Testing odd-ring XXZ polynomials independently¶

XXZ family · Supported models · Odd-ring continuation · Polynomial equations

The internal check_odd_wronskian<Real> in xxz_wronskian.hpp supplies an independent coefficient test for odd periodic rings at \(-1\lt \Delta \le 0\). It does not change the public solver domain. Its consistent status is a numerical statement, not an unconditional physical-state or ground-state certificate.

Why another equation?¶

Bajnok et al., section 3 relate XXZ admissibility to a second solution of the Baxter equation, using a quantum Wronskian. Their construction assumes generic \(q =\exp (i \,\gamma)\), not a root of unity. For odd N the second solution is a Laurent polynomial; even N can require an additional logarithmic term. Their finite, distinct-root hypotheses and singular-pair condition matter: solving a cleared Bethe equation alone is not enough.

The identity to check is

\[ \begin{aligned} (t-t^{-1})^N&=P(tq^{1/2})Q(tq^{-1/2})-P(tq^{-1/2})Q(tq^{1/2}),\\{} q&=e^{i\gamma},\qquad \Delta=\cos\gamma,\qquad t=e^\lambda. \end{aligned} \]

The code restricts N to odd values. It does not try to detect whether a floating-point coupling is mathematically a root of unity, and never treats numerical consistency as proof that the theorem's hypotheses hold.

From the solver's coefficients to the Wronskian¶

Our scaled rapidity is \(z =a \,\tanh (\lambda)\), where \(a =\sqrt{(1+\Delta)/(1-\Delta)}\). Thus

\[ X=t^2=\frac{a+z}{a-z},\qquad z=a\frac{X-1}{X+1}. \]

For a monic input polynomial F(x) with \(z =o +s \,x\), homogenized Horner evaluation constructs

\[ \begin{aligned} G(X)&=[s(X+1)]^M F\!\left(\frac{(a-o)X-(a+o)}{s(X+1)}\right),\\{} Q(t)&=t^{-M}G(t^2). \end{aligned} \]

This transformation avoids extracting colliding roots and does not divide by the endpoint values \(F ((+-a -o)/s)\). A common positive scale in the Horner factors limits intermediate growth. The largest absolute coefficient of G is normalized to one. The returned q[j] multiply \(t ^{2\,j -M }\).

Write the unknown second solution as

\[ P(t)=\sum_{j=0}^{N-M}\frac{p[j]}{2i}\,t^{2j-(N-M)}. \]

The Wronskian becomes a real rectangular linear system. A pair of indices j,l contributes to row \(k =j +l\) with coefficient

\[ q[l]\sin\!\left(\frac{(2j-2l-N+2M)\gamma}{2}\right). \]

There are N+1 equations and N-M+1 unknowns. Complete pivoting selects a square subset; Uni20 solves the original selected equations, together with an identity right-hand side to estimate the inverse infinity norm. Every one of the N+1 equations is then checked. This is not a least-squares fit, and no normal equations or lower-precision conversions are used.

The binomial right-hand side is divided by its central coefficient, which is generated by ratios without forming a large binomial integer. The returned \(p\) includes that normalization. residual_norm is the maximum absolute coefficient discrepancy against this unit-normalized RHS. It is not divided by a possibly enormous P; large cancelling terms must not make an inaccurate identity appear more accurate.

The statuses distinguish consistent, inconsistent, ill_conditioned, and nonfinite. A pivot or reciprocal condition estimate at/below \(64\,\epsilon\) is inconclusive. Invalid arguments throw. A caller supplies the coefficient-residual tolerance explicitly when the default \(256\,\epsilon\) is insufficient; the checker never loosens it automatically. Continuation and Wronskian residuals have different conditioning and units. In particular, a converged continuation can fail a tight Wronskian check because the coordinate transform amplified coefficient errors.

The phantom-root exception has a nonzero eigenvector¶

Popkov, Zhang and Klümper, equations (4), (5), (11), and (12), construct periodic spin helices and their fixed-Sz projections. Their Hamiltonian uses Pauli matrices and subtracts the polarized energy. In our spin-1/2 convention their zero energy is \(N \,\Delta /4\).

For our odd-ring collision, choose

\[ \mathrm{pitch}=\pi+\frac\pi N,\qquad \Delta_c=\cos(\mathrm{pitch})=-\cos(\pi/N). \]

The pitch closes around the ring because N*pitch is an integer multiple of \(2\,\pi\). In the sector with M down spins, the projected state has amplitude

\[ \psi(j_1,\ldots,j_M)=\frac{e^{i\,\mathrm{pitch}\,(j_1+\cdots+j_M)}}{\sqrt{\binom NM}}. \]

on each ordered set of occupied sites. Every amplitude has equal nonzero magnitude. Local divergence terms telescope on the ring, giving an eigenvector with energy \(N \,\Delta_{c} /4\). The opposite pitch gives its reflected partner.

The spin-helix API and polynomial diagnostic now construct these states explicitly and recognize the continued all-phantom polynomial separately from this Wronskian test.

Our collision polynomial has all z roots at \(-a =-\tan (\pi /(2\,N))\), so its Laurent transform is proportional to \(t ^M\): an endpoint coefficient vanishes. These are infinite conventional rapidities, outside the finite-root Wronskian test. Failure of that test here must not be interpreted as an unphysical state. The returned endpoint_ratio records the smaller endpoint coefficient relative to the largest coefficient, without imposing a universal cutoff. Nor does existence of the helix eigenvector alone prove sector or global minimality.

Regression coverage and remaining work¶

Native fp64, long-double, and fp128 tests cover:

Continued N=3,5,7,9 folded sectors at three generic negative couplings.
N=13,17,21, \(\lvert S^z \rvert =1/2\), at Delta=-0.999 with the default Wronskian tolerance.
The analytic five-site two-magnon polynomial, affine-coordinate invariance, and direct evaluation of the Laurent identity at complex t.
Rejection of the singular pair \(z =+-i\) on odd rings. Its cleared TQ equation is polynomial, but its physicality condition would require \(1=(-1)^N\).
A perturbed polynomial whose exact momentum constraint is preserved but whose Wronskian is inconsistent.
Vanishing endpoint coefficients at the phantom collision, and direct spin-basis Hamiltonian action on both projected helices in every sector through N=9, with no double-precision ED oracle for those eigenvectors.
The physical vacuum at Delta=-1/2: \(q ^{3}=1\) makes the Laurent ansatz inconclusive even without occupied roots. Root-of-unity exceptions are not limited to the ground-branch collision.

A GCC 13 probe of the same larger branches at Delta=-0.97 illustrates the conditioning issue. For N=21 the Wronskian residual is approximately 7.75e-13 (fp64), 4.11e-15 (long double), and 7.86e-30 (fp128). All are small in absolute terms, but exceed the default \(256\,\epsilon\) in their own types; the checker reports inconsistent at that tolerance. The reciprocal condition estimate is about 9.79e-6 and the endpoint ratio about 1.05e-9. These are numerical observations, not universal error bounds or grounds for labelling the continued state unphysical. At Delta=-0.5 the same probe returns ill_conditioned, consistent with the root-of-unity restriction, rather than silently accepting a failed factorization.

The helper is not yet a continuation acceptance gate. Larger-chain error amplification, other root-of-unity configurations, finite repeated roots, and a precise admissibility policy still need work. Even an admissible eigenstate can be an excited state. The internal all-sector scan now compares every followed branch and refuses to select an incomplete minimum. Physical-state acceptance and reliable sector-minimum tracking remain requirements for public odd-ring negative-Delta support. The regular-state test now provides a complementary Gaudin-based criterion that does not require generic q, but still excludes infinite rapidities, collisions, and exact strings.