Jain-2/5 parent Hamiltonian: Structure of zero modes, dominance patterns, and zero mode generators

We analyze general zero mode properties of the parent Hamiltonian of the unprojected Jain-2/5 state. We characterize the zero mode condition associated to this Hamiltonian via projection onto a four-dimensional two-particle subspace for given pair angular momentum, for the disk and similarly for the spherical geometry. Earlier numerical claims in the literature about ground-state uniqueness on the sphere are substantiated on analytic grounds, and related results are derived. Preference is given to second-quantized methods, where zero mode properties are derived not from given analytic wave functions, but from a “lattice” Hamiltonian and associated zero mode conditions. This method reveals new insights into the guiding-center structure of the unprojected Jain-2/5 state, in particular, a system of dominance patterns following a “generalized Pauli principle,” which establishes a complete one-to-one correspondence with the edge mode counting. We also identify one-body operators that function as generators of zero modes.

Figure 1

Composite fermion occupancy patterns and resulting dominance patterns. Three different cases are shown. Level diagrams show composite fermion occupancies, followed by a more symbolic composite fermion occupancy pattern and the associated dominance pattern as explained in text. (a) Corresponds to the densest (minimum angular momentum) zero mode for odd particle number, followed by the two configurations corresponding to the doubly degenerate densest zero modes for even particle number (b) and (c). Note that only the dominance patterns manifestly encode the total angular momentum of the state. More general dominance patterns consistent with Lemmas 1–6, and thus in one-to-one correspondence with zero modes (see text), are shown in Table 1.

Figure 2

Same as Fig. 1, but for the sphere, where the first excited LL has one more orbital at both maximum and minimum $L_z$, for both electrons and composite fermions. Shown (bottom line) is the resulting unique dominance pattern for a sphere satisfying $2s=\frac{5}{2}N-4$, where $2s$ is the number of flux quanta penetrating the sphere.