Mean-Field Theory of Fluid-Solid Transition of Fermion Systems

概要

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The condition of fluid-solid transition of fermion system at 0K is surveyed within the mean-field theory for three and two spatial dimensions. Fermions are supposed to interact through two-body central potentials. The effective action S_<eff> for a mean field M( p^^~ ) of density waves (DWs) of a mode p^^~=(p, 0) is constructed by the functional integration method. In order to obtain S_<eff>, functional integrations are performed over modes with high wavevectors (fast modes) of mean fields as well as fermionic fields. A necessary condition for solidification is that S_<eff> has a non-vanishing stationary point M(p^^~)=<M(p^^~)>, thereby yielding the renormalized effective interaction at <M(p^^~)>. Another requirement for solidification is that the effective interaction for mean field should get weaker under the progressive integration of fast modes. These two requirements are shown to lead to a simple equation as a solidification condition (SC). The SC is applied to two and three dimensional (2D and 3D) electrons and helium-3. It is shown that electrons form the Wigner crystal above r_s ≈ 43 and ≈ 37 in three and two dimensions, respectively. The critical Fermi wavevector k_F of 3D helium-3 for fluid-solid transition is shown to be about 0.89×10^8 cm^<-1>. A prediction is made that the 2D helium-3 will solidify at k_F≈0.69×10^8 cm^<-1>.
理論物理学刊行会の論文
1995-06-25