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Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences

The geometric phase for chaotic systems

Jonathan M. Robbins, Michael Berry

Published Mar 9, 1992• 40 citations
Publisher

Abstract

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.

DOI
10.1098/rspa.1992.0039

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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.
We consider the geometric phase for a family of quantum/classical Hamiltonians in which the effect of changing parameters is simply to induce unitary/canonical transformations. In this case the classical limit of the geometric phase is easily obtained, even when the classical motion is chaotic. The results agree with those previously obtained for general chaotic families, but may be expressed in a simpler form, not involving time integrals of correlation functions. It is also straightforward to establish some results which are problematic in the general case, for example the form of periodic orbit corrections, and the closeness of the classical 2-form. If the parameters are regarded as dynamical variables, evolving slowly so as to maintain adiabaticity, they are subject to geometric magnetism, but not, in contrast to the general case, deterministic friction and Born-Oppenheimer forces. Examples including families of translated and rotated systems are discussed.
We study the concepts of adiabatic driving and geometric phases of classical integrable systems under the Koopman-von Neumann formalism. In close relation to what happens to a quantum state, a classical Koopman-von Neumann eigenstate will acquire a geometric phase factor $exp\left\{ iΦ\right\} $ after a closed variation of the parameters λλλ in its associated Hamiltonian. The explicit form of ΦΦΦ is then derived for integrable systems, and its relation with the Hannay angles is shown. Additionally, we use quantum formulas to write a classical adiabatic gauge potential that generates adiabatic unitary flow between classical eigenstates, and we explicitly show the relationship between the potential and the classical geometric phase.
Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.
We study the geometric phase factors underlying the classical and the corresponding quantum dynamics of a driven nonlinear oscillator exhibiting chaotic dynamics. For the classical problem, we compute the geometric phase factors associated with the phase space trajectories using Frenet-Serret formulation. For the corresponding quantum problem, the geometric phase associated with the time evolution of the wave function is computed. Our studies suggest that the classical geometric phase may be related to the the difference in the quantum geometric phases between two neighboring eigenstates.
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.