Abstract. It is shown that the investigation of the conductivity in a single crystal of a normal metal with a complicated Fermi surface in strong magnetic fields B can reveal integral topological characteristics which are determined by the topology of open-ended quasiclassical electron trajectories. Specifically, in the case of open-ended trajectories of the general position there always exists a direction η orthogonal to B in which the conductivity approaches zero for large B, and this direction lies in some integral (i.e., generated by two reciprocal-lattice vectors) plane that remains stationary for small variations of the direction of B.In the present letter we describe topological effects arising in the study of the conductivity tensor of normal metals in strong magnetic fields. We assume that conductivity is described well in the quasiclassical approximation for the one-electron problem with some dispersion law (p), which is periodic in the quasimomentum space with periods equal to reciprocal-lattice vectors. The only vestiges of quantum mechanics in the theory are a transition from Euclidean momentum space E = R 3 to the first Brillouin zone B, which is a three-dimensional torus T 3 related with the reciprocal lattice, and the form of the function (p). In this approximation the electrons follow quasiclassical trajectories which, as functions of the time t, are solutions of the systemThe system (1) is a Hamiltonian system with the Hamiltonian H(x, p) = (p) and, where, and so on. The trajectories of the system (1) in momentum space are given by the intersection of the surfaces of constant energy (p) = const with planes perpendicular to the magnetic field:The theory described above works well up to the comparatively high values of the magnetic field B which we shall study. For comparison we point out that the upper limit on the magnetic field is determined by the applicability of the quasiclassical approximation ω B F , while the effects due to the topology of the trajectories come into play for ω B > τ −1 , where τ is the free time of flight of the electrons (see [1]-[3]).
We investigate theoretically soliton excitations and dynamics of their formation in strongly correlated systems of ultracold bosonic atoms in two and three dimensional optical lattices. We derive equations of nonlinear hydrodynamics in the regime of strong interactions and incommensurate fillings, when atoms can be treated as hard core bosons. When parameters change in one direction only we obtain Korteweg-de Vries type equation away from half-filling and modified KdV equation at half-filling. We apply this general analysis to a problem of the decay of the density step. We consider stability of one dimensional solutions to transverse fluctuations. Our results are also relevant for understanding nonequilibrium dynamics of lattice spin models. PACS numbers: 03.75.Be, 32.80.Pj, 42.50.Vkwhich corresponds to the small values of J z and the density n ∼ 1/2 in the pattern. The solutions we considered in the opposite situationare unstable from the point of view of the two-dimensional modulations.[106] Let us say now that the analogous considerations can be performed also in the case of equation (39) so the results formulated above can be used also in the limit µ 0 → 0.
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