Classical field descriptions of one-dimensional systems

Abstract

In this paper we discuss the limitations of classical field treatments of one-dimensional systems in the static approximation. Two exactly solvable Hamiltonians, the ferromagnetic Ising model, and its extension to a zero-width half-filled band, are studied after their transformation to a classical field form via the Hubbard–Stratonovich identity. The more usual two-field transformation consists of using one field to describe the divergent order parameter and another to represent the nondivergent modes. The fluctuations in this latter one are usually neglected and this is shown to lead to incorrect thermodynamic behavior throughout the critical region, which is unusually large in one-dimensional systems, and even beyond to the high temperature limit. Any limited expansion of the free energy is further seen to lead to incorrect treatment of the amplitude fluctuations. A rigorous treatment of both fields is required. Alternately, a one-field transformation can assure a simpler approach although all terms in the free energy expansion must be retained. The findings are extrapolated to other known Hamiltonians: Hubbard, Peierls and spin-Peierls, and Bardeen–Cooper–Schrieffer (BCS) superconductivity. The Peierls case is examined in some detail because the usual one-field free energy functional is not obtained by a straightforward use of the Hubbard–Stratonovich transformations. As for the BCS Hamiltonian, it is seen to be in a special class because both symmetry fields are equally divergent and are automatically treated on an equal footing.