IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 40 (2007) 13075–13089 doi:10.1088/1751-8113/40/43/015 Fractional monodromy in systems with coupled angular momenta M S Hansen 1 , F Faure 2 and B I Zhilinski´ ı 3 1 Institute of Mathematics, Technical University of Denmark, 2800 Kgs Lyngby, Denmark 2 Institut Fourier, Universit´ e Joseph Fourier, BP 74, 38402 Saint-Martin d’H` eres, Cedex, France 3 Universit´ e du Littoral, UMR du CNRS 8101, 59140 Dunkerque, France E-mail: M.S.Hansen@mat.dtu.dk, frederic.faure@ujf-grenoble.fr and zhilin@univ-littoral.fr Received 27 February 2007, in final form 12 September 2007 Published 9 October 2007 Online at stacks.iop.org/JPhysA/40/13075 Abstract We present a one-parameter family of systems with fractional monodromy, which arises from a 1:2 diagonal action of a dynamical symmetry SO(2). In a regime of adiabatic separation of slow and fast motions, we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi-quantum spectra. PACS numbers: 03.65.Sq, 02.40.Yy (Some figures in this article are in colour only in the electronic version) 1. Introduction In this paper we consider a simple one-parameter family of Hamiltonian which is a slight generalization of the well-known example of spin–orbit coupling. This latter model has been the object of several studies [1, 6–8], demonstrating the presence of integer monodromy for some interval of parameter values. We remind here that Hamiltonian monodromy is a generic property of classical integrable systems, intensively studied and popularized by Cushman [5] and described in detail by Duistermaat in 1980 [9]. In a classical dynamical system with two degrees of freedom, the Hamiltonian monodromy can typically appear in one-parameter families through the Hamiltonian Hopf bifurcation [10]. It was shown later that there is a correspondence between the appearance of monodromy within a one-parameter family of classical Hamiltonians and the redistribution of bands in the spectrum of the associated quantum problem [6, 11]. The appearance of Hamiltonian monodromy in the classical system also indicates the presence of a topological bifurcation in a semi-quantum description [12]. Our model has fractional monodromy which is a recent generalization of the integer monodromy concept [13–16]. This is the first example of a system with this property on a compact phase space. 1751-8113/07/4313075+15$30.00 © 2007 IOP Publishing Ltd Printed in the UK 13075