Constructive eigenfunction mapping for the one-dimensional charge-pair Hubbard model and the origin of odd-lattice-size level crossings in the Hubbard chain

In 2020 M. J. Martins solved, by a nested Bethe ansatz, a one-dimensional model of spin-1/2 fermions in which the Hubbard hopping is replaced by a nearest-neighbor triplet (p-wave) charge-pairing term plus the on-site Hubbard interaction. The solution left two concrete structures unfinished. First, for odd lattice sizes $L$ the energy spectra of the charge-pair Hamiltonian and of an equivalent coupled-XY spin-chain representation are expected to coincide (a conjecture from the matching Bethe equations, verified by exact diagonalization, i.e. full numerical diagonalization of the finite Hamiltonian, for $L=3,5,7$ ), but no constructive relation between the two models' eigenfunctions was found. Second, exact diagonalization of the standard repulsive Hubbard chain on odd lattices reveals level crossings among the lowest states in different particle-number sectors (subspaces of fixed electron numbers $(N_\uparrow,N_\downarrow)$ for the two spin species) — for $L=5$ between the ground states of sectors $(3,2)$ and $(3,3)$ , for $L=7$ between those of $(3,4)$ and $(3,3)$ — a pattern apparently governed by the parity of $(L-1)/2$ , whose origin and large- $L$ fate are unexplained. Both questions remain unresolved.

Background

The one-dimensional Hubbard model — spin-1/2 fermions on a ring of $L$ sites with nearest-neighbor hopping and on-site repulsion $U$ — has been exactly solved since Lieb and Wu (1968) by the Bethe ansatz, a wave function built from plane waves whose amplitude relations encode two-body scattering. Its half-filled (one electron per site) finite-size theory is thoroughly developed (Essler, Frahm, Göhmann, Klümper and Korepin 2005), almost exclusively for even $L$ . The model's integrability was later understood algebraically through Shastry's construction (1986, 1988): the Hubbard Hamiltonian commutes with the transfer matrix — the row-to-row partition-function generator of a two-dimensional classical vertex model, here of a special free-fermion type — and is equivalent to a quantum spin chain.

In 2020 M. J. Martins introduced and solved a Hubbard variant in which the hopping is replaced by a nearest-neighbor triplet (p-wave) charge-pairing potential:

$H_c=\sum_{j=1}^{L}\sum_{\alpha=\uparrow,\downarrow}\left[c_\alpha(j)c_\alpha(j+1)+c^\dagger_\alpha(j+1)c^\dagger_\alpha(j)\right]+U\sum_{j=1}^{L}\left(n_\uparrow(j)-\tfrac{1}{2}\right)\left(n_\downarrow(j)-\tfrac{1}{2}\right),$

with periodic boundary conditions. The pairing term creates or annihilates pairs of same-spin electrons on neighboring sites (a triplet pairing of p-wave-superconductivity type); there is no single-electron hopping. For any $L$ the model conserves the total $y$ -component of spin and the $x$ -component of the pseudo-spin (charge) SU(2) generators, the on-site pairing and density operators that form a second SU(2) algebra. A site-by-site unitary transformation that mixes particles and holes of the two spin species maps $H_c$ to a Hubbard-like Hamiltonian with imaginary, asymmetric hopping, whose conserved charges become the particle numbers $N_\uparrow,N_\downarrow$ of the transformed fermions, so the Hilbert space decomposes into the same $(N_\uparrow,N_\downarrow)$ sectors as the Hubbard model. The nested Bethe ansatz (a two-level Bethe ansatz in which the scattering amplitudes of one particle species are themselves solved by a second Bethe ansatz) then yields Lieb-Wu-type Bethe equations with extra twist phases $e^{i\pi L/2}$ depending on the lattice size: for $L$ a multiple of 4 the twists are unity and the charge-pair spectrum coincides exactly with the Hubbard spectrum, while for odd $L$ the twists compensate the frustration introduced by the lattice parity. Notably, the particle-hole spectral identity $E(N_\uparrow,N_\downarrow)=E(L-N_\uparrow,L-N_\downarrow)$ holds for every $L$ , whereas in the Hubbard model it requires a bipartite (even- $L$ ) lattice. In the repulsive regime the charge (Mott) gap — the energy cost of inserting charge — can be defined for both parities of $L$ and both converge to the Lieb-Wu half-filled value, while the gapless spin excitations carry the conformal dimensions (the scaling exponents governing finite-size corrections in one-dimensional critical systems) of the isotropic Heisenberg chain — with a boundary twist, i.e. half-integer vorticity in the sense of Alcaraz, Barber and Batchelor (1988), when $L$ is odd. Shastry's construction carries over to this model: the transfer matrix of a special free-fermion eight-vertex model with one vanishing weight commutes with $H_c$ , and the covering quantum spin chain consists of two coupled XY-type chains (spin-1/2 chains with purely transverse exchange) with Ising coupling $\frac{U}{4}\sum_j\sigma^z_j\tau^z_j$ and lattice-parity-dependent boundary terms.

Two parts of this solution remain unfinished. First, for odd $L$ the energy spectra of the charge-pair Hamiltonian and of the coupled spin chain are expected to coincide — a conjecture from the matching Bethe equations, confirmed by exact diagonalization (full numerical diagonalization of the finite Hamiltonian matrix) for $L=3,5,7$ — but the paper's canonical and Jordan–Wigner transformations (the latter the standard exact mapping between spin-1/2 operators and fermions in one dimension) reproduce only the bulk of the spin-chain Hamiltonian, and no constructive relation between the two models' eigenfunctions could be given. Second, while comparing spectra at odd $L$ , exact diagonalization of the standard repulsive Hubbard chain revealed level crossings among its lowest states in different particle-number sectors (subspaces of fixed electron numbers $(N_\uparrow,N_\downarrow)$ for the two spin species) — for $L=5$ between the ground states of sectors $(3,2)$ and $(3,3)$ , for $L=7$ between those of $(3,4)$ and $(3,3)$ — a pattern that seems to depend on the parity of $(L-1)/2$ and that existing finite-size analyses, concentrated on even $L$ , do not explain.

Problem Statement

The problem is to complete the parity-resolved exact theory of the one-dimensional charge-pair Hubbard model and of its standard Hubbard-model counterpart at the two places where the 2020 Bethe-ansatz solution stopped — the eigenfunctions and the odd-lattice-size low-lying spectra. It comprises two linked deliverables:

(1) Constructive eigenfunction correspondence. Give an explicit, constructive relation between the eigenfunctions of the charge-pair Hamiltonian $H_c=\sum_{j,\alpha}\left[c_\alpha(j)c_\alpha(j+1)+c^\dagger_\alpha(j+1)c^\dagger_\alpha(j)\right]+U\sum_j\left(n_\uparrow(j)-\tfrac{1}{2}\right)\left(n_\downarrow(j)-\tfrac{1}{2}\right)$ and those of the equivalent coupled-spin-chain representation (two coupled XY-type chains with Ising coupling $\frac{U}{4}\sum_j\sigma^z_j\tau^z_j$ and lattice-parity-dependent boundary terms), for odd lattice sizes $L$ , where equality of the two energy spectra is expected from the matching Bethe equations and known numerically (exact diagonalization for $L=3,5,7$ ) but no eigenvector relation was found. The relation should be valid for arbitrary odd $L$ rather than case by case and should cover the factorized reference states (the simple product states on which the Bethe ansatz constructs the eigenvectors). It may be given, for example, as an explicit unitary or intertwining operator on the many-body Hilbert space — an operator that maps every eigenstate of one Hamiltonian to an eigenstate of the other with the same energy — or as an explicit state-by-state correspondence between the two sets of Bethe wave functions, matching Bethe quantum numbers and amplitudes sector by sector.

(2) Origin and pattern of odd-size Hubbard level crossings. For the repulsive one-dimensional Hubbard model on an odd number $L$ of sites, determine which lowest states in which particle-number sectors $(N_\uparrow,N_\downarrow)$ (subspaces of fixed numbers of up- and down-spin electrons) cross as the on-site coupling $U$ increases, explain why the crossing pattern depends on the parity of $(L-1)/2$ (observed: for $L=5$ the crossing is between the ground states of sectors $(3,2)$ and $(3,3)$ ; for $L=7$ between those of $(3,4)$ and $(3,3)$ ), and establish the large- $L$ behavior of these low-lying states. An acceptable answer derives the crossing pattern from the Bethe-ansatz solution or controlled numerics and yields verifiable predictions — the crossing sectors and couplings for further odd $L$ , and the scaling of the near-degenerate low-lying energies as $L\to\infty$ .

The two deliverables are joint components of one objective: lifting the exact solution from the level of energies to wavefunctions and to the odd-parity sector structure of the two models, where the second question was uncovered precisely by the odd- $L$ spectral comparison that the charge-pair model's solution invites. The broader direction of extending exact results to the charge-pair model in spatial dimensions $\geq 2$ without a bipartite-lattice assumption is outside this problem's completion boundary.

References

M. J. Martins, 'Integrability of spin-1/2 fermions with charge pairing and Hubbard interaction', Nuclear Physics B 958 (2020) 115109. DOI: 10.1016/j.nuclphysb.2020.115109; arXiv:2006.01730, https://arxiv.org/abs/2006.01730
E. H. Lieb and F. Y. Wu, 'Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension', Physical Review Letters 20 (1968) 1445-1448. DOI: 10.1103/PhysRevLett.20.1445, https://doi.org/10.1103/PhysRevLett.20.1445
B. S. Shastry, 'Exact Integrability of the One-Dimensional Hubbard Model', Physical Review Letters 56 (1986) 2453-2455. DOI: 10.1103/PhysRevLett.56.2453, https://doi.org/10.1103/PhysRevLett.56.2453
B. S. Shastry, 'Decorated star-triangle relations and exact integrability of the one-dimensional Hubbard model', Journal of Statistical Physics 50 (1988) 57-79. DOI: 10.1007/BF01022987, https://doi.org/10.1007/BF01022987
F. C. Alcaraz, M. N. Barber and M. T. Batchelor, 'Conformal invariance, the XXZ chain and the operator content of two-dimensional critical systems', Annals of Physics 182 (1988) 280-343. DOI: 10.1016/0003-4916(88)90015-2, https://doi.org/10.1016/0003-4916(88)90015-2
E. H. Lieb, 'Two theorems on the Hubbard model', Physical Review Letters 62 (1989) 1201-1204. DOI: 10.1103/PhysRevLett.62.1201, https://doi.org/10.1103/PhysRevLett.62.1201
F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper and V. E. Korepin, 'The One-Dimensional Hubbard Model', Cambridge University Press (2005). ISBN 978-0-521-80262-8; DOI: 10.1017/CBO9780511534843, https://doi.org/10.1017/CBO9780511534843