Statistical mechanics of a paramagnetic chain

The transfer matrix method is used to find the exact partitiorJ function in the thermodynamic limit of a two-level system coupled to a one-dimensional elastic, cyclically closed chain of atoms. The number of two-level objects ($\dfrac12$ spins) is equal to the number of degrees of freedom of the chain (the number of modes). The spin system is in a transverse field $\omega_0$, and the chain in a linear potential $\alpha\varphi^2$. The following results are obtained. At $\omega_0=0$, the problem is equivalent to the one-dimensional Kac model with antiferromagnetic exchange interaction between the spins. There is no phase transition in the system at any $\omega_0$ in contrast to the case with a finite number of field modes. The interaction of the spins with the lattice suppresses the Curie paramagnetism, and in the limit $T\to0$ the transverse susceptibility of the system remains finite.