the NS and SS have cross point, there may occur a phase transition between them. Figure 9 plots the free energy as a function of temperature for different J₂ values when \eta =0.5 and J_{1b}=0.8. It is seen from Figs. 9(a) and (b) that, as the value of J₂ increases from 0.25 to 0.39, the difference of the free energies between the two states gradually becomes small, especially near critical temperature, see the curves in Fig. 9(b). It seems to tell us when J₂ approaches 0.4, the free energy curves of the two states may cross. Nevertheless, numerical calculations indicate that at J_2\to 0.4, . This case is not shown here. In fact, Fig. 8(a) clearly shows at \eta =0.5 and J_2=0.4. Therefore, F_{\text{N}\text{S}}(T) is always less than F_{\text{S}\text{S}}(T) at . This means that in this case NS is more stable. For the case of J_2>0.4, one should distinguish three cases. (1) , F_{\text{N}\text{S}}(T) is still less than F_{\text{S}\text{S}}(T), see the curves with J_2=0.401 in Fig. 9(c). In this case, NS is more stable. At J_2=0.4088, when T\to 0, the free energies of the two states is negligible, i.e., F_{\text{N}\text{S}}(0^{+})=F_{\text{S}\text{S}}(0^{+}). (2) When , the free energy curves of the two states cross. The examples are the curves with J_2=0.43 in Fig. 9(c). At cross point, a first-order phase transformation between NS and SS may occur. Below the cross point, SS is more stable. Above the cross point, NS is more stable. (3) When J_2>0.4574, F_{\text{N}\text{S}}(T) is always greater than F_{\text{S}\text{S}}(T). The examples are the curves in Fig. 8(d). In this case, SS is more stable. Fig. 8 reflects a fact that, as the value of J₂ increases, the value of F_{\text{N}\text{S}}(0^{+}) is rising, while F_{\text{S}\text{S}}(0^{+}) is decreasing. So on the whole, the curve of F_{\text{N}\text{S}} is rising, and F_{\text{S}\text{S}} is down.

Let us make a summary of our results for the case of J_2\ne J_{1b}/2.

(i) The higher the critical temperature, the lower the F(0^{+}), and the larger the difference between the critical temperature of NS and SS, the larger the difference between F_{\text{N}\text{S}}(0^{+}) and F_{\text{S}\text{S}}(0^{+}).

(ii) For , F_{\text{N}\text{S}}(T) is always less than F_{\text{S}\text{S}}(T), i.e., in this case, NS is always more stable.

(iii) When J_2>J_{1b}/2, one should distinguish three cases. (a) When , F_{\text{N}\text{S}}(T) is still less than F_{\text{S}\text{S}}(T). (b) When , F_{\text{N}\text{S}}(0^{+}) becomes greater than F_{\text{S}\text{S}}(0^{+}), and the free energy curves of the two states have a cross point, at which a first-order phase transformation between NS and SS may occur. (c) When J_2>J_{\mathrm{2,}c2}, F_{\text{N}\text{S}}(T) is always greater than F_{\text{S}\text{S}}(T). The J_{\mathrm{2,}c1} and J_{\mathrm{2,}c2} values depend on the values of η and J_{1b}.

4 Concluding remarks

In this paper, a possible phase transition of the 2D frustrated anisotropic J_1a–J_1b–J₂ Heisenberg antiferroamgnetic model with spin-1/2 has been studied by means of the double-time Green’s function method. Our results have shown that under the condition of the NS and SS have the same critical temperature as long as J_2=J_{1b}/2. For the case of J_2\ne J_{1b}/2, our results indicate that these two states can also exist, while they have different critical temperatures. Thus a possible phase transformation between NS and SS with the case of J_2=J_{1b}/2 and J_2\ne J_{1b}/2 has been also discussed, respectively.

In order to discuss explicitly which state is more stable, free energy as a function of temperature is calculated.

When J_2=J_{1b}/2, our results show that the effect of J_{1b} and η on the free energies of the NS and SS may be divided into four regions, see Fig. 8. In region I, the difference between the free energies of the NS and SS is negligible, i.e., F_{\text{N}\text{S}}=F_{\text{S}\text{S}}. In this case, the system can be in either the NS or SS, a coexistence of them. In region II, , NS is more stable. In region III, a first-order phase transition between NS and SS can occur at a temperature. Below temperature of the cross point, NS is more stable. Above temperature of the cross point, SS is more stable. For region IV, F_{\text{N}\text{S}}>F_{\text{S}\text{S}}, SS is more stable.

For the case of J_2\ne J_{1b}/2, when , the NS is always more stable than the SS below T_{\text{N}\mathrm{,}\text{S}\text{S}}. But for J_2>J_{1b}/2, there are three cases. (i) When J₂ takes value in the vicinity of J_{1b}/2, the NS is more stable. (ii) When the J₂ value increases further, a first-order phase transformation between the two states may occur. Therefore, in this case, when the temperature approaches zero, the SS is more stable. Near T_{\text{N}\mathrm{,}\text{N}\text{S}}, the NS is more stable. (iii) When the J₂ value continues increasing, the SS is more stable.

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