Publications (7)43.83 Total impact

Article: Reply: Apel and Bychkov
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ABSTRACT: A Reply to the Comment by Alexander V. Khaetskii. 
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ABSTRACT: We study the relaxation of a spin I that is weakly coupled to a quantum mechanical environment. Starting from the microscopic description, we derive a system of coupled relaxation equations within the adiabatic approximation. These are valid for arbitrary I and also for a general stationary nonequilibrium state of the environment. In the case of equilibrium, the stationary solution of the equations becomes the correct Boltzmannian equilibrium distribution for given spin I. The relaxation towards the stationary solution is characterized by a set of relaxation times, the longest of which can be shorter, by a factor of up to 2I, than the relaxation time in the corresponding Bloch equations calculated in the standard perturbative way. Comment: 4 pages, Latex, 2 figures  [Show abstract] [Hide abstract]
ABSTRACT: We study the spin relaxation in an interacting twodimensional electron gas in a strong magnetic field for the case that the electron density is close to filling just one Landau sublevel of one spin projection, i.e., for filling factor near one. Assuming the relaxation to be caused by scattering with phonons, we derive the kinetic equations for the electron's spindensity which replace the Bloch equations in our case. These equations are nonlinear and their solution depends crucially on the filling factor and on the temperature of the phonon bath. In the limit of zero temperature and for filling factor 1, the solution relaxes asymptotically with a power law inversely proportional to time, instead of following the conventional exponential behavior. Comment: 4 pages, 1 figure  [Show abstract] [Hide abstract]
ABSTRACT: We can introduce the generalized momentum P i qLaq X i and quantize the problem, assuming the standard commutation relations P i X i i" h to be fulfilled. This yields the cyclotron frequency of the skyrmion as a whole " ho s 2g and the minimum energy E s g. Thus, in experiments the skyrmion should exhibit a cyclotron resonance at a frequency 2g. The quantity g is determined as the exchange energy per electron, in the case of a completely filled Landau level g e 2 l B 2p p X The energy g must be added to expression (5) obtained above for the total energy. It is also interesting to find a term with the Hopf invariant in the action which, according to modern views, determines the skyrmion statistics [11]. For this purpose we should calculate the terms containing one timedependent O l t and two spacedependent O l in the expansion of the action in terms of O l . Calculations up to the third order are rather cumbersome and require the consideration of numerous diagrams, the secondorder diagrams also contributing, since their nonlocal character in time and space must be taken into account [9, 10]. We present only the final result corresponding to thèfermion' character of vortexskyrmions: S H pH Y H 1 2p 2 e ljm O t O j Â O l d 2 r dt Y 7 where e ljm is the unit antisymmetric thirdrank tensor. The integer Hopf invariant H is expressed in terms of O l [12]. This result does not coincide with that obtained within the method of Landau functions projected onto the zero level [7] which is a sum of several spatial derivatives. At the same time, formula (7) has a standard form and agrees with that suggested in note [13]. In conclusion we emphasize once again that the solution of differential equations of the Hartree ± Fock approximation is necessary, since it determines the discrepancy between our results and those obtained by projection onto a single Landau level, when the differential form of the kinetic energy in the SchroÈ dinger equation is replaced by a constant energy. It is of special importance in the calculation of the thermodynamic energy of the vortexskyrmion where the additional term À" ho c a2Q appears, which can lead to spontaneous appearance of vortices and a rearrangement of the ground state in a rather strong magnetic field. 
Article: Apel and Bychkov Reply
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ABSTRACT: A Reply to the Comment by G. E. Volovik and V. M. Yalcovenko.  [Show abstract] [Hide abstract]
ABSTRACT: We study interacting electrons in two dimensions moving in the lowest Landau level under the condition that the Zeeman energy is much smaller than the Coulomb energy and the filling factor is one. In this case, Skyrmion quasiparticles play an important role. Here, we present a simple and transparent derivation of the corresponding effective Lagrangian. In its kinetic part, we find a nonzero Hopf term the prefactor of which we determine rigorously. In the Hamiltonian part, we calculate, by means of a gradient expansion, the SkyrmionSkyrmion interaction completely up to fourth order in spatial derivatives. Comment: 4 pages, Latex
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19962000

PhysikalischTechnische Bundesanstalt
Brunswyck, Lower Saxony, Germany
