Free-end spin-1 biquadratic chains and Temperley–Lieb models¶

Biquadratic/TL family · Supported models · Precision and CLI controls

The worked negative-sign tutorial plots small-chain sectors and follows a complex-root singlet to longer chains.

For the opposite sign, --ferromagnetic, see ferromagnetic excitations: an exact one-defect band, targeted two-/three-defect bound droplets, sign-aware real-root scans and complex-root Q-system levels. Real-root scattering windows select a manageable high-label subset on long odd/even chains. The conventions and default modes below describe the antiferromagnetic sign.

bethe-biquadratic-obc calculates the singlet ground state, TL module minima, and restricted real-root excitations of the even-length, free-end spin-1 pure biquadratic chain, N>=2. A separate Q-system mode also handles complex-root levels and budgeted spectrum searches; --singlet-excitation separately targets a low-lying complex-root singlet on long chains. The Hamiltonian is

The coefficient is -1, the S operators are spin 1, and there is no added constant. A bond has energies -4, -1, -1 in total spin 0, 1, 2 respectively. This is neither the Takhtajan–Babujian point nor the ULS point: both of those also have a bilinear term.

bethe-biquadratic-obc 64
bethe-biquadratic-obc 4 --roots --precision long-double
# With binary128 enabled:
bethe-biquadratic-obc 64 --precision fp128

Two useful checks are \(E (2)=-4\) and E(4)=-(15+sqrt(17))/2=-9.561552812808830274910704927.... The report includes physical, TL, and reference XXZ energies, convergence diagnostics, and CPU time. --roots prints the auxiliary XXZ rapidities and integer labels. There is no lattice momentum for these open chains.

Excited levels and their multiplicities¶

# Lowest energy in each TL module (not each physical-spin sector):
bethe-biquadratic-obc 16 --sectors
bethe-biquadratic-obc 64 --sectors --threads 4
bethe-biquadratic-obc 16 --through-lines 2
# Lowest 10 supported real-root levels in ell=2, including its minimum:
bethe-biquadratic-obc 16 --excitations 10
bethe-biquadratic-obc 16 --through-lines 4 --excitations all
# A specific Bethe level (its root count determines ell):
bethe-biquadratic-obc 8 --quantum-numbers 1,2,5 --roots

Each numerical level carries a TL through-line label ell and a multiplicity per TL eigenvector. For even ell=0,2,4,6,... these multiplicities are 1,8,55,377,... in the spin-1 chain. In particular, the ell=2 multiplicity space is one spin-1 triplet plus one spin-2 quintuplet, not eight multiplets and not a single spin-1 multiplet. The code reports the total multiplicity; the shared spin-content API and optional --spin-content table separately decompose the multiplicity space into physical SU(2) spins when its dimension fits uint64. Coincident levels are not numerically merged: sum their weights if a genuine degeneracy has been established. A decimal energy tolerance alone is not a reliable criterion for doing so.

--excitations defaults to ell=2; use --through-lines to select any even ell in [0,N]. Gaps are relative to the global singlet ground state, not the selected module's minimum. COUNT limits retained levels, not work: every candidate in the selected family is solved before energy ordering. The --max-candidates guard (default 10000) checks the combinatorial count before allocating roots or solving a reference state. Failed candidates are excluded, reported, and cause exit status 2; a failed ground reference makes gaps unavailable even if some candidate levels converge.

all does not mean the complete spectrum. For M=(N-ell)/2 roots the family consists of increasing integer labels selected from 1,...,N-M, giving choose(N-M,M) candidates. Complex-root levels are missing. For example, N=4, ell=0 has two TL eigenvalues, but only its ground state is in this real-root family. The other singlet energy \((-15+\sqrt{17})/2\) is not returned by --excitations; use the Q-system mode below. By contrast, the zero- and one-root modules are complete here: for N=4, ell=2 the three energies are \(-6-\sqrt{2}, -6, -6+\sqrt{2}\), each with multiplicity 8. The first has gap 2.147339250435735226109016203....

--sectors, --excitations, and --quantum-numbers are separate modes.

--threads N overlaps independent solves in --sectors and --excitations; one worker is the default. Sector output remains in increasing through-line order, with gaps formed only after the singlet reference has finished. Each Newton solve and its continuation remain sequential. A failed reference still makes every sector gap unavailable, and numerical failures retain exit status 2. Selected single states, the Q-system search and the complex singlet continuation do not accept this option. See the ferromagnetic modes for concurrent bound-cluster scans. Explicit labels determine ell and cannot be combined with --through-lines; use --quantum-numbers none for the zero-root ell=N level. Multiplicities use checked uint64 arithmetic: ell>=46 at d=3 is reported as overflow (>uint64), never wrapped, rounded, or silently replaced by 1. Energy calculations can still converge when this integer count is unavailable.

Including complex roots¶

# Both four-site singlets, including the complex-root level:
bethe-biquadratic-obc 4 --q-spectrum --roots
# Search all nine levels of the six-site ell=2 module (multiplicity 8 each):
bethe-biquadratic-obc 6 --q-spectrum --through-lines 2
# Eight-site singlets benefit from both extra attempts and extra precision:
bethe-biquadratic-obc 8 --q-spectrum --max-attempts 12000 --precision long-double
# One selected branch, using a monic polynomial seed rather than real labels:
bethe-biquadratic-obc 4 --q-seed 1.5,-1.3333333333333333 --roots
# Typed exports use the same Uni20 table/output machinery:
bethe-biquadratic-obc 4 --q-spectrum --roots --json levels.json

--q-spectrum searches one TL module, default ell=0, including real and complex roots. It uses a deterministic multistart search, validated through N=8, not exact diagonalization. --max-attempts bounds the work (default 4000). Completeness is checked numerically against choose(N,M)-choose(N,M-1); it is not a rigorous certificate. If any levels remain missing, the program returns exit 2 and explicitly labels the results as incomplete discoveries, not necessarily the lowest levels. Higher precision can resolve strings that fp64 cannot; a larger search budget only helps with state discovery.

--q-seed c0,c1,... selects one branch of Q(x)=x^M+c[M-1]*x^(M-1)+...+c[0], with \(x =\cosh (2u)=\cos (\alpha)\). Its degree determines ell, so do not supply --through-lines; none selects the vacuum. This is an expert interface, not an excitation rank or string label. Neither Q-system mode has a site cutoff; larger chains are experimental. Removing the cutoff does not guarantee convergence or accurate root recovery at every size. The Q-system modes do not combine with the real-label modes or --sectors.

The physical energy shift and multiplicity are unchanged. Gaps still use the global ground reference. Named tables are states, reference, q_coefficients, and (with --roots) roots; complex values have separate real/imaginary columns, and unavailable diagnostics are null. A failed selected solve is retained as an explicitly unverified estimate, with no gap. See the Q-system guide for equations, native-precision checks, API usage and the remaining large-chain work.

A complex-root singlet on long chains¶

bethe-biquadratic-obc 128 --singlet-excitation
bethe-biquadratic-obc 512 --singlet-excitation --precision long-double --roots

This separate solver targets one positive-deviation two-string above a real sea in ell=0, with physical multiplicity one. It reproduces the lowest excited singlet in the small-chain tests and follows that branch to long chains, without an exhaustive spectrum or global first-excitation claim. The deviation is retained logarithmically even when rounded roots look like an exact string. It supports native and explicit MPFR precision and typed exports; see the two-string guide for equations, budgets, output coordinates and scope. This mode does not combine with other state selection or scan options.

The algebra connects spectra, not physical spin labels¶

Define e_i=(S_i.S_(i+1))^2-1=3*P_(singlet,i,i+1). These operators satisfy the Temperley–Lieb relations with loop weight lambda=3:

Another representation of the same open-chain algebra uses spin-1/2 XXZ operators. For general lambda=2*Delta>2, our reference Hamiltonian is

At lambda=3 this gives Delta=3/2, end-field coefficient sqrt(5)/4, and E_b=2*E_ref-7*(N-1)/4. The end fields belong to the auxiliary spin-1/2 representation; the physical spin-1 chain still has free ends. The existing bethe-xxz-obc solves the zero-end-field XXZ model and cannot be used with just an energy shift to obtain these results.

The equivalence is within TL modules. A module labelled by ell through-lines has different multiplicities in different spin-chain representations. For the generic singlet-projector spin-chain representation of local dimension d,

For spin 1, d=3 and the sequence is 1, 3, 8, 21, 55, ...; only ell with the same parity as N occurs. For spin 1/2, d=2 and \(m_{\ell} =\ell +1\). Consequently an XXZ spin multiplet's degeneracy is not the biquadratic degeneracy. Nor should ell/2 be relabelled as the physical spin-1 total spin: the multiplicity space can contain several SU(2) multiplets. An accidental degeneracy shared by different TL modules adds their contributions. These formulas do not claim to cover arbitrary TL representations such as RSOS models.

The present ground state lies in the ell=0 module and is a single physical singlet. In particular, do not attach a factor of two for dimerization to a finite even open chain. Its ends select the dimer pattern.

Bethe equations and numerical branch¶

Following Albertini, Eqs. (6)–(10), set \(\Delta =\cosh (\eta)\). A module with ell through-lines uses M=(N-ell)/2 real roots in (0,pi). Its lowest state has consecutive labels \(I_{i} =i\), i=1,...,M; other supported states leave holes in the label window:

Sending the largest root to pi gives \(I =N -M +1\), but that endpoint has a vanishing Bethe wavefunction and is excluded. Thus 1<=I_1<...<I_M<=N-M. The finite regular roots select quantum-group highest weights, so the corresponding TL module is ell=N-2M: counting every auxiliary XXZ Sz sector again would duplicate descendants. Its full module dimension is choose(N,M)-choose(N,M-1), generally larger than the real-family count.

The reflected sum must retain its phase branch when alpha_i+alpha_j>pi. The solver uses x_i=Theta(alpha_i;eta/2)/2 in (0,pi/2), with \(\alpha_{i} =2\,\mathrm{atan2} (\tanh (\eta /2)\,\sin (x_{i}),\cos (x_{i}))\). In this coordinate the energy contribution is \(-(\Delta +\cos (2\,x_{i}))\). Its direct excitation shift is accumulated as \(-[(\Delta -1)+2\,\cos (x_{i})^{2}]\), avoiding cancellation near Delta=1 and the ferromagnetic low-energy edge. The TL adapter uses this direct shift, not a difference of extensive total energies. No complex boundary root is needed for this reference model's even ground state. Specified real-root states (--quantum-numbers) also accept odd N; this does not enable the odd-chain AF ground/sector or one-spinon-band helpers.

Newton iteration uses an analytic Jacobian and a line search preserving strictly ordered interior roots. The seed is the bare driving phase \(x_{i} =\pi \,I_{i} /(2\,N)\), not an approximate thermodynamic result. Dense storage is O(N^2), and the partial-pivot solve costs O(N^3) per update. It uses the shared native-precision Newton helper with recoverable small-pivot failure, without changing Uni20's process-wide error policy.

residual_norm is the maximum absolute logarithmic equation residual divided by 2N. The default tolerance is 32 epsilon of the selected type; it is not an energy-error bound. Failed solves retain mutually consistent roots, energy, residual, and accepted-update count. A zero iteration budget evaluates the seed. The CLI returns 2 for an unconverged result and 1 for invalid input; it never silently changes precision or relaxes the tolerance.

Library layers¶

#include <bethe/biquadratic.hpp>
auto state = bethe::biquadratic::ground_state<long double>(64);
if (!state.reference.converged) { /* state.energy is an incomplete estimate */ }
auto sector = bethe::biquadratic::sector_ground_state<long double>(64, 2);
auto levels = bethe::biquadratic::real_excitations<long double>(16, 2, {.count=10});
if (!levels.converged()) { /* failed candidates and/or reference; inspect diagnostics */ }
// Each level has state.energy, state.through_lines, state.multiplicity and optional gap.

// The generic even, zero-through-line TL ground state, lambda>2:
auto tl = bethe::temperley_lieb::open_ground_state(64, 3.0L);
// Exact representation multiplicity, nullopt if uint64_t would overflow:
auto copies = bethe::temperley_lieb::spin_chain_multiplicity(3, 4); // 55

The three layers have separate responsibilities:

• xxz_quantum_group.hpp: the reference equations, roots, energy and numerical diagnostics, for Delta>1.
• temperley_lieb.hpp: the TL energy normalization and spin-chain representation multiplicities, for lambda>2.
• biquadratic.hpp: the physical spin-1 Hamiltonian and its additive energy shift.

All numerical layers support fp64, native long double, enabled fp128 and explicit MPFR. The multiplicity helper uses checked integer arithmetic, not floating point. sector_ground_state, solve_real, and real_excitations are available in both the TL and physical biquadratic layers. TL calls additionally take a loop weight; their energies do not include the physical spin-1 shift. The generic TL layer leaves representation multiplicities to its caller. The GroundState/OpenGroundState type aliases retain the original ground API.

Arbitrary precision¶

The C++ real-root APIs also accept uni20::mpreal with explicit controls:

using Real = uni20::mpreal;
auto precision = uni20::Precision::bits(256);
bethe::SolverOptions<Real> controls(precision);
auto state = bethe::biquadratic::ground_state<Real>(16, controls);
auto selected = bethe::biquadratic::real_excitations<Real>(8, 4, {.count=6}, controls);

All existing frontend modes also accept --precision bits-N or --precision digits-N when built with UNI20_ENABLE_MPREAL=ON. This includes real-root solves and scans, analytic ferro levels, the selected singlet, all supported bound-cluster families, and Q-system seeds and spectrum searches.

bethe-biquadratic-obc 16 --sectors --threads 4 --precision bits-160
bethe-biquadratic-obc 4 --singlet-excitation --roots --precision digits-100
bethe-biquadratic-obc 12 --ferromagnetic --bound-triples 2 --threads 2 --precision bits-160
bethe-biquadratic-obc 4 --q-seed 1.5,-1.3333333333333333333333333333333333 --precision bits-256

Newton updates use Uni20's normal dispatch: MPLAPACK when available, or its CPU MPFR LU implementation otherwise. Analytic ferro helpers take an explicit Precision argument. TL loop weights and reference XXZ Delta carry the same finite precision as the controls. Unset, exact and mixed precision library inputs are rejected before numerical work. CLI tolerances and polynomial seeds are parsed directly at the requested precision. Root reconstructions, energy references and exported values retain that precision as well.

Existing bounded scans and owning Async sweeps preserve precision independently for each request. --threads has the same scope in every arithmetic format; single selected states and Q-system searches remain direct. Failed iterates retain their diagnostics and have no verified gap. More bits do not enlarge the state family, guarantee a complete Q-system search, or replace convergence and candidate-budget checks.

Validation and future slices¶

Tests build the physical spin-1 bond matrix and square it independently. For N=2,4,6 they reconstruct its entire ED spectrum from independently diagonalized XXZ sectors and TL multiplicities. They also verify the unique ground state and explicitly detect the wrong answer when end fields are omitted. Further tests check the original multiplicative Bethe equations, the analytic Jacobian, exact two-/four-site energies in each native precision, longer chains through N=128, and CLI/failure contracts. Excitation tests isolate TL modules by subtracting adjacent auxiliary Sz spectra, and match every returned real-root level for N=2,4,...,10 without reusing an ED eigenvalue. They verify module minima, eightfold physical degeneracies, exact one-root energies and gaps at native precision, candidate limits, failed-reference behavior, and the singlet missing from the real-root family. The separate Q-system tests recover that singlet and reconstruct the full physical N=2,4,6 spectrum from Bethe solutions with their TL multiplicities. The targeted two-string solver is checked against the lowest excited singlet through N=10 and against the original complex equations on chains through N=512, retaining native precision and the logarithmic deviation.

Odd free-end chains need their own one-domain-wall/spinon branch. Their low-lying states describe motion of that defect, not just a factor-two choice of dimer pattern. Periodic chains require sector-dependent XXZ twists and periodic representation bookkeeping. Neither is enabled here, nor are large-chain complex-root enumeration, physical-spin sector scans, general boundary fields, or thermodynamics.

The original spectral mapping is due to Barber–Batchelor. For the open-chain normalization and real-root equations see Albertini above; for TL modules and multiplicities see Aufgebauer–Klümper, Secs. 2.3 and 3. The Q-system and regularized two-string solver use Bajnok et al., Sec. 5. These references also appear with --references.