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Stability Conditions and the Single Mode Approximation in FQHE

A thermodynamic stability criterion for the spontaneous breaking of the translation invariance of many particle systems is derived. It simply requires the positive character of the wavevector dependent dielectric function as generalising the same condition for macroscopic dielectric constants. Its application to the Single Mode Approximation (SMA) for the description of the collective modes of the filling factor 1/m Laughlin states is considered. The results indicate that the stability condition is violated by the SMA for all the relevant states m=3,5,7,9 in a wavevector neighborhood of the magneto-roton minima. These conclusions are in qualitative agreement with similar results obtained from the solution of the Bethe-Salpeter equation at 1/3 filling factor for both composite fermions and phenomenologically described electrons in the Laughlin state.