Spin-1/2 XXZ Heisenberg cupolae: magnetization process and related enhanced magnetocaloric effect

The magnetization process and adiabatic demagnetization of antiferromagnetic spin-1/2 XXZ Heisenberg clusters with the shape of Johnson's solids (triangular cupola, square cupola and pentagonal cupola) are examined using the exact numerical diagonalization depending on a relative strength of the exchange anisotropy. It is demonstrated that XXZ Heisenberg cupolae display at least one more magnetization plateau, which is totally absent in Ising cupolae. The novel magnetization plateaux extend over a wider range of magnetic field with increasing of a quantum part of the XXZ exchange interaction at the expense of the original plateaux present in the Ising limiting case. It is shown that XXZ Heisenberg triangular and pentagonal cupola exhibits in the proximity of all magnetization jumps an enhanced magnetocaloric effect in a vicinity to zero magnetic field, making these magnetic systems a promising refrigerant for cooling down to absolute temperature.

In addition, antiferromagnetic spin clusters open the way towards several applications. One of them could be mentioned: the magnetic cooling based on the magnetocaloric effect [27,28,29,30]. An enhanced magnetocaloric effect was found in the vicinity of magnetization jumps, around which the rapid change of temperature was observed upon the variation of the magnetic field during the process of adiabatic demagnetization. Moreover, it was shown in our recent study [31] that the absence of zero magnetization plateau causes an enhanced magnetocaloric effect when shutting down the external magnetic field during the process of adiabatic demagnetization. It was found that the Ising octahedron, dodecahedron and cuboctahedron exhibit this feature and they are promising candidates for reaching ultra-low temperatures [17,31]. Unfortunately, by considering (xy) part of the exchange interaction the temperature achieves only finite values by switching off the magnetic field during the process of adiabatic demagnetization due to the presence of zero magnetization plateau in the magnetization process of the XXZ Heisenberg octahedron, dodecahedron and cuboctahedron. Heisenberg spin clusters with the absence of zero magnetization plateau as a promising candidates for future application in magnetic refrigeration has been much less studied. Therefore, it appears worthwhile to investigate the low-temperature magnetization process and isentropy lines in the field-temperature plane of the spin-1/2 XXZ Heisenberg cupolae, for two of which we are able to predict the absence of zero magnetization plateau in the zero-temperature magnetization curve. It should be pointed out, that for spin-1/2 clusters the fractional values, at which the magnetization plateau can occur are given by the formula where N is the total number of spins of a given cluster ( N 2 denotes the total spin S T ) and n accounts for the total number of antiparallel spins against the external magnetic field. It is clear from Eq. (1) that the existence of the zero magnetization plateau is possible only if N = 2n, which follows that N is an even number.
The organization of this paper is as follows. Spin-1/2 XXZ Heisenberg cupolae are introduced in Sec. 2, where the used method of calculation is also given. The most interesting results for the ground-state phase diagram, magnetization process and isentropy lines in the field-temperature plane are discussed in Sec. 3. Finally, several concluding remarks are mentioned in Sec. 4.

Model and methods
Let us consider the spin-1/2 XXZ Heisenberg clusters with the shape of Johnson's cupolae (see Fig. 1), which are defined through the following Hamiltonian whereŜ α i denotes the spatial projections (α = x, y, z) of the spin-1/2 operator placed at ith vertex of Johnson cupolae, the first summation accounts for the antiferromagnetic interaction J > 0 between the nearest-neighbor spins and ∆ ∈ 0; 1 represents anisotropy parameter in the XXZ Heisenberg interaction (i.e. ∆ = 0 corresponds to the Ising model and ∆ = 1 corresponds to the isotropic Heisenberg model). The second summation accounts for the Zeeman energy of magnetic moments in the external magnetic field h > 0, N denotes the total number of spins, and finally N e is the total number of edges of a given cupola. To obtain exact diagonalization data for the magnetization and entropy of the spin-1/2 XXZ Heisenberg cupolae we have adapted the subroutine fulldiag from the Algorithms and Libraries for Physics Simmulations (ALPS) project [40]. This approach allows a rigorous calculation of the magnetization process and magnetocaloric properties of the model under investigation.

Results and discussion
Let us start our discussion with the ground-state phase diagram of the spin-1/2 XXZ Heisenberg triangular cupola, which is depicted in Fig. 2 in the ∆−h/J plane. There are five distinct phases in the ground-state phase diagram of the XXZ Heisenberg triangular cupola for any ∆ = 0, which are delimited by displayed level-crossing fields and which   differ by value of the total magnetization normalized with respect to its saturation value. Contrary to this, five-ninth plateau is missing in the ground-state phase diagram of the spin-1/2 Ising triangular cupola (∆ = 0). To bring an insight into how the particular ground states are manifested at finite temperatures we depict in Fig. 3 the isothermal magnetization as a function of magnetic field for several values of the anisotropy parameter ∆ at two different temperatures (k B T /J = 0.001 and k B T /J = 0.1). It can be noticed from Fig. 3(a) that the lowtemperature magnetization curve at k B T /J = 0.001 strongly resembles itself in the ground state, showing almost abrupt magnetization jumps at the level-crossing fields. As one can see from Fig. 3(a), the low-temperature magnetization curve of spin-1/2 Ising triangular cupola exhibits beside saturation value indeed three intermediate plateaux at one-ninth, one-third and seven-ninth of the saturation magnetization. In addition, one more five-ninth plateau can be observed in low-temperature magnetization curve of the spin-1/2 XXZ Heisenberg triangular cupola for any ∆ = 0, which extend over a wider range of magnetic field with increasing of anisotropy parameter ∆. For a comparison, the magnetization process of the spin-1/2 XXZ Heisenberg cupola at the moderate temperature k B T /J = 0.1 is depicted in Fig. 3(b). As expected, magnetization plateaux and jumps become gradually smoother upon rising temperature. It can be found from Fig. 3(b) that all magnetization plateaux except wide one-third plateau at relatively small values of anisotropy parameter ∆ and one-ninth plateau at sufficiently high values of ∆ are reflected by inflection points.
Next, let us also examine magnetocaloric properties of the spin-1/2 XXZ Heisenberg triangular cupola, which can be especially fascinating in the proximity of the magnetization jumps. A few isentropy lines are displayed in Fig. 4(a) for the spin-1/2 Ising triangular cupola in the h/J − k B T /J plane. The plotted isentropy lines can alternatively be viewed as the adiabatic temperature response with respect to varying external magnetic field. The Ising triangular cupola exhibits a giant magnetocaloric effect just above (below) of four magnetization jumps, where a sharp increase (decrease) of temperature occurs upon lowering of the magnetic field. The observed giant magnetocaloric effect in a vicinity of zero magnetic field makes this frustrated spin system a promising refrigerant enabling cooling down to absolute zero temperature quite similarly to the Ising octahedron, dodecahedron and cuboctahedron [31,17].
The adiabatic demagnetization of the XXZ Heisenberg triangular cupola with relatively strong anisotropy parameter ∆ = 0.5 is displayed in Fig. 4(b), which shows a contour plot of entropy as a function of temperature and magnetic field. A giant magnetocaloric effect can be found in the proximity of all magnetization jumps of the spin-1/2 XXZ Heisenberg triangular cupola (see Figs. 4(a) and 4(b)). Moreover, an absence of zero-magnetization plateau for all values of anisotropy parameter ∆ brings on the possibility of effective cooling during the process of adiabatic demagnetization in contrast to the spin-1/2 XXZ Heisenberg octahedron, dodecahedron and cuboctahedron [31,17], which exhibit a giant magnetocaloric effect when switching off the external magnetic field only for the Ising limiting case ∆ = 0.
A few isentropy lines of the spin-1/2 isotropic Heisenberg triangular cupola (∆ = 1) are plotted in Fig. 4(c). As one can see from Fig. 4(c), the density plot of the entropy of isotropic Heisenberg triangular cupola is in qualitative concordance with the XXZ triangular cupola at the a relatively strong anisotropy parameter ∆ = 0.5.
The ground-state phase diagram of the spin-1/2 XXZ Heisenberg square cupola is shown in Fig. 5 in the ∆ − h/J plane. The Ising square cupola (∆ = 0) exhibits three intermediate plateaux at zero, one-third and two-thirds of saturation magnetization in zero-temperature magnetization process. When introducing xy part of the XXZ exchange interaction ∆ > 0, two more magnetization plateaux start to evolve in a wider interval of magnetic fields. In addition, zero-temperature magnetization process of the XXZ Heisenberg square cupola exhibits one more plateau if anisotropy parameter reaches value ∆ ≈ 0.395.
To confirm this statement, we have depicted in Fig. 6 the isothermal magnetization curves of a spin-1/2 XXZ Heisenberg square cupola for several values of the anisotropy parameter ∆ and two different temperatures. As one can see from Fig. 6(a), the low-temperature magnetization curve indeed exhibits three magnetization plateaux if The influence of higher temperature on the magnetization process of the XXZ Heisenberg square cupola is shown in Fig. 6(b). At moderate temperature k B T /J = 0.1 only those intermediate plateaux are still visible, which were found in low-temperature magnetization curve in relatively wide interval of magnetic fields. Furthermore, it is clear from Fig. 7 that the magnetocaloric response of the spin-1/2 XXZ Heisenberg square cupola depends on the choice of the exchange anisotropy ∆. If ∆ = 0 Ising square cupola exhibits a steep change of temperature around three different magnetic fields and if ∆ = 0.5 (∆ = 1) the XXZ Heisenberg square cupola shows an enhanced magnetocaloric effect in the proximity of five (six) level-crossing fields.
Last, but not least, the ground-state phase diagram of the spin-1/2 XXZ Heisenberg pentagonal cupola is depicted in Fig. 8 in the ∆ − h/J plane. It can be seen from Fig. 8 that the zero-temperature magnetization curve of an Ising pentagonal cupola (∆ = 0) should exhibit three magnetization jumps since h/J = 0.5, 1.5 and 2.0, which determine rise and fall of the one-fifteenth, one-third and eleven-fifteenths magnetization plateau. On the other hand, the spin-1/2 XXZ Heisenberg pentagonal cupola with ∆ = 0 should display in addition four further lowest-energy eigenstates, which correspond to the one fifth, seven-fifteenths and eleven-fifteenths magnetization plateaux in the zerotemperature magnetization curve.
The above-mentioned findings can be corroborated by the isothermal magnetization curves, which are depicted in Fig. 9(a) for a spin-1/2 XXZ Heisenberg pentagonal cupola at low temperature k B T /J = 0.001. As one can see from Fig. 9(a), the low-temperature magnetization process of Ising pentagonal cupola exhibits three intermediate magnetization plateaux if ∆ = 0 and seven intermediate magnetization plateaux whenever ∆ = 0. It can be found from Fig. 9(b) that the moderate temperature k B T /J = 0.1 is sufficient to suppress intermediate plateaux except the widest one-fifth and one-third plateau at relatively weak and relatively strong exchange anisotropies, respectively. For completeness, a few isentropy lines of the spin-1/2 XXZ Heisenberg pentagonal cupola are shown in Fig. 10 for three different values of the exchange anisotropy ∆, which demonstrate magnetocaloric response of temperature achieved upon adiabatic change of the magnetic field. The absence of a zero magnetization plateau in the magnetization process of the spin-1/2 XXZ Heisenberg pentagonal cupola (see Fig. 9(a)) gives rise to an enhanced magnetocaloric effect in the vicinity of zero magnetic field. It means that an experimental realization of the spin-1/2 XXZ Heisenberg pentagonal cupola should be a promising refrigerant similar to experimental realization of the spin-1/2 XXZ Heisenberg triangular cupola. Moreover, if ∆ = 0, Ising pentagonal cupola shows an enhanced magnetocaloric effect if magnetic field is fixed to three values, in which magnetization jumps in zero-temperature magnetization curve occur (h/J = 0.5, 1.5 and 2.0). In addition, if ∆ = 0.5 and ∆ = 1 the spin-1/2 Heisenberg pentagonal cupola exhibits steep change of temperature around eight values of magnetic field including the zero magnetic field.

Conclusion
The present work deals with the ground-state phase diagram, magnetization curves and isentropy lines of the spin-1/2 XXZ Heisenberg cupolae (triangular, square, pentagonal), which have been studied by the use of the exact numerical diagonalization. It has been shown that the mentioned spin-1/2 XXZ Heisenberg cupolae exhibit three intermediate magnetization plateaux in the limiting Ising case. Moreover, it was demonstrated that quantum (xy) part of the XXZ exchange interaction induces the appearance of additional intermediate magnetization plateaux, which absent in the Ising limit. While most of the novel magnetization plateaux are present at arbitrary non-zero anisotropy parameter ∆ > 0, there may be an exception as evidenced by one-half plateau of a spin-1/2 XXZ Heisenberg square cupola.
In the present work the magnetocaloric effect of the spin-1/2 XXZ Heisenberg cupolae was also studied. A giant magnetocaloric effect has been detected in the proximity of all magnetization jumps. There is the difference between an even site number (square) cupola and odd site number (triangular and pentagonal) cupolae, when considering the magnetization process. By the adiabatic demagnetization, the former one shows the small temperature rise when switching off the magnetic field, while for the latter one it is possible to reach zero temperature due to the absence of zero magnetization plateau. Thus, it becomes evident that the absence of zero magnetization plateau in magnetic clusters is a necessary condition to achieve a possible cooling down to absolute zero temperature. The odd-site number (triangular and pentagonal) cupolae with a half-integer total spin along the magnetic field fits well for this description.
Hence, the experimental realization of the antiferromagnetic finite-size magnetic spin-1/2 clusters with odd total number of spins are promising candidates for future application in magnetic refrigeration.