Some exact calculations on a chain of spins 1/2

Abstract

The X-Y model of a linear chain of spins 1/2, introduced by Lieb, Schultz and Mattis, is studied in the presence of a magnetic field h along the axis. In section A the Hamiltonian is diagolalized in terms of fermion operators. In section B the magnetization along the axis is calculated for arbitrary field and temperature. We find that there is no spontaneous magnetization and that only at zero temperature there is a phase transition of the second kind, the magnetic susceptibility being of the form c·In|h — hc| in the neighborhood of a critical field hc. In section C we derive an expression for the time-dependent correlation function (β, t) of the -components of spins separated by an arbitrary number R of lattice sites. Starting from this expression we will show that there is no long-range order of the components of the spins in the absence of the field and that in the presence of a field the long-range order corresponds with the magnetization. Furthermore we discuss the time-dependent autocorrelation function of the component of one single spin, of the total magnetization of the chain, and the possibility that Im (β, t) satisfies a kind of wave equation, for this special case, as has been proposed by Ruygrok for more general cases. In section D the isolated chain is assumed to be in thermal equilibrium in a certain magnetic field, after which the field is suddenly changed by an arbitrary amount, and an exact expression is derived for the temporal development of the component of the magnetization, which is found to reach an equilibrium value as time goes to infinity. In section E this exact time development is compared with the exact solution of the Kubo formalism and it is proved analytically that the Kubo formula holds for high temperatures and small perturbation, but some numerical calculations show that the development into powers of the perturbation converge very slowly. We conclude with a short discussion of the approach of Mazur and Terwiel to the relaxation of more general spin systems, in connection with our exact results on the chain.