D-grading and quasifree evolution
Heide Narnhofer
TL;DR
The paper generalizes the spin–Fermi correspondence to $d$-grading by constructing a gauge-invariant quasilocal subalgebra ${\\cal A}$ and its crossed-product extension ${\\cal M}_{\\gamma}$ with $\\gamma^d=1$, enabling quasifree time evolutions. It demonstrates a continuous extension of the shift and builds a controlled time evolution from gauge-invariant local Hamiltonians, proving norm-asymptotic abelianness on ${\\cal A}$ for evolutions with absolutely continuous one-particle spectrum $h(p)$ and establishing related KMS-like states. It then relates $d$-grading to $kd$-grading, showing the benefits and limitations of extending the grading while preserving locality, and finally extends the framework to higher-dimensional lattices, preserving the transfer of automorphisms between spin and Fermi-type algebras. The results provide a structured method to connect spin and Fermi systems under generalized grading while controlling asymptotic and locality properties, with potential implications for symmetry breaking and phase structure in lattice models.
Abstract
Generalizing the relation between spin-systems and Fermi-systems on the lattice we construct for a spin-system with dimension d an algebra for which quasifree time-evolutions exist. With appropriate assumptions the gauge invariant subalgebra common for both algebras is invariant under this time-evolution and on this subalgebra is norm-asymptotically abelian.