Thermodynamics of antiferromagnetic solids in magnetic fields
Section snippets
Motivation
The literature on the thermodynamic properties of antiferromagnets in three spatial dimensions is considerable. Low-temperature representations for the free energy density, staggered magnetization, and other observables describing quantum Heisenberg antiferromagnets have been derived, e.g., in Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Various authors have furthermore discussed how an external magnetic field influences the low-temperature physics of antiferromagnets (see Refs. [11]
Free energy density: Two-loop representation
The paradigmatic microscopic description of antiferromagnets is based on the Heisenberg model, whose Hamiltonian in the simplest situation, where only the nearest neighbor spin interactions are taken into account, takes the form where the summation in the first term extends over nearest neighbor spin pairs on a bipartite three-dimensional lattice. The exchange constant defines the fundamental energy scale of the system. The first term is
Dressed magnons and interaction free energy density
Naively, one might expect that the first line of our result (2.8) for the free energy density, that is its one-loop part, corresponds to a gas of free magnons, while all the rest captures magnon–magnon interactions. That would, however, be premature: the magnons get dressed by self-energy corrections even at . Part of the thermal two-loop free energy density can then be accounted for as the free energy density of such dressed, yet noninteracting, magnons. Whatever is left can be considered
Low-temperature series
The effective field theory expansion of the free energy density, Eq. (2.8), is valid at low temperatures and in weak external fields. More precisely, the quantities have to be small compared to a characteristic scale inherent in the underlying microscopic system. In the present case of the Heisenberg antiferromagnet, the thermal scale is given by the Néel temperature . The actual definition of low temperature and weak field is somewhat arbitrary. To be concrete, here we choose
Conclusions
Antiferromagnets subjected to magnetic and staggered fields can be addressed straightforwardly with the systematic effective Lagrangian method. Starting from the two-loop representation of the partition function, we have discussed the low-temperature behavior of antiferromagnets in a configuration of mutually orthogonal external magnetic and staggered fields.
To have a clear picture of what “interaction” means in the free energy density – and any other thermodynamic quantity derived from
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
Tomas Brauner acknowledges financial support from the ToppForsk-UiS program of the University of Stavanger and the University Fund, Grant No. PR-10614.
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