1 Introduction

Antiferromagnetic materials demonstrate a special demeanor in an applied magnetic field depending on the temperature. The material shows no response to the external magnetic field at very low temperatures, since the antiparallel ordering of atoms is strictly maintained. Some atoms break free of the antiparallel arrangement and align with the external magnetic field at higher temperatures. This alignment remains until a critical temperature called as the Néel temperature [1]. Above this temperature, thermal agitation progressively prevents alignment of the atoms with the magnetic field, so that the weak magnetism produced in the material by the alignment of its atoms continuously decreases as temperature is increased.

In addition, when the spins are chosen to lie along a given axis, which is usually taken as the z-axis, the problem is relatively easier in comparison with the two- or three-dimensional cases. The latter leads to the quantum spin models which is much more difficult and may be approached by using the Schrödinger equations or Heisenberg models. The difficulty arises from the nonlinearity of the considered equations [2,3,4,5,6,7] in addition to the non-commutativity of the spin operators.

Some examples for the two-dimensional models usually include the bilinear interaction parameter in the z-direction and the transverse magnetic field in the x-direction which were considered in the effective-field theory (EFT) [8,9,10,11,12,13]. Besides, some other parameters were also included in this model, such as the addition of the longitudinal crystal field in the EFT [14], the longitudinal magnetic field by combining the pair approximation with the discretized path-integral representation [15], random longitudinal crystal-field interactions in terms of an expansion technique for cluster identities of localized spin systems [16], the longitudinal crystal and external magnetic fields [17] and the effect of random transverse crystal field [18] in terms of a mean-field approximation (MFA), the Blume–Capel (BC) model with longitudinal random crystal and transverse magnetic fields in the MFA [19, 20] and with both transverse field and transverse crystal field in a stochastic series expansion quantum Monte Carlo method [21]. The Blume–Emery–Griffiths (BEG) model was also used by including the transverse fields effects [22] in the MFA, the random transverse field with the EFT [23] and a new effective correlation MFA in a transverse crystal field [24]. The Oguchi approximation (OA) was also considered in a few works. The spin-1/2 anisotropic Heisenberg model was studied by taking into account the Dzyaloshinskii–Moriya (DM) interaction parameter [25] and by the consideration of the second-nearest-neighbor exchange interactions [26]. Both exchange interaction and single-ion anisotropy effects were examined in the mixed spin-1 and spin-1/2 case [27], its compensation temperature behavior [28] and the investigation of magnetic susceptibility [29], in the mixed spin-3/2 and spin-1/2 model [30] and with random crystal field [31] and, in the mixed spin-2 and spin-1/2 for the square [32] and simple cubic lattices [33].

The DM interaction is an antisymmetric exchange interaction with a contribution to total magnetic exchange interaction between two neighboring magnetic spins as a source of weak ferromagnetic behavior in an antiferromagnet. The spin-1/2 model with the inclusion of the DM interaction (DMI) was established in a precise mapping correspondence to the simple spin-1/2 Ising model on Kagome lattice [34], in the EFT with the study of the critical and reentrant behaviors for the antiferromagnet model in a longitudinal magnetic field [35], the MFA in the study of the potassium jarosite compound KFe\(_3\)(OH)\(_6\)(SO\(_4\))\(_2\) for the antiferromagnetic XY model [36], in the matrix-product-state method to examine the quantum phase transitions and the ground-state phase diagram for the Heisenberg–Ising alternating chain [37], the magnetization of triangular-lattice antiferromagnets Ba\(_3\)CoSb\(_2\)O\(_9\) and CsCuCl\(_3\) with a \(120^{\circ }\) spin structure in the ab plane [38], in a combined analytical and density matrix renormalized group study of the antiferromagnetic XXZ chain in a transverse magnetic field [39], the pair XYZ Heisenberg interaction and quartic Ising interactions was exactly solved by establishing a precise mapping relationship with the corresponding zero-field eight-vertex model [40] and with antiferromagnetic exchange interactions in the presence of a longitudinal external magnetic field by employing the usual MFA [41].

In order to see the effects of crystal field on this system, the spin-1 models were also taken under consideration. A few examples may be given as: the two-spin cluster MF method on the simple cubic lattice [42], the Schwinger boson MF theory for one-dimensional antiferromagnet [43], the infinite time-evolving block decimation method on ground-state phase diagrams of alternating chains [44], the local moments on the pyrochlore lattice with a generic interacting spin model on a pyrochlore lattice [45], the chain in the presence of an external magnetic field by using a combination of numerical and analytical techniques [46] and the two-dimensional Heisenberg model with Ising anisotropy and dipolar interaction under zero and finite magnetic fields using a Monte Carlo method [47]. The quantum-memory-assisted entropic uncertainty relations in two-qutrit spin-1 Heisenberg XYZ and XXX chains under homogeneous magnetic fields were studied [48]. The spin-1 SU(3)-models in the seven-parameter manifold of translation- and reflection-invariant Hamiltonians with nearest-neighbor couplings were considered [49]. The quantum coherence was studied in a spin chain with both symmetric exchange and antisymmetric DM couplings [50]. The XYZ Heisenberg model was considered from the standpoint of the quantum inverse problem method [51]. The negativity as a measure of thermal entanglement was studied for a two-qutrit spin-1 anisotropic Heisenberg XYZ chain with the DM interaction in a homogeneous magnetic field [52]. As we can see from these works, the thermal variations and the DMI effects on the magnetization and magnetic phase diagrams have not been considered in detail.

Recently, the spin-1 XYZ model with the DMI effects was examined in the MFA by introducing the spin operators for the ferromagnetic case, i.e., \(J_z>0.0\) [53] and very interesting behaviors for the magnetization and phase diagrams were observed. In this work, the antiferromagnetic case with \(J_z<0.0\) is considered for the Hamiltonian consisting of the DMI parameter \(\Delta _m\), the crystal field \(D_z\) along the z-axis and the external magnetic field in all three dimensions with \(H_x=H_y=H_z=H\) for the coordination number \(q=6.0\) corresponding to the simple cubic lattice. The thermal variations of the total magnetization and its components are investigated in detail to obtain the phase diagrams on the \((H/|J_z|, T/|J_z|)\), \((D_z/|J_z|, T/|J_z|)\) and \((\Delta _m/|J_z|, T/J_z)\) planes for given values \(\Delta _m\), \(D_z\) and H.

The remainder of this work is structured as follows: Sect. 2 involves our approach to the formulation in the MFA, the thermal variations of net magnetization are illustrated in Sect. 3, Sect. 4 contains the phase diagrams, and the last section includes a brief summary and conclusions.

2 The formulation

The spin-1 XYZ model Heisenberg Hamiltonian in terms of the bilinear exchange interaction parameter (\(J_z\)) between the nearest-neighbor (NN) spins, the crystal field (\(D_z\)) along the z-axis, the DMI (\(\Delta _m\)) and external magnetic field components (\(H_x, H_y, H_z\)) is given as:

$$\begin{aligned} {\mathcal {H}}= & {} -J_z \sum _{\langle i,j\rangle } S_{i}^z S_{j}^z- \sum _{i} D_z \left( S_{i}^z\right) ^2-\Delta _m \sum _{\langle i,j\rangle } \left( S_{i}^x S_{j}^y-S_{i}^y S_{j}^x\right) \nonumber \\&-H_x \sum _{i} S_{i}^x-H_y \sum _{i} S_{i}^y-H_z \sum _{i} S_{i}^z, \end{aligned}$$
(1)

where \(\langle i,j\rangle \) points out the summation over the NN spins, \(J_z<0\) for the antiferromagnetic (AFM) interaction and \(S_{i}^x, S_{i}^y\) and \(S_{i}^z\) are the components of spin-1 operator at site i which are presented in the matrix forms as:

$$\begin{aligned} S_{i}^x=\frac{1}{\sqrt{2}} \left( \begin{array}{ccc} 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 \\ \end{array} \right) ,\quad S_{i}^y=\frac{1}{\sqrt{2}} \left( \begin{array}{ccc} 0 &{}\quad -i &{}\quad 0 \\ i &{}\quad 0 &{}\quad -i \\ 0 &{}\quad i &{}\quad 0 \\ \end{array} \right) , \quad S_{i}^z= \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -1 \\ \end{array} \right) . \end{aligned}$$
(2)

In order to surmount the difficulty of studying the quantum mechanical model, we approach to the problem by using a mean-field approximation. Even if this approach does not lead to the exact solutions, one can get at least some qualitative results. Therefore, the Hamiltonian \({\mathcal {H}}\) in Eq. (1) can be written in the MF form as:

$$\begin{aligned} -\beta {\mathcal {H}}_{\mathrm{MFA}}=-\beta \sum _{i} {\mathcal {H}}_{\mathrm{MFA}}^{(i)} \end{aligned}$$
(3)

in which

$$\begin{aligned} -\beta {\mathcal {H}}_{\mathrm{MFA}}^{(i)}= & {} \beta q J_z M_z S_{i}^z+\beta D_z \left( S_{i}^z\right) ^2+\beta q \Delta _m \left( M_y S_{i}^x-M_x S_{i}^y\right) \nonumber \\&+\beta \left( H_x S_{i}^x+H_y S_{i}^y+H_z S_{i}^z\right) \end{aligned}$$
(4)

where \(M_{\mu }=\langle S_{j}^{\mu }\rangle \) are the magnetization components with \({\mu }=x, y, z\), \(\beta =1/(kT)\), k is being the Boltzmann constant and set to 1.0 for convenience and q is the number of the NNs.

The matrix representation of \(-\beta {\mathcal {H}}_{\mathrm{MFA}}^{(i)}\) is achieved by using the spin operators, i.e., Eq. (2), and found as:

$$\begin{aligned} -\beta {\mathcal {H}}_{\mathrm{MFA}}^{(i)}=\left( \begin{array}{ccc} \beta (D_z+H_z+q J_z M_z)&{}\quad H_{12} &{}\quad 0 \\ H_{21}\ &{}\quad 0 &{}\quad H_{23}\ \\ 0 &{}\quad H_{32} &{}\quad \beta (D_z-H_z-q J_z M_z) \\ \end{array} \right) \end{aligned}$$
(5)

in which \(H_{12}=H_{23}=\frac{\beta }{\sqrt{2}} [(H_x-i H_y)+ q \Delta _m (i M_x + My)]\) and \(H_{32}=H_{21}=H_{12}^*=H_{23}^*\), i.e., the last two are the complex conjugates of first two. The eigenvalues of this matrix are real and are attained from the below third degree equation

$$\begin{aligned} \lambda ^3+a_2 \lambda ^2+a_1 \lambda +a_0=0 \end{aligned}$$
(6)

where the coefficients are found as:

$$\begin{aligned} a_0= & {} \beta ^3 D_z \left[ H_x^2+H_y^2+2 q \Delta _m (H_x M_y-H_y M_x)+q^2\Delta _m^2 \left( M_x^2+M_y^2\right) \right] , \nonumber \\ a_1= & {} \beta ^2 \left[ D_z^2-H_x^2-H_y^2-H_z^2+2 q \Delta _m (H_y M_x -H_x M_y)-2 q H_z J_z M_z\right. \nonumber \\&\left. - q^2 \Delta _m^2 \left( M_x^2-M_y^2\right) - q^2 J_z^2 M_z^2\right] ,\nonumber \\ a _2= & {} -2 \beta D_z. \end{aligned}$$
(7)

The eigenvalues are reached by using the mathematical identity given as:

$$\begin{aligned} \lambda _{n+1}=2y \cos \left[ \frac{1}{3}\left\{ \arccos \left( -\frac{G}{2y^3}\right) +2n\pi \right\} \right] -\frac{a_2}{3}, n=0,1,2. \end{aligned}$$
(8)

with \(y=(-W)^{1/2}\), \(W=(3 a_1-a_2^2)/9\) and \(G=(27 a_0-9 a_1 a_2+2 a_2^3)/27\).

The partition function can be calculated by using these three eigenvalues obtained from the MFA Hamiltonian as:

$$\begin{aligned} Z_i= & {} \hbox {Tr}_{(i)} \exp \left[ -\beta {\mathcal {H}}_{\mathrm{MFA}}^{(i)}\right] \nonumber \\= & {} \sum _{n=1}^3e^{\lambda _n}=e^{\lambda _1}+e^{\lambda _2}+e^{\lambda _3}. \end{aligned}$$
(9)

The free energy of the model is found from the well-known definition by the usage of the partition function as:

$$\begin{aligned} f=-\frac{1}{\beta } \ln Z_i \end{aligned}$$
(10)

which will be utilized to find the formulations of the order-parameters. The dipolar moments or magnetization components, \(M_{\mu }=\langle S_{j}^{\mu }\rangle \) with \({\mu }=x, y, z\) are found from

$$\begin{aligned} M_{\mu }= & {} \langle S_i^\mu \rangle =-\frac{\partial f}{\partial H_\mu }=\frac{1}{\beta } \frac{\partial \ln Z_i}{\partial H_\mu }=\frac{\hbox {Tr}_{(i)} \left[ S_i^z \exp \left( -\beta {\mathcal {H}}_{\mathrm{MFA}}^{(i)}\right) \right] }{Z_i} \nonumber \\= & {} \frac{1}{\beta } \left[ \frac{\frac{\partial \lambda _1}{\partial H_\mu }e^{\lambda _1}+\frac{\partial \lambda _2}{\partial H_\mu }e^{\lambda _2}+\frac{\partial \lambda _3}{\partial H_\mu }e^{\lambda _3}}{e^{\lambda _1}+e^{\lambda _2}+e^{\lambda _3}}\right] \nonumber \\= & {} \frac{1}{\beta '} \left[ \frac{\frac{\partial \lambda _1}{\partial h_\mu }e^{\lambda _1}+\frac{\partial \lambda _2}{\partial h_\mu }e^{\lambda _2}+\frac{\partial \lambda _3}{\partial h_\mu }e^{\lambda _3}}{e^{\lambda _1}+e^{\lambda _2}+e^{\lambda _3}}\right] . \end{aligned}$$
(11)

where \(\beta '=\beta J_z\) and \(h_\mu =H_\mu /J_z\) which are the reduced temperature and external magnetic field components.

It should be noted that the lattice must be divided into two sublattices A and B to count the AFM interactions, i.e., \(J_z<0.0\). Thus, the magnetization components for the sublattices must be written as:

$$\begin{aligned} M^A_{\mu }= & {} M^A_{\mu }\left( M^B_{x}, M^B_{y}, M^B_{z}\right) , \nonumber \\ M^B_{\mu }= & {} M^B_{\mu }\left( M^A_{x}, M^A_{y}, M^A_{z}\right) . \end{aligned}$$
(12)

Since the magnetization vector is given as , the sublattice magnetizations can be obtained from

$$\begin{aligned} M_{A}=\sqrt{\sum _{\delta =x,y,z} \left( M^A_\mu \right) ^2}=\sqrt{\left( M^A_x\right) ^2+\left( M^A_y\right) ^2+\left( M^A_z\right) ^2} \end{aligned}$$
(13)

and

$$\begin{aligned} M_{B}=\sqrt{\sum _{\delta =x,y,z} \left( M^B_\mu \right) ^2}=\sqrt{\left( M^B_x\right) ^2+\left( M^B_y\right) ^2+\left( M^B_z\right) ^2}. \end{aligned}$$
(14)

Finally, the total magnetization of the system, i.e., the net magnetization, is given as:

$$\begin{aligned} M_{\mathrm{Tot}}=M_A-M_{B}. \end{aligned}$$
(15)

After having calculated the magnetization components and net magnetization for the sublattices A and B, we are now ready to study their thermal variations for given values of \(D_z\), \(\Delta _m\), the external magnetic field components chosen as \(H=H_x=H_y=H_z\) and the coordination number \(q=6\) corresponding to the simple cubic lattice. An iterative procedure is followed for the calculation of our order-parameters with given values of the system parameters under temperature variations. The obtained results are presented in the next section.

3 Thermal variations of net magnetization

The usual approach to obtain the phase diagrams involves the study of thermal changes of magnetization. It is well known that when magnetization goes to zero continuously, it is called as the second-order phase transition (SOPT), and when it presents a discontinuity, it is the first-order phase transition (FOPT), the lines of which enclose a region of the ordered phase, i.e., the AFM phase since \(J_z<0.0\). In addition to these phase regions, this work also reveals a region of random or oscillatory behavior for magnetization which is well confined by the borders and it is to be called as the random phase regions. The temperatures at which these phase transitions take place are denoted as \(T_c\), \(T_t\) and \(T_{r}\) for the SOPT, FOPT and random cases, respectively. A model can display any combinations of these phase transitions or the same ones more than once. When the latter occurs, the reentrant behavior appears in the phase diagrams.

Before presenting the thermal variations of \(M_{\mathrm{Tot}}\), it should be noted we have combined together two solutions of \(\pm M_{\mathrm{Tot}}\) whenever it is possible which allows us to distinguish the symmetric and anti-symmetric solutions. In Fig. 1, the topological thermal variation plots of the total magnetization, \(M_{\mathrm{Tot}}\), are presented for given values of (\(\Delta _m, H, D_z\)) when \(q=6\) and \(J_z=-1.0\). Figure 1a and b shows the existence of one \(T_c\), while Fig. 1c and d presents two \(T_c\)’s. Figure 1e shows that the magnetization lines enclosing the AFM phase at lower temperatures, before termination they start oscillating randomly at the \(T_r\) and this randomness ends at the second \(T_r\). Afterward, the system enters into the PM phase. Figure 1f first shows the AFM phase at lower temperatures which terminates at the \(T_c\). Then the system enters into the PM phase. With the increasing temperature, the random phase appears at \(T_r\) which also terminate at the second \(T_r\) with the entrance of the PM phase again. It is shown in Fig. 1g that the random phase starts from zero and finishes at the \(T_r\), then \(M_{\mathrm{Tot}}\) separates the AFM phase which expires at the \(T_c\). Figure 1h and i shows the existence of \(T_t\)’s. The first one gives a \(T_t\) and then a \(T_c\) and the last one gives only a \(T_t\). In Fig. 1j and k, the random phase starts at zero temperature and terminates at the \(T_r\)’s. \(M_{\mathrm{Tot}}=0.0\) appears at lower temperatures and then, the random phase starts from the first \(T_r\) with increasing temperature and disappears at the second \(T_r\) which leads to the reentrant behavior as seen in Fig. 1l. \(M_{\mathrm{Tot}}\) shows itself in an oscillatory fashion in Fig. 1m which terminates at the \(T_r\). When the random solutions are zoomed closer, this oscillatory behavior is observed in some cases; therefore, they are not distinguished from each other in this work. Figure 1n first shows a \(T_t\) and then two \(T_r\)’s. The final plot, Fig. 1o shows a \(T_r\) then a \(T_t\) and then a \(T_c\), i.e., all types of phase transitions coexist together.

Fig. 1
figure 1

Thermal variations of total magnetization for given values of (\(\Delta _m , H, D_z\)) as a \((0.6,3.0,-2.0)\), b \((0.3,3.0,-1.0)\), c (0.75, 3.0, 0.0), d (1.0, 4.25, 0.5), e \((0.77,3.0,-3.0)\), f \((1.0,4.0,-2.8)\), g \((0.5,0.5,-2.6)\), h \((0.25,1.0,-2.5)\), i \((0.25,0.5,-3.5)\), j \((0.6,3.0,-3.0)\), k \((0.94,3.0,-2.0)\), l \((1.0,4.0,-3.1)\), m \((1.0,0.2,-2.0)\), n \((1.0,4.0,-2.25)\) and o (1.0, 4.0, 0.5)

Full size image

In the next section, we combine all these solutions to construct the phase diagrams on the possible planes of our system parameters. The combinations of these points make up the possible phase lines and their combinations lead to the some critical points.

4 The phase diagrams

We are now ready to construct the phase diagrams on the \((H/|J_z|, T/|J_z|)\), \((D_z/|J_z|, T/|J_z|)\) and \((\Delta _m/|J_z|, T/|J_z|)\) planes for given values of \(\Delta _m\), H and \(D_z\), under the conditions \(H_x=H_y=H_z=H\) and \(q=6\) with \(J_z\) taken as a scaling factor. It is found that the model exhibits SOPT, FOPT and random phase lines which are denoted by the solid, dashed and dotted-dashed lines and called as \(T_c\)-, \(T_t\)- and \(T_r\)-lines, respectively. These lines can combine at some critical points in doubles or triples: It is well known that the tricritical point is the point at which the \(T_c\)- and \(T_t\)-lines merge together is called here as TCP\(_1\) in shorthand notation. Similarly, the point at which the \(T_c\)- and \(T_r\)-lines combined together is called as TCP\(_2\), while the meeting point of \(T_t\)- and \(T_r\)-lines is called as TCP\(_3\). They are denoted with the solid, empty and dashed circles in the phase diagrams, respectively. The bicritical point is the combination point of two same type and one other type of phase lines. When two \(T_t\)-lines and a \(T_c\)-line, two \(T_r\)-lines and a \(T_c\)-line and two \(T_r\)-lines and a \(T_t\)-line do this called as the BCP\(_1\), BCP\(_2\) and BCP\(_3\) and shown with the solid, empty and dashed squares, respectively. In addition, the critical end point is observed when a critical line is terminated on another critical line. When this happens with \(T_c\)-line ending on \(T_t\)-line, \(T_c\)-line terminating on \(T_r\)-line and \(T_t\)-line finishing on \(T_r\)-line are called as CEP\(_1\), CEP\(_2\) and CEP\(_3\) and denoted with the solid, empty and dashed ellipses, respectively. In addition, the combination points of the \(T_c\)-, \(T_t\)- and \(T_r\)-lines denoted with a star. The final type of critical point observed in this work is termination of the \(T_c\)- and \(T_t\)-lines on the line of \(T_r\)-line denoted with a cross.

Before the presentations of phase diagrams, the possible effects of our system parameters should be mentioned briefly. As \(D_z\) gets low negative enough the lowest spin value, i.e., the zero state of spin-1, is preferred. External magnetic field tries to align the spins in the given direction ferromagnetically. The bilinear exchange interaction parameter \(J_z<0\) favors the antiferromagnetic alignment of the spins. The temperature competes with all these parameters to destroy all possible orders, i.e., it forces the spins to orient themselves randomly leading to the paramagnetic phase. \(\Delta _m\) leads to spin-canting which is a special condition which occurs when antiparallel magnetic moments are deflected from the antiferromagnetic plane, resulting in a weak net magnetism. All these competitions should lead to very rich magnetic phase diagrams with various phase transitions and critical points.

Fig. 2
figure 2

The phase diagrams on the \((H/|J_z|, T/|J_z|)\) planes for given values of \(D_z\) when a \(\Delta _m=0.25\), b\(_1\)–b\(_2\) \(\Delta _m=0.5\), c \(\Delta _m=0.75\) and d\(_1\)–d\(_2\) \(\Delta _m=1.0\)

Full size image

The first phase diagrams are obtained on the \((H/|J_z|, T/|J_z|)\) planes for given values of \(D_z\) when \(\Delta _m=0.25, 0.5, 0.75\) and 1.0. Figure 2a is mapped for \(\Delta _m=0.25\) shows that the \(T_c\)-lines start from higher temperatures for higher \(D_z\) at \(H=0.0\) and they are lowered as H increases which eventually terminate at higher H for higher \(D_z\). The \(T_t\)- and \(T_r\)-lines are seen at lower H and temperatures for negative \(D_z\). It is clear that \(T_c>T_t>T_r\) and the magnetic fields satisfy for each phase \(H_c>H_t>H_r\). When \(D_z=-2.0\), in addition to the \(T_c\)-line we see a combination \(T_t\)- and \(T_r\)-lines combined at the CEP\(_3\) for low T and H values. For \(D_z=-2.5\) the \(T_c\)- and \(T_t\)-lines unite at the BCP\(_1\). The critical line is only \(T_t\) type for \(D_z=-3.0\). The next one, Fig. 2b\(_1\)–b\(_2\) is obtained for \(\Delta _m=0.5\). The first one shows that only the \(T_c\)-lines exist for \(-1.0\le D_z\le 2.0\). When \(D_z=-2.0\), the \(T_r\)-line is also seen at low H and T in addition to \(T_c\)-line at higher H and T. The latter exhibits not only the BCP\(_1\) but also the CEP\(_3\) for \(D_z=-2.5\). The lower portion of the \(T_t\)-line for \(\Delta _m=0.25\) is now turned into a \(T_r\)-line for \(\Delta _m=0.5\). For \(D_z=-2.56\), one sees two separated lines with \(T_c\)- and \(T_t\)-lines combined at TCP\(_1\) and only a \(T_r\)-line. The lines are only the \(T_r\)-lines for other values of \(D_z\). The reentrant behavior at higher H for all the \(T_r\)-lines is also clear. Figure 2c is obtained for \(\Delta _m=0.75\) shows again the \(T_c\)-lines exist from \(D_z=2.0\) to \(-1.0\). The reentrant behavior is also seen for these \(T_c\)-lines except for \(-1.0\). When \(D_z=-2.0\) and \(-2.5\) the BCP\(_2\) is observed. Only the \(T_r\)-lines are seen when \(D_z=-3.0\) and \(-3.5\) which display reentrant behavior also. In Fig. 2d, we see that the \(T_c\)-lines terminate at TCP\(_1\) for \(D_z=2.0, 0.0\) and 1.0, the \(T_t\)-line for 1.0 also merges with a \(T_r\)-line at TCP\(_3\). For \(D_z=-0.5\), we see all the phase lines with a BCP\(_2\), a TCP\(_1\) and CEP\(_3\). When \(D_z=-1.0\), we see TCP\(_1\). BCP\(_2\) is observed for \(D_z=-2.0\). Again only a \(T_r\)-line is seen for \(D_z=-3.0\). The \(T_c\)-lines always present some reentrant behavior. It is clear from these figures that as \(\Delta _m\) increases the critical lines persist to higher H values.

Fig. 3
figure 3

The phase diagrams on the \((D_z/|J_z|, T/|J_z|)\) planes for given values of H when a \(\Delta _m=0.25\), b \(\Delta _m=0.5\), c \(\Delta _m=0.75\) and d \(\Delta _m=1.0\)

Full size image
Fig. 4
figure 4

The phase diagrams on the \((\Delta _m/|J_z|, T/|J_z|)\) planes for given values of H when a \(H=0.5\), b \(H=1.5\), c \(H=3.0\) and d \(H=4.0\)

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The next phase diagrams are calculated on the \((D_z/|J_z|, T/|J_z|)\) planes for given values of H when \(\Delta _m=0.25\), 0.5, 0.75 and 1.0. Figure 3a is obtained for \(\Delta _m=0.25\). Only the \(T_c\)-lines are observed for \(H=3.0\) and 4.0 which start from lower \(D_z\) for lower H at zero T and they become constant temperature lines for higher \(D_z\) values. When \(H=2.0\), we see the combination of \(T_c\)- and \(T_t\)-lines combined at the TCP\(_1\). The BCP\(_1\) is seen for \(H=1.0\). The \(T_c\)-, \(T_t\) and \(T_r\)-lines emerge at the critical point indicated with a star for \(H=0.5\). For \(\Delta _m=0.5\), the \(T_c\)-lines for \(H=3.0\) and 4.0 moves to lower \(D_z\) values at zero temperature. This means increasing \(\Delta _m\) shifts the \(T_c\)-lines to lower \(D_z\). For \(H=2.0\), critical lines are the same as in Fig. 3b but now a bulging towards negative \(D_z\) values appears. The \(T_c\)- and \(T_r\)-lines combine at the BCP\(_2\) for both \(H=0.5\) and 1.0. The case with \(\Delta _m=0.75\) is shown in Fig. 3c. Now for \(H=2.0\) we see that TCP\(_1\) transforms into BCP\(_2\). When \(H=3.0\), all types of the phase lines exit which are combined at BCP\(_2\) and BCP\(_3\). The bulging in \(T_c\)-line for \(H=4.0\) is also apparent. The last plot on this plane is calculated for \(\Delta _m=1.0\) exhibits BCP\(_2\) for \(H=0.5-3.0\). Now \(H=4.0\) case is very complicated with half closed loops of \(T_t\)- and \(T_r\)-lines located around \(D_z=0.0\) as seen in Fig. 3d. The other parts of the critical lines consist of \(T_c\)-line terminating at BCP\(_2\), then the \(T_r\)-lines enclose a region before combining and afterward the \(T_c\)-, \(T_t\)- and \(T_r\)-lines combine at the critical point indicated with the star.

The final phase diagrams, i.e., Fig. 4, are mapped on the \((\Delta _m/|J_z|, T/|J_z|)\) planes for given values of \(D_z\) when \(H=0.5, 1.5, 3.0\) and 4.0. In these figures, we always see the \(T_c\)-lines for \(D_z=2.0\) and 1.0 whatever the values of other parameters are. The temperatures of these lines decrease as H increases. They are not disturbed when H is small, i.e. for 0.5 and 1.5, since the \(T_c\)-lines stay straight. When \(H=3.0\), the temperatures of these lines are further lowered at small \(\Delta _m\) values. They are further lowered when \(H=4.0\). Actually, \(D_z=1.0\) curve makes a deep at lower \(\Delta _m\) but as \(\Delta _m\) increases it’s temperature rises. Thus, the magnetic field is strong enough to overcome when \(\Delta _m\) is small but it eventually surrender to the effects of \(\Delta _m\). For \(D_z=0.0\) line we see that the \(T_c\)-lines behave as in the \(D_z=2.0\) and 1.0 cases. When \(H=4.0\), the \(T_c\)-line starts from zero temperature at about \(\Delta _m=0.63\), then rises as \(\Delta _m\) increases. The \(T_t\)- and \(T_r\)-lines are also observed for higher \(\Delta _m\) with \(T_c>T_t>T_r\). The \(T_c\)-lines for \(D_z=-1.0\) and \(-2.0\) are similar with the previous \(T_c\)-lines, but now they terminate on their BCP\(_2\)’s. For \(D_z=-2.0\), we also see \(T_t\)-line merged together with the \(T_r\)-lines at BCP\(_3\) for lower \(\Delta _m\). The case with \(D_z=-3.0\) and \(-4.0\) starts with the \(T_t\)-lines at nonzero temperatures then combine at the CEP\(_3\) with their \(T_r\)-lines, which starts from zero temperatures at some mid-values of \(\Delta _m\) and increase with increasing \(\Delta _m\). As H increases they move to higher \(\Delta _m\) and disappear for \(H=4.0\). Figure 4b also shows the phase lines for \(D_z=-2.5\) in which the \(T_r\)-lines are similar with the previous case. In addition, we see a \(T_c\)-line emerging from BCP\(_2\) and terminating at the BCP\(_1\). Then the \(T_t\)-line terminates on CEP\(_3\). Finally, in the last figure we see the \(T_c\)- and \(T_t\)-lines combine at the TCP\(_1\), the \(T_c\)- and \(T_t\)-lines terminate on the same point on a \(T_r\)-line indicated with a cross at higher \(\Delta _m\).

After having given all the details of the thermal variations of the net magnetization and the phase diagrams on possible planes, we are now ready to conclude this work in the last section.

5 Summary and conclusions

As a result of our investigation for the AFM spin-1 XYZ model by using a mean-field approach with the introduction of the spin operators on the model for the simple cubic lattice have revealed compelling behaviors for the total magnetization. It includes three different types of phase lines including the \(T_c\)-, \(T_t\)- and \(T_r\)-lines with three different phase regions, i.e., AFM, PM and random phase. In addition, various combinations of these lines have led to eleven types of critical points including the TCP, BCP and CEP, each of which are obtained in triples, and the critical points indicated with star and cross. The reentrant behavior was also observed in the \(T_c\)- and \(T_r\)-lines. It should be noted that negative crystal field values lead to interesting critical behaviors for our model as in the well-known Blume–Capel model. As we mentioned earlier as \(D_z\) becomes more negative the spin zero state of spin-1 is favored at lower temperatures. The external magnetic field tries to align the spins ferromagnetically by competing with \(D_z\) and with \(J_z\), which favors AFM phase leading to zero magnetization, thus interesting critical behaviors are observed. It is also clear that as \(\Delta _m\) increases toward the value of 1.0, the ordered phase regions move to higher magnetic field values as seen from the bulging of the phase lines which causes the existence of reentrant behavior. It should be noted that this is an approximate approach to the spin-1 XYZ model and it was inspired by the two-dimensional works mentioned in the references, thus, it only leaves us finding the eigenvalues of the model to calculate the partition function. This model has not been investigated so far; therefore, we cannot give any quantitative comparisons. As a last word, we hope that this work gets the attention of scientists for its deeper studies.