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从立方晶格点群对称性涌现Pauli代数

卓冰 蒋

Published Jun 7, 2026• 0 citations
Publisher

Abstract

我们证明泡利代数——由反对易关系 {σ^a, σ^b} = 2δ^{ab} I 定义——从简立方晶格的 O_h 点群对称性中必然涌现。张量积 T_{1u} ⊗ T_{1u} 的 T_{1g} 子空间包含三个反对称生成元,满足 so(3) ≅ su(2) 的李代数。该代数的二维不可约表示正是泡利代数。推导仅使用标准群论,不要求任何关于动力学或量子化的假设。

DOI
10.5281/zenodo.20576944

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This paper reports on a comprehensive study of the long-range interaction of the K39(4s)+39K(4p) asymptotic system. We present a detailed discussion of the R-dependent angular momentum couplings and correlation between the Hund’s case (a) and case (c) molecular states. Analytical expressions for the 16 adiabatic Hund’s case (c) long-range potential curves are derived including the higher order dispersion forces and the effects of retardation. Experimentally, six Hund’s case (c) long-range molecular states (0u+, 1g, and 0g− dissociating to the 4 2S1/2+4 2P3/2 asymptote and 0u+, 1g, and 0g− to the 4 2S1/2+4 2P1/2 limit) are observed with rovibrational resolution by photoassociative spectroscopy of ultracold K39 atoms in a high density magneto-optical trap (MOT). Among the six observed long-range states, the upper 0g− “pure long-range” state has negligible short-range chemical exchange contributions and the measured molecular binding energies (v=0–26) are used to precisely determine the long-range potential constants of the 4s+4p interaction. We determine: C3Π=8.436(14) a.u., C3Σ=16.872(28) a.u., C6Π=6272(94) a.u., and C6Σ=9365(141) a.u.. Molecular constants for the three special pure long-range states, the 0g− and 1u (dissociating to the 4 2P3/2 limit and with potential minimum) and the 1u (dissociating to the 4 2P1/2 and with potential maximum), are reported. The internal consistency of the theoretical model used in this work is confirmed by the excellent agreement between the long-range potential curve of the 1g state obtained in present work (from the 0g− state) and the long-range portion of the RKR potential curve of the 1 1Πg state previously determined by conventional molecular spectroscopy. The radiative lifetime of the K 4p state derived from the dipole–dipole interaction constant C3Π is also in excellent agreement with a recent fast-beam measurement.
We give a computer-assisted proof that the first twelve excitations of the neutral SU(3) lattice gauge Hamiltonian on a single cube form two isolated six-dimensional spectral subspaces. Their representations under spatial symmetry and charge conjugation are A+ ⊕ E+ ⊕ T + and A− ⊕ E− ⊕ T − . Complete electric- 1g g 1u 1u u 1g energy windows, rational Ritz bounds, and full Wilson-output Gram matrices yield certified eigenvalue counts for the untruncated operator. The main theorems use interior thresholds with larger comparison-form margins and give separations of at least 0.78482728 between the clusters and 0.72784037 above the second cluster. A separate refinement improves these values to 0.89782728 and 0.85184037. The neutral gap lies in [4.415605103, 4.978456040]. All multiplicities are retained, and selected matrix entries are audited with an independent Haar-integration algorithm. No ordering of the irreducible components within either cluster, volume-uniform gap, or continuum limit is asserted.