Exhaustive Construction of Generalized Parent Hamiltonians for Matrix Product States

Abstract:Characterizing local Hamiltonians that host interesting many-body states as eigenstates has been an important problem in quantum many-body physics. For matrix product states (MPS), there is a well-known construction of parent Hamiltonians that have the MPS as the frustration-free ground state. However, recent results on quantum many-body scars have shown that such a construction does not exhaust all possible Hamiltonians that have the MPS as an eigenstate. In this work, we develop an exhaustive construction of generalized parent Hamiltonians that have a given translation-invariant MPS as an eigenstate. These go beyond the conventional projector-based construction of frustration-free parent Hamiltonians, where each local Hermitian term has the MPS as an eigenstate, by allowing non-Hermitian telescoping terms that cancel out with periodic boundary conditions (PBC). We derive a local tensor equation that establishes a correspondence between inequivalent classes of generalized parent Hamiltonians, i.e., those that cannot be related using linear combinations with frustration-free Hamiltonians, and the MPS transfer matrix spectrum. We illustrate the connection to intrinsic physical properties, including non-Abelian symmetries and zero correlation length, and further extend this construction to block-injective MPS. Our framework hence provides a systematic bridge between generalized parent Hamiltonians and the local structure of quantum states.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)

arXiv:2611.00203 [quant-ph]

(or arXiv:2611.00203v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2611.00203 arXiv-issued DOI via DataCite (pending registration)