General-relativistic spin system

The models of spin systems defined on the Euclidean space provide powerful machinery for studying a broad range of condensed matter phenomena. While the nonrelativistic 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 nonlinear 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 semiclassical 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.

Figure 1

Illustration of the precession of a spin vector $\overrightarrow{S}$ around the ground state $\overrightarrow{S}=(S,0,0)$, which corresponds to $(\phi ,{\pi }_{\phi })=(0,0)$. For the precession angle tending to zero, the dynamics is approximately described by harmonic oscillations at the $(\phi ,{\pi }_{\phi })$ phase space. In general, our parametrization maps a hemisphere into a nonsquare rectangle (cf. Fig. 2).

Figure 2

A sample of phase space trajectories [Eq. (19)] on the hemisphere with $\phi \in [-\frac{\pi }{2}\sqrt{\frac{S}{m}},\frac{\pi }{2}\sqrt{\frac{S}{m}}]$, for $S=1$, $m=1$; the arrows denote the direction of time.

Figure 3

Plot of the $\phi =G(\xi )$ function composed of two branches: $G_{+}(\xi )$ for $\xi \in [-{\xi }_c,{\xi }_c]$ and $G_{-}(\xi )$ for $\xi \in ({\xi }_{\min },-{\xi }_c)\cup ({\xi }_c,{\xi }_{\max }]$. We see that this function is increasing monotonically and covers the whole range of variability of $\phi$.