Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus

We analyze the Moore-Read Pfaffian state on a thin torus. The known sixfold degeneracy is realized by two inequivalent crystalline states with a four- and twofold degeneracy, respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension $2^{n-1}$ for $2n-k$ holes and $k$ particles at fixed positions and assign each a charge $\pm e∕4$. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model Hamiltonian and a small number of particles support the presence of a stable phase with degenerate vacua and quarter-charged domain walls also away from the thin-torus limit. A spin-chain Hamiltonian encodes the degenerate vacua and the various domain walls.

Figure 1

(Color online) Phase diagram for a half-filled Landau level on the torus as a function of $L_1$ and the pseudopotential parameter $\delta V_1$ ($\delta V_1=0$ corresponds to Coulomb interaction in the lowest LL). Results are obtained using exact diagonalization for eight electrons. The Pfaffian phase is found in the central part (solid lines) of the diagram. This is the region where the six lowest-lying states are those that continuously approach the crystals $A$ and $B$. Within the dashed lines, these states are almost degenerate—they differ in energy by less than 10% of the gap to the next state. In the Pfaffian phase we observe quasiparticles and quasiholes of charge $\pm e∕4$. In the top left the ground state is $101010\cdots$ and the phase in the top right is the gapless Luttinger liquid phase described in Ref. 10.