Distinct Types of Parent Hamiltonians for Quantum States: Insights from the W State as a Quantum Many-Body Scar

The construction of parent Hamiltonians that possess a given state as their ground state is a well-studied problem. In this work, we generalize this notion by considering simple quantum states and examining the local Hamiltonians that have these states as exact eigenstates. These states often correspond to Quantum Many-Body Scars (QMBS) of their respective parent Hamiltonians. Motivated by earlier works on Hamiltonians with QMBS, in this work we formalize the differences between three distinct types of parent Hamiltonians, which differ in their decompositions into strictly local terms with the same eigenstates. We illustrate this classification using the W state as the primary example, for which we rigorously derive the complete set of local parent Hamiltonians, which also allows us to establish general results such as the existence of asymptotic QMBS, and distinct dynamical signatures associated with the different parent Hamiltonian types. Finally, we derive more general results on the parent Hamiltonian types that allow us to obtain some immediate results for simple quantum states such as product states, where only a single type exists, and for short-range-entangled states, for which we identify constraints on the admissible types. Altogether, our work opens the door to classifying the rich structures and dynamical properties of parent Hamiltonians that arise from the interplay between locality and QMBS.