Solution of the antiferromagnetic Ising model with multisite interaction on a zigzag ladder

We consider the antiferromagnetic spin-$1/2$ Ising model with multisite interaction in an external magnetic field on an infinite zigzag ladder. The model is solved exactly by using the transfer matrix method. Using the exact expression for the total magnetization per site, the magnetic properties of the model are investigated in detail. The model exhibits the formation of magnetization plateaus for low temperatures, and it is shown that their properties depend strongly on the strength of the multisite interaction. All possible ground states of the model are found and discussed. The existence of nontrivial singular ground states is proven and exact explicit expressions for them are found. The macroscopic degeneracy of the ground states is investigated and discussed.

Figure 1

The structure of the infinite triangular zigzag ladder.

Figure 2

The total magnetization per site as a function of the external magnetic field for $\alpha =0$ and for various values of the reduced temperature.

Figure 3

The total magnetization per site as a function of the external magnetic field for $\alpha =-0.25$ and for various values of the reduced temperature.

Figure 4

The total magnetization per site as a function of the external magnetic field for $\alpha =0.25$ and for various values of the reduced temperature.

Figure 5

The total magnetization per site as a function of the external magnetic field for $\alpha =-0.75$ and for various values of the reduced temperature.

Figure 6

The total magnetization per site as a function of the external magnetic field for $\alpha =0.75$ and for various values of the reduced temperature.

Figure 7

The total magnetization per site as a function of the external magnetic field for $\alpha =-1.5$ and for various values of the reduced temperature.

Figure 8

The total magnetization per site as a function of the external magnetic field for $\alpha =-1.5$ and for various values of the reduced temperature.

Figure 9

The total magnetization per site as a function of the reduced temperature for $\alpha =0.25$ as well as for various values of the external magnetic field $H/\left|J\right|$.

Figure 10

The total magnetization per site as a function of the reduced temperature for $\alpha =0.75$ as well as for various values of the external magnetic field $H/\left|J\right|$.

Figure 11

The total magnetization per site as a function of the reduced temperature for $\alpha =1.5$ as well as for various values of the external magnetic field $H/\left|J\right|$.

Figure 12

The total magnetization per site as a function of the reduced temperature for $\alpha =0.5$ as well as for various values of the external magnetic field $H/\left|J\right|$.

Figure 13

The total magnetization per site as a function of the reduced temperature for $\alpha =1$ as well as for various values of the external magnetic field $H/\left|J\right|$.

Figure 14

The properties of the ground states of the model. The solid lines denote the positions of the singular ground states with total magnetization $m\approx -0.6115$ (left solid line) and $m\approx 0.6115$ (right solid line), the dashed lines denote positions of the singular ground states with $m\approx -0.3967$ (left dashed line) and $m\approx 0.3967$ (right dashed line), the dash-dotted lines correspond to the singular ground states with $m\approx -0.1593$ [left (upper) dash-dotted line] and $m\approx 0.1593$ [right (lower) dash-dotted line], and the dotted lines describe the singular ground states with $m=-1/3$ (left dotted line) and $m=1/3$ (right dotted line). The two filled circles determine the positions of the singular ground states with $m\approx -0.4472$ (left filled circle) and $m\approx 0.4472$ (right filled circle) and the two filled squares show the positions of the singular ground states with $m\approx -0.2230$ (left filled square) and $m\approx 0.2230$ (right filled square).

Figure 15

The magnetization of the model as a function of the external magnetic field for $T=0$ and $\alpha =0$. The singular ground states at $H/\left|J\right|=\mp 4$ have the values of magnetization $m\approx \mp 0.611492$ and the singular ground states at $H/\left|J\right|=\mp 1$ have the values of magnetization $m\approx \mp 0.159320$.