Characterization of the set of “ergodic directions” in Novikov's problem of quasi-electron orbits in normal metals

Abstract

In this paper we give a beautiful characterization of the ‘ergodic directions’ in Novikov’s problem of semiclassical orbits of quasi-electrons in a normal metal. Since the end of the 1950s physicists have known that the qualitative behavior of conductivity in normal metals under the influence of a strong homogeneous magnetic field is dictated by the topological properties of the orbits of the electron quasi-momentum. However, from the topological point of view there was no essential progress in the study of this phenomenon until the 1980s, when Novikov [1] noted that a beautiful topological structure lies behind this problem. His students have studied the basic properties of this structure. From the mathematical point of view, the problem represents the simplest example of a Poisson dynamical system [1], that is, a pair (f, (T3, { · , · }H)), where the phase space (the quasimomentum space) possesses a ‘magnetic’ Poisson structure {pi, pj}H = eijkH, and the Hamiltonian (the Fermi function) f is a periodic function. For such a pair the multivalued function (1-form) g(pi) = piH is a Casimir operator, and all the solutions at a given ‘energy’ level c are obtained as the leaves of the foliation of ω = dg on the surface M2 c = {f = c}. In spite of the simplicity of the model, the topological properties of the trajectories of the quasi-momentum have a highly non-trivial structure. The following picture emerged from the fundamental results of Zorich [2] and Dynnikov [3], [4]: every function f induces on the direction space RP2 of the field ω two functions cm, cM : RP 2 → R such that the set cm(ω) = cM (ω) is a disjoint union of open sets Si, each of which is marked by some element li of the latticeH2(T,Z). Moreover, if there are at least two distinct sets Si (which will be assumed in the following), then there are infinitely many of them, and the complement of their union admits a fractal structure. The meaning of the elements li is as follows: if cm(ω) < c < cM (ω), then ω belongs to one of the regions Si0 and induces on M 2 c open trajectories having the strong asymptotic direction ω × li0 , but when c < cm(ω) or cM (ω) < c, then all trajectories are closed. The set given by the condition cm(ω) = cM (ω) is the union of the boundaries B = ⋃ i ∂Si and the set E of socalled ‘ergodic directions’, that is, directions ω that induce on M2 c open trajectories filling the components of genus larger than 2 [5].