Low-temperature susceptibility of the classical frustrated ferromagnetic spin chain

Background. Low-dimensional magnets with competing (frustrated) interactions have attracted much attention last years because these systems have many unusual magnetic properties which are important for application. One of the interest class of such compounds is cuprates consisting of edge-sharing chains with CuO${}$plaquets with the ferromagnetic (F) and the antiferromagnetic (AF) exchange interactions between Cu${}$ magnetic ions. Of particular interest are the cuprates for which the frustration parameter is close to the critical value corresponding to the quantum phase transition. A minimal model describing the magnetic properties of these cuprates is so-called F-AF chain with the F interaction of nearest-neighbor spins and the AF interaction of next-nearest-neighbor ones. The aim of this work is to study low-temperature magnetic properties of the F-AF chain with the frustration parameter which is close to the critical value. We focus our attention on the behavior of the magnetic susceptibility and the correlation functions in this point. Materials and methods. The study of the low-temperature thermodynamics of the quantum F-AF chain is a complicated problem. However, there are reasons to expect that the behavior of the low-temperature magnetic properties is universal for the quantum and the classical F-AF chains. Therefore, we consider the classical F-AF chain for which the exact analytical calculations can be provided. They are based on the transfer matrix method adapted to the systems with the competing interactions. Results. The partition function of the classical F-AF chain with the critical frustration parameter in the low-temperature limit can be reduced to the Schrodinger equation for the quantum particle in the special potential. The eigenvalues and the eigenfunctions of this equation are found. As a result we obtain exact low-temperature asymptotic of the pair correlation functions and the magnetic susceptibility. In the T$\to$ 0 limit the susceptibility diverges as T${}$ and the correlation length as T${}$. So, the critical indices of the susceptibility and correlation length are 4/3 and 1/3 correspondingly. Conclusions. The obtained results demonstrate strong influence of the frustration effects on the magnetic properties of the F-AF chain. In particular, the critical index of the susceptibility is changed from 2 to 4/3 when the frustration parameter is changed from zero to the critical value and from 1 to 1.3 for the correlation length. It is noted that the low-temperature asymptotic with the same critical indices have been obtained for the quantum F-AF chain by spin-wave method. This testifies the universality of the behavior of the magnetic properties of both quantum and the classical F-AF chains.