Flux Quantization in a One-Dimensional Model

A one-dimensional model of interacting electrons is studied to determine whether such a system, in thermal equilibrium, can exhibit flux quantization. The free energy and current of the system are calculated and shown to be periodic functions of the flux enclosed in a ring-shaped sample with period $\frac{\mathrm{hc}}{e}$. The Maxwell equations provide a second relation between the current and flux. It is found that at finite temperatures, the equations for the current $I$ and the flux ${\Phi }_B$ have only the trivial solution $I={\Phi }_B=0\mathrm{}$ in the limit of macroscopic systems. Therefore, there is no flux quantization. The free energy is calculated by a generalization of the method of Tomonaga. This method describes the Fermi system in terms of an equivalent set of bosons which represent the collective modes of the Fermi gas. The major results of the generalization are the appearance of trilinear terms in the equivalent boson Hamiltonian and effects of a vector potential.