Half-Filled Lowest Landau Level on a Thin Torus

We solve a model that describes an interacting electron gas in the half-filled lowest Landau level on a thin torus, with radius of the order of the magnetic length. The low-energy sector consists of noninteracting, one-dimensional, neutral fermions. The ground state, which is homogeneous, is the Fermi sea obtained by filling the negative energy states, and the excited states are gapless neutral excitations out of this one-dimensional sea. Although the limit considered is extreme, the solution has a striking resemblance to the composite fermion description of the bulk $\nu =1/2$ state—the ground state is homogeneous and the excitations are neutral and gapless. This suggests a one-dimensional Luttinger liquid description, with possible observable effects in transport experiments, of the bulk state where it develops continuously from the state on a thin torus as the radius increases.

Figure 1

Phase diagram for $\nu =1/2$ on a torus, where one circumference, $L_1$, varies while the other is infinite. For a thin torus, $L_1\sim 5$ magnetic lengths, the $\nu =1/2$ system is that of noninteracting neutral one-dimensional fermions (dipoles). The question mark indicates whether the state develops continuously into the bulk $\nu =1/2$ state or whether there is a phase transition. Based on the similarities of the state at short $L_1$ and the bulk state, such as a homogeneous ground state and neutral gapless excitations, we conjecture that there is no phase transition. This suggests a one-dimensional description of the bulk $\nu =1/2$ state as a Luttinger liquid rather than as a free two-dimensional Fermi gas. (For very short $L_1$ a crystalline state determined by electrostatics alone is obtained.)