Mixed finite and infinite XXZ roots¶

XXZ family · Supported models · Odd-ring continuation · Spin helices · Regular-state test

The internal reduce_phantom_polynomial in xxz_phantom.hpp removes a specified one-sided cluster of infinite rapidities and checks the remaining, twisted finite-root problem. This is a necessary-equation diagnostic, not a nonzero-state lifting test or a ground-state solver. Reduction alone does not make an unresolved entry in the sector scan acceptable; a separate numerical witness checks the dressed amplitude.

The extra condition left by infinite roots¶

Write \(\Delta =\cos (\gamma)\) with \(0\lt \gamma \lt \pi\). Suppose M roots consist of p phantom roots at the same infinity and r=M-p finite roots. Popkov, Zhang and Klümper, equation (15) and the following reduced equations, give a commensurability condition and a twisted finite-root problem. In our momentum and endpoint conventions:

\[ \begin{aligned} \chi&=\pm1,\qquad z_{\mathrm{phantom}}=\chi\sqrt{\frac{1+\Delta}{1-\Delta}},\\{} e^{ik_{\mathrm{phantom}}}&=e^{i\chi\gamma},\qquad e^{i(N-2r)\gamma}=1,\\{} e^{iNk_j}&=e^{-2i\chi p\gamma}\prod_{l\ne j}S(k_j,k_l). \end{aligned} \]

The integer p counts roots at one endpoint, not an arbitrary mixture of the two infinities. The commensurability condition depends on the number of finite roots r; testing only whether \(\exp (i \,N \,\gamma)=1\) would miss mixed states. At odd N and minimal \(\lvert S^z \rvert =1/2\), a single phantom root has \(N -2\,r =3\), hence the familiar Delta=-1/2 exception.

For our minimal-|Sz| continuation, successive candidate clusters occur at \(\Delta =-\cos (\pi /(2\,p +1))\). This equation-level observation does not identify every possible singular branch or establish a sector minimum.

Deflation and the rotated polynomial equations¶

For input F(x) with \(z =o +s \,x\), monic synthetic division removes p copies of \(x -(z_{\mathrm{phantom}} -o)/s\). The code reconstructs the full polynomial and compares every coefficient, with error

\[ \max_j\frac{|\mathrm{original}_j-\mathrm{reconstructed}_j|} {\max(1,|\mathrm{original}_j|,|\mathrm{reconstructed}_j|)}. \]

Thus a guessed multiplicity is not accepted just because F is small at an endpoint. A different chirality or an excessive multiplicity must pass the same complete-factorization check. The reduced coefficients retain the original affine coordinate; no individual finite roots are extracted.

The polynomial engine now has evaluate_rotated(coefficients, Delta, rotation, jacobian). It takes the same parity-selected real or imaginary component as before, but of rotation*G, where G=(1+i*z)^N*K_- mod F. For a boundary phase phi,

\[ \mathrm{rotation}=e^{i\phi/2},\qquad \mathrm{rotation}\,G-\sigma\,\overline{\mathrm{rotation}\,G}=0,\qquad \sigma=(-1)^{N-M-1}. \]

Phantom reduction uses rotation=exp(-i*chi*p*gamma). Integer powers of \(\Delta +i \,\chi \,\sqrt{1-\Delta ^{2}}\) generate the required phases without an inverse trigonometric function. The separate commensurability residual is checked before normalizing radial roundoff in the rotation.

The analytic Jacobian includes this fixed rotation; it still differentiates the coefficient-dependent polynomial reduction. evaluate remains exactly the zero-twist call. Energy and total-momentum formulas are unchanged, but momentum_defect is specifically a zero-twist check and must not be applied as a quantization criterion to the reduced twisted state.

The regular-state diagnostic accepts the same optional rotation. A diagonal twist multiplies the algebraic vacuum eigenvalues by constant nonzero phases; the logarithmic derivative and the regular norm argument are otherwise unchanged. This applies to the reduced state, not to the singular map that adds phantom roots back.

Tolerances and returned evidence¶

regular_reduced_equations requires a successful factorization, the commensurability condition, and a regular_on_shell reduced state. Other statuses distinguish no_endpoint_factor, phase_mismatch, unresolved_reduced_state, and nonfinite. Invalid arguments throw. The function supports \(0\lt p \lt M\) and \(-1\lt \Delta \le 0\); the all-phantom construction already has its own helix implementation.

The output retains the finite coefficients, rotation, reconstruction and commensurability errors, finite-root regularity result, and the difference between full and reduced energies. Infinite roots contribute zero energy on the exact limiting solution. That energy difference is an additional diagnostic, not an independently certified energy-error bound.

The factor/phase tolerance defaults to \(128\,\epsilon\); the reduced coefficient-residual tolerance defaults to \(32\,\epsilon\). Neither is loosened automatically, and the requested coupling is never replaced. Deflation and the changed residual normalization can amplify roundoff: for example, the N=21, p=9 fp64 audit has a reduced residual about 2.65e-14, above its default 7.11e-15 threshold. It correctly stays unresolved at that tolerance. A separate regression requests \(512\,\epsilon\) for the larger-cluster audit and checks that the coefficients do not change. This is not a hidden fallback or evidence of exact commensurability.

An independent mixed-state wavefunction¶

For r=1, our coordinate-ansatz deduction gives an explicit useful check. Put k0=chi*gamma, \(M =p +1\), and choose the finite momentum k satisfying exp(i*N*k)=exp(-2*i*p*k0). On ordered occupied sites, with rank a starting at zero, the unnormalized wavefunction is

\[ \psi(j_0,\ldots,j_p)=e^{ik_0\sum_a j_a} \sum_{a=0}^{p}e^{2ik_0a+i(k-k_0)j_a}. \]

The rank factor is the amplitude ratio for moving the finite magnon past a phantom. Native-precision tests check this formula by applying the spin Hamiltonian directly, without reusing the reduced Bethe residual. Both chiralities and the regular finite-momentum modes through N=9 are covered, with multiple phantom roots. Endpoint finite momenta and the infinite polynomial coordinate at k=0 are explicitly excluded from this regular finite-root check.

Additional tests compare the rotated coefficient equations with direct rootwise products (including conjugate roots), their Jacobian with finite differences, and the one-magnon twist with exact momentum quantization. Continuation/deflation audits cover all mixed multiplicities in the minimal-|Sz| branch at N=5,7,9,13,17,21; p=1 passes the default tolerance. Deliberately off-shell finite roots and incorrect commensurability are rejected independently of successful endpoint deflation.

A general coordinate-space dressing map now constructs the lifted amplitudes from a finite-state callback. Full Hamiltonian-identity tests and nonzero lifts of continued two-finite-root states provide additional independent checks. The map also has a genuine kernel, exhibited explicitly in the tests. A bounded numerical witness search now checks amplitudes against uncertainty estimates; it is not a proof of general nonzero lifting. The remaining singular-root families, and reliable sector-minimum tracking remain work to do before public odd-ring ground-state integration.