A topographic exploration of the von Neumann entropy surface for a spin-1 (three-level) nuclear paramagnet

Paramagnetic spin systems are typically analyzed using classical statistical mechanics, and the discussion is usually limited to the simplest, two-level, isolated spin-1/2 system. Here, I extend the analysis of these paramagnetic spin systems in two significant ways: first, I analyze a three-level, isolated spin-1 system, and second, I use quantum statistical mechanics for the analysis. In this way, I provide an introduction to quantum statistical mechanics, including the density operator and von Neumann entropy, but I do so using paramagnetic spin systems, which are quite familiar to students from their study of classical statistical mechanics. The high-field equilibrium density operator ρ̂eq of an isolated spin-1 system can be reincarnated as an angle-axis {Θ,ẑ} parameterized rotation operator ρ̂eq≃exp(βIz)=D̂(j)(Θ,ẑ)=exp(−iΘIz), where the modulus of the imaginary rotation angle Θ = iβ is the normalized inverse spin temperature |Θ| = β = ℏω0/(kBTS). Using the Chebyshev polynomial operators of a discrete real variable fλ(j)(Iz) as the density operator expansion basis, I show that all of the most significant quantum statistical mechanical properties of this paramagnetic spin-1 system, the quantum partition function Z=Tr[exp(βIz)], the average energy ⟨H⟩, the equilibrium von Neumann entropy SνNeq, the spin temperature TS, and the vector (dipole) and tensor (quadrupole) polarizations, can all be expressed in terms of the generalized characters χλ(j)(Θ)≡χλ(j)(iβ) of irreducible representations of the rotation group SO(3). Expanding the density operator ρ̂ in spin polarization moments (or statistical tensors) ρ10(1) and ρ20(1) defines the von Neumann entropy surface SνN(ρ10, ρ20), whose topography defined on the two-dimensional polarization moments landscape is used (i) to elicit the relation between the thermal equilibrium polarization moments ρ10(β) and ρ20(β), and (ii) to visualize the connection between entropy and spin temperature.