OFFSET
2,1
COMMENTS
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.
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, e.g., [De Pasquale et al. 2008, Franchini 2017]. The Hamiltonian is H = Sum_{j=1..n} (sigma_x(j) * sigma_x(j+1) + sigma_y(j) * sigma_y(j+1)), with periodic boundary conditions sigma(n+1) = sigma(1). By the Jordan-Wigner transformation, the system maps to free fermions [Jordan and Wigner 1928], where the quantization of momenta k depends on the parity of the total number of fermions, |K|. If the number of fermions |K| is odd, the momenta are quantized as k = 2*j*Pi/n (Periodic boundary conditions for fermions), e.g., [De Pasquale et al. 2008, Franchini 2017]. If the number of fermions |K| is even, the momenta are quantized as k = (2*j + 1)*Pi/n (anti-periodic boundary conditions for fermions), e.g, [De Pasquale et al. 2008, Franchini 2017]. 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.
LINKS
Ilario Bonacina, Nicola Galesi, and Massimo Lauria, On vanishing sums of roots of unity in polynomial calculus and sum-of-squares, Comput. Complex. 32 (2023) 12.
Fabio Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer, 2017, pages 1-8.
Yongao Hu, Felix Gerken, and Thore Posske, Hidden Twisted Sectors and Exponential Degeneracy in Root-of-Unity XXZ Heisenberg Chains, arXiv:2602.15098 [cond-mat.stat-mech], 2026. See pp. 5, 12, references.
P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Z. Physik, 47 (1928), 631-651.
T. Lam and K. Leung, On vanishing sums of roots of unity, J. of Alg. 224 (2000), 91-109.
A. De Pasquale, G. Costantini, P. Facchi, et al., XX model on the circle, Eur. Phys. J. Spec. Top. 160 (2008), 127-138.
FORMULA
a(p) = 2 <=> p is prime.
Conjecture: a(2*p) = 2*(6^((p-1)/2)+1) for odd prime p.
EXAMPLE
The XX Heisenberg chain with four sites has an a(4)=10-fold degenerate nullspace.
MATHEMATICA
energies[momenta_, n_] :=
Chop@Total[Cos[N[ \[Pi] (2 momenta + 1/2 ((-1)^(Length[momenta]) + 1))/n]]];
a[n_] := Module[{qlist, energyList}, qlist = (Subsets[Range[n], n]);
energyList = (energies[#, n] &) /@ qlist;
Count[energyList, 0]];
Table[a[n], {n, 2, 10}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Thore Posske, Jan 09 2026
EXTENSIONS
a(28)-a(34) from Sean A. Irvine, Jan 24 2026
STATUS
approved