The integer spectrum of a superintegrable quantum field theory
Abstract
In our previous article CMP 407 (2026) 131, we explored the rich algebraic structures arising from a specific quantum many-body Hamiltonian quartic in the bosonic creation-annihilation operators. This Hamiltonian originated from the works of Gérard and Grellier on classical turbulent energy transfer in a Lax-integrable setting, and its quantum version respects hierarchies of polynomial conservation laws all of which possess purely integer spectra, as does the Hamiltonian itself. The latter feature, known from numerical studies, has previously remained unproved. Here, we present an elementary solution of the quantum Gérard-Grellier Hamiltonian. The eigenstates are labelled by partitions and the corresponding eigenvalues admit simple explicit expressions in terms of the principal hook lengths. The key ingredient in this construction is a specific basis we introduce, inspired by the classical multisoliton coherent states, in which the Hamiltonian is upper-triangular. The expressions for the eigenvalues then immediately follow, while the full eigenbasis can be recovered via a Gram-Schmidt procedure.
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BibTeX
@misc{clerck2026integer,
title = {{The integer spectrum of a superintegrable quantum field theory}},
author = {Marine De Clerck and Oleg Evnin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05015},
eprint = {2611.05015},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05015},
}
RIS
TY - UNPB
TI - The integer spectrum of a superintegrable quantum field theory
AU - Marine De Clerck
AU - Oleg Evnin
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05015
UR - https://arxiv.org/abs/2611.05015
AB - In our previous article CMP 407 (2026) 131, we explored the rich algebraic structures arising from a specific quantum many-body Hamiltonian quartic in the bosonic creation-annihilation operators. This Hamiltonian originated from the works of Gérard and Grellier on classical turbulent energy transfer in a Lax-integrable setting, and its quantum version respects hierarchies of polynomial conservation laws all of which possess purely integer spectra, as does the Hamiltonian itself. The latter feature, known from numerical studies, has previously remained unproved. Here, we present an elementary solution of the quantum Gérard-Grellier Hamiltonian. The eigenstates are labelled by partitions and the corresponding eigenvalues admit simple explicit expressions in terms of the principal hook lengths. The key ingredient in this construction is a specific basis we introduce, inspired by the classical multisoliton coherent states, in which the Hamiltonian is upper-triangular. The expressions for the eigenvalues then immediately follow, while the full eigenbasis can be recovered via a Gram-Schmidt procedure.
ER -