Abstract
The models of spin systems defined on Euclidean space provide powerful machinery for studying a broad range of condensed matter phenomena. While the non-relativistic effective description is sufficient for most of the applications, it is interesting to consider special and general relativistic extensions of such models. Here, we introduce a framework that allows us to construct theories of continuous spin variables on a curved spacetime. Our approach takes advantage of the results of the non-linear field space theory, which shows how to construct compact phase space models, in particular for the spherical phase space of spin. Following the methodology corresponding to a bosonization of spin systems into the spin wave representations, we postulate a representation having the form of the Klein-Gordon field. This representation is equivalent to the semi-classical version of the well-known Holstein-Primakoff transformation. The general-relativistic extension of the spin wave representation is then performed, leading to the general-relativistically motivated modifications of the Ising model coupled to a transversal magnetic field. The advantage of our approach is its off-shell construction, while the popular methods of coupling fermions to general relativity usually depend on the form of Einstein field equations with matter. Furthermore, we show equivalence between the considered spin system and the Dirac-Born-Infeld type scalar field theory with a specific potential, which is also an example of k-essence theory. Based on this, the cosmological consequences of the introduced spin field matter content are preliminarily investigated.
Highlights
The models of spin systems, for instance the Ising model, Heisenberg model, XY model [1], or Hubbard model [2], provide a theoretical description of such phenomena as magnetism, superconductivity, and topological phase transitions [3]
Following the methodology corresponding to a bosonization of spin systems into the spin wave representations, we postulate a representation having the form of the Klein-Gordon field
It would be interpreted as the interaction term of a continuous distribution of internal magnetic moments with an external homogeneous magnetic field oriented along the x axis
Summary
The models of spin systems, for instance the Ising model, Heisenberg model, XY model [1], or Hubbard model [2], provide a theoretical description of such phenomena as magnetism (ferromagnetism and antiferromagnetism), superconductivity, and topological phase transitions [3]. In the case of spin models describing the ground-based laboratory experiments, spins are considered to be attached to given space points, i.e., forming a fixed lattice. The analogous continuous spin system approximations play a powerful role in theoretical condensed matter physics, e.g., in the theory of topological phase transitions [3] The system in such a case becomes a spin field S⃗ ðxÞ defined on some spatial hypersurface Σ. The exact matching of phase spaces (of a standard scalar field theory and a spin field model) is obtained by taking the large spin limit (S → ∞). The second type of excitations relates to microscopic (quantum) properties of the spin lattice and is relevant for very short wavelengths, comparable with the lattice spacing In both cases, in the simplest realization, only the nearest-neighbor interactions are considered, forming a net of oscillators.
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