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

## TL;DR

This paper investigates the zero mode structure of the Jain-2/5 parent Hamiltonian, revealing new insights into its guiding-center patterns, dominance structures, and zero mode generators using second quantized methods.

## Contribution

It provides an analytic characterization of zero modes, dominance patterns, and generators for the Jain-2/5 state, advancing understanding beyond previous numerical studies.

## Key findings

- Zero mode conditions characterized analytically for disk and spherical geometries.
- Identification of dominance patterns following a generalized Pauli principle.
- Discovery of one-body operators that generate zero modes.

## Abstract

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.