Magnetization Plateaus at Low Temperature in a Triangular Spin Tube

Abstract

We studied some magnetic behaviors of the Ising model in a triangular spin tube using the effective field theory (EFT) with correlations. We examined the effects of the exchange anisotropy $J^{\prime} = \frac{{J_{ \bot } }}{{J_{//} }}$, and temperature T on the magnetic properties of the spin ½ Ising model in a triangular spin tube, particularly for isotherm magnetization, spontaneous magnetization, and magnetic susceptibility. We investigated the magnetic properties of the triangular spin ½ tube in detail. At low temperature, magnetization curves of a triangular spin ½ tube of the Ising model with the intra- triangular J_┴, and inter-triangular J_// interaction exchange are examined within the behavior magnetic. It is shown that the developed model satisfactorily describes all pronounced features of the low-temperature magnetization process and the magneto-thermodynamics such as abrupt changes of the isothermal magnetization curves, a giant magnetocaloric effect. Among the most notable features of zero-temperature magnetization curves, one could mention fractional magnetization plateaus, quantum spin liquids, and macroscopic magnetization jumps, which can be found first of all in frustrated quantum Heisenberg spin models.

Appendix 1

The coefficients $P_{0}$,$P_{1}$,$P_{2}$,$P_{3}$ and $P_{4}$ used in Eq. (10) are given by:

$$\begin{gathered} P_{0} = \tanh (2(A + B) + C) + \tanh ( - 2(A + B) + C) + \tanh (2(A - B) + C) + \tanh ( - 2(A - B) + C) + 2\tanh (2A + C) \hfill \\ + 2\tanh ( - 2A + C) + 2\tanh (2B + C) + 2\tanh ( - 2B + C) + 4\tanh (C) \hfill \\ \end{gathered}$$

$$P_{1} = 4\tanh (2(A + B) + C) - 4\tanh ( - 2(A + B) + C) + 4\tanh (2A + C) - 4\tanh ( - 2A + C) + 4\tanh (2B + C) - 4\tanh ( - 2B + C)$$

$$P_{2} = 6\tanh (2(A + B) + C) + 6\tanh ( - 2(A + B) + C) - 2\tanh (2(A - B) + C) - 2\tanh ( - 2(A - B) + C) - 8\tanh (C)$$

$$P_{3} = 2\tanh (2(A + B) + C) - 2\tanh ( - 2(A + B) + C) - 2\tanh (2A + C) + 2\tanh ( - 2A + C) - 2\tanh (2B + C) + 2\tanh ( - 2B + C)$$

$$\begin{gathered} P_{4} = \tanh (2(A + B) + C) + \tanh ( - 2(A + B) + C) + \tanh (2(A - B) + C) + \tanh ( - 2(A - B) + C) - 2\tanh (2A + C) - 2\tanh ( - 2A + C) \hfill \\ - 2\tanh (2B + C) - 2\tanh ( - 2B + C) + 4\tanh (C), \hfill \\ \end{gathered}$$

with:

$$A = \frac{{J_{ \bot } }}{2}$$

$$B = \frac{{J_{//} }}{2}$$

$$C = \frac{\beta h}{2}$$

$$\beta = \frac{1}{{k_{B} T}}$$

where $k_{B}$ is the Boltzmann constant and T is the absolute temperature.

The coefficients $\left( {\frac{{\partial P_{0} }}{\partial h}} \right)$,$\left( {\frac{{\partial P_{1} }}{\partial h}} \right)$,$\left( {\frac{{\partial P_{2} }}{\partial h}} \right)$,$\left( {\frac{{\partial P_{3} }}{\partial h}} \right)$ and $\left( {\frac{{\partial P_{4} }}{\partial h}} \right)$ used in Eq. (13) are given by the following:

$$\left( {\frac{{\partial P_{0} }}{\partial h}} \right) = \frac{\beta }{2}\left( {\frac{2}{{{\text{Cosh}}^{2} \left( {2\left( {A + B} \right)} \right)}} + \frac{2}{{{\text{Cosh}}^{2} \left( {2\left( {A - B} \right)} \right)}} + \frac{4}{{{\text{Cosh}}^{2} \left( {2A} \right)}} + \frac{4}{{{\text{Cosh}}^{2} \left( {2B} \right)}}} \right)$$

$$\left( {\frac{{\partial P_{1} }}{\partial h}} \right) = 0$$

$$\left( {\frac{{\partial P_{2} }}{\partial h}} \right) = \frac{\beta }{2}\left( {\frac{12}{{{\text{Cosh}}^{2} \left( {2\left( {A + B} \right)} \right)}} - \frac{4}{{{\text{Cosh}}^{2} \left( {2\left( {A - B} \right)} \right)}}} \right)$$

$$\left( {\frac{{\partial P_{3} }}{\partial h}} \right) = 0$$

$$\left( {\frac{{\partial P_{4} }}{\partial h}} \right) = \frac{\beta }{2}\left( {\frac{2}{{{\text{Cosh}}^{2} \left( {2\left( {A + B} \right)} \right)}} + \frac{2}{{{\text{Cosh}}^{2} \left( {2\left( {A - B} \right)} \right)}} - \frac{4}{{{\text{Cosh}}^{2} \left( {2A} \right)}} - \frac{4}{{{\text{Cosh}}^{2} \left( {2B} \right)}}} \right)$$

The coefficients $T_{0}$,$T_{1}$,$T_{2}$,$T_{3}$ and $T_{4}$ used in Eq. (17) are given by:

$$T_{0} = \tanh (2(A + B) + C) - \tanh ( - 2(A + B) + C) + \tanh (2(A - B) + C) - \tanh ( - 2(A - B) + C) + 2\tanh (2A + C) - 2\tanh ( - 2A + C)$$

$$T_{1} = 8\tanh (2(A + B) + C) + 8\tanh ( - 2(A - B) + C) + 8\tanh (2A + C) + 8\tanh ( - 2A + C)$$

$$T_{2} = 24\tanh (2(A + B) + C) + 8\tanh ( - 2(A + B) + C) - 8\tanh (2(A - B) + C) - 24\tanh ( - 2(A - B) + C)$$

$$T_{3} = 32\tanh (2(A + B) + C) + 32\tanh ( - 2(A - B) + C) - 32\tanh (2A + C) - 32\tanh ( - 2A + C)$$

$$\begin{gathered} T_{4} = 16\tanh (2(A + B) + C) - 16\tanh ( - 2(A + B) + C) + 16\tanh (2(A - B) + C) - 16\tanh ( - 2(A - B) + C) - 32\tanh (2A + C) \hfill \\ + 32\tanh ( - 2A + C) \hfill \\ \end{gathered}$$

The coefficients $T^{\prime}_{0}$,$T^{\prime}_{1}$,$T^{\prime}_{2}$,$T^{\prime}_{3}$ and $T^{\prime}_{4}$ used in Eq. (18) are given by the following:

$$T_{0}^{^{\prime}} = \tanh (2(A + B) + C) + \tanh ( - 2(A + B) + C) - \tanh (2(A - B) + C) - \tanh ( - 2(A - B) + C) + 2\tanh (2B + C) - 2\tanh ( - 2B + C)$$

$$T_{1}^{^{\prime}} = 8\tanh (2(A + B) + C) + 8\tanh ( - 2(A - B) + C) + 8\tanh (2B + C) + 8\tanh ( - 2B + C)$$

$$T_{2}^{^{\prime}} = 24\tanh (2(A + B) + C) - 8\tanh ( - 2(A + B) + C) + 8\tanh (2(A - B) + C) - 24\tanh ( - 2(A - B) + C)$$

$$T_{3}^{^{\prime}} = 32\tanh (2(A + B) + C) + 32\tanh ( - 2(A - B) + C) - 32\tanh (2B + C) - 32\tanh ( - 2B + C)$$

$$\begin{gathered} T_{4}^{^{\prime}} = 16\tanh (2(A + B) + C) + 16\tanh ( - 2(A + B) + C) - 16\tanh (2(A - B) + C) - 16\tanh ( - 2(A - B) + C) - 32\tanh (2B + C) \hfill \\ + 32\tanh ( - 2B + C) \hfill \\ \end{gathered}$$