Energy levels and their degeneracies for two-ring Ising chains of spins-1/2 with NN and NNN couplings: spin frustration of ferromagnetic and antiferromagnetic orders

In an Ising linear chain of spins-1/2 with the periodic conditions in NN and NNN interactions as a two-ring structure of two different coupling constants, all or half spins contribute to the NNN couplings, depending on whether the total number of spins is odd or even. In this article, we formulate precise predictions on the values, degeneracy factors, and the number of allowed energy levels for the two-ring Ising chains with any arbitrary number of spins in two different forms for odd and even number of spins. We confirm the validity of our formulations firstly by comparing them with the data obtained from the manual search for small finite-size two-ring Ising chains. Secondly, the validity of the formulations is verified by comparing the obtained partition functions by energy levels with the other forms obtained from the change of variable method. We study analytically the ground levels and show that they exhibit the maximal spin frustration when the ferromagnetic and antiferromagnetic orders are demanded via the NN and NNN coupling constants, respectively. Finally, we show that the low-temperature behavior of the entropy and specific heat capacity per spin in the maximally frustrated regime matches with the analytical results. Graphic abstract

Automated summary

This article studies the Ising chains with two different coupling constants for odd and even number of spins, providing precise predictions on values, degeneracy factors, and number of allowed energy levels. The validity of the formulations is confirmed by comparing them with data obtained from manual search and other results obtained from the change of variable method. The low-temperature behavior of entropy and specific heat capacity per spin in the maximally frustrated regime matches with the analytical results.