A three-state vertex model whose Boltzmann weights lie on a genus-five algebraic curve, based on an R-matrix not of difference form, generates an integrable periodic spin-one chain H(U)\mathrm{H}(\mathrm{U}) with a single coupling U\mathrm{U}, exactly solved by algebraic Bethe ansatz in 2017. For U≥23\mathrm{U}\ge 2\sqrt{3} the chain is gapped, all Bethe roots are real, and the thermodynamic limit was obtained analytically (root density from a singular integral equation, ground-state energy, and an exact lowest-gap expression supported by numerical evidence). For 0≤U<230\le\mathrm{U}<2\sqrt{3} the extrapolated excitation gap vanishes and the roots reorganize into real roots plus complex two-strings, but the numerical Bethe equations become ill-conditioned and no string hypothesis could be formulated; the negative-coupling half is accessible only through an unproven spectral correspondence. The open problem is to construct the analytic thermodynamic-limit description of this massless phase over the full regime −23≤U≤23-2\sqrt{3}\le\mathrm{U}\le 2\sqrt{3}: the string hypothesis, the root-density integral equations, the bulk ground-state energy and excitation dispersion, and the finite-size corrections that determine the central charge and scaling dimensions of the continuum limit.

Background

An integrable two-dimensional vertex model is built from an RR-matrix, a 9×99\times 9 matrix (for three edge states) whose entries are the Boltzmann weights attached to the four edges of a lattice vertex, and whose integrability follows from the Yang–Baxter equation — the associativity condition that makes the row-to-row transfer matrices T(λ)=Tr0[R01(λ,μ)⋯R0L(λ,μ)]\mathrm{T}(\lambda)=\mathrm{Tr}_0[\mathrm{R}_{01}(\lambda,\mu)\cdots\mathrm{R}_{0L}(\lambda,\mu)] commute for all spectral parameters λ\lambda. Almost all exactly solved vertex models use an RR-matrix of difference form, R(λ,μ)=R(λ−μ)\mathrm{R}(\lambda,\mu)=\mathrm{R}(\lambda-\mu). A rare exception, introduced by M. J. Martins, is a family of U(1)U(1)-invariant three-state RR-matrices that are non-additive (they depend on λ\lambda and μ\mu separately) and whose weights are constrained to lie on an algebraic curve of genus five — the set of allowed weight configurations is parametrized by the points of that curve, which admits degree-two maps onto elliptic (genus-one) curves. Expanding the transfer matrix around a regular point of this RR-matrix yields a commuting family of local Hamiltonians; the first non-trivial one, with periodic boundary conditions, is the spin-one chain (Nuclear Physics B 924 (2017) 728–744, Eq. 19)

H(U)=∑j=1L{−e−iπ/62Sj+Sj+1−−eiπ/62Sj−Sj+1+−i2(Sj+Sjz)Sj+1−+i2Sj−(Sj+1+Sj+1z)+U2(Sjz)2},\mathrm{H}(\mathrm{U})=\sum_{j=1}^{L}\left\{-\frac{e^{-i\pi/6}}{2}S_j^{+}S_{j+1}^{-}-\frac{e^{i\pi/6}}{2}S_j^{-}S_{j+1}^{+}-\frac{i}{2}\left(S_j^{+}S_j^{z}\right)S_{j+1}^{-}+\frac{i}{2}S_j^{-}\left(S_{j+1}^{+}S_{j+1}^{z}\right)+\frac{\mathrm{U}}{2}\left(S_j^{z}\right)^{2}\right\},

where S±S^{\pm} and SzS^{z} are the 3×33\times 3 spin-one generators. The chain is non-Hermitian, but its low-lying eigenvalues are real for generic U\mathrm{U}, and for even LL a rotation of the odd sites by π\pi about the zz-axis gives the exact unitary equivalence H(U)=−H(−U)\mathrm{H}(\mathrm{U})=-\mathrm{H}(-\mathrm{U}). Because the chain is U(1)U(1)-invariant, the Hilbert space splits into magnetization sectors labeled by nn, and the algebraic Bethe ansatz expresses every eigenvalue through Bethe roots kjk_j obeying

eikjL=∏i=1L−nsin⁡(kj−π/6) e−iπ/3−sin⁡(ki−π/6) eiπ/3+iU/2sin⁡(kj−π/6) eiπ/3−sin⁡(ki−π/6) e−iπ/3−iU/2,En(U)=−∑j=1L−n2cos⁡(kj+π/6)+nU2.e^{ik_j L}=\prod_{i=1}^{L-n}\frac{\sin(k_j-\pi/6)\,e^{-i\pi/3}-\sin(k_i-\pi/6)\,e^{i\pi/3}+i\mathrm{U}/2}{\sin(k_j-\pi/6)\,e^{i\pi/3}-\sin(k_i-\pi/6)\,e^{-i\pi/3}-i\mathrm{U}/2},\qquad E_n(\mathrm{U})=-\sum_{j=1}^{L-n}2\cos(k_j+\pi/6)+n\frac{\mathrm{U}}{2}.

The standard route from such equations to physics is the thermodynamic limit L→∞L\to\infty: one postulates a string hypothesis, i.e. that the Bethe roots organize into regular patterns of complex-conjugate pairs (strings) whose real centers become continuously distributed with densities σ(k)\sigma(k); taking logarithms of the Bethe equations and expanding differences of neighboring roots then yields linear integral equations for the densities, from which follow the ground-state energy per site, the dispersion of low-lying excitations, and — through conformal finite-size scaling, where the 1/L1/L correction to the ground-state energy is −πcv/(6L)-\pi c v/(6L) with vv the characteristic velocity — the central charge cc and scaling dimensions of the continuum (conformal field theory) limit.

For the present chain this program was completed only in the massive regime U≥23\mathrm{U}\ge 2\sqrt{3}: all roots are real, the density σ(k)\sigma(k) solves a singular integral equation, the ground-state energy per site follows, finite-size corrections converge exponentially, and an exact expression for the lowest energy gap was found with strong analytical and numerical support. Across the whole interval 0≤U<230\le\mathrm{U}<2\sqrt{3} the extrapolated gap is consistent with zero (massless phase), the roots coexist as real roots and two-strings, and — because the numerical solution of the Bethe equations becomes extremely sensitive to the initial guess — the authors state they could not formulate the string hypothesis for this regime. For U≤0\mathrm{U}\le 0 two types of two-strings appear already at small LL and no root-distribution proposal was possible either; the exact diagonalization data instead reveal the empirical correspondence E1(U)−E1(−U)=(U/2)LE_1(\mathrm{U})-E_1(-\mathrm{U})=(\mathrm{U}/2)L for the lowest level of the n=1n=1 sector (verified to machine precision up to L=12L=12), with an analogous ground-state relation holding up to small deviations that appear to vanish as LL grows, suggesting the two halves of the massless regime −23≤U≤23-2\sqrt{3}\le\mathrm{U}\le 2\sqrt{3} mirror each other. The boundary points Uc=±23\mathrm{U}_c=\pm 2\sqrt{3} are themselves special: there the genus-five curve factorizes into two cubic (genus-one) components, and the finite-size spectral-reality threshold U(L)\mathrm{U}(L) of the non-Hermitian Hamiltonian extrapolates to U(∞)≈23\mathrm{U}(\infty)\approx 2\sqrt{3}, a coincidence the source leaves unexplained. Within the classification of three-state U(1)U(1)-invariant Hamiltonians solvable by coordinate Bethe ansatz (Crampe–Frappat–Ragoucy), this Hamiltonian appears as the generalized Bariev Hamiltonian; that classification work derived spectra but no thermodynamic limits.

Problem Statement

Construct the analytic thermodynamic-limit description of the massless phase of the spin-one chain H(U)\mathrm{H}(\mathrm{U}) defined in the background, over the full regime −23≤U≤23-2\sqrt{3}\le\mathrm{U}\le 2\sqrt{3} in which the extrapolated excitation gap vanishes. Concretely, determine: (1) the string hypothesis for this regime — the classification of Bethe-root configurations (real roots and complex strings, with the relations fixing their centers and imaginary parts) that solve the Bethe equations stated in the background as L→∞L\to\infty, for the ground state and low-lying states, on both sides U>0\mathrm{U}>0 and U<0\mathrm{U}<0; (2) the linear integral equations for the thermodynamic root densities derived from the Bethe equations under that hypothesis; (3) the ground-state energy per site e0(U)e_0(\mathrm{U}) (equivalently the bulk free energy of the underlying genus-five vertex model) and the dispersion relation and characteristic velocities of the low-lying excitations, including whether the gap closes everywhere in the interval; and (4) the leading finite-size corrections to the energy levels, fixing the central charge and the scaling dimensions (the operator content of the continuum limit) as functions of U\mathrm{U}. The U>0\mathrm{U}>0 half may be treated directly; for the U<0\mathrm{U}<0 half the completion boundary is the same — components (1)–(4) must be delivered for U<0\mathrm{U}<0 as well, either by a parallel direct analysis or by proving the observed spectral correspondence E1(U)−E1(−U)=(U/2)LE_1(\mathrm{U})-E_1(-\mathrm{U})=(\mathrm{U}/2)L and extending it to the relevant low-lying levels. The correspondence is a route to the U<0\mathrm{U}<0 half, not a substitute for it, and a proof of the lowest-level energy identity alone does not complete that half. These deliverables are the standard, jointly necessary components of one objective — the exact thermodynamic characterization of the massless phase — each feeding the next: the string hypothesis is required to write the density equations, the densities give the bulk energy and dispersion, and the dispersion underlies the finite-size coefficients and scaling dimensions.

References

  1. 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://arxiv.org/abs/1706.01525.
  2. M. J. Martins, 'Integrable three-state vertex models with weights lying on genus five curves', Nuclear Physics B 874 (2013) 243–278, DOI 10.1016/j.nuclphysb.2013.05.014, arXiv:1303.4010, https://arxiv.org/abs/1303.4010
  3. N. Crampe, L. Frappat, E. Ragoucy, 'Classification of three-state Hamiltonians solvable by the coordinate Bethe ansatz', Journal of Physics A: Mathematical and Theoretical 46 (2013) 405001, DOI 10.1088/1751-8113/46/40/405001, arXiv:1306.6303, https://arxiv.org/abs/1306.6303
  4. T. Fonseca, L. Frappat, E. Ragoucy, 'R-matrices of three-state Hamiltonians solvable by coordinate Bethe ansatz', Journal of Mathematical Physics 56 (2015) 013503, DOI 10.1063/1.4905893, arXiv:1406.3197, https://arxiv.org/abs/1406.3197
  5. M. J. Martins, 'The spectrum properties of an integrable G2G_2 invariant vertex model', Nuclear Physics B 989 (2023) 116131, DOI 10.1016/j.nuclphysb.2023.116131, arXiv:2301.05963, https://arxiv.org/abs/2301.05963