Massive free-end XXZ: ground-state boundary root¶

XXZ family · Supported models · Open XXZ guide

The library module bethe/xxz_open_massive.hpp implements sector minima at \(\Delta \gt 1\), for either chain parity and either sign of Sz. It uses the same spin-1/2 Hamiltonian, J=1 and zero boundary fields as the gapless open solver. The existing bethe-xxz-obc executable and the three ground-state functions in bethe::xxz::open select this module for Delta>1. The gapless and exact XXX paths are unchanged; no new executable is needed.

#include <bethe/xxz_open.hpp>

auto state = bethe::xxz::open::ground_state<long double>(16, 3.0L);
auto sector = bethe::xxz::open::sector_ground_state(
    17, 3.0L, uni20::half_int::parse("-1/2"));

All arithmetic, including the boundary-root equation and Newton matrix, uses the selected real type: fp64, long-double, optional fp128 or explicit MPFR. See finite XXZ precision controls. Check converged, status, and residual_norm before using an energy. This module does not implement excited states, boundary fields, or general complex strings.

Why extending the real interval is insufficient¶

Write \(M =N /2-\lvert S^z \rvert\) and \(\Delta =\cosh (\eta)\). Bulk roots use \(z =\tan (\lambda)/\tanh (\eta /2)\). When \(M \lt N /2\), the ground state has M positive bulk roots. For even N with Sz=0, its last root must be tracked separately: as anisotropy grows it passes through infinity in z and becomes purely imaginary. This is not a singularity of the energy.

The two-site example makes the issue explicit:

\[ z_B^2=\frac{1+\Delta}{3-\Delta},\qquad E_0=-\frac12-\frac\Delta4. \]

The root is real below Delta=3, infinite at Delta=3, and imaginary above it. Changing a range check on an all-real solver cannot represent all three cases. The transition anisotropy depends on chain length; 3 is not a general cutoff.

The reflection equations follow Mei and Bolech, Eqs. (11)-(12). The massive boundary-root interpretation and exponentially small finite-size deviations are discussed by Grijalva, De Nardis, and Terras, Sec. 4.3.2. Their Pauli-matrix Hamiltonian at zero boundary fields is divided by four and shifted by \((N -1)\,\Delta /4\) to obtain ours. The regularization below is an algebraic implementation of the finite-size equations, not an ideal-string or thermodynamic approximation.

Regularized coordinate¶

For the distinguished root, define

\[ r=\frac{\Delta-1}{\Delta+1},\qquad y=\frac1{z_B^2},\qquad w=-\log(1+y/r^2),\qquad y=r^2\mathrm{expm1}(-w). \]

The ground-state branch has \(y \gt -r ^{2}\). Negative w describes a real z_B; w=0 is its crossing through infinity; positive w describes an imaginary z_B. Large positive w retains the deviation from the boundary pole even when \(y\) rounds to \(-r ^{2}\). Do not reconstruct w from that rounded value.

GroundState::rapidities and quantum_numbers contain only bulk roots and their labels 1,...,M-1 when a boundary root is present, or 1,...,M otherwise. The optional boundary_root stores w as log_distance, y as inverse_square, and the last label M. The label tracks continuation from the real branch; it does not assert that the continued root is real.

Coupled equations¶

Set \(P =1+1/\Delta\), \(Q =1/\Delta -1\). For bulk roots, the half scattering phases are evaluated with atan2, retaining their winding. A bulk root z sees the distinguished root and its reflection through their combined phase

\[ S_B(z,y)=\mathrm{atan2}\!\left(2z(Py-Q),(P^2-z^2)y+1-Q^2z^2\right). \]

Near the pole at strong anisotropy, both arguments in this expression suffer cancellation. The implementation expands them in \(1/\Delta\) and \(r ^{2}\,\exp (-w)\) before evaluation; it does not evaluate the displayed differences naively.

The bulk residual used in the solver is

\[ \begin{aligned} R_i=\frac1N\biggl[&2N\arctan z_i-2\arctan(rz_i)-\pi I_i\\{} &-\sum_{\substack{j\ne i\\j\ \mathrm{bulk}}} \left\{\mathrm{atan2}(z_i-z_j,P-Qz_iz_j) +\mathrm{atan2}(z_i+z_j,P+Qz_iz_j)\right\}-S_B(z_i,y)\biggr]. \end{aligned} \]

Omit \(S_{B}\) if there is no distinguished root. Direct and reflected self-scattering are both excluded.

For even N=2M, the unregularized equation for the last root vanishes at \(1/z_{B} =0\) regardless of Delta. Divide out that vanishing factor before taking the limit, or infinity becomes a spurious solution. Define the analytic continuation

\[ \begin{aligned} A(y,a)&=\begin{cases} \arctan(\sqrt y\,a)/\sqrt y,&y\gt 0,\\{} a,&y=0,\\{} \mathrm{artanh}(\sqrt{-y}\,a)/\sqrt{-y},&y\lt 0, \end{cases}\\{} a_j&=\frac{P+iz_j}{1-iQz_j},\\{} R_B&=\frac1N\left[-NA(y,1)+A(y,1/r)+\sum_{\mathrm{bulk}}\mathrm{Re}A(y,a_j)\right]. \end{aligned} \]

The implementation evaluates the real parts without complex transcendental functions. In particular, for w>0 let t=sqrt(-expm1(-w)); then r*A(y,1/r)=(w/2+log1p(t))/t. This retains the boundary-pole logarithm when t rounds to one. Small-argument formulas are used near the crossing to avoid manufacturing a tiny residual through cancellation.

The energy is

\[ E=\left(\frac{N-1}{4}-M\right)\Delta +\sum_{\mathrm{bulk}}\frac{z_j^2-1}{z_j^2+1}+\frac{1-y}{1+y}. \]

with the last term omitted if there is no boundary root. Evaluate its denominator as \(1+y =4/\Delta /(1+1/\Delta)^{2}+r ^{2}\,\exp (-w)\) near the pole.

Iteration and diagnostics¶

Damped Newton uses bulk angles atan(z) and w, with a central-difference Jacobian at the selected working precision. Continuation starts at min(Delta,1.1) and increases \(\Delta -1\) by a factor of 1.25 per converged stage. Ordered bulk roots and, while real, a largest distinguished root are enforced during backtracking. After the crossing, continuation carries w relative to the moving pole rather than rounding away the small deviation.

Each accepted Newton update consumes the single global iteration budget; there is no hidden unbudgeted seed solve. On exhaustion, energy and residual are still evaluated for the requested Delta at the returned roots. root_delta records the continuation stage reached. status distinguishes convergence, budget exhaustion, a stalled line search, and a continuation step that rounds back to the current anisotropy (continuation_stalled). The last case requires more working precision: a converged stage uses no Newton iterations, so an iteration budget alone cannot stop a repeated stage. If the initial continuation anisotropy itself rounds to one, the solver rejects the request before constructing the singular seed. Unconverged CLI energies remain explicitly labelled estimates, with exit status 2 for a stalled continuation.

The residual is \(\max (\lvert R_{i} \rvert,\lvert R_{B} \rvert)\), with the boundary entry absent when appropriate. Its boundary component is regularized, unlike the old all-real logarithmic residual. The default tolerance is 32 native epsilons. It is not an energy-error bound, or a bound on the exponentially small ground/first-excited splitting; this solver returns only the ground branch. Work is O(M^3) per Newton update, with O(M^2) dense workspace.

Command-line output¶

bethe-xxz-obc 16 --delta 3 --roots --precision long-double
bethe-xxz-obc 5 --delta 2 --sectors --roots --format plain

--roots writes bulk roots and separate boundary_roots tables, linked to states by state_id. The boundary schema includes quantum_number, inverse_square, log_distance, and kind. Its kind describes the returned coordinate (real, imaginary, or infinity), not an exact symbolic determination of the finite-chain crossing. Pretty output uses separate bulk and boundary tables and retains full-precision numeric tokens even on narrow terminals. A boundary-only state is not labeled fully polarized.

Massive reports name the regularized residual convention and show Root Delta. The typed state table includes root_delta, converged, and descriptive status columns, distinguishing iteration limits from stalled line searches. CSV/TSV and JSON exports preserve these diagnostics and selected-precision coordinates; see output schemas. A nonconverged calculation exits with status 2; invalid requests exit with status 1. Neither output mode invents momentum.

Validation and remaining scope¶

Typed tests check every magnetization sector against independent small-chain exact diagonalization through N=9, original complex reflection equations, analytic two- and three-site energies in native precision, the root crossing, spin reversal, exhausted-budget consistency, and longer chains whose boundary deviation is smaller than machine epsilon. Further tests check continuity to XXX without snapping Delta to one, and strong-coupling root limits through Delta=10^9. The near-crossing residual test specifically detects cancellation that can be hidden by double-precision ED.

API tests verify dispatch, spin reversal, and exact preservation of the gapless solver's values. CLI tests exercise all precision modes, roots, sector tables, narrow-terminal formatting, CPU time, and exhausted budgets. The old all-real API retains its restricted excitation family. Massive excitation scans and the first-excited splitting remain unimplemented.