Negative XXZ anisotropy: ground-root engine and remaining work¶

XXZ family · Supported models · Periodic XXZ · Open XXZ

The free-end ground-state API and frontend now support \(-1\lt \Delta \lt 0\), for either parity of N and every physical Sz sector. The periodic API and frontend support the same interval on even rings. The bethe::xxz::detail::negative_ground_roots engine in xxz_negative.hpp solves ground-state sectors for \(-1\lt \Delta \lt 0\) on even periodic rings and free-end chains of either parity. It supports fp64, long-double, optional fp128 and explicit MPFR through the finite XXZ interfaces. Negative-Delta odd rings are still not public. Their remaining physical-state and lowest-state selection work is deferred until a concrete need arises; see the restart checklist. The internal implementation and its tests are retained. Excitations and \(\Delta \le -1\) require separate work.

The Hamiltonian is unchanged:

\[ \begin{aligned} H&=\sum_{\mathrm{bonds}}\left(S_i^xS_j^x+S_i^yS_j^y+\Delta S_i^zS_j^z\right), \qquad S=\frac12,\\{} M&=\frac N2-|S^z|. \end{aligned} \]

Sources and geometry¶

The periodic equations follow Kozlowski, Eqs. (0.4), (0.7), with the Pauli Hamiltonian divided by four. That paper explicitly assumes even length; its ground-state identification cannot simply be applied to odd rings. The open equations follow the reflection equations of Mei and Bolech, Eqs. (11)-(12), with the normalization already derived in the open guide. The rank subtraction and residual scaling below are our algebraic/numerical implementation of these equations.

Why subtract the phase ranks?¶

For negative Delta the ground roots approach a shrinking interval as Delta approaches -1. A small residual in the unscaled equations can then mean only that the interval is small, rather than that the roots are accurate. Define

\[ s=\sqrt{\frac{1+\Delta}{1-\Delta}},\qquad b=\sqrt{1+\Delta}\sqrt{1-\Delta},\qquad a=-\Delta,\qquad z_i=s\tanh\lambda_i. \]

The engine works with finite ordered lambda, positive for free ends. On this branch the scattering denominators are positive. For distinct roots define the complementary half-scattering phase

\[ \begin{aligned} p(x)&=\frac{\arctan(s\tanh x)}s,\\{} h(x)&=\frac{\mathrm{sgn}(x)}s\mathrm{atan2}\!\left(b,a|\tanh x|\right). \end{aligned} \]

The original half-scattering phase is -sign(x)*pi/2 + s*h(x). Its rank-dependent constant cancels the consecutive ground labels analytically. For PBC those labels are \(I_{i} =i -(M -1)/2\), with zero-based i. For OBC they are \(I_{i} =i +1\); the positive reflected roots and the boundary phase contribute the remaining constants. This yields the following equations, all divided by N and s relative to the original logarithmic equations:

\[ \begin{aligned} R_i^{\mathrm{PBC}}&=\frac1N\left[Np(\lambda_i)-\sum_{j\ne i}h(\lambda_i-\lambda_j)\right],\\{} R_i^{\mathrm{OBC}}&=\frac1N\left[2Np(\lambda_i)-\frac2s\arctan\!\left(\frac s{\tanh\lambda_i}\right) -\sum_{j\ne i}\{h(\lambda_i-\lambda_j)+h(\lambda_i+\lambda_j)\}\right]. \end{aligned} \]

Self-scattering, including the reflected self, is excluded. These formulas apply to the ordered ground branch, not arbitrary excitation labels. The analytic Jacobian uses

\[ h'(x)=-\frac{(1-\Delta)a\,\mathrm{sech}^2x}{b^2+a^2\tanh^2x}. \]

The periodic diagonal is \(N \,\mathrm{sech} (\lambda_{i})^{2}/(1+z_{i} ^{2})\) minus the sum of direct h' terms. The open diagonal doubles that driving term, adds \(2\,\mathrm{sech} (\lambda_{i})^{2}/(\tanh (\lambda_{i})^{2}+s ^{2})\), and subtracts direct and reflected h' terms. Off-diagonal entries are direct h', minus reflected h' for OBC. Every Jacobian entry is divided by N, matching R.

The energy is evaluated without subtracting \(1-\tanh (\lambda)^{2}\):

\[ E=\frac{\mathrm{bonds}\,\Delta}{4} -\sum_i\frac{(1+\Delta)\mathrm{sech}^2\lambda_i}{1+z_i^2}, \qquad \mathrm{bonds}=\begin{cases}N,&\mathrm{PBC},\\N-1,&\mathrm{OBC}.\end{cases} \]

Numerical contract¶

The initial roots are the analytic free-fermion roots at Delta=0, expressed in lambda. Newton iteration uses a dense Uni20 solve and a residual-decreasing line search that preserves root order and the physical domain. Each accepted step consumes one iteration. There is no hidden continuation budget.

The result retains both z (rapidities) and lambda (log_rapidities), labels, energy, accepted iteration count, and status. residual_norm is \(\max \lvert R_{i} \rvert\) in the scaled equations above, evaluated at the returned roots. It is not interchangeable with the old nonnegative solver's \(\max \lvert F_{i} \rvert /N\) diagnostic. Always check converged; an exhausted budget or stalled line search returns the last iterate, not a claimed solution. Odd periodic input is rejected.

The public free-end result retains both coordinate arrays and carries GroundResidualConvention::negative_rank_scaled, including on failure and for the polarized vacuum. Its root_delta equals the requested Delta; this engine uses a direct solve, not coupling continuation. The CLI reports the residual convention and appends lambda to the negative-Delta root table. Its default ground-state sector remains Sz=0 (even N) or Sz=1/2 (odd N); the odd-periodic caveat below does not apply to an open chain.

The periodic result is now a distinct xxz::GroundState<Real>, with the same two coordinate arrays and its own residual/status tags; real excitation states remain xxz::RealState<Real>. There is no implicit conversion between them. On an even ring, the symmetric labels give momentum zero for even M and pi for odd M. All periodic negative-Delta entry points consistently reject odd N, including polarized sectors and scans, pending the odd-ring work.

Tests independently check all sectors through N=9 for OBC and even PBC at four negative couplings, using spin-basis exact diagonalization. Periodic momentum is checked against a joint energy/translation spectrum. Additional tests cover the original unscaled equations, finite-difference Jacobians, spin reversal, analytic N=2 and N=3 energies, invalid input, and iteration budgets. Native-precision checks reach \(\Delta =-1+128\,\epsilon\) through N=64, and ensure that the zero-step seed does not falsely pass the scaled residual.

An independent first-order check uses the staggered rotation to the isotropic ferromagnet at Delta=-1. In its symmetric fixed-Sz state, \(\lt S^z_{i} \,S^z_{j} \ge [(N -2M)^{2}-N]/[4\,N \,(N -1)]\) for distinct sites, fixing the energy slope as Delta increases. This distinguishes the sector minimum from merely returning the polarized limiting energy.

The initial internal-engine checkpoint passed all 393 tests in the GCC 13 Debug/fp128 build and all 272 in the Clang 20 Release build without MPLAPACK. Sixteen and eleven of those cases, respectively, belong to this new engine and odd-ring audit.

Free-end public integration adds typed tests of every returned sector, global selection, native-precision energies near both Delta=-1 and Delta=0, residual reconstruction, exhausted budgets, and the separation from excitation scans. The free-end checkpoint passed 445 GCC/fp128 cases and 307 Clang cases. The CLI checks also pass in an applications-only build using the published Uni20 pin, without a sibling checkout or GoogleTest.

The even-periodic integration checkpoint passes 457 GCC 13 Debug/fp128 tests and 315 Clang 20 Release tests, including its additional typed API/translation checks and full-precision CLI cases. The periodic CLI also passes against the pinned Uni20 build. Shared XXZ definitions now live below the solvers in xxz_common.hpp, allowing both real and polynomial engines to remain independent of the public ground-state dispatcher.

Why odd rings are not public¶

An odd ring is not bipartite. Two issues need explicit handling:

The real-root continuation can encounter colliding roots and require complex conjugate pairs. A real-coordinate range change is insufficient; regularized complex-root variables or a polynomial formulation are needed. The polynomial building block now supplies coefficient equations, analytic derivatives, and observables. An internal adaptive odd-ring driver now follows these coefficients with momentum, conditioning, and energy-concavity checks; selected sector energies agree with independent spin-basis calculations through 21 sites. An independent Wronskian diagnostic and explicit phantom-point eigenvectors now support the admissibility work, but do not yet provide a general acceptance policy. It is not yet a production odd-ring solver.
The global minimum need not lie in the smallest-|Sz| sector. An independent ED regression for N=5, Delta=-0.9 finds the fully polarized energy -1.125 below the |Sz|=1/2 sector minimum (approximately -0.9898034892). The internal all-sector candidate scan now compares every folded branch and does not select an incomplete minimum. Its result is separate from physical-state validation.

The second observation alone rules out routing all negative couplings through the existing ground_state wrapper's sector choice. The internal work is retained for a future implementation, but no complete odd-ring phase diagram is claimed by the small-system audit.

Deferred odd-ring work: restart checklist¶

Decision: leave negative-Delta odd periodic ground states unsupported in the public API and CLI, and revisit only when those results are needed. This does not defer negative-Delta even rings or free ends, which are already supported. It does not remove the separate explicit spin-helix API. Other models can proceed independently; there is no commitment to an experimental odd-ring frontend or to completing every singular-state family first.

Preserved checkpoint¶

Through commit d7fc028, the internal implementation has polynomial continuation, all-sector candidate comparison, regular-state and Wronskian diagnostics, explicit helix states, mixed-phantom dressing and bounded nonzero-amplitude checks, native-precision root recovery, and independent free-sea/helix variational upper bounds. Follow the continuation guide for equations, test coverage, counterexamples, and links to the individual modules. Existing regression tests remain part of the normal suite; deferral is not permission to break these components.

Issues to resolve before public ground-state support¶

Lowest-state selection within a sector. The driver follows one free-sea branch at fixed momentum. A physical, converged eigenstate need not be the sector minimum. Continuity, concavity, and variational bounds reject some wrong branches but do not establish that no lower branch or momentum sector was missed. The documented N=17 branch jump is a concrete regression, not evidence of a complete classification.
A single physical-state acceptance policy. Regularity, helix matching, and mixed-phantom witnesses cover different cases. Generic-q Wronskian assumptions cannot be applied indiscriminately at roots of unity; repeated roots, exact strings, and other singular limits can remain unresolved. Witness budgets and conditioning limits must remain visible, and an unresolved check must not be interpreted as a zero or absent state.
Recoverable numerical failures and practical costs. The continuation and Wronskian code still call Uni20's ordinary dense solve, whose default singular-matrix policy can abort. An expected singularity needs a safe failure path before public exposure, without changing process-global policy. Establish a useful size/precision envelope: polynomial conditioning and exponential subset work in the optional wavefunction checks matter.
Ground-state versus candidate API semantics. Keep polynomial coefficients and affine coordinates authoritative; complex-root recovery is optional and may be unresolved. Extend result/status and residual tags rather than squeezing this representation into real-root arrays. Compare every required magnetization sector for a global ground state, retaining failed or unresolved sectors instead of selecting from an incomplete set. The current internal lowest_index compares converged branch energies; it is not by itself a public ground-state acceptance flag.

Where to resume¶

First specify the actual requested lengths, magnetization sectors, couplings, and accuracy. Then audit branch selection over that domain against independent spin-basis spectra: dense coupling sweeps on small rings, selected larger sparse calculations, momentum competitors, and neighborhoods of known collisions. Use the existing tests/reference_xxz_odd_ed.py and typed tests as the starting point. Cross-check precision and continuation step sensitivity. This investigation should determine whether the present branch is adequate or whether competing branches or a more systematic Q-system treatment are needed; adding another necessary acceptance inequality alone does not settle it.

A future numerical release need not wait for a new general completeness theorem. It does need a defensible state-selection method, discriminating validation over its stated scope, and honest unresolved/failure behaviour. Only after that decision should API/CLI integration and any experimental opt-in be designed. Negative-Delta excitation scans remain separate work.