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arXiv:2411.12403 · Journal of Physics A Mathematical and Theoretical

Trace formula for quantum chaotic systems with geometrical symmetries and spin

Vaios Blatzios, Christopher H Joyner, Sebastian Müller, Martin Sieber

Published May 28, 2025• 0 citations

Abstract

Abstract 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 (Robbins 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.

DOI: 10.1088/1751-8121/adddfe

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In the reference (B. Gremaud, Phys. Rev. E 65, 056207 (2002)) flrst-order ~ corrections to Gutzwiller's trace formula for systems with a smooth potential were presented. Here we present an extension of that theory to systems with discrete symmetries. We apply the method to the two-dimensional hydrogen atom in a uniform magnetic fleld and exploit the C4v-symmetry in the calculation of the correction terms. The numerical results for the semiclassical values are compared with values extracted from exact quantum mechanical calculations. The comparison shows a very good agreement.

Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.

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Abstract The geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two-form through a surface in the system’s parameter space. We obtain the classical limit of this two-form in a form applicable to systems whose classical dynamics is chaotic. For integrable systems the expression is equivalent to the Hannay two-form. We discuss various properties of the classical two-form, derive semiclassical corrections to it (associated with classical periodic orbits), and consider implications for the semiclassical density of degeneracies.

In general, trace formulas relate the density of states for a given quantum mechanical system to the properties of the periodic orbits of its classical counterpart. Here we report for the first time on a semiclassical description of microwave spectra taken from superconducting billiards of the Limaçon family showing mixed dynamics in terms of a generalized trace formula derived by Ullmo et al. [Phys. Rev. E 54, 136 (1996)]. This expression not only describes mixed-typed behavior but also the limiting cases of fully regular and fully chaotic systems and thus presents a continuous interpolation between the Berry-Tabor and Gutzwiller formulas.