Shastry's covering vertex model makes the one-dimensional Hubbard model a Yang-Baxter integrable lattice model, but its R-matrix is not of difference form: it depends on two independent spectral variables, and its weights lie on an Abelian surface birational to a product of two non-isomorphic elliptic curves. Consequently the inversion-relation method, which for difference-form models converts the local crossing and unitarity properties of the R-matrix into closed functional equations for transfer-matrix eigenvalues and, with analyticity assumptions, into the free energy and excitation dispersion, has never been available for this paradigmatic model. Martins (2016) uniformized the Lax-operator weights on their genus-one spectral curve by ratios of theta functions and derived the corresponding local crossing and unitarity-type relations, explicitly posing as an open problem their promotion to global functional relations: finite-size exact inversion identities for the transfer matrix, the analyticity assumptions still to be made, and a Bethe-ansatz-independent route to the spectrum and thermodynamics.

Background

The one-dimensional Hubbard model, electrons on a chain with nearest-neighbor hopping and on-site repulsion U\mathrm{U}, is the paradigmatic exactly solved correlated-electron system: Lieb and Wu diagonalized it by a coordinate Bethe ansatz in 1968. Its integrability also has a second, algebraic face constructed by Shastry. The Hubbard Hamiltonian commutes with the row-to-row transfer matrix (the trace of a product of local Boltzmann-weight matrices across a lattice row) of a solvable two-dimensional covering vertex model whose weights are encoded in a 16×1616\times16 R-matrix solving the Yang-Baxter equation, the cubic consistency condition underlying exactly solvable vertex models.

In most exactly solved vertex models the R-matrix depends only on the difference of two spectral parameters, and Baxter's inversion-relation method then works: two local properties of the R-matrix, crossing (a relation between the R-matrix and its transpose at a shifted argument) and unitarity (the product of the R-matrix with its inverse-space counterpart being proportional to the identity), are promoted into global functional equations, or inversion identities, for the transfer-matrix eigenvalues; with suitable analyticity assumptions on the eigenvalues these identities determine the spectrum, the bulk free energy and the dispersion of low-lying excitations without any Bethe ansatz. Shastry's R-matrix, however, is not of difference form: R(ω1,ω2)R(\omega_1,\omega_2) depends on two independent spectral variables, one attached to each of the two interpenetrating free-fermion sublattices of the covering model, and this is the structural reason the inversion-relation machinery has never been applied to it.

Martins (2016) analyzed this obstruction geometrically. The Lax operator (the local Yang-Baxter building block acting on one site and an auxiliary space) of the covering model lives on a genus-one, i.e. elliptic, spectral curve

E‾2:(x2+y2)2−U x y c2−c4=0,\overline{\mathrm{E}}_2:\quad (x^2+y^2)^2-\mathrm{U}\,x\,y\,c^2-c^4=0,

which he uniformized, i.e. parametrized by a single complex spectral parameter λ\lambda, through ratios of Jacobi theta functions (entire quasi-periodic functions standard in elliptic-function theory) with modulus k=U/(4i)k=\mathrm{U}/(4\mathrm{i}), so that the two independent local weights become the theta-function ratios x(λ)/c(λ)x(\lambda)/c(\lambda) and y(λ)/c(λ)y(\lambda)/c(\lambda). This yielded relativistic-type local relations for the Lax operator L12(λ)\mathrm{L}_{12}(\lambda): a crossing relation L12(λ)=M L12(K(k)−λ)t2 M−1\mathrm{L}_{12}(\lambda)=\mathrm{M}\,\mathrm{L}_{12}(\mathbf{K}(k)-\lambda)^{t_2}\,\mathrm{M}^{-1}, where K(k)\mathbf{K}(k) is the complete elliptic integral of the first kind, t2t_2 denotes transposition on the second tensor factor and M\mathrm{M} is a charge-conjugation matrix; and a unitarity-type relation L12(λ) L12(−λ)=[x(λ)/c(λ)]2[x(−λ)/c(−λ)]2 I\mathrm{L}_{12}(\lambda)\,\mathrm{L}_{12}(-\lambda)=[x(\lambda)/c(\lambda)]^2[x(-\lambda)/c(-\lambda)]^2\,\mathrm{I}, notably without permutation of the two spaces, since the Lax operator is not parity-reversal invariant for generic U\mathrm{U}. He further showed that the R-matrix weights lie on an Abelian surface (a two-dimensional analogue of an elliptic curve) birational, i.e. related by rational maps, to the product of two non-isomorphic elliptic curves, one isomorphic to the Lax-operator curve and the other only fourfold isogenous to it, geometrically explaining the non-difference form.

The transfer-matrix spectrum itself is already known by other means: the algebraic Bethe ansatz, the standard eigenvalue construction for Yang-Baxter transfer matrices, was adapted to the Hubbard covering model by Martins and Ramos (1998), and the original Lieb-Wu equations solve the underlying quantum chain. What is missing, and explicitly acknowledged in the 2016 paper, is the functional-relation route: whether the local crossing and unitarity relations of the uniformized Lax operator can be promoted to global functional relations, in particular finite-size exact inversion identities, for the transfer-matrix eigenvalues, together with the analyticity assumptions on the theta-ratio weights under which such identities determine the spectrum and the thermodynamics. The paper names two relevant guidelines: Shastry's own earlier attempt through decorated star-triangle relations (1988) and the fusion procedure, a method of building composite R-matrices for bound states, as adapted to R-matrices without difference form by Beisert, de Leeuw and Nag (2015).

Problem Statement

Convert the local inversion relations of the theta-uniformized Lax operator of Shastry's covering vertex model of the one-dimensional Hubbard model into a functional-relation solution of its transfer-matrix spectrum. Concretely, establish the following linked results, which together realize the inversion-relation program for a non-difference-form R-matrix:

  1. Finite-size inversion identities: closed functional equations relating products of transfer-matrix eigenvalues at shifted arguments of the uniformizing spectral parameter λ\lambda (for example λ\lambda, −λ-\lambda and K(k)−λ\mathbf{K}(k)-\lambda), valid for arbitrary chain length and coupling U\mathrm{U}, derived from the crossing relation and the unitarity-type relation of the Lax operator, or from an equivalent algebraic construction adapted to the non-difference form such as the fusion procedure of Beisert-de Leeuw-Nag.

  2. Analyticity structure: the analyticity assumptions on the theta-function ratios x(λ)/c(λ)x(\lambda)/c(\lambda) and y(λ)/c(λ)y(\lambda)/c(\lambda), in particular their zero and pole distribution and quasi-periodicity domains, under which these identities determine the transfer-matrix spectrum.

  3. Thermodynamics: the derivation from the identities plus analyticity input of the thermodynamic-limit free energy and the dispersion relation of the low-lying excitations of the covering vertex model.

A solution must show that the identities and analyticity input together pin down the transfer-matrix spectrum independently of the Bethe ansatz and yield the thermodynamic-limit free energy and the dispersion relation of the low-lying excitations, and must be consistent with the free-fermion limit U→0\mathrm{U}\to 0, in which the weights degenerate to trigonometric free-fermion form, and with the known algebraic-Bethe-ansatz eigenvalues and the known thermodynamic Bethe ansatz results for the model. A rigorous refutation, that is, a proof that no consistent inversion-identity and analyticity structure of this type exists for the non-difference-form Hubbard transfer matrix, so that the spectrum and thermodynamics cannot be obtained by this route, would equally resolve the question.

References

  1. M. J. Martins, "Algebraic geometry methods associated to the one-dimensional Hubbard model", Nuclear Physics B 907 (2016) 479-494. DOI: 10.1016/j.nuclphysb.2016.04.019; arXiv:1602.00523. https://doi.org/10.1016/j.nuclphysb.2016.04.019
  2. 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
  3. B. Sriram 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
  4. B. Sriram 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
  5. R. J. Baxter, "The inversion relation method for some two-dimensional exactly solved models in lattice statistics", Journal of Statistical Physics 28 (1982) 1-41. DOI: 10.1007/BF01011621. https://doi.org/10.1007/BF01011621
  6. M. J. Martins and P. B. Ramos, "The quantum inverse scattering method for Hubbard-like models", Nuclear Physics B 522 (1998) 413-470. DOI: 10.1016/S0550-3213(98)00199-0; arXiv:solv-int/9712014. https://doi.org/10.1016/S0550-3213(98)00199-0
  7. N. Beisert, M. de Leeuw and P. Nag, "Fusion for the one-dimensional Hubbard model", Journal of Physics A: Mathematical and Theoretical 48 (2015) 324002. DOI: 10.1088/1751-8113/48/32/324002; arXiv:1503.04838. https://doi.org/10.1088/1751-8113/48/32/324002
  8. M. J. Martins, "Algebro-geometric approach for a centrally extended U_q[sl(2|2)] R-matrix", Nuclear Physics B 917 (2017) 193-208. DOI: 10.1016/j.nuclphysb.2017.01.030; arXiv:1610.01125. https://doi.org/10.1016/j.nuclphysb.2017.01.030
  9. M. J. Martins, "The spectrum of a vertex model and related spin one chain sitting in a genus five curve", Nuclear Physics B 924 (2017) 728-744. DOI: 10.1016/j.nuclphysb.2017.09.026; arXiv:1706.01525. https://doi.org/10.1016/j.nuclphysb.2017.09.026
  10. 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://doi.org/10.1016/j.nuclphysb.2020.115109
  11. J.-M. Maillet, G. Niccoli and L. Vignoli, "Separation of variables bases for integrable gl(M|N) and Hubbard models", SciPost Physics 9(4) (2020) 060. DOI: 10.21468/SciPostPhys.9.4.060; arXiv:1907.08124. https://doi.org/10.21468/SciPostPhys.9.4.060
  12. M. de Leeuw and V. Regelskis, "An algebraic approach to the Hubbard model", Physics Letters A 380 (2016) 645-653. DOI: 10.1016/j.physleta.2015.12.013; arXiv:1509.06205. https://doi.org/10.1016/j.physleta.2015.12.013
  13. M. de Leeuw, C. Paletta, A. Pribytok, A. L. Retore and P. Ryan, "Yang-Baxter and the Boost: splitting the difference", SciPost Physics 11(3) (2021) 069. DOI: 10.21468/SciPostPhys.11.3.069; arXiv:2010.11231. https://doi.org/10.21468/SciPostPhys.11.3.069
  14. A. Melikyan, "On integrability of the one-dimensional Hubbard model", Physics Letters B 847 (2023) 138291. DOI: 10.1016/j.physletb.2023.138291; arXiv:2310.12232. https://doi.org/10.1016/j.physletb.2023.138291