Trace formula for quantum chaotic systems with geometrical symmetries and spin

We derive a Gutzwiller-type trace formula for quantum chaotic systems that accounts for both particle spin precession and discrete geometrical symmetries. This formula generalises previous results that were obtained either for systems with spin (Bolte and Keppeler 1998 Phys. Rev. Lett. 81 1987; 1999 Ann. Phys., NY 274 72) or for systems with symmetries (Robbin 1989 Phys. Rev. A 40 2128–36; Lauritzen 1991 Phys. Rev. A 43 603; Seligman and Weidenmüller 1994 J. Phys. A: Math. Gen. 27 7915–23), but not for a combination of both. The derivation requires not only a combination of methodologies for these two settings, but also the treatment of new effects in the form of double groups and spin components of symmetry operations. The resulting trace formula expresses the level density of subspectra associated to irreducible representations of the group of unitary symmetries in terms of periodic orbits in the system's fundamental domain. We also derive a corresponding expression for the spectral determinant. In a follow-up paper (Blatzios et al 2025 in preparation) we will show that our formula allows to study the impact of geometrical symmetries and spin on spectral statistics.