BOSONIZATION OF INTERACTING FERMIONS IN ARBITRARY DIMENSION BEYOND THE GAUSSIAN APPROXIMATION

We use our recently developed functional bosonisation approach to boso nize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d = 1 the finite curvature of the energy dispe rsion at the Fermi surface gives rise to interactions between the boso ns. In higher dimensions scattering processes describing momentum tran sfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian a pproximation. We derive an explicit expression for the leading correct ion to the bosonized Hamiltonian and the irreducible self-energy of th e bosonic propagator that takes the finite curvature as well as around -the-corner processes into account. In the special case that around-th e-corner scattering is negligible, we show that the self-energy correc tion to the Gaussian propagator is negligible if the dimensionless qun atities (q(c)/k(F))F-d(0)[1+F-0](-1)mu/nu(alpha)\partial derivative nu (alpha)/partial derivative mu are small compared with unity for all pa tches alpha. Here q(c) is the cutoff of the interaction in wave-vector space, k(F) is the Fermi wave-vector, mu is the chemical potential, F -0 is the usual dimensionless Landau interaction parameter, and nu(alp ha) is the local density of states associated with patch alpha. We als o show that the well known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d = 1, exists also in d > 1, provide d (1) the interaction cutoff q(c) is small compared with k(F), and (2) the energy dispersion is locally linearized at the Fermi surface. Fin ally, we suggest a systematic method to calculate corrections to the R PA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosoniz ed Hamiltonian.