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.
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BibTeX
@misc{wang2026exhaustive,
title = {{Exhaustive Construction of Generalized Parent Hamiltonians for Matrix Product States}},
author = {He-Ran Wang and Lorenzo Gotta and Sanjay Moudgalya},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00203},
eprint = {2611.00203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00203},
}
RIS
TY - UNPB
TI - Exhaustive Construction of Generalized Parent Hamiltonians for Matrix Product States
AU - He-Ran Wang
AU - Lorenzo Gotta
AU - Sanjay Moudgalya
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00203
UR - https://arxiv.org/abs/2611.00203
AB - 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.
ER -