Exact solutions for the ferromagnetic and antiferromagnetic two-ring Ising chains of spin-[formula omitted

Abstract

We study analytically the Ising linear chains of spin-12 with the periodic conditions in nearest and next nearest neighbour interactions. The couplings for any of the nearest and next nearest neighbours are considered uniform but not necessarily equivalent to each other. The chain involves two rings of spins with different coupling constants. Therefore all or half spins contribute to interaction with next nearest neighbours depending on whether the number of spins in the chain is odd or even, respectively. We derive two exact classical solutions for the partition function of the chain in two different forms for odd and even number of spins. Besides, the pair correlations between nearest and next nearest neighbours are calculated. The influence of coupling constant corresponding to the next nearest-neighbour interactions of the ferromagnetic and antiferromagnetic chains on the thermodynamic quantities such as entropy, energy and specific heat capacity per spin at the low-temperatures is analysed. Especially, it is shown that when the nearest and next nearest neighbour interactions are of the ferromagnetic and antiferromagnetic types respectively, specific heat capacity for the chain with both odd and even number of particles exhibits the spin frustration.