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Bethe-State Counting and Witten Index

by Hongfei Shu, Peng Zhao, Rui-Dong Zhu, Hao Zou

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Submission summary

Authors (as registered SciPost users): Hao Zou
Submission information
Preprint Link: scipost_202303_00019v2  (pdf)
Date accepted: 2023-07-20
Date submitted: 2023-06-19 09:11
Submitted by: Zou, Hao
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We count the physical Bethe states of quantum integrable models with twisted boundary conditions using the Witten index of 2d supersymmetric gauge theories. For multi-component models solvable by the nested Bethe ansatz, the result is a novel restricted occupancy problem. For the SU(3) spin chain and the t-J model, we propose formulae for the solution count on singular loci in the space of twist parameters.

List of changes

1. In the abstract: “We count the Bethe states … ” to “We count the physical Bethe states …”
2. At the end of the first paragraph of Introduction: we added a footnote [7]: “Not all the Bethe states correspond to the eigenstates of the Hamiltonian, see, for example, [5, 6]. In this paper, we will only count the physical Bethe states. ”
3. In paragraph starting with “A complementary approach …” in the Introduction: we changed “whether Bethe states can be …” to “whether physical Bethe states can be … ”
4. In the paragraph below Equation (9): we changed “We remark that the Bethe states of …” to “We remark that the eigenstates of … ”
5. In the first paragraph of Section IV: we added a word “physical” in the second sentence.

Published as SciPost Phys. 15, 103 (2023)

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