bibkey: posske2026a392387
authors: Thore Posske
year: 2026
title: "OEIS A392387, Nullspace dimension of the periodic XX Heisenberg chain with n > 1 sites"
doi: null
url: https://oeis.org/A392387
claim: "The entry counts the subsets K of {1,...,n} whose cosines at the angles (2j + (1 + (-1)^|K|)/2)Pi/n, j in K, sum to zero, identifies this number with the dimension of the zero-energy subspace of the periodic spin-1/2 XX Heisenberg chain on n sites, and conjectures a(2p) = 2(6^((p-1)/2)+1) for odd primes p."
strata_touched:
  - D5/S3/Quantum/SpinChains/XXChainNullspaceCount
license: citation-only
triage: anchor

OEIS A392387

The NAME of A392387 is

Nullspace dimension of the periodic XX Heisenberg chain with n > 1 sites.

The COMMENTS state

a(n) with n>1 is the dimension of the zero-energy subspace (nullspace) of the periodic spin-1/2 XX Heisenberg chain on n sites … By the Jordan-Wigner transformation, the system maps to free fermions … If the number of fermions |K| is odd, the momenta are quantized as k = 2jPi/n … If the number of fermions |K| is even, the momenta are quantized as k = (2*j + 1)*Pi/n … The sequence counts the number of subsets M of these momenta such that the total energy E = Sum_{m in M} cos(m) is 0.

and

Also, the number of subsets K of {1,…,n} such that the sum of cosines of the angles in {(2j + (1 + (-1)^|K|)/2 )*Pi/n | j in K} is zero.

The FORMULA section records

Conjecture: a(2p) = 2(6^((p-1)/2)+1) for odd prime p.

The entry cites Y. Hu, F. Gerken and T. Posske, Hidden Twisted Sectors and Exponential Degeneracy in Root-of-Unity XXZ Heisenberg Chains, arXiv:2602.15098 (2026), whose appendix on the XX model (Δ = 0) rephrases the nullspace degeneracy as the number of combinations of distinct angles whose cosines sum to zero and calls this, equivalently the counting of certain purely imaginary sums of distinct N-th and 2N-th roots of unity, “an unsolved mathematical problem”.

Verified locator

• URL: https://oeis.org/A392387
• Version and location: OEIS A392387, revision 52 (2026-02-20), COMMENTS and FORMULA; author Thore Posske, 2026-01-09.
• Related source: arXiv:2602.15098, appendix “Δ = 0: XX Model”.