A regular-state test without extracting Bethe roots¶

XXZ family · Supported models · Odd-ring continuation and scans · Wronskian diagnostic

The internal check_regular_polynomial<Real> in xxz_regularity.hpp tests a sufficient regular-case nonzero-vector criterion using the solver's polynomial coefficients. It supplements the quantum Wronskian and does not require its generic-anisotropy assumption. Singular limiting states are a different problem; they are not declared unphysical when this test cannot resolve them.

The norm formula and our deduction¶

Korepin's historical account, section IV, equation (23), expresses the scalar product of an on-shell XXZ Bethe vector and its algebraic dual as a product of one-particle and scattering factors times the determinant of the logarithmic Bethe-equation Jacobian. The roots are taken distinct. A nonzero scalar product implies a nonzero ket even without identifying the algebraic dual with its Hermitian conjugate.

Our deduction is that it is enough to verify, on an exact solution:

Conventional rapidities are finite and distinct modulo their period.
The one-particle factors and both pair-scattering factors are nonzero.
The full logarithmic Bethe Jacobian is nonsingular.

Then the regular algebraic Bethe construction produces a nonzero eigenvector. This is a state-existence test, not a lowest-energy theorem. Its numerical implementation is not an interval-arithmetic certificate: finite residuals and rank indicators only test numerical versions of the hypotheses, at the caller's precision and requested residual tolerance.

Expressing those conditions in polynomial coordinates¶

The input is the monic polynomial F(x), with physical rapidity \(z =o +s \,x\). Let \(a =\sqrt{(1+\Delta)/(1-\Delta)}\). The conventional multiplicative rapidity is

\[ e^{2\lambda}=\frac{a+z}{a-z}. \]

Roots at z=+-a therefore require infinite-rapidity limits. The one-particle singularities \(\lambda =+-i \,\gamma /2\), where \(\Delta =\cos (\gamma)\), map to z=+-i. Homogeneous Horner evaluation tests F at these points without forming large coordinate ratios. Each *_margin for a value is its magnitude divided by the coefficient-magnitude bound evaluated at max(1,abs(x)). The unit-radius floor also detects small endpoint values without cancellation, such as \(F (x)=x ^M\) near x=0; a cancellation-only ratio would misleadingly return one. Real coefficients make the two driving values at +-i conjugates, so only one needs evaluation.

The remaining root conditions are checked in the quotient algebra C[x]/F:

Multiplication by F' is invertible exactly when F has distinct roots.
Multiplication by the self-scattering-removed K_- is invertible exactly when it has no common root with F. At a root x_i,

\[ K_-(x_i)=\prod_{j\ne i} \left[1+\Delta-(1-\Delta)z_iz_j-i\Delta(z_i-z_j)\right]. \]

Thus the second matrix detects vanishing pair-scattering factors. The opposite-sign factor is covered by conjugation for a real polynomial. PolynomialBetheSystem::scattering_remainder exposes K_- modulo F before the driving factor is multiplied in; it uses the same divided-Horner identity that removes self-scattering without a numerical division.

For the Jacobian test, use all M coefficient equations, not the reduced M-1 equations and imposed momentum used in the continuation driver. At an exact solution, evaluating their first variations at the roots is a Vandermonde change of basis. The root-to-coefficient map is nonsingular for distinct roots. Dividing the rootwise equations by their nonzero driving and scattering factors, and changing from x to lambda, then gives the logarithmic Bethe Jacobian. All these changes are invertible under the preceding conditions. Therefore nonsingularity of the full coefficient Jacobian is equivalent to the determinant condition needed here.

This deduction deliberately excludes collisions, infinite rapidities, singular driving roots, and exact strings; a limiting construction is required when a prefactor or change of variables vanishes. A root-of-unity anisotropy by itself is not a reason to reject a regular finite-root state. This differs from applying the generic-q Wronskian theorem outside its stated hypotheses.

Numerical rank, not a terminal linear solve¶

All arithmetic stays in fp64, native long double, or fp128. Complex multiplication matrices use the real block representation

\[ \begin{pmatrix} \mathrm{Re}A&-\mathrm{Im}A\\{} \mathrm{Im}A&\mathrm{Re}A \end{pmatrix}. \]

After normalizing the largest matrix entry to one, complete-pivot elimination returns

\[ \frac{\mathrm{smallest\_absolute\_pivot}} {\mathrm{matrix\_order}\,\max(1,\mathrm{element\_growth})}. \]

This is a rank-resolution indicator, not a reciprocal condition number or a rigorous error bound. Zero/nonfinite pivots or arithmetic give an unresolved result. Every value/rank margin must exceed \(64\,\epsilon\), and the full polynomial residual must satisfy the caller's tolerance (default \(32\,\epsilon\)). The routine does not loosen tolerances to obtain a pass.

An exact singular matrix is an expected input here. Uni20's ordinary dense solve uses a terminal singular-matrix error policy, which aborts under its default configuration. This rank calculation does not invoke that solve or mutate Uni20's process-wide error policy. The singular-matrix regressions run under the unmodified default policy in all supported scalar types.

The result distinguishes regular_on_shell, off_shell, exceptional_or_unresolved, ill_conditioned, and nonfinite. Invalid arguments throw. A small endpoint or resultant margin is not proof of an exact singularity: numerical resolution can also be inadequate. Uncomputed rank margins stay zero and an unavailable residual stays infinite.

Use in the sector scan and tests¶

An optional final rotation argument applies the same criterion to the constant-twist finite-root equations. It defaults to one; the sector scan still uses the periodic, zero-twist check. This extension tests the finite reduced state, not its singular phantom dressing.

Each equation-converged entry in the internal all-sector scan now carries an optional regularity result alongside its Wronskian diagnostic. regular_states_complete says whether every entry passed this regular criterion. Neither diagnostic changes the candidate minimum or substitutes for sector-minimum tracking. Failed continuations have neither diagnostic.

Tests cover all folded sectors through N=9 at five couplings; selected N=13,17,21 branches at Delta=-0.97 and -0.999; and explicit repeated, infinite, singular-driving, and exact-string roots. Root-of-unity tests separate regular vacua/one-magnon states from infinite-root states.

An independent native-precision five-site check extracts the two roots from an analytic quadratic only in the test, constructs the coordinate Bethe wavefunction, verifies it is nonzero, and applies the spin Hamiltonian directly. This covers both real and conjugate-pair roots. A separate affine coordinate check compares K_- against the explicit pair factor above.

An explicit spin-helix construction now handles the all-phantom collision separately. Public odd-ring integration still requires a policy for the other exceptional states this regular criterion excludes, together with sector-minimum tracking. A singular regular-root norm formula alone is never a reason to discard a state.