Spin-lattice coupling as environmental effect

The exchange constant in a magnetic system results from the overlap of orbitals, so it depends upon the ion distance and yields a coupling with the lattice vibrations: this can strongly influence the statistical mechanics of magnets. As the fast lattice dynamics allows for a rapid thermalization, phonon-induced spin correlations can be reasonably neglected, so that phonons can be regarded as a thermalized environment (or bath) for the spins. It is therefore natural to treat it within a Caldeira-Leggett scheme, i.e., by the influence action obtained after tracing-out the bath’s degrees of freedom, whose contribution can be accounted for within a known semiclassical approach. In this way extended phase diagrams, accounting for the environmental interaction, can be drawn for the magnetic phase transitions. Real magnetic systems, having the spins sitting on the sites of a crystal lattice, are necessarily interacting with its vibration modes. As the faster dynamics allows for a rapid thermalization of the lattice, one expects that phonons could be considered as a quantum environment (or bath, or reservoir) for each spin. Such a coupled system is naturally approached by the system plus reservoir model [1, 2], in a suitably generalized version including a bath interaction for the momenta [3]. In this way one can study if environmental effects could be displayed by magnetic materials. Although a microscopic derivation of the spin-phonon coupling gives a nonlinear bath-spin interaction, in order to derive a first qualitative picture we consider here the linear case, which gives rise to a quadratic influence action. The exchange coupling between two spins sitting on neighboring ions arises from the fermionic character of electrons, and is essentially due to the overlap of orbitals, so that its strength J = J(r) depends on the separation r of the ions. This dependence is usually neglected when dealing with magnetic models, but in principle it can be very important: for instance, the gain in magnetic energy can even lead to a distorted lattice configuration, as in the spin-Peierls transition [4, 5]. To derive the microscopic form of the coupling, consider the bond connecting lattice sites i and j, J ( |ri−rj| ) ≃ J + J ′ d·(ri−rj) , (1) where d ≡ j−i is the equilibrium distance. The overall quantum Hamiltonian must therefore International Conference on Magnetism (ICM 2009) IOP Publishing Journal of Physics: Conference Series 200 (2010) 022069 doi:10.1088/1742-6596/200/2/022069 c © 2010 IOP Publishing Ltd include the magnetic and the (harmonic) lattice contribution,