Excitations of the ferromagnetic biquadratic chain¶

Biquadratic/TL family · Supported models · Biquadratic/TL conventions · Output and exports

For plots, downloadable examples and an MPS-oriented introduction, see the defect and droplet tutorial.

The ferromagnetic option uses H=+sum (S_i.S_(i+1))^2, the opposite sign to this tool's default. The spins are 1 and the physical chain has free ends. There is no bilinear term or added constant. With \(e_{i} =3\,P_{\mathrm{singlet}}\),

\[ H_F=(N-1)+\sum_i e_i,\qquad E_0=N-1,\qquad E-E_0=\sum_i e_i\ge0. \]

The ground space is known exactly. The focus here is the positive-energy structure above it: the one-defect band, levels in other TL modules, and the complex-root states that real-root scans miss. All modes support fp64, native long double, enabled fp128 and explicit MPFR (bits-N/digits-N), with the shared screen/JSON/CSV/TSV output. See the biquadratic precision controls.

A first exploration¶

# Complete one-defect band, analytically (odd N also supported):
bethe-biquadratic-obc 64 --ferromagnetic --one-defect
bethe-biquadratic-obc 65 --ferromagnetic --one-defect --json band.json
# Target bound-pair modes on a long chain (not all two-defect states):
bethe-biquadratic-obc 128 --ferromagnetic --bound-pairs 8
# Three-defect droplets, not all three-defect states:
bethe-biquadratic-obc 128 --ferromagnetic --bound-triples 8
# Mixed bound pair plus a separate defect, a 6-by-6 label window:
bethe-biquadratic-obc 128 --ferromagnetic --pair-defects 8 --mixed-window 6
# Overlap independent candidates in the same family:
bethe-biquadratic-obc 128 --ferromagnetic --pair-defects 8 --mixed-window 6 --threads 4
# Real-root scattering near the two-defect low-energy edge:
bethe-biquadratic-obc 129 --ferromagnetic --through-lines 125 --excitations all --real-window 8
# Small two-defect module, including complex-root states:
bethe-biquadratic-obc 6 --ferromagnetic --q-spectrum --through-lines 2 --roots
# Compare it with the restricted real-root family in the same module:
bethe-biquadratic-obc 6 --ferromagnetic --excitations all --through-lines 2 --roots
# Selected real-root level; labels and roots retain the auxiliary XXZ convention:
bethe-biquadratic-obc 8 --ferromagnetic --quantum-numbers 7 --roots

Both scans default to \(\ell =N -2\) in ferromagnetic mode. Set \(\ell =N -4\) to explore two defects, \(\ell =N -6\) for three, etc. The number \(M =(N -\ell)/2\) counts TL singlet defects, not physical spin lowerings. Through-lines are not a physical SU(2) total-spin quantum number. The diagrammatic through-lines are the unpaired strands; the remaining sites form M non-crossing singlet arcs, so N=ell+2M. Some TL literature calls the through-lines themselves "defects"; our M-defect terminology instead counts singlet insertions M. These are algebraic sectors, not counts of fixed local singlet bonds in an eigenstate.

--threads N controls concurrency for --excitations, --bound-pairs, --bound-triples, --bound-quartets, --pair-defects, --triple-defects and --two-pair-states. The default is one worker. Labels are generated in their usual order and copied into each solve; continuous scheduler workers take the next candidate as soon as they finish. Each candidate's Newton steps stay sequential. Results preserve mode order or the existing gap/label ranking, including the first failed candidate in enumeration order, regardless of completion order. The option does not change the selected physical family.

All admitted solves finish before an execution error propagates or exports open. Table output stays on the calling thread. At most the requested number of candidate solves are active, capped by the candidate count; requests do not form an unbounded queue. As before, bound-cluster scans retain candidate states until final ranking, so memory still grows with --max-candidates and the root count. Selected --pair-defect, --triple-defect, --two-pairs, analytic bands and Q-system modes do not accept --threads. Use one BLAS/OpenMP thread per solve when benchmarking; small scans may be faster with the default.

Without state-selection options, --ferromagnetic reports only the exact ground-space reference and the positive spectral gap. --through-lines N-2 alone selects the exact first positive level. For ell=N-4, use the targeted --bound-pairs family, --bound-triples for ell=N-6, or --bound-quartets for ell=N-8. These select bound branches, not a general module-minimum search; other nontrivial module minima still require a Q-system investigation. --sectors and the AF-specific --singlet-excitation are rejected with this sign, not silently reinterpreted.

What the one-defect band means¶

Take a singlet on bond i and fully polarized spins elsewhere, and call the state |i>, for i=1,...,N-1. These states span one copy of the \(\ell =N -2\) module. They are independent but not orthogonal: adjacent states overlap by 1/3. Direct action of the singlet projectors gives

Thus coefficients proportional to \(\sin (\pi \,j \,i /N)\) give

\[ \begin{aligned} k_j&=\frac{\pi j}{N},\quad j=1,\ldots,N-1,\\{} \mathrm{gap}_j&=3+2\cos k_j,\qquad E_j=N-1+\mathrm{gap}_j. \end{aligned} \]

The program emits decreasing j, hence increasing energy. Its wave_number column is this OBC standing-wave coordinate, not a translation eigenvalue. The gap is evaluated separately from the extensive energy; subtracting two large printed total energies is not needed. These are exact expressions evaluated at the chosen floating-point precision, not a Newton calculation.

Every TL eigenvector carries the spin-1 representation multiplicity \(m _{N -2}\), where \(m_{0} =1\), \(m_{1} =3\), and \(m _{\ell +1}=3\,m_{\ell} -m _{\ell -1}\). This includes other physical-spin states beyond the polarized-background representative. It is not one SU(2) multiplet. If levels from other modules coincide, their multiplicities must be added separately.

The global positive gap, not just a one-defect variational estimate, is

\[ \Delta_N=3-2\cos(\pi/N)\longrightarrow1. \]

This follows by TL equivalence from Koma-Nachtergaele, Proposition 2, rescaling their XXZ Hamiltonian by \(2\,\Delta =3\). It is the gap above the entire ground manifold. Physical one-spin-lowering states on the fully polarized background remain inside that manifold and cost zero energy. They are not the positive-energy band tabulated here.

The full band has N-1 rows. --max-candidates bounds its allocation (default 10000); raise it explicitly for larger bands. Analytic modes accept odd and even N>=2. --roots is not available in analytic modes; use --excitations all --roots for the equivalent one-root Bethe family.

Why one band is not the full low-energy spectrum¶

For N=4, the ground energy is 3 with multiplicity 55. All the distinct positive-energy levels, in increasing physical energy, are:

Energy	TL defects M	Through-lines ell	Multiplicity	Root family
\(6-\sqrt{2}\)	1	2	8	One real root
\((15-\sqrt{17})/2\)	2	0	1	Complex roots
`6`	1	2	8	One real root
\(6+\sqrt{2}\)	1	2	8	One real root
\((15+\sqrt{17})/2\)	2	0	1	Two real roots

The two-defect singlet already interleaves the one-defect band. A real-root scan of ell=0 returns only the higher singlet, even with all. To see the missing level:

bethe-biquadratic-obc 4 --ferromagnetic --q-spectrum --through-lines 0 --roots
bethe-biquadratic-obc 4 --ferromagnetic --excitations all --through-lines 0

The Q-system's first state now has complex roots. q_coefficients and roots use the reordered state_id; an exported polynomial can be supplied to --q-seed to follow that branch. This is a practical entry point to studying complex-root families, not an automatic classification of bound states.

For the selected two-string family, the bound-pair solver now follows modes directly on long chains, including both signs of the finite string deviation. --bound-pairs COUNT|all fixes ell=N-4, supports odd and even N>=4, and excludes scattering states. The lowest branch approaches E-E0=5/3, below the two-separated-defect threshold 2.

The three-string solver adds three-defect droplets in ell=N-6, with a complex finite-size deviation. --bound-triples COUNT|all supports odd/even N>=6. Its lowest branch approaches E-E0=2, below 3 for three separated defects and 8/3 for a separated pair plus one defect. Neither targeted family enumerates scattering.

Real-root scattering windows select the high-I edge of the existing real family with --real-window WIDTH. They make few-defect long-chain scans practical without enumerating every combination. all then means all combinations within the chosen window, not all excited states; mixed string-plus-real-root branches remain excluded.

Ordering, references, and coverage¶

Sign reversal leaves the auxiliary XXZ equations and roots unchanged. It reverses physical energies and TL energies:

\[ E_F=-E_{\mathrm{AF}}=-2E_{\mathrm{ref}}+\frac{7(N-1)}4,\qquad \mathrm{tl\_energy}=\sum_i e_i=E_F-(N-1). \]

The reference_energy column still denotes the unchanged auxiliary XXZ energy. The separate reference table denotes the global physical ground reference. Real-family scans select in ferromagnetic energy order before retaining COUNT levels. Their exact vacuum reference cannot fail because of a Newton iteration budget, although excited-state solves can. Failed states have no published gap and scans exclude them from ranking.

Q-system searches include complex roots, retaining the existing budget, admissibility and numerical completeness checks. Validation covers even N<=8; larger systems are experimental. An incomplete search is a set of discoveries, not necessarily the lowest levels, even after sorting. Selected real roots and ferro real-family/window scans accept odd/even N, as do the targeted pair/triple solvers, analytic band and ground space. Q-system modes and AF ground/sector helpers still require even N.

Multiplicities use checked uint64 arithmetic, becoming unavailable on overflow rather than approximate. This does not invalidate an energy or gap. See Zhou et al. for the ground-space structure and Fibonacci degeneracies.

Library and next steps¶

#include <bethe/biquadratic_ferromagnetic.hpp>
namespace ferro = bethe::biquadratic::ferromagnetic;
auto first = ferro::one_defect_level<long double>(65, 64);
auto gap = ferro::spectral_gap<long double>(65);
auto real = ferro::real_excitations<long double>(6, 2, {.count=10});
auto complex = ferro::qsystem::spectrum<long double>(6, 2);
auto pair = ferro::bound_pair<long double>(128, 1);
auto triple = ferro::bound_triple<long double>(128, 1);
auto scattering = ferro::real_excitations_window<long double>(129, 125, 8);
auto mixed = ferro::pair_defect<long double>(128, 125, 123);
auto pair_two_real = ferro::pair_with_real_roots<long double>(128, std::vector<std::size_t>{123, 124}, 121);
auto two_pairs = ferro::two_bound_pairs<long double>(128, {121, 122});
// Inspect real.converged() and complex.complete() before interpreting coverage.

Targeted two-, three-, and four-defect bound families are available on long chains. Real-root scattering windows and pair-plus-defect states and CLI scans are also available. One pair with several real roots is supported through selected labels or --pair-defects ... --real-defects R scans. A two-bound-pair solver and CLI scans are also available (--two-pairs J1,J2 or --two-pair-states COUNT|all, with optional --pair-window WIDTH). A selected-state triple-plus-real-root solver and CLI scans add another four-defect family (--triple-defect I,J or --triple-defects COUNT|all with optional --mixed-window WIDTH). A selected-state bound-quartet solver and CLI completes the five four-defect topologies, jointly checked against complete module ED through N=10. Use --bound-quartets COUNT|all for the selected four-string family. Higher-defect families remain open work. The physical-spin decomposition is now available through the shared library and optional --spin-content table for multiplicity spaces whose total dimension fits uint64. Other open work is state-dependent spectral weights/form factors, and periodic twists with genuine momentum labels. Energies and multiplicities alone do not determine which branches an operator or a chosen iMPS ground state couples to. None of those additional capabilities is implied here.