Microscopic derivation of the effective Lagrangian for skyrmions in an interacting two-dimensional electron gas at small g -factor
and quantize the problem, assuming the standard commuta-tion 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 time-dependent O l t and two space-dependent 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 second-order diagrams also contributing, since their non-local character in time and space must be taken into account [9, 10]. We present only the final result corresponding to thèfermion' character of vortex-sky-rmions:  e ljm O t O j Â O l d 2 r dt Y 7 where e ljm is the unit antisymmetric third-rank 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 vortex-skyrmion 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.