Explicit periodic XXZ spin-helix states¶

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

periodic_spin_helix<Real> in xxz_spin_helix.hpp constructs a special family of exact eigenstates without solving Bethe equations. It supports either ring parity, every physical magnetization, and all integer windings w=0,...,N-1. It is not a ground-state solver or an arbitrary-coupling excitation solver. There is no separate command-line mode yet.

From winding to an eigenstate¶

For the usual \(J =1\) spin-1/2 Hamiltonian, choose

\[ k=\frac{2\pi w}{N},\qquad \Delta_w=\cos k,\qquad E_w=\frac{N\Delta_w}{4}. \]

Popkov, Zhang and Klümper, equations (4), (5), and (12), give the spin-helix construction and its fixed-magnetization projections. Their Pauli-matrix Hamiltonian subtracts the polarized energy; the energy above restores our spin-1/2 convention.

In the sector with \(M =N /2-S^z\) down spins, the unnormalized amplitude is

\[ \psi(j_1,\ldots,j_M)=e^{ik(j_1+\cdots+j_M)},\qquad 0\le j_1\lt \cdots\lt j_M\lt N. \]

The state is nonzero: every amplitude has modulus one, and its squared norm is binomial(N,M). A normalized amplitude would divide by the square root of this number. The implementation leaves that factor out to avoid underflow and does not allocate an exponentially large wavefunction.

The local Hamiltonian action cancels around the closed ring. One can also check this directly on a bit configuration: the numbers of clockwise and counterclockwise domain walls agree, and each pair contributes a factor \(\cos (k)-\Delta_{w} =0\) to (H-E_w)psi. Thus every magnetization projection has the same energy. This establishes an eigenstate, not its position in the spectrum.

#include <bethe/xxz_spin_helix.hpp>
#include <array>

auto const helix = bethe::xxz::periodic_spin_helix<long double>(
    7, 4, uni20::from_twice(std::int64_t{1}));  // Sz=1/2, M=3
std::array<std::size_t, 3> const occupied{0, 2, 5};
auto const amplitude = helix.unnormalized_amplitude(occupied);
// helix.delta = -cos(pi/7), helix.energy = 7*helix.delta/4
// helix.momentum_index = (4*3) mod 7 = 5

The momentum convention is the same as the other periodic APIs: P=2*pi*momentum_index/N. Translating the arguments of the wavefunction forward by one site multiplies it by \(\exp (i \,P)\); active translation of the ket has the inverse phase. The reflected winding \(N -w\) reverses the chirality. Negative Sz is stored as requested, not folded to its partner.

Integer phase sums and products are reduced modulo N before conversion to Real. This avoids both integer-product overflow and large trigonometric arguments. The arithmetic is native fp64, long double, or fp128. The ferromagnetic \(w =0\) and even-ring \(w =N /2\) endpoints are supported by this explicit construction even though other XXZ APIs have narrower domains.

Recognizing the exceptional polynomial¶

For \(-1\lt \Delta_{w} \le 0\), the helix has a repeated scaled root

\[ z_w=\cot(k/2)=\frac{\sin k}{1-\cos k}. \]

These are infinite conventional rapidities. In affine coordinates \(z =o +s \,x\), its monic polynomial is \((x -(z_{w} -o)/s)^M\). The internal check_helix_polynomial compares all coefficients with that polynomial and independently checks the coupling. Finding an endpoint root, matching only the energy or momentum, or recognizing a root-of-unity anisotropy is not enough.

Its compatible status means numerical agreement at the requested tolerance, default \(128\,\epsilon\). The coefficient error is the maximum of |c_j-expected_j|/max(1,|c_j|,|expected_j|). This is coordinate-dependent and is not an interval certificate or a bound on extracted roots. Other statuses distinguish a different coupling, a different polynomial, and nonfinite arithmetic. Invalid inputs throw.

No input Delta is snapped. The result retains the helix's own coupling and energy, the absolute coupling mismatch, and

\[ \mathrm{eigenvector\_defect\_bound}=\frac N4|\Delta_{\mathrm{input}}-\Delta_w|. \]

Apart from floating-point roundoff, this bounds ||(H(Delta_input)-E_w)psi_w||/||psi_w||: the difference of the two Hamiltonians is (Delta_input-Delta_w)*sum Sz_j*Sz_(j+1), whose operator norm is at most \(N \,\lvert \Delta_{\mathrm{input}} -\Delta_{w} \rvert/4\). This bound refers to the explicit helix and its energy, not an arbitrary vector reconstructed from nearby polynomial coefficients.

The odd-ring sector scan tries winding \((N +1)/2\) when its regular-state test is unresolved. At \(\Delta_{w} =-\cos (\pi /N)\), this recognizes the followed all-phantom branch. Each entry retains the separate regularity and optional helix results; state_checks_complete summarizes whether every entry passes one of these checks or has a resolved mixed-phantom numerical witness. It neither changes the continued coefficients/energies nor supplies a missing continuation result. regular_states_complete retains its stricter, regular-case meaning.

Tests and remaining coverage¶

Native-precision tests apply the spin Hamiltonian directly to every winding and magnetization through seven sites, including both parity classes and the isotropic endpoints. They also check momentum under translation, large-integer phase arithmetic, affine polynomial matching, rejection of perturbed polynomials, opposite chiralities and different couplings, and the requested tolerance. Continued all-phantom polynomials are recognized in every folded sector at N=3,5,7,9,13,17,21, in all three scalar types.

Mixed finite/infinite roots, other singular configurations, and reliable sector-minimum tracking remain separate tasks. A mixed-root reduction now checks the finite twisted problem and its separate commensurability condition, without treating that as a general nonzero-lift test. For example, an odd-ring continued state at Delta=-1/2 need not be an all-phantom helix. It must not be accepted just because the coupling is a root of unity. General public negative-Delta odd-ring ground-state support remains unavailable.