Algebraic Geometry for Spin-Adapted Coupled Cluster Theory
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912880657432576 |
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| author | Faulstich, Fabian M. Sverrisdóttir, Svala |
| author_facet | Faulstich, Fabian M. Sverrisdóttir, Svala |
| contents | We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We give an explicit description of the SU(2)-invariant (spin singlet) many-body space by identifying it with an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. We define spin-adapted truncation varieties via embeddings of graded subspaces of this ring, and we identify the CCS truncation variety with the Veronese square of the Grassmannian. Compared to the spin-generalized formulation, this approach yields a substantial reduction in dimension and degree, with direct computational consequences. In particular, the CC degree of the truncation variety -- governing the number of homotopy paths required to compute all CC solutions -- is reduced by orders of magnitude. We present scaling studies demonstrating asymptotic improvements and we exploit this reduction to compute the full solution landscape of spin-adapted CC equations for water and lithium hydride. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16646 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic Geometry for Spin-Adapted Coupled Cluster Theory Faulstich, Fabian M. Sverrisdóttir, Svala Chemical Physics Algebraic Geometry Representation Theory Quantum Physics We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We give an explicit description of the SU(2)-invariant (spin singlet) many-body space by identifying it with an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. We define spin-adapted truncation varieties via embeddings of graded subspaces of this ring, and we identify the CCS truncation variety with the Veronese square of the Grassmannian. Compared to the spin-generalized formulation, this approach yields a substantial reduction in dimension and degree, with direct computational consequences. In particular, the CC degree of the truncation variety -- governing the number of homotopy paths required to compute all CC solutions -- is reduced by orders of magnitude. We present scaling studies demonstrating asymptotic improvements and we exploit this reduction to compute the full solution landscape of spin-adapted CC equations for water and lithium hydride. |
| title | Algebraic Geometry for Spin-Adapted Coupled Cluster Theory |
| topic | Chemical Physics Algebraic Geometry Representation Theory Quantum Physics |
| url | https://arxiv.org/abs/2601.16646 |