Spin-1 Takhtajan–Babujian chains¶

Overview | Supported models | Precision and CLI controls

bethe-tb-pbc calculates the zero-field singlet ground state of the periodic spin-1 Takhtajan–Babujian (TB, also Babujan–Takhtajan) chain for even L>=4:

\[ H=\sum_j\left[\mathbf S_j\cdot\mathbf S_{j+1}-(\mathbf S_j\cdot\mathbf S_{j+1})^2\right]. \]

The S operators are spin 1. The bilinear coefficient is 1, with no additive constant. A polarized bond has energy 0; its total-spin-0, 1, 2 eigenvalues are -6, -2, 0. This normalization is J=4 in Vlijm–Caux, Eq. (1.2), whose Hamiltonian has a J/4 prefactor. It differs from the often-used coefficients cos(theta), sin(theta) at theta=-pi/4 by a factor sqrt(2).

This is not the generic bilinear spin-1 Heisenberg chain, and not the SU(3) ULS point, whose biquadratic term has the opposite sign. Odd lengths, other spin representations, magnetization sectors, excitations, fields and open boundaries are not yet supported here.

The separate bethe-tb-dispersion frontend now supplies thermodynamic spinon lines and two-/four-spinon bounds, with spin labels and two-site-cell folding. These do not enumerate finite-ring excited roots.

First calculations¶

bethe-tb-pbc 16
bethe-tb-pbc 6 --roots --precision long-double
# With binary128 enabled:
bethe-tb-pbc 64 --precision fp128

The four-site ground energy is \(-11-\sqrt{41}\), approximately -17.40312423743284868648821767. Ground momentum is zero for this even-ring family. The report includes both phase and modulus residuals, CPU time, and accepted Newton updates. --roots prints string centres and deviations, then the real and imaginary parts of every rapidity.

Why complex roots are necessary¶

Relative to the all-up spin-1 reference, the singlet has L spin lowerings, and hence L rapidities. They form L/2 conjugate pairs:

\[ \lambda_{j,\pm}=x_j\pm i(1/2+\delta_j). \]

An ideal two-string sets delta_j=0. That is not the finite-chain solution: the original equations then contain a singular within-pair scattering factor. Here both x_j and delta_j are solved, retaining positive finite deviations. Even the four-site energy changes appreciably if the imaginary parts are replaced by +/-1/2 after solving for the centres.

The original complex equations and energy, in our normalization, are

\[ \left(\frac{\lambda_j+i}{\lambda_j-i}\right)^L =\prod_{k\ne j}\frac{\lambda_j-\lambda_k+i}{\lambda_j-\lambda_k-i}, \qquad E=-4\sum_j\frac1{1+\lambda_j^2}. \]

Conjugate contributions make E real. Translation obeys exp(iP)=product_j (lambda_j+i)/(lambda_j-i); reflection and conjugation give P=0 for this filled even-ring sea. Tests verify that product independently as well as translation in the spin basis.

Finite-deviation equations and branch selection¶

The filled two-string sea has consecutive centred string labels \(I_{j} =j -(L /2-1)/2\), j=0,...,L/2-1. These are not the quantum numbers of the individual complex roots. In the relation (3.9) of Vlijm–Caux, the ordered string-sign sum cancels I_j, giving \(J _++J _-=0\). The half-odd-integer root labels and Eq. (3.11) select positive deviations, \(J _+=-1/2, J _-=1/2\). Coinciding root quantum numbers across different strings do not imply coinciding rapidities.

For clarity, the actual equations are recorded here. Define \(A (a,b)=\mathrm{atan2} (a,b)\), \(B (a,b)=\log (a \,a +b \,b)/2\), \(d_{\mathrm{jk}} =x_{j} -x_{k}\), and widths

\[ w_{jk}=(2+\delta_j+\delta_k,-\delta_j-\delta_k,1+\delta_j-\delta_k,1-\delta_j+\delta_k), \qquad \mathrm{signs}=(+1,-1,+1,-1). \]

Our normalized residuals are

\[ \begin{aligned} F_j&=A(x_j,3/2+\delta_j)+A(x_j,1/2-\delta_j) -\frac1L\sum_{k\ne j}\sum_{a=1}^{4}A(d_{jk},w_{jk}[a]),\\{} G_j&=B(x_j,3/2+\delta_j)-B(x_j,1/2-\delta_j) -\frac{\log(1+\delta_j)-\log\delta_j}{L}\\{} &\quad-\frac1L\sum_{k\ne j}\sum_{a=1}^{4}\mathrm{signs}[a]B(d_{jk},w_{jk}[a]). \end{aligned} \]

These specialize Eqs. (3.8) and (3.10) to the ground-state two-string sea. The atan2 branch matters when the second width is negative. Its first argument is nonzero for distinct centres, so no branch value at (0,0) is needed. The self-pair modulus is evaluated from delta directly, avoiding subtraction of two nearly equal imaginary parts. Positive deviations below 1/2 keep this branch away from the driving poles at lambda=+/-i.

Reflection symmetry removes redundant equations: centres occur as +/-x, deviations are equal in each reflected pair, and a central string at x=0 has an identically zero phase equation. There are L/2 real Newton variables. The seed uses the thermodynamic density \(1/(2\,\cosh (\pi \,x))\) and the asymptotic deviation estimate in Eq. (3.18). Only the seed is approximate: convergence is assessed using the finite-size equations above.

Library and numerical contract¶

#include <bethe/takhtajan_babujian.hpp>
auto state = bethe::takhtajan_babujian::ground_state<long double>(16);
if (!state.converged) { /* state.energy is an incomplete estimate */ }

The result retains centres, deviations, exact half-integer string labels, adjacent conjugate rapidities, energy, momentum, residuals, status and update count. Computation and reporting retain fp64, long double, optional fp128 or explicit MPFR precision. There is no precision fallback or tolerance relaxation.

Real MPFR Newton corrections use normal Uni20 dispatch, including CPU LU. Construct controls at a finite working precision; custom tolerances must have that same precision:

using R = uni20::mpreal;
auto p = uni20::Precision::bits(256);
bethe::takhtajan_babujian::SolverOptions<R> options(p);
auto state = bethe::takhtajan_babujian::ground_state<R>(16, options);

The CLI accepts bethe-tb-pbc 16 --precision bits-256 --roots, or digits-N. Constants, central zero roots, Newton corrections, sums and incomplete states retain the requested precision without changing ambient MPFR settings. Rapidities use uni20::complex<Real>: native types retain their usual complex representation, while MPFR uses Uni20's owning MPC scalar. Structured output keeps separate real and imaginary columns. Disabled scalar support is rejected before exports are opened.

Newton uses an analytic real Jacobian, recoverable partial-pivot linear solves, and a residual-decreasing line search preserving ordered centres and positive, representable deviations. Dense storage is O(L^2), factorization O(L^3). Allocation-overflow checks are not a promise that very large chains fit in available memory. Singular/tiny pivots return an unresolved status without changing Uni20's error policy.

residual_norm=max(max|F_j|,max|G_j|) defaults to a tolerance of 32 epsilon. It is an equation-residual target, not an energy-error bound. A zero iteration budget evaluates the seed only. Incomplete solves retain the final iterate and report iteration limit, stalled, or ill-conditioned. The CLI exits 0 on convergence, 2 for an incomplete solve, 1 for invalid or unsupported input. Convergence identifies the prescribed ground-state branch; it is not a scan or certificate of the entire spectrum.

Validation includes the original multiplicative complex equations through L=128 in every scalar type; independent spin-1 matrix construction and ground-state energy/translation checks at L=4,6,8; the native-precision four-site analytic energy; finite-difference Jacobian checks; symmetry, budget, precision-I/O and invalid-input tests. The spin-basis ED oracle deliberately uses fp64, separately from native-precision analytic/root checks. MPFR checks compare the four-site energy against a 768-bit evaluation of \(-11-\sqrt{41}\), test the original complex equations, and compare every state field in owning mixed-precision Async grids with direct solves. Each grid task owns its controls and roots; Newton iterations within a solve remain sequential.

The next extension is state selection with broken two-strings. Real roots, three-strings, singular solutions and their quantum-number branches need explicit treatment; changing I_j in this ground-only solver is insufficient.

Independent even sizes compose through the existing owning Async grid; the nested sweep example retains complete centers, deviations, conjugate roots and failure diagnostics. Coupled Newton updates and the single-request CLI remain serial. MPFR needs explicit controls, constants and Newton precision plus a finite-deviation/root-range audit. Scheduling does not extend the solver's prescribed singlet family.