Characteristics of an Ising-Like Model with Ferromagnetic and Antiferromagnetic Interactions

1. Introduction

Magnetic properties of multilayer superlattices have been subject to intensive theoretical and experimental study. Special interest is given to multilayer structures consisting of thin layers of various ferromagnetic and antiferromagnetic materials [1,2,3]. An example of such structures are designs with alternating ferromagnetic and non-ferromagnetic layers. In these structures, the thickness of the non-ferromagnetic layer can be chosen in such a way that the long-range exchange interaction between the ferromagnetic layers is antiferromagnetic [4]. This sort of structures has a gigantic magnetoresistance. Ferromagnetic layered structures with antiferromagnetic inter-layer interactions are theoretically investigated in [5,6] in the framework of the Heisenberg model. These studies reveal that this kind of system can have many distinct phases: ferromagnetic, antiferromagnetic, paramagnetic, and a whirling phase. Papers [7,8] consider the effect of weak interaction between two square Ising lattices on the phase transitions in this system. It is shown that if a structure has alternating layers of different ferromagnetic materials, there might exist a compensation temperature, which is a temperature below the critical value, at which the full magnetization of the lattice is zero. The conditions for the appearance of the compensation effect and the critical behavior of such systems near the compensation temperature are investigated in [9,10,11,12,13,14,15]. In recent years, great consideration has been given to the study of spin systems with competing ferromagnetic and antiferromagnetic interactions, or with interactions between different groups of spins [16,17,18,19]. This sort of systems usually has a complex landscape of free energy containing many local minima separated from the global minimum by a deep potential barrier.

This paper considers a mean field approximation for the simplest Ising-like model having not only the global minimum, but also a local minimum of energy. Spins are divided into two subgroups (sub-ensembles). The intra-group interaction of spins is ferromagnetic, while spins from different sub-ensembles can interact in both ferromagnetic and antiferromagnetic manner. With antiferromagnetic inter-group interaction, the ground state is the ferromagnetic ordering within sub-ensembles and antiferromagnetic ordering among spins of different sub-ensembles. Yet there is also a local energy minimum corresponding to a complete ferromagnetic order.

To compare the mean-field model characteristics with real lattices, we carried out a Monte Carlo simulation of a layered model with ferromagnetic intra-layer interactions and antiferromagnetic inter-group interactions. This model corresponds to realistic ferromagnetic structures with gigantic magnetoresistance. The simulation results qualitatively agree with the predictions of our model.

The paper is organized as follows. State equations for the mean-field model are given in chapter 2. Chapter 3 investigates the critical properties of the system and derives the values of the critical exponents. The effect of an external magnetic field on the temperature dependencies of magnetization is considered in chapter 4. Chapter 5 compares the results of the Monte Carlo simulation with the analytical results obtained in the previous chapters. The discussion and our conclusions are given in Chapter 6. Concluding we would like to mention that in our paper [20] we for the first time used a similar technique.

2. Basic expressions

2.1. Equations of state and critical temperature.

Let us consider a system of $N$ spins with a connection matrix $\widehat{J}$. The system is divided into two groups I and II holding $p_{1}N$ and $p_{2}N$ spins, respectively ($N=p_{1}N+p_{2}N$, $p_1+p_2=1$). The inter-spin connection in group I is defined by a quantity $J_{11}$, in group II by $J_{22}$, and cross-connections between the spins of group I and group II are defined by a quantity $J_{12}$:

$$J_{11}=\frac{q_{11}}{N},J_{12}=\frac{q_{12}}{N},J_{22}=\frac{q_{22}}{N}.$$

(1)

For definiteness, in what follows we assume that spin interaction within each group is of ferromagnetic nature, interaction between spins from different groups is antiferromagnetic, and the direction of the external magnetic field $H$ is positive:

$$q_{11}\ge 0,q_{22}\ge 0,q_{12}\le 0,H\ge 0$$

The case $q_{12}>0$ in the absence of the external field was investigated earlier in [21,22].

The mean-field approximation for the Hamiltonian of this system can be written as

$$E_N=-\left(J_{11}{\displaystyle \sum _{i,j=1}^{N{p}_1}{S}_i{S}_j}+J_{22}{\displaystyle \sum _{i,j=1}^{N{p}_2}{S^{\prime }}_i{S^{\prime }}_j}+J_{12}{\displaystyle \sum _{i=1}^{N{p}_1}{\displaystyle \sum _{j=1}^{N{p}_2}{S}_i{S^{\prime }}_j}}\right)-H\left({\displaystyle \sum _{i=1}^{N{p}_1}{S}_i}+{\displaystyle \sum _{j=1}^{N{p}_2}{S^{\prime }}_j}\right),$$

(2)

where $S$ and $S\text{'}$are the values of the spins from groups I and II, respectively. The sums in (2) are taken over all the spins of the corresponding groups. Introducing partial magnetizations of the two groups, $m_1$ and $m_2$:

$$m_1=\frac{1}{N{p}_1}{\displaystyle \sum _{i=1}^{N{p}_1}{S}_i},m_2=\frac{1}{N{p}_2}{\displaystyle \sum _{j=1}^{N{p}_2}{S^{\prime }}_j},$$

we can represent the energy of the system per one spin, $E=E_N/N$, as follows:

$$E=-\frac{1}{2}\left(q_{11}{p}_1^2{m}_1^2+2q_{12}{p}_1{p}_2{m}_1{m}_2+q_{22}{p}_2^2{m}_2^2\right)-H(p_1{m}_1+p_2{m}_2),$$

(3)

and the full magnetization of the system becomes: $M=p_1{m}_1+p_2{m}_2$.

The statistical sum of this spin system is determined as:

$$Z={\displaystyle \sum _{n_1=0}^{N{p}_1}{\displaystyle \sum _{n_2=0}^{N{p}_2}\left(\begin{array}{c}N{p}_1\\ n_1\end{array}\right)}}\left(\begin{array}{c}N{p}_2\\ n_2\end{array}\right)e^{-NKE},$$

(4)

where $K=1/T$ is the inverse temperature, $n_1$ and $n_2$ denote, respectively, the number of downward spins (-1) in groups I and II; $n_1$ and $n_2$ are related to partial magnetizations by the following expressions:

$$n_1=N{p}_1\frac{1+m_1}{2},n_2=N{p}_2\frac{1+m_2}{2}.$$

Let us define the free energy per one spin as $F=\ln Z/N$. Given that for large $N$ the sum (4) is dominated by the terms with magnetization close to the equilibrium value at a given temperature, and using Stirling’s formula, we get

$$F=p_1{S}_1+p_2{S}_2+KE,$$

(5)

where

$$S_i=\frac{1+m_i}{2}\ln \frac{1+m_i}{2}+\frac{1-m_i}{2}\ln \frac{1-m_i}{2},i=1,2.$$

The equations of state ($\partial F/\partial m_1=0$, $\partial F/\partial m_2=0$) take the form:

$$\begin{array}{c}\frac{1}{2K}\ln \frac{1+m_1}{1-m_1}=q_{11}{p}_1{m}_1-q_{12}{p}_2{m}_2+H\\ \frac{1}{2K}\ln \frac{1+m_2}{1-m_2}=q_{22}{p}_2{m}_2-q_{12}{p}_1{m}_1+H.\end{array},$$

(6)

Let us determine the critical temperature of a phase transition, $K_{с}$. Assuming $H=0$ and $m_1\to 0$, $m_2\to 0$ ($K\to K_{с}$), in the case of $q_{12}=0$ from (6), we get an obvious result: a system split into two non-interacting subsystems has two critical points:

$$K_c^{(1)}=\frac{1}{q_{11}{p}_1},K_c^{(2)}=\frac{1}{q_{22}{p}_2}$$

(7)

Expressions (7) fully agree with the mean-field theory [8] because in the notations of (1) and (2), the quantities $q_{11}{p}_1$ and $q_{22}{p}_2$ are the numbers of the nearest neighbors in groups I and II.

In the general case of $q_{12}\ne 0$, the system has one critical point, which we can find from (6):

$$K_{с}=\frac{2}{q_{11}{p}_1+q_{22}{p}_2+\sqrt{{\left(q_{11}{p}_1-q_{22}{p}_2\right)}^2+4p_1{p}_2{q}_{12}^2}}$$

2.2. Energy minima. Critical value$H_c$.

To proceed further, it is necessary to investigate the presence of energy minima to which the system can converge when the temperature approaches zero. It is easy to see that the energy (3) can be at its minimum only in configurations where $m_1=\pm 1$ and $m_2=\pm 1$. The stability of such states (the presence of the minimum) is only possible when two conditions are simultaneously met:

$$H_1{m}_1>0\mathrm{and}{H}_2{m}_2>0$$

(8)

where $H_1$ and $H_2$are the local fields acting on the spins of groups I and II, respectively:

$$H_1=q_{11}{p}_1{m}_1+q_{12}{p}_2{m}_2+H,H_2=q_{22}{p}_2{m}_2+q_{12}{p}_1{m}_1+H.$$

(9)

When there is no external magnetic field ($H=0$), the unconditionally stable configurations are $S_{+-}=S(m_1=+1,m_2=-1)$ and $S_{-+}=S(m_1=-1,m_2=+1)$, because spins of both groups are directed along the local fields acting on them: $H_1{m}_1=q_{11}{p}_1+\left|q_{12}\right|p_2>0$ and $H_2{m}_2=q_{22}{p}_2+\left|q_{12}\right|p_1>0$. The stability conditions (8) for configurations $S_{++}=S(m_1=+1,m_2=+1)$ and $S_{--}=S(m_1=-1,m_2=-1)$ are $H_1{m}_1=q_{11}{p}_1-\left|q_{12}\right|p_2>0$ and $H_2{m}_2=q_{22}{p}_2-\left|q_{12}\right|p_1>0$. Simple calculations show that these configurations correspond to energy minima only when two conditions are met simultaneously:

$$q_{11}{q}_{22}>q_{12}^2\mathrm{and}\frac{\left|q_{12}\right|}{q_{11}+\left|q_{12}\right|}<p_1<\frac{q_{22}}{q_{22}+\left|q_{12}\right|}.$$

(10)

It follows from (3) that if $H=0$, the configurations $S_{+-}$ and $S_{-+}$ correspond to a doubly degenerate global energy minimum, while the configurations $S_{++}$ and $S_{--}$ are local minima if (10) is met.

When $H\ne 0$, the picture becomes more complicated – the number of minima can vary from one to four. This situation is illustrated in Figure 1, which demonstrates how configuration energies change with the growing strength of the field $H$. We are interested in the energy values in the configurations $S_{++}$($m_1=1,m_2=1$), $S_{+-}$($m_1=1,m_2=-1$) and $S_{-+}$($m_1=-1,m_2=1$):

$$E_{++}=-\frac{1}{2}\left(q_{11}{p}_1^2+q_{22}{p}_2^2\right)+\left|q_{12}\right|p_1{p}_2-H$$

$$E_{+-}=-\frac{1}{2}\left(q_{11}{p}_1^2+q_{22}{p}_2^2\right)-\left|q_{12}\right|p_1{p}_2-H(p_1-p_2)$$

$$E_{-+}=-\frac{1}{2}\left(q_{11}{p}_1^2+q_{22}{p}_2^2\right)-\left|q_{12}\right|p_1{p}_2-H(p_2-p_1)$$

The energy $E_{--}$ in the configuration $S_{--}$($m_1=-1,m_2=1$) is of no interest, because it is a minimum only when $H$ is very small.

Let us introduce the critical value of the magnetic field as follows:

$$H_c=\left|q_{12}\right|p_{\max },p_{\max }=\max (p_1,p_2).$$

(11)

Let $p_1>p_2$. Then, from comparing $E_{++}$, $E_{+-}$ and $E_{-+}$, we can conclude:

• when $H<H_c$, the antiferromagnetic configuration $m_1=1,m_2=-1$ is the ground state of the system, and the ferromagnetic configuration $m_1=1,m_2=1$ is its local minimum;
• when $H>H_c$, the ground state is the ferromagnetic state $m_1=1,m_2=1$, and the antiferromagnetic configuration $m_1=1,m_2=-1$ is its local minimum.

If $p_1<p_2$, we should swap indices 1 and 2 in these expressions. The aforesaid is summarized in Table 1.

2.3. Balanced system. “Critical” value $q_{12}=q_{12}^{\ast }$.

Further analysis reveals that the thermodynamic characteristics of balanced and unbalanced systems are considerably different. We call a system balanced if it has the effective number of spin “nearest neighbors” in the first sub-ensemble is equal to that in the second sub-ensemble. This equality can be expressed as:

$$q_{11}{p}_1+q_{12}{p}_2=q_{22}{p}_2+q_{12}{p}_1.$$

(12)

Indeed, a spin from the first sub-ensemble has $q_{11}{p}_1$ neighbors from this same sub-ensemble and $q_{12}{p}_2$ neighbors from the other sub-ensemble. The same is true for a spin from the second sub-ensemble. We use the term “effective” number of neighbors because the neighbors with antiferromagnetic interaction make a negative contribution to this number.

The balance (12) is possible only when a certain relationship between the interaction parameters holds. For a symmetric system with $p_1=p_2$ and $q_{11}=q_{22}$, it follows from (12) that the system is balanced for any $q_{12}$. In a more general case, the balance condition (12) can be rewritten as $q_{12}=q_{12}^{*}$, where $q_{12}^{*}<0$ is a certain “critical” value of antiferromagnetic interaction:

$$q_{12}^{*}=\frac{q_{11}{p}_1-q_{22}{p}_2}{p_1-p_2},q_{12}^{*}<0.$$

(13)

Below we will use notations $q_{12}=q_{12}^{*}$ or $q_{12}\ne q_{12}^{*}$ with $q_{12}^{*}$ defined in (13) to indicate whether a system is balanced or unbalanced.

Note that a system can be balanced only within a specific range of parameters $p_{1,2}$ and $q_{11},q_{22}$. Indeed, it follows from (13) that the condition $q_{12}^{*}<0$ can be written as:

$$q_{12}^{*}<0if\{\begin{array}{ccc}\frac{q_{22}}{q_{11}+q_{22}}\le p_1<\frac{1}{2}& ,& q_{11}\ge q_{22}\\ \frac{1}{2}<p_1\le \frac{q_{22}}{q_{11}+q_{22}}& ,& q_{11}\le q_{22}.\end{array}$$

(14)

Relationship (14) is illustrated as a phase diagram in Figure 2.

3. Critical Behavior

Let us introduce the relative temperature $t$ as follows

$$t=\frac{T-T_c}{T_c}=\frac{K_c-K}{K}$$

We will consider the behavior of various physical quantities near the critical temperature, paying special attention to the difference between critical exponents in the general case ($q_{12}\ne q_{12}^{*}$) and critical exponents in the case of the “critical” antiferromagnetic interaction ($q_{12}=q_{12}^{*}$).

3.1. Spontaneous magnetization. Critical exponent$\beta$.

• The case $q_{12}\ne q_{12}^{*}$. Let us consider the equation of state (6) when $H=0$ and $m_{1,2}\to 0$. Let us expand it in small parameters $m_{1,2}$ with an accuracy up to the terms of the order of $m_{1,2}^3$ and $t$. Then, for partial magnetizations near the critical temperature ($K>K_c$), we get:

  $$m_1^2=-3t{D}_1,m_2^2=-3t{D}_2,$$

  (15)

  where

  $$\begin{array}{l}{D}_1=\frac{p_2(1-K_c{q}_{22}{p}_2)\left[2-K_c(q_{11}{p}_1+q_{22}{p}_2)\right]}{p_1{(1-K_c{q}_{11}{p}_1)}^2+p_2{(1-K_c{q}_{22}{p}_2)}^2},\hfill \\ D_2=\frac{p_1(1-K_c{q}_{11}{p}_1)\left[2-K_c(q_{11}{p}_1+q_{22}{p}_2\right]}{p_1{(1-K_c{q}_{11}{p}_1)}^2+p_2{(1-K_c{q}_{22}{p}_2)}^2}.\hfill \end{array}$$
  The full spontaneous magnetization $M_0$ is defined as:

  $$M_0=p_1{m}_1+p_2{m}_2=\pm \sqrt{-3t}\left(p_1\sqrt{D_1}-p_2\sqrt{D_2}\right).$$

  (16)
  Here we take into account the fact that with $H=0$ and $q_{12}<0$ the inequality $m_1{m}_2<0$ holds.
• The case $q_{12}=q_{12}^{*}$. Here $p_1\sqrt{D_1}=p_2\sqrt{D_2}$, and the parenthesized expression from (16) becomes zero. It means that in the expressions resulted from the expansion of the equation of state (6), we must retain the terms of the order of $t^2$. Then the spontaneous magnetization near the critical temperature is described as

  $$M_0=\pm {(-t)}^{\frac{3}{2}}\frac{\sqrt{3}{p}_1{p}_2(p_1-p_2)}{K_c\left|q_{12}^{*}\right|{\left(1-3p_1{p}_2\right)}^{3/2}}.$$

  (17)

Thus, in an unbalanced system ($q_{12}\ne q_{12}^{*}$), the critical exponent $\beta =1/2$, which agrees with the conventional mean-field theory. In a balanced system ($q_{12}=q_{12}^{*}$), the critical exponent takes a “non-classical” value $\beta =3/2$.

3.2. Jump in heat capacity ($H=0$). Critical exponent $\alpha$.

At the critical point, the heat capacity $С$ experiences a finite jump. Indeed, when $t>0$, the heat capacity $С=0$, and when $t\to 0^{-}$, the quantity $С=-KdE/dK$ can be easily found using (3) and (23). Then for the heat capacity jump at the critical point, we have:

$$\Delta C=\underset{t\to 0^{-}}{\lim }C=\frac{3}{2}{K}_c{p}_1{p}_2\frac{{\left(q_{11}{p}_1-q_{22}{p}_2\right)}^2+4p_1{p}_2{q}_{12}^2}{p_1{\left(1-K_c{q}_{11}{p}_1\right)}^2+p_2{\left(1-K_c{q}_{22}{p}_2\right)}^2}.$$

(18)

When $q_{12}=q_{12}^{*}$, the expression (18) takes the form:

$$\Delta C=\frac{3}{2K_c}\frac{p_1{p}_2}{p_1^3+p_2^3}$$

Since the heat capacity tends to a finite value at the critical point, the classical definition of the critical exponent α loses sense. Therefore, we will use its alternative definition [23]:

$$F_{+}(K)-F_{-}(K)\sim t^{2-\alpha }\mathrm{when}t\to 0,$$

(19)

where $F_{+}(K)$ and $F_{-}(K)$ are the free energies for $K<K_c$ and $K>K_c$, respectively. To compute the exponent $\alpha$ from (19), the functions $F_{+}(K)$ and $F_{-}(K)$ must be analytically extended to the $K$-axis beyond their domain. Using the expression for free energy (5) and taking into account the equation of state (6) and expressions for spontaneous magnetization (23), we get:

$$F_{+}(K)-F_{-}(K)=-\frac{3t^2}{4}\left[2K_c\left(q_{11}{p}_1^2{D}_1-2q_{12}{p}_1{p}_2\sqrt{D_1{D}_2}+q_{22}{p}_2^2{D}_2\right)+p_1{D}_1^2+p_2{D}_2^2\right].$$

(20)

Comparing (20) with (19), we get $\alpha =0$, which agrees with the classical mean-field model. Note that this result is independent of the magnitude of the antiferromagnetic interaction, i.e., it holds both for $q_{12}=q_{12}^{*}$ and $q_{12}\ne q_{12}^{*}$.

Typical curves of spontaneous partial magnetization $m_{1,2}(K)$ and energy variance ${\sigma }^2(K)=С(K)/K$ as functions of temperature without an external field are shown in Figure 3.

3.3. Susceptibility $\chi$ ($H=0$). Critical exponents $\gamma$ and ${\gamma }^{\prime }$

Let us consider the behavior of the susceptibility of the system near the critical point, given that $H=0$. We define the full and partial susceptibilities as follows:

$$\chi =\frac{\partial M(K,H)}{\partial H}=p_1{\chi }_1+p_2{\chi }_2,{\chi }_1=\frac{\partial m_1(K,H)}{\partial H},{\chi }_2=\frac{\partial m_2(K,H)}{\partial H}$$

Differentiating the equations of state (6) with respect to $H$ and solving the resulting equations for ${\chi }_1$ and ${\chi }_2$ in view of (15), we get the following expressions:

$$\{\begin{array}{ccc}{\chi }_1=\frac{t+\left[1-K_c{p}_2(q_{22}-q_{12})\right]}{t\sqrt{{(q_{11}{p}_1-q_{22}{p}_2)}^2+4p_1{p}_1{q}_{12}^2}}& & \\ & & ift>0\\ {\chi }_2=\frac{t+\left[1-K_c{p}_1(q_{11}-q_{12})\right]}{t\sqrt{{(q_{11}{p}_1-q_{22}{p}_2)}^2+4p_1{p}_1{q}_{12}^2}}& & \end{array},$$

(21)

and

$$\{\begin{array}{ccc}{\chi }_1=-K_c\frac{[1-K_c{p}_2(q_{22}-c)]-t(1-3D_2)}{t\left[(1-K_c{q}_{11}{p}_1)(1-3D_1)+(1-K_c{q}_{22}{p}_2)(1-3D_2)\right]}& & \\ & & ift<0\\ {\chi }_2=-K_c\frac{[1-K_c{p}_1(q_{11}-c)]-t(1-3D_1)}{t\left[(1-K_c{q}_{11}{p}_1)(1-3D_1)+(1-K_c{q}_{22}{p}_2)(1-3D_2)\right]}& & \end{array},$$

(22)

• When $q_{12}\ne q_{12}^{*}$, the terms in the square brackets in the numerators of (21)-(22) are not zero, and the quantity $t$ in the numerators can be neglected. In this event the susceptibility of the system $\chi =p_1{\chi }_1+p_2{\chi }_2$ takes the classical form:

  $$\chi =\frac{1-K_c{p}_1{p}_2(q_{11}+q_{22}-2q_{12})}{\left|t\right|\sqrt{{(q_{11}{p}_1-q_{22}{p}_2)}^2+4p_1{p}_1{q}_{12}^2}}ift>0ort<0,$$

  (23)

  holding true for both $t>0$and $t<0$.
• When $q_{12}=q_{12}^{*}$, we have $\left[1-K_c{p}_2(q_{22}-q_{12}^{*})\right]=\left[1-K_c{p}_1(q_{11}-q_{12}^{*})\right]=0$. Then ${\chi }_1$ and ${\chi }_2$ in (21) and (22) become constant values, and full susceptibility $\chi =p_1{\chi }_1+p_2{\chi }_2$ takes the same “non-classical” form for both $t>0$and $t<0$:

  $$\chi =\frac{1}{\left|q_{12}^{*}\right|}.$$

  (24)

From (23) and (24) we can derive critical exponents:

$$\{\begin{array}{cc}\gamma ={\gamma }^{\prime }=1& if{q}_{12}\ne q_{12}^{*}\\ \gamma ={\gamma }^{\prime }=0& if{q}_{12}=q_{12}^{*}\end{array}$$

It should be noted that in a balanced system (12) (at $q_{12}=q_{12}^{*}$), partial susceptibilities ${\chi }_1$ and ${\chi }_2$ experience a finite jump at the critical point; yet $K$-dependence of the full susceptibility$\chi$ remains continuous.

3.4. Scaling hypothesis. Critical exponent $\delta$.

According to the scaling hypothesis, the field $H$ is a homogeneous function of variables $M^{1/\beta }$ and $t$ near the critical point. Let us consider how the critical exponent $\delta$ and the form of the scaling function rely on the interaction parameters.

1. The case $q_{12}\ne q_{12}^{*}$ (unbalanced system). Expanding the equations of state (6) in terms of the small parameters $m_1$, $m_2$, $t$ and extracting full magnetization $M=p_1{m}_1+p_2{m}_2$, we get:

$$K_{c}H=M^3{R}_1+tM{R}_2$$

where

$$R_1=\frac{{\left(1-K_c{q}_{11}{p}_1\right)}^2{p}_1+{\left(1-K_c{q}_{22}{p}_2\right)}^2{p}_2}{3p_1{p}_2{\left[1-K_c{p}_1{p}_2(q_{11}+q_{22}-2q_{12})\right]}^2},R_2=\frac{K_c\sqrt{{(q_{11}{p}_1-q_{22}{p}_2)}^2+4p_1{p}_2{q}_{12}^2}}{1-K_c{p}_1{p}_2(q_{11}+q_{22}-2q_{12})}.$$

(25)

It is easy to see that the scaling hypothesis in this case is confirmed, because if $q_{12}\ne q_{12}^{*}$, it follows from (16) that $\beta =1/2$. Indeed, expression (25) can be represented in the classical form $K_{c}H=M{\left|M\right|}^{\delta -1}{h}_s\left(t{\left|M\right|}^{-1/\beta }\right)$ with the critical exponent $\delta =3$ and the scaling function $h_s(x)$ of the form:

$$h_s(x)=R_1+R_{2}x,\mathrm{where}x=t/M^2.$$

2. The case $q_{12}=q_{12}^{*}$ (balanced system (12)). Performing similar calculations with regard to the relations $1-K_c{p}_1\left(q_{11}+\left|q_{12}^{*}\right|\right)=1-K_c{p}_2\left(q_{22}+\left|q_{12}^{*}\right|\right)=0$, which take place when $q_{12}=q_{12}^{*}$, we get:

$$K_{c}H=\{\begin{array}{lll}{K}_c\left|q_{12}^{*}\right|M\hfill & ,\hfill & t>0\hfill \\ K_c\left|q_{12}^{*}\right|M-{\left|t\right|}^{3/2}{R}_3\hfill & ,\hfill & t<0\hfill \end{array},\mathrm{where}{R}_3=\frac{\sqrt{3}{p}_1{p}_2\left|p_1-p_2\right|}{{(1-3p_1{p}_2)}^{3/2}}.$$

(26)

In this event (17) gives$\beta =3/2$. Correspondingly, expression (26) can be rewritten in the classical form $K_{c}H=M{\left|M\right|}^{\delta -1}{h}_s\left(t{\left|M\right|}^{-1/\beta }\right)$, with the critical exponent $\delta =1$ and the scaling function $h_s(x)$ of the form:

$h_s(x)=\{\begin{array}{lll}{K}_c\left|q_{12}^{*}\right|\hfill & ,\hfill & t>0\hfill \\ K_c\left|q_{12}^{*}\right|-{\left|x\right|}^{3/2}{R}_3\hfill & ,\hfill & t<0\hfill \end{array}$ , where $x=t/M$.

It is easy to see that with $q_{12}=q_{12}^{*}$, the Rushbrooke relation, $\alpha +2\beta +\gamma \ge 2$, the Widom relation, $\gamma \ge \beta (\delta -1)$, and the Griffiths relation, $\beta (\delta +1)\ge 2-\alpha$, hold as strict inequalities. At the same time, the monograph [23] claims that, being a consequence of the scaling hypothesis, these relations should turn into strict equalities. Thus, the validity of the scaling hypothesis remains in question when $q_{12}=q_{12}^{*}$.

Summing up the above, we should note that the critical exponents of the model under consideration correspond to the parameters of the classical mean-field model provided that $q_{12}\ne q_{12}^{*}$, i.e. when the effective numbers of neighbors in different sub-lattices are not equal (see Table 2). However, in a balanced system, when condition (12) is met (i.е. $q_{12}=q_{12}^{*}$), the critical exponents $\beta$, $\gamma$ and $\delta$ take “non-classical” values. Then the relation $\alpha +2\beta +\gamma =2$, which is a consequence of the scaling hypothesis, is violated.

At the end of this section let us show how functions of certain physical quantities change when $q_{12}=q_{12}^{*}$. Figure 4a presents the temperature dependence of susceptibility in the absence of an external field $\chi (K,0)$. We can see that when $q_{12}\ne q_{12}^{*}$, the susceptibility diverges at the critical point, just as in the case of the classical mean-field model. When $q_{12}=q_{12}^{*}$, the susceptibility has a finite critical value defined by expression (24). At the “critical” values of antiferromagnetic interactions and critical temperature, the field-dependences of partial magnetizations $m_1(K_c,H)$and $m_2(K_c,H)$ also change (Figure 4b). If $q_{12}\ne q_{12}^{*}$ and the field $H$ is small, the system is in the antiferromagnetic state ($m_1{m}_2<0$), and it makes a cross-over to the ferromagnetic state ($m_1{m}_2>0$) when the field increases. If $q_{12}=q_{12}^{*}$, the system is in the ferromagnetic state at any value of $H$, and its partial magnetizations are equal ($m_1=m_2$).

4. Dependences of Physical Quantities on Temperature and External Field

In this section, we investigate how the physical parameters of the system depend on temperature in the presence of an external field. We find that these dependences change dramatically when the antiferromagnetic interaction parameter $q_{12}$ takes a “critical” value $q_{12}=q_{12}^{*}$. In the first subsection, we deal with the situation when $q_{12}\ne q_{12}^{*}$. In the second subsection, we consider a symmetric configuration $q_{11}=q_{22},$$p_1=p_2=1/2$, which is a particular case of a system with “critical” antiferromagnetic interaction.

4.1. Unbalanced system ($q_{12}\ne q_{12}^{*}$).

As it follows from the equations of state (6), at $H\ge H_c$ the right sides of the both equations of state are necessarily positive and, therefore, when $m_{1,2}>0$ the system is in the ferromagnetic phase at any temperature. That is why the change of phase is only possible when the external field is weak, $H<H_c$. It follows from the expressions (9) for the local fields that for small $K$ the local fields $H_{1,2}>0$ and, therefore, when $m_{1,2}>0$ the both sub-groups are in the ferromagnetic phase. If $H<H_c$, when $m_1{m}_2<0$ there is a temperature $K_p(H)$, at which the transition into the antiferromagnetic phase occurs. If $q_{12}\ne q_{12}^{*}$, there are four fundamentally different behaviors of partial magnetizations as functions of temperature,$m_{1,2}(K)$, as determined by the characteristics of the model and by the strength of the external field $H$ (Figure 5 and Figure 6).

Before we proceed to further analysis, let us consider the behavior of partial magnetizations at high temperatures, i.e. at $K\to 0$. Differentiating equations (6) with respect to $K$ and retaining only the quantities of the first order of smallness, we get the following expressions for the derivatives ${\dot{m}}_1$ and ${\dot{m}}_2$ at $K\to 0$:

$$\begin{array}{ccc}{\dot{m}}_1>{\dot{m}}_2& if& p_1\left(q_{11}+\left|q_{12}\right|\right)>p_2\left(q_{22}+\left|q_{12}\right|\right),\\ {\dot{m}}_2>{\dot{m}}_1& if& p_1\left(q_{11}+\left|q_{12}\right|\right)<p_2\left(q_{22}+\left|q_{12}\right|\right).\end{array}$$

(27)

This implies that with growing $K$, the partial magnetization $m_1$ increases faster than $m_2$ if the effective number of neighbors in the lattice I is greater than that in the lattice II, and vice versa. For definiteness, in this subsection we will assume that $p_1\left(q_{11}+\left|q_{12}\right|\right)>p_2\left(q_{22}+\left|q_{12}\right|\right)$. If the opposite is true, all considerations will hold up to a permutation of the indices 1 and 2.

• The case $H<H_c$. Let us first consider the case $H<H_c$, when the ground state is antiferromagnetic ($m_1{m}_2=-1$ at $K\to \infty$).

Let $p_1\left(q_{11}+\left|q_{12}\right|\right)>p_2\left(q_{22}+\left|q_{12}\right|\right)$. Then, in accordance with (27), for$K<<1$ the magnetization $m_1$ grows with $K$ faster than $m_2$. In this case, the magnetization $m_1$ will keep growing as $K$ increases, while the magnetization $m_2$ will initially grow and then will drop to negative values, passing through a zero value at a particular temperature $K=K_p$. At further growth of $K$, there are two distinct scenarios for the curves development in the $K>K_p$ region:

• If $p_1>p_2$, the partial magnetization $m_1$ will monotonously increase, and the partial magnetization $m_2$ will decrease, tending to 1 and -1, correspondingly (Figure 5a). There is no phase transition in this case.
• If $p_1<p_2$, the system’s ground state is the configuration with $m_1=-1$, $m_2=1$. At a certain temperature, a first-order phase transition will occur, as a result of which the magnetization $m_1$ will become negative, and the magnetization $m_2$ will be positive (Figure 5b). This transition will be accompanied by an abrupt change in the measurable parameters: magnetization $M(K)$, internal energy $U(K)$, and heat capacity $C(K)$.

Note that the full magnetization of the system $M=p_1{m}_1+p_2{m}_2$ always remains positive, and in the limit $K\to \infty$ we have$M=\left|p_1-p_2\right|$ independently of the relationship between $p_1$ and $p_2$.

The temperature $K_p$ of the phase change, i.e. the temperature at which $m_2$ passes zero, can be easily found: at $m_2=0$ the second of equations (6) reduces to $\left|q_{12}\right|p_1{m}_1=H$. Substituting this relation in the first of equations (6), we get:

$$K_p=\frac{\left|q_{12}\right|}{2H(q_{11}+\left|q_{12}\right|)}\ln \frac{p_1\left|q_{12}\right|+H}{p_1\left|q_{12}\right|-H},m_2(K=K_p)=0,m_1(K=K_p)=\frac{H}{\left|q_{12}\right|p_1}.$$

(28)

When the external field is weak ($H\ll p_1\left|q_{12}\right|$), the temperature of the transition into the antiferromagnetic phase approaches the quantity

$$K_p(H\to 0)=\frac{1}{p_1(q_{11}+\left|q_{12}\right|)}$$

(29)

Let us compare (29) with the temperature $K_{с}$. It is easy to see that inequality $K_p(H\to 0)\le K_{с}$ is identical to condition (27), i.e., when the field is weak, the temperature $K_p$ always tends to a value less or equal to $K_{с}$. The two temperatures can be equal, $K_p(H\to 0)=K_{с}$, only if $q_{12}=q_{12}^{*}$.

Moreover, the expression (28) is correct given that $p_1\left|q_{12}\right|>H$. It means that $m_2$ can pass zero only if the external field is relatively weak, or if the transition into the antiferromagnetic phase is an abrupt process. Otherwise,$m_2$ cannot pass zero, and $m_{1,2}\left(K\right)\ge 0$ for any $K$.

• If $H>H_c$, the behavior of the temperature dependences is quite predictable, without any peculiarities (see Figure 6). The partial magnetizations are positive at any $K$. In particular, if the field $H$ is close enough to the critical value $H_c$, the magnetization $m_2$ decreases with the growth of $K$in a certain temperature range, but it never becomes negative (Figure 6a). When $H\gg H_c$, the partial magnetizations $m_1$ and $m_2$ are increasing functions of $K$ over the whole temperature range. When $H>H_c$, the ground state of the system is the ferromagnetic state: $M=m_1=m_2=1$, $K\to \infty$.

Summarizing the results of subsection 4.1, we can highlight the following properties of an unbalanced ($q_{12}\ne q_{12}^{*}$) system: Firstly, the presence of magnetic field suppresses a second-order phase transition in the system. Secondly, if $0<H<H_c$, a first-order phase transition can occur for a certain relation between parameters (27) and $p_{1,2}$, as illustrated in Figure 5b.

4.2. Balanced system ($q_{12}=q_{12}^{*}$).

If the condition of equal effective numbers of neighbors (12) is met, there is a solution to the equations of state (6): $m_1=m_2$. This solution does not always correspond to the free energy minimum, and the behavior of partial magnetizations is so diverse that the analysis of the general case ($q_{11}\ne q_{22}$,$p_1\ne p_2$) is not within the scope of this paper.

Here we consider the simplest (symmetrical) case, when

$$q_{11}=q_{22}=q\mathrm{and}{p}_1=p_2=\frac{1}{2}.$$

(30)

Note that even in this simplest case the behavior of partial magnetizations can be fairly diverse (Figure 7). Although the below formulae are true only for symmetric systems (30), general patterns hold true for all systems with $q_{12}=q_{12}^{*}$.

We recall that in the symmetric model the equality $q_{12}=q_{12}^{*}$ is true for all negative $q_{12}$. The critical temperature in this case is determined as:

$$K_c=\frac{2}{q+\left|q_{12}\right|}$$

and equations of state (6) take the form:

$$\begin{array}{c}\frac{1}{K}\ln \frac{1+m_1}{1-m_1}=q{m}_1-\left|q_{12}\right|m_2+2H,\\ \frac{1}{K}\ln \frac{1+m_2}{1-m_2}=q{m}_2-\left|q_{12}\right|m_1+2H.\end{array}$$

(31)

As we see, these equations have the same coefficients and admit a solution $m_1=m_2$. Indeed, analysis of (31) shows that over a certain temperature interval $0\le K\le K_S$ partial magnetizations $m_1=m_2=m$ is defined by the equation:

$$\frac{1}{K}\ln \frac{1+m}{1-m}=m\left(q-\left|q_{12}\right|\right)+2H$$

(32)

Magnetization $m$ is positive because the external magnetic field $H$ is positive. If the field is smaller than a certain value$H_{s\max }$, at a temperature $K=K_s$ the curve $m=m(K)$ experiences a “soft” splitting into two diverging partial magnetizations $m_1(K)$ and $m_2(K)$ (Figure 7а).

• Soft splitting and merging. Let us determine the “soft” splitting point $K_S$. Quantities $m_1=m_2=m_S$ at this point are derived from (32) at $K=K_S$. Let us consider the behavior of magnetizations $m_{1,}{}_2$ in a small vicinity of $K_S$, introducing the small deviation

  $$t_s=\frac{T-T_s}{T_s}=\frac{K_s-K}{K},\left|t_s\right|\to 0$$

In this case, we seek partial magnetizations in the form $m_1=m_2=m_S-\delta$ when $K<K_s$ and in the form $m_{1,2}=m_S\pm {\delta }_{1,2}$ when $K>K_s$, where $\delta \to 0$ and ${\delta }_{1,}{}_2\to 0$. Let us substitute $m_{1,2}$ of this form into (31) and (32) and carry out expansion in small parameters $\delta$, ${\delta }_{1,}{}_2$ retaining terms up to ${\delta }_{1,2}^3$. Equating the terms of the same order of smallness, we get the expression for the splitting point:

$$K_s=\frac{K_c}{1-m_s^2},$$

(33)

as well as the expressions for small deviations from $m_S$:

$$\begin{array}{l}\delta =t_s\kappa ,\hfill \\ {\delta }_{1,}{}_2=\sqrt{-t_s{\kappa }_1}\pm {\kappa }_2{t}_s,\hfill \end{array}$$

(34)

where

$$\begin{array}{l}\kappa =\frac{\left(q-\left|q_{12}\right|\right)m_s+2H}{2\left|q_{12}\right|}\hfill \\ {\kappa }_1=3\left(1-m_s^2\right)\left(\frac{\left|q_{12}\right|-q{m}_s^2-2H{m}_s}{\left|q_{12}\right|-3q{m}_s^2}\right)\hfill \\ {\kappa }_2=\frac{(1+3m_s^2){\kappa }_1-3{(1-m_s^2)}^2}{6m_s(1-m_s^2)}\hfill \end{array}$$

(35)

2.

Splitting point and “bubble” formation. The quantity $m_S$ is a solution to equation (32) at $K=K_s$. This equation allows for a simple graphical solution if we rewrite it in the form:

$$H=R(m_s)\equiv \frac{1}{2}{m}_s\left(\left|q_{12}\right|-q\right)+\frac{1}{4}\left(\left|q_{12}\right|+q\right)(1-m_s^2)\ln \frac{1+m_s}{1-m_s}$$

(36)

One example of such a solution for $\left|q_{12}\right|<q$, $\left|q_{12}\right|=q$ and $\left|q_{12}\right|>q$ is given in Figure 8. The function $R(m_s)$ has one maximum at $m_s=m_{\max }$, where $m_{\max }$ is the solution to the equation $\partial R(m)/\partial m=0$:

$$m_{\max }\ln \frac{1+m_{\max }}{1-m_{\max }}=\frac{2\left|q_{12}\right|}{q+\left|q_{12}\right|}.$$

(37)

Correspondingly, equation (36) can have a solution only when $H\le H_{\max }$, where

$$H_{\max }=\frac{(\left|q_{12}\right|-q{m}_{\max }^2)}{2m_{\max }}.$$

(38)

If $\left|q_{12}\right|=q$, the right wing of the curve $R(m_s)$ is on the abscissa axis. In the case $\left|q_{12}\right|<q$, the curve is lower than the abscissa. As one can see, if $\left|q_{12}\right|\le q$, the equation (36) may have two solutions. The first one (${m^{\prime }}_s\le m_{\max }$) corresponds to the splitting point of the curves $m_1(K)$ and $m_2(K)$, the second solution (${m^{″}}_s\ge m_{\max }$) corresponds to the merging point, and formation of a “bubble” is shown in Figures 7b and 7c.

If $\left|q_{12}\right|>q$, the right wing of the curve $R(m_s)$ behaves as $R=\left(\left|q_{12}\right|-q\right)/2$ and is above the abscissa. Therefore, for $0<H\le \left(\left|q_{12}\right|-q\right)/2$, equation (36) has only one solution ($m_s\le m_{\max }$), which is responsible for the splitting, whereas for $\left(\left|q_{12}\right|-q\right)/2<H\le H_{\max }$ there are two solutions ${m^{\prime }}_s\le m_{\max }$ and ${m^{″}}_s\ge m_{\max }$, i.e., a “bubble” can form (as in Figures 7b and 7c).

Let us determine the asymptotic values of $m_{\max }$ and $H_{\max }$. In the case of weak antiferromagnetic interaction ($\left|q_{12}\right|<<q$), it follows from (37)-(38) that

$$m_{\max }\simeq {\left|\frac{q_{12}}{q}\right|}^{1/2},H_{\max }\simeq \frac{{\left|q_{12}\right|}^{3/2}}{2\sqrt{q}},$$

(39)

and in the limit case $\left|q_{12}\right|>>q$ we have:

$$m_{\max }\simeq 0.8336,H_{\max }=\frac{\left|q_{12}\right|}{2m_{\max }}\simeq 1.2H_c.$$

(40)

As we see, a soft splitting can be observed when $H$ is varied within a wide enough range. Note that the width of this range grows with $\left|q_{12}\right|$.

If soft splitting is not feasible and the antiferromagnetic state ($H<H_c$) is the ground state, then passing through the critical point is accompanied by a jump of partial magnetizations (Figure 7b). If the condition of a “bubble” formation is met, a jump in magnetization occurs after the merging point at $H<H_c$(Figure 7c).

Generally, the temperature behavior of magnetizations is very diverse (Figure 7), depending strongly on the relative magnitudes of the ratio of $\left|q_{12}\right|/q$ and $H$. This aspect of the model will be investigated in more detail in future papers.

3.

Field-controlled phase transition at $K=K_s$. A second-order phase transition occurs at the point of soft splitting / merging and is accompanied by a jump of heat capacity. The magnitude of the heat capacity jump, $\Delta C$, under the splitting can be easily calculated by differentiating expression (3) for the energy, with the account of expression (35):

$$\Delta C=\frac{3\left(q+\left|q_{12}\right|\right){\left(\left|q_{12}\right|-q{m}_s^2-2H{m}_s\right)}^2}{4\left|q_{12}\right|\left(\left|q_{12}\right|-3q{m}_s^2\right)}.$$

(41)

At the point of merging of the curves $m_1(K)$ and $m_2(K)$, $\Delta C$is determined by the same expression, but with a different value of $m_s$.

It follows from what has been said that $K_s$ determined by (33) is nothing else but a magnetic field-dependent critical point: $K_s=K_c(H)$. In other words, in a symmetrical system, the external field does not suppress a phase transition, but only shifts it towards larger temperatures: $K_c(H\ne 0)>K_c(H=0)\equiv K_c$. The range of the critical point values can be quite large. As follows from (33), given strong antiferromagnetic interaction ($\left|q_{12}\right|>>q$) the phase transition at the soft-splitting point can be greatly shifted: with the field changing from $H=0$ to $H=H_{\max }~1.2H_c$ ($m_s$ changing from $m_s=0$ to $m_s=m_{\max }~0.83$) $K_s$ changes from $K_s=K_c$ to $K_s\simeq 3.3K_c$. The variations at the merging point (if any) cover a yet wider range, where the case of $m_s\to 1$, (i.e. K_s →∞) can take place. For instance, Figure 7f demonstrates the case when with $H\simeq 0.999H_c$ a “bubble” forms, and the system has two critical points: one at the splitting point at $K_s\simeq 1.4K_c$, the second at the merging point at $K_s\simeq 8.3K_c$. We have deliberately chosen such interaction parameters that reveal the richness of the system: besides two second-order phase transitions, it has two first-order phase transitions (the curve $m_2(K)$ undergoes two abrupt changes).

4.

Critical exponents of a phase transition at the point of soft splitting. Let us consider critical behaviour of the involved physical quantities near the temperature $K_s$.

We start with pointing out that the exponent ${\alpha }_s=0$, since, in accordance with equation (41), at the splitting point the heat capacity experiences a finite jump. We will define other critical exponents as follows:

$$m_s-M(K)\sim {(-t_s)}^{{\beta }_s}\mathrm{with}{t}_s\to 0^{-},$$

$$m_s-M(K)\sim t_s{}^{{{\beta }^{\prime }}_s}\mathrm{with}{t}_s\to 0^{+},$$

$$\chi (K)\sim {(-t_s)}^{-{\gamma }_s}\mathrm{with}{t}_s\to 0^{-},$$

$$\chi (K)\sim t_s{}^{-{{\gamma }^{\prime }}_s}\mathrm{with}{t}_s\to 0^{+},$$

$$M(K_s,H)-m_s\sim {\left(H-H_s\right)}^{1/{\delta }_s}\mathrm{with}H-H_s\to 0^{+},$$

$$m_s-M(K_s,H)\sim {\left(H_s-H\right)}^{1/{{\delta }^{\prime }}_s}\mathrm{with}H-H_s\to 0^{-},$$

where $H_s$ is the magnitude of the external field at which the phase transition happens.

It follows from equation (34) that in the vicinity of splitting point, the full magnetisation is determined by the following equations:

$$M=m_s+{\kappa }_2{t}_s,t_s<0,M=m_s-\kappa t_s,t_s>0.$$

(42)

It follows from (42) that the critical exponents ${\beta }_s={{\beta }^{\prime }}_s=1$. For $H\to 0$ ($m_s\to 0$) we have ${\kappa }_2\to 0$, and hence we should take into account the terms of the order of $t_s^{3/2}$in the expansion of the full magnetization $M$over $t_s$. This agrees with the results obtained earlier in section 3.1. We also point out that at large values of $\left|q_{12}\right|$ there is such a value of the magnetic field $H^{\prime }<H_{\max }$beyond which ${\kappa }_2$ becomes negative, so that the magnetization increases when the temperature rises above the splitting point.

We differentiate the equations of state (31)-(32) over $H$and, making use of equations (33)-(34), we obtain for the susceptibilities at the critical points:

$$\begin{array}{l}\chi =\frac{1}{\left|q_{12}\right|},t_s\to 0^{+}\\ \chi ={\left[\left|q_{12}\right|+\frac{2{\kappa }_1{m}_s^2(q+\left|q_{12}\right|)}{\left(1-m_s^2+2m_s\right)\left(1-m_s^2\right)-(3m_s^2+1){\kappa }_1}\right]}^{-1},t_s\to 0^{-}\end{array}$$

(43)

It follows from (43) that the susceptibility has a finite jump in the critical point. Its magnitude depends on $m_s$, and hence on the magnitude of the field $H$. In the absence of the external field the jump disappears. From equation (43) we find the critical exponents ${\gamma }_s={{\gamma }^{\prime }}_s=0$.

We will use equation (32) to estimate the exponent ${\delta }_s$. Let us look at how the magnetization changes in response to a small increase of the external field. We introduce the notations $\Delta M=M-m_s$ and $\Delta H=H-H_s$, and account only for the first order terms in $\Delta M$, we find at the temperature $K_s$:

$$\Delta H=\left|q_{12}\right|\Delta M.$$

(44)

Equation (44) is valid also for negative values of $\Delta H$and $\Delta M$, therefore, we get for the critical exponents: ${\delta }_s={{\delta }^{\prime }}_s=1$.

We see that in this case the relationships $\alpha +2\beta +\gamma =2$, $\gamma =\beta (\delta -1)$and $\beta (\delta +1)=2-\alpha$, which are the consequences of the scaling hypothesis, are satisfied in the form of equalities. Except for$\beta$, all other critical exponents at the soft splitting point are equal to their values in the absence of the external field.

5.

Restrictions in the soft splitting. Above we analyzed the processes of soft splitting and merging in the assumption that these regimes could be reached. However sometimes this is not possible.

Firstly, the field $H$ should be such that the condition $K_s<K_p$ is met, i.e., the curves $m_{1,2}(K)$ must split before one of them passes zero.

Secondly, the field cannot be too big, $H<H_{\max }$, otherwise the soft splitting will be replaced by an abrupt jump-like change of partial magnetizations $m_{1,2}(K)$.

Thirdly, the splitting at point $K_s$ is possible if ${\delta }_1$ and ${\delta }_2$ take real values. It follows from (35) that ${\delta }_1,{\delta }_2\in ℝ$ at $t_s<0$, if

$$\left(\left|q_{12}\right|-q{m}_s^2-2H{m}_s\right)\left(\left|q_{12}\right|-3q{m}_s^2\right)>0.$$

(45)

Expression (45) is a necessary condition for the soft splitting point to exist. The quantities ${\delta }_1$ and ${\delta }_2$ must be real numbers at $t_s>0$ for $K_s$ to be the point of merging. Correspondingly, the necessary condition for the point of merging to exist is as follows:

$$\left(\left|q_{12}\right|-q{m}_s^2-2H{m}_s\right)\left(\left|q_{12}\right|-3q{m}_s^2\right)<0.$$

(46)

Examination of (46) in the light of (36)-(40) shows that merging of the curves $m_1(K)$ and$m_2(K)$, (i.e., the formation of a “bubble”) usually takes place only when the magnitude of $H$ is very close to $H_c$.

5. The Layered Model. Computer Simulation.

Let us compare the results of the mean-field model with the results of the computer simulation of lattices with a finite interaction radius. We want to make sure that in the case of “critical” antiferromagnetic interaction $q_{12}=q_{12}^{*}$, which takes place if the condition (12) is satisfied, the critical parameters indeed take non-classical values presented in Table 2. To this end, let us consider a three-dimensional cubic lattice consisting of alternating two-dimensional layers of spins (see Figure 9). The intra-layer spin interaction is described by the interaction constants $J_{11}>0$ and $J_{22}>0$ for even and odd layers, respectively. The interlayer interaction is antiferromagnetic, and it is described by the constant $J_{12}<0$. The interaction only between the nearest neighbors is taken into account.

This model is described by the Hamiltonian

$$E_N=-\left(J_{11}{\displaystyle \sum _{\langle i,j\rangle }{S}_i{S}_j}+J_{22}{\displaystyle \sum _{\langle i,j\rangle }{S^{\prime }}_i{S^{\prime }}_j}+J_{12}{\displaystyle \sum _{\langle i,j\rangle }{S}_i{S^{\prime }}_j}\right)-H\left({\displaystyle \sum _{i=1}^{N_1}{S}_i}+{\displaystyle \sum _{j=1}^{N_2}{S^{\prime }}_j}\right)$$

where $\langle i,j\rangle$ stands for a set of nearest neighbors. The relationship between the interaction parameters of the layered model and the parameters of the mean-field model is defined by the expression:

$$q_{11}=8J_{11},q_{22}=8J_{22},q_{12}=4J_{12}$$

Condition (12) under which $q_{12}=q_{12}^{*}$ is set for the simplest case of $p_1=p_2$, $q_{11}=q_{22}$.

The Metropolis algorithm is used to calculate the thermodynamic parameters for the lattices of the size $N=L\times L\times L$ where $L$ is the dimension of the lattice varying from 6 to 64. The linear dimension $L$ is always an even number to secure the equality $p_1=p_2$. The case of $p_1\ne p_2$ was also modeled preliminarily, but the results of the modeling showed that in this case it is impossible to achieve the equality of the moduli of partial magnetizations, $\left|m_1\right|=\left|m_2\right|$, over a wide temperature range even if the condition (12), the effective neighbor numbers being equal, is met. A possible cause for this is the effect of boundaries. Therefore, to make it possible to study “critical” antiferromagnetic interactions occurring in the balanced model ($q_{12}=q_{12}^{*}$), the modeling is carried out for the case of $p_1=p_2$ only.

1.

Figure 10 shows temperature dependences of partial magnetizations of the layered model when $q_{11}=q_{22}$. The general behavior of these dependences is similar to the dependences obtained in the mean-field model for a balanced system (c). Depending on the magnitude of the external field, we can observe both “soft” (Figure 10b) and “hard” (Figure 10c) splitting. However, we failed to observe magnetization “bubbles” in this model.

When the lattice dimension is big enough, and the external field $H$is close to, but less than, the critical magnitude$H_{с}$, the system does not proceed into the antiferromagnetic phase by cooling, instead, it stays in the local energy minimum (Figure 10d). We observe the same situation for a fully connected lattice (the analogue of the mean-field model), if we use the Metropolis algorithm to compute the thermodynamic parameters.

To avoid misunderstanding, we should add that “hard” splitting accompanied by an abrupt change of magnetization (shown in Figure 10c) was observed only with relatively small lattice dimensions ($L\le 24$). Small lattice size makes it possible to overcome the energy barrier between the local and global minima. When dimensions are large ($L=32,64$) and $K\to \infty$, the system always transits into the ferromagnetic state with the magnetization $M=1$, which is a local minimum at $H<H_c$.

2.

Let us consider the temperature dependence of the susceptibility $\chi =\chi (K)$ in the layered model in the absence of the external field: $H=0$. If $q_{11}=q_{22}$, i.e.$q_{12}=q_{12}^{*}$, the form of the curve $\chi (K)$ is independent of the lattice dimension, and the susceptibility takes a finite value at the critical point (Figures 11a, 11b). With large $\left|q_{12}\right|$, the curve $\chi (K)$ does not have a maximum (Figure 11a). It means that the critical exponents$\gamma ={\gamma }^{\prime }=0$, which agrees with the numbers in Table 2.

If $q_{11}\ne q_{22}$, the susceptibility peak, as expected, is observed at the critical temperature (Figures 11c, 11d). The height of the peak increases with the lattice dimensions $L$, which means that the susceptibility diverges at the critical point. This finding agrees with the classical critical exponents $\gamma ={\gamma }^{\prime }=1$ .

3.

To get the value of the critical exponent δ, we need to evaluate the critical temperature $K_c$. The temperature dependences of the Binder cumulants [24] for lattices of different dimensions were built for this purpose. The asymptotic value of $K_c$ for the case of $L\to \infty$ is determined as a point of intersection of these dependences. We define the Binder cumulants for partial and full magnetization in the following way:

$$g_1=1-\frac{\langle m_1^4\rangle }{3{\langle m_1^2\rangle }^2},g_2=1-\frac{\langle m_2^4\rangle }{3{\langle m_2^2\rangle }^2},g=1-\frac{\langle M^4\rangle }{3{\langle M^2\rangle }^2}.$$

When $q_{11}=q_{22}$, in the absence of the external field the full magnetization $M$ is zero. For this reason, we had to use partial magnetization cumulants to determine the critical temperature (Figure 12a). The results obtained with the help of the cumulants $g_1$ и $g_2$ gave approximately the same value of $K_c$. This allowed us to determine the exponent $\delta =1$ to within 10^-3 (Table 3).

If $q_{11}\ne q_{22}$, the cumulants $g_1$ and $g_2$ give different estimates of $K_c$. At the same time, the cumulants $g$ for the full magnetization do not intersect at one point (Figure 12b). For this reason, we evaluated $K_c$ roughly as the middle of the interval where the cumulant $g$ experiences a jump. As can be seen from Table 3, such inaccuracy in determination of $K_c$ results in large deviations of the exponent $\delta$.

To evaluate the exponent $\delta$, we built relationships between $\ln M$and $\ln H$ (Figure 13).

If $q_{11}=q_{22}$, the relationships between $\ln M$and $\ln H$ for lattices of different dimensions $L$ merge into one line already at a sufficiently small $H$ (see Figure 13a). The slope of this relationship was used to evaluate $\delta$. The exponent $\delta$ was found to be independent of the ratio $q/\left|q_{12}\right|$, being roughly equal to one (see Table 3). Its deviations from one are about 10^-3, which is comparable to the error of our evaluation. Thus, the value of the exponent $\delta$ in this case fully agrees with its value $\delta =1$in the mean-field model for $q_{12}=q_{12}^{*}$.

If $q_{11}\ne q_{22}$, the curves $\ln M$ as functions of $\ln H$ become linear at certain $H$, whose magnitude decreases with growing $L$ (see Figure 13b). The parameter $\delta$ measured by the curve slope differs considerably from the expected value $\delta =3$ in all the investigated configurations (Table 3), and depends on the parameters of the model.

4.

Let us consider how the critical exponents change with the increase of the external field in a balanced layered model. We estimated the splitting temperature $K_s$ by plotting the temperature dependencies of the following cumulants:

$$g_s=1-\frac{\langle {\left(m{}_1-m_2\right)}^4\rangle }{3{\langle {\left(m{}_1-m_2\right)}^2\rangle }^2}.$$

The temperature $K_s$ was determined as the point of intersection of the cumulants for lattices with various dimensions (Figure 14). We point out that the cumulants $g_s$ are usually used for determining the critical temperature of an antiferromagnetic phase transition [25]. The magnetization at the splitting point,$m_s$, was calculated as the magnitude of the local maximum of magnetization for the lattice of the maximum size ($L=32$).

To determine the critical exponents ${\beta }_s$ and ${{\beta }^{\prime }}_s$ we plotted the dependencies of $\ln (m_s-M)$ on $\ln \left|t_s\right|$ for the lattices with various dimensions (Figure 15). These curves become linear when $t_s<0$, and their slope allowed us to estimate the exponent ${\beta }_s$ (see Table 4). For $t_s>0$ the dependencies of $\ln (m_s-M)$ on $\ln t_s$ are nonlinear for the whole range of $t_s$, making it impossible to estimate the value of the exponent ${{\beta }^{\prime }}_s$.

The temperature dependencies of the susceptibility $\chi$ near the splitting point are shown in Figure 16. As one can see, when the external field is present, the susceptibility curves depend on the lattice size. The accuracy or our simulations is insufficient to unambiguously attribute this observation to either a divergence of the susceptibility at the temperature $K_s$, or to its finite jump.

In order to determine the critical exponents ${\delta }_s$ and ${{\delta }^{\prime }}_s$, we plotted the dependencies of $\ln \left|\Delta M\right|$ on $\ln \left|\Delta H\right|$ for the lattices with the dimensions ranging from $L=12$ to $L=32$ (Figure 17). Independently on the sign of $\Delta H$, these curves have a linear asymptotic, the slope of which allowed us to extract the critical exponents (Table 4).

The results of these simulations are shown in Table 4. The splitting temperature $K_s$ differs substantially from the one obtained in the framework of the mean-field model (MFM). The corresponding numbers we present in the column 2 of Table 4. However, when in Eq. (33) we use the values of $K_{с}$ and $m_s$ obtained as a result of this modelling, we obtain $K_s$ rather accurately (compare three numbers in the column 5 of Table 4). The obtained critical exponent of the layered model ${\beta }_s$ takes values in a range of 0.5-0.6, which differs significantly from its value from the mean-field model (${\beta }_s=1$). The value of the exponent ${\beta }_s$ decreases with the increase of the external field. However, similarly to the mean-field model, the value of ${\beta }_s$exceeds considerably the values of the exponents of partial magnetization, ${\beta }_{m_{1,2}}\approx 0.3$, measured in the absence of the external field (in the mean-field model, ${\beta }_{m_{1,2}}=1/2$). For small values of the external field, the values of the exponents ${\delta }_s$ and ${{\delta }^{\prime }}_s$ are approximately equal to 1. With the increase of the external field the exponent ${\delta }_s$increases, while ${{\delta }^{\prime }}_s$ decreases.