No Go Theorems in Interacting

In a Galilean invariant system, for which the Landau theory of Fermi-liquids was originally designed, the (velocity-independent) interactions are the same in the moving frame as in the stationary frame. Therefore interactions do not renormalize the current operator. It then follows that, among other things, instability due to interactions to a state with uniform current is not possible. If it were not forbidden, it would amount to a dipolar deformation of the Fermi-surface, which would be the ` = 1 spin-symmetric Pomeranchuk instability. Such a state occurring spontaneously is also forbidden more generally than Galilean invariance by gauge-invariance or the proposition that a photon cannot acquire mass, sometimes called Elitzur’s theorem. Such a state was proposed by Heisenberg as representing the superconducting state[1]. In response F. Bloch is said to have stated, but never published, that such a state is impossible. The proof of this Bloch’s theorem, for a Hamiltonian with any one-particle potential (for example a periodic lattice) and multi-particle interactions which are purely density dependent, was provided by D. Bohm in an elegant and simple variational calculation [2]. The argument is simply that for any variational wave-function carrying uniform current, one can construct another with energy decreasing linearly with lower current carried. Consistent with the Bloch-Bohm theorem, supercurrents only flow in metastable states or in response to variations in magnetic field as required by the Meissner-Ochsenfeld effect in superconductors. The argument against uniform current-carrying states can also be made from conservation laws and Landau theory for velocity independent Hamiltonians which are not Galilean invariant. There is no contribution using the incoherent parts of the single-particle Green’s function for conserved quantities