Critical three-state Potts levels¶

The worked tutorial plots the charged branch and vacuum finite-size scaling, with downloadable data and charge/momentum checks.

bethe-potts-pbc computes the finite-ring vacuum and a selected charged one-hole branch at every lattice momentum. These are solutions of finite-size Bethe equations, not energies obtained by inserting conformal dimensions into an asymptotic formula. The branch supplies a useful discrete-symmetry benchmark for MPS excitation calculations.

bethe-potts-pbc 16 --json potts.json --csv potts.csv
bethe-potts-pbc 64 --momentum-index 1 --precision long-double
bethe-potts-pbc 32 --branch charged --charge -1 --tsv charged.tsv
bethe-potts-pbc 64 --threads 8 --csv levels.csv --no-retain --quiet
bethe-potts-pbc 128 --branch ground --precision fp128
bethe-potts-pbc 16 --precision bits-256 --threads 4 --json high-precision.json
bethe-potts-pbc --references

fp64 is the default; native long-double and optional MPLAPACK-enabled fp128 arithmetic are supported. bits-N and digits-N select explicit MPFR precision when Uni20's GMP/MPFR/MPC scalar family is enabled; these Potts solves do not need MPLAPACK. Build instructions and output conventions apply.

Precision reaches the anisotropy, twists, auxiliary roots, energy map, momentum, velocity, bulk energy and failed-iterate diagnostics. It does not change the selected physical family or remove iteration limits. The residual tolerance controls the logarithmic equations; it is not an energy-error bound. Independent two-site algebraic energies and a 150-digit diagonalization of the three-site clock Hamiltonian test accuracy beyond binary128.

Library calls with uni20::mpreal take explicit bethe::SolverOptions<uni20::mpreal> constructed from uni20::Precision::bits(N). The analytic helpers accept the same precision: velocity<uni20::mpreal>(p) and bulk_energy_density<uni20::mpreal>(p). Owning Async grids can mix requests at different precisions, retaining each request's controls and complete result until its final completion value is ready; they do not change ambient MPFR settings.

--threads N overlaps independent charged momentum solves after computing one vacuum reference. One worker is the default; vacuum-only and single-momentum requests remain direct serial calculations. Each coupled Bethe solve remains sequential, with its own roots and scratch. Charge-conjugate rows share one solve, preserving the original state counting.

The shared Async grid retains at most 128 active or uncollected results, with concurrency capped by that window and the selected momentum count. Rows stream on the calling thread in momentum order, with +1 before -1 when both charges are requested. Numerical failures and missing-reference statuses remain explicit rows. --no-retain discards delivered rows. CPU time includes the vacuum solve and the complete ordered level grid, including row output. See measurements and limits.

Hamiltonian and physical labels¶

For a ring of L clock sites, use

Here \(X|a\rangle=|a+1\pmod3\rangle\) and \(Z|a\rangle=\omega^a|a\rangle\), with \(\omega=e^{2\pi i/3}\). The exchange and critical transverse field are both one. The two oriented bonds are both included when L=2. Off-critical, chiral, antiferromagnetic and nonperiodic Potts chains are outside this frontend.

charge is q=0,+1,-1, the exponent in the eigenvalue \(\omega^q\) of \(\prod_j X_j\). k labels physical translation by one clock site, with momentum \(p=2\pi k/L\). Charge conjugation gives equal energies at q=+1 and q=-1, at the same momentum. Spatial reflection relates k and L-k. These are distinct symmetries; do not conflate their multiplicities.

The Hamiltonian is \(\sqrt3/2\) times the normalization of Dasmahapatra et al.. In our units the exact thermodynamic energy density and velocity are

Why use an auxiliary XXZ problem?¶

The Potts and twisted-XXZ Hamiltonians represent the same Temperley–Lieb generators, with loop weight \(\sqrt3\). The auxiliary XXZ ring has 2L spin-half sites and \(\Delta=\sqrt3/2\). With the usual spin-half XXZ energy including its ferromagnetic reference constant, the map is

This follows by writing the clock Hamiltonian as \(-\sqrt3\sum_{j=1}^{2L}e_j+2L\) and using the XXZ realization of the same generators. One clock translation advances two auxiliary sites. The map and its representation-theory qualifications are discussed in Fukai et al., Sections 2.2–2.3 and Nichols, Section 6.

Let \(\gamma=\pi/6\) and store the L real auxiliary roots in the bounded coordinate \(z_j=\tanh\lambda_j/\tan(\gamma/2)\). The shared XXZ solver solves

Roots stay strictly inside the finite-real branch. No finite-size string deviations are discarded. The internal twist parameter extends the shared iteration; it does not widen the public XXZ explicit-label solver's supported window or turn it into an arbitrary-twist spectrum enumerator.

Physical-root selection and counting audit¶

An arbitrary converged twisted-XXZ solution is not a physical Potts level. At this root of unity, auxiliary representations contain submodules which must be excluded or combined differently in the Potts representation. Even the small-ring decomposition is not simply the entire zero-magnetization XXZ spectrum. A regression explicitly constructs an excluded level in the vacuum twist sector by adding a root at negative infinite rapidity to a zero-twist XXZ state with one fewer root. Its Bethe equations hold and its energy is unchanged, but that energy is absent from every charge/momentum block of the four-site clock Hamiltonian. Root convergence is not a physical-state test.

The public selection is fixed as follows:

Use 0≤h<L, not 0≤h≤L: the second endpoint has the same physical momentum and energy as h=0 and is not another level of either charge. The retained family contains 2L charged levels, or 2L+1 rows including the vacuum. At even L, k=L/2 is its own reflection partner; it still has two charge copies, not four. Summing the occupied labels gives \(\sum I_j=-h\), consistent with the physical momentum map above.

This is a selected family, not a complete decomposition of the Potts Hilbert space of dimension \(3^L\). The charged roots at L=4 reproduce the real-root examples and their Potts identifications in Fukai et al., Tables 3 and 11. Independent clock-Hamiltonian tests resolve both charges and all momenta for L=2 through 6. The family is the lowest charged level in each of those tested blocks; the API does not claim a proof of a global sector minimum for arbitrary L. Convergence alone is never used to classify an arbitrary auxiliary solution as physical.

The direct Potts Bethe equations use a different set of complex roots, including a paired-root ground sea and auxiliary “ghost” species involved in counting. Their bookkeeping is not interchangeable with these L auxiliary XXZ roots. The audit of those species in Dasmahapatra et al., Section 2 and Appendix A, is the reason this frontend does not offer generic direct-root enumeration. Neutral excited levels, other charged families, arbitrary root labels and full-spectrum --excitations all remain unimplemented.

Output and finite-size checks¶

The levels table has branch, charge, k, momentum, energy, gap, x_scaled, residual, iterations, and status. All gaps subtract the separately calculated finite-ring vacuum, even with --branch charged. --branch all means all supported families, not all physical excitations. --momentum-index and --charge filter only the charged rows; the vacuum remains the reference and is output unless --branch charged is requested.

The scaled gap is an estimator,

The vacuum approaches central charge c=4/5. The charged k=0 state approaches scaling dimension x=2/15; k=1,L-1 approach its first descendants at x=17/15. Finite-size corrections are retained, not fitted away. At fixed physical momentum the branch tends to \(3\sqrt3\sin(p/2)\) for \(0\le p\le2\pi\). Energies alone provide no spectral weights or guarantee that a particular observable couples to a level.

The normalized residual is \(\max_i|F_i|/(2L)\), in radians. The default tolerance is 32 machine epsilons; --max-iterations limits sweeps per solve. A zero budget evaluates the initial roots but does not manufacture a level. Unconverged energies are null/empty; gaps also require a converged vacuum. Incomplete numerical output exits 2, invalid arguments exit 1. Invalid inputs are checked before opening or overwriting exports. The default site budget is 512, adjustable with --max-sites; each sweep costs O(L²), and a full momentum scan requires L charged solves. Charge conjugation reuses a solve.

Validation includes exact L=2 expressions, independent 75-digit clock-matrix references at L=3, all momentum/charge blocks through L=6, reflected roots, exponentiated Bethe equations, conformal scaling at L=16,32,64, and explicit failure/serialization contracts. Separate sparse-oracle audits at L=8 and 9 match all selected levels to within 1.4e-13 in double precision. The independent oracle can be rerun without invoking the Bethe code:

python3 scripts/reference_potts.py --sites 3 --digits 75
OPENBLAS_NUM_THREADS=1 python3 scripts/reference_potts.py --sites 8 --backend scipy

The C++ entry points are potts::ground_state<Real>(L) and potts::charged_one_hole<Real>(L, charge, k) in bethe/potts.hpp. The result keeps auxiliary roots and half-integer labels for inspection. Its energy is a last-iterate diagnostic unless converged is true; the frontend withholds unconverged observables.