Shows that XXZ chain, transverse-field Ising model, Levin-Wen string nets, and Kitaev quantum doubles are a single ISA family parameterised by the beta-deformation. At beta=1/2 the BKT transition is a TWIST failure. Walker-Wang models are identified as H2 (BIND-mandatory). Haah codes require twisted coefficients and lie outside the standard H2 tier (fracton sector). The quantum group parameter q = exp(i*pi*beta) encodes the semiring interpolation between tropical (beta=infinity) and metaplectic (beta=0).
Shows that XXZ chain, transverse-field Ising model, Levin-Wen string nets, and Kitaev quantum doubles are a single ISA family parameterised by the beta-deformation. At beta=1/2 the BKT transition is a TWIST failure. Walker-Wang models are identified as H2 (BIND-mandatory). Haah codes require twisted coefficients and lie outside the standard H2 tier (fracton sector). The quantum group parameter q = exp(ipibeta) encodes the semiring interpolation between tropical (beta=infinity) and metaplectic (beta=0).
The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.
The new class of exactly solvable models of one-dimensional spin chains is found. Interactions of spins with crystalline electric fields (CEF) and spin-spin coupling between neighboring sites are taken into account for those models. Spin-spin coupling includes higher-order spin operators (the Stevens operators) and describes the interaction between z th components of the spins. We show that essential parts of the Hamiltonians of these models can be written in terms of exactly solvable transverse field Ising model (TFIM), in which certain combinations of the parameters of spin-spin interactions play the role of the TFIM exchange constant, and components of the CEFs play the role of external transverse magnetic field of the TFIM. The obtained results are valid for any values of spin (or total moments for rare-earth ions) and for the wide class of CEFs. The requirement for the exact solvability is that the space of single spin states should be split by the CEF into the subspaces with dimensions ≤ 2 (i.e., the only singlets and doublets are allowed). Cases of integer and half-integer spins are considered. For finite-size chains with integer spins and open ends we show numerically that energy spectra always contain the states whose energies do not depend on the CEF (the magnetic field for the TFIM). The energies (per spin) of these states decay as L − 1 , where L is the chain length. This behavior qualitatively distinguishes these states from known Majorana edge states and strong zero modes, where the energy decreases exponentially in L .
This manuscript develops a geometric theory of quantum phase transitions that does not begin from a preassigned order parameter, symmetry-breaking pattern, or topological invariant. The basic object is a phase-sensitive state kernel extracted from a ground state, thermal state, band projector, correlation matrix, or quantum geometric tensor. A kernel-resolved phase is defined operationally as a maximal parameter region on which the principal phase-sensitive direction of this kernel can be transported continuously with finite accumulated projective variation. The central diagnostic is the rupture index\kappa=\alpha(1-\beta)=\alpha-\gamma,where \alpha is the local closing exponent of the relevant spectral or kernel separation, \gamma is the independently measured exponent of the transverse carrier amplitude, and \beta=\gamma/\alpha is the relative carrier-amplitude exponent. The theorem-bearing class treated in this paper is the finite-dimensional free-fermion / two-level Dirac-mass class. The transverse-field Ising chain, the SSH chain, and the Kitaev chain are developed as the first model realizations. In all three cases the local channel is a nondegenerate Dirac mass closure with\alpha=1,\qquad \gamma=0,\qquad \beta=0,\qquad \kappa=1.The numerical section is presented as an implementation check of the common local theorem, not as an independent proof of three unrelated mechanisms. A next-harmonic deformation of the SSH model verifies stability under nonuniversal velocity and curvature corrections, while a gapped noncritical control confirms saturation rather than an artificial rupture peak. The paper explicitly delimits BKT, quadratic-touching, multicritical, infinite-dimensional-kernel, interacting-kernel, and non-gap-controlled cases as frontier problems rather than classifying them under the present theorem.
We show that every exactly solvable quantum spin model with a phase transition is a different β-schedule of the Origami instruction-set architecture (ISA). The unifying equation is q = exp(iπβ): the quantum group deformation parameter q underlying the XXZ chain, the transverse-field Ising model (TFIM), Levin–Wen string-net models, and Kitaev's quantum doubles is identically the Gibbs ISA inverse temperature. This gives a single β-axis on which every topological phase of matter has a precise address. The main results are: (i) The XXZ Bethe-ansatz two-magnon R-matrix is the FLIP eigenvalue at q = exp(iπβ), with the BKT transition as a TWIST failure at β = 1/2 (exact, by cos(π/2) = 0). (ii) The 1D XY model and the 2D Kitaev toric code share β = 1/4 and quantum group SU(2)₂ — the same ISA programme in different geometries. (iii) For non-Abelian string-net phases in the SU(2)k family, the quantum dimension d{1/2}(β) = 2 cos(πβ) is an exact phase label, verified for k = 1 through 15. (iv) The cohomological chain H⁰/H¹/H² = Islands/Flatland/The Deep corresponds to classical/toric-code/Walker–Wang codes. Fracton phases require twisted cohomology H*(M; L) beyond standard H². The β-ladder (Table I) places every major model on a single axis: β = 0 (Heisenberg SU(2)), β = 1/5 (Fibonacci anyons, SU(2)₃), β = 1/4 (Ising anyons / toric code, SU(2)₂), β = 1/3 (semion / XY model, SU(2)₁), β = 1/2 (BKT transition, TWIST failure). All results are verified numerically in experiments x446a–x446d. This paper belongs to the Origami ISA series. Paper 443 (Planck's Constant in Disguise, doi:10.5281/zenodo.20752384) establishes the semiring deformation; Paper 445 (Kitaev Menagerie, doi:10.5281/zenodo.20752352) treats the Kitaev models as a special case; Paper 447 (Opcode Rosetta Stone, doi:10.5281/zenodo.20761260) extends the ISA dictionary to twelve models. Keywords exactly solvable models, quantum groups, XXZ model, BKT transition, topological phases, string-net models, toric code, Walker-Wang model, fracton phases, instruction set architecture, beta deformation, Yang-Baxter equation, Bethe ansatz, quantum dimension, Luttinger liquid, cohomology, tensor networks, quantum computing
Ordre topologique à température finie dans le modèle string-net et le modèle du double quantique L'ordre topologique est un ordre quantique particulier qui apparaît dans les systèmes quantiques gappés et fortement interactifs, et qui ne peut pas être décrit par un paramètre d'ordre local et une brisure spontanée de symétrie. En deux dimensions et à température nulle, cet ordre est caractérisé par une dégénérescence de l'état fondamental dépendant de la topologie de la variété, de l'intrication à longue portée et la présence de quasi-particules avec des nombres quantiques et des statistiques d'échange fractionnaires (également appelées anyons). Cette thèse étudie l'ordre topologique à température finie au moyen de deux modèles jouets exactement solubles : le modèle de string-net (réseau de cordes) de Levin et Wen et le modèle du double quantique de Kitaev. L'accent principal est mis sur le modèle de string-net, qui réalise tous les ordres topologiques doublés achiraux, c'est-à-dire tous les ordres topologiques décrits par un centre de Drinfeld. Ce modèle prend comme entrée une catégorie de fusion unitaire et produit le centre de Drinfeld correspondant en sortie. Dans un premier temps, nous dérivons une formule pour les dégénérescences spectrales du modèle, qui dépendent à la fois de la topologie et de l'ordre topologique considéré. En particulier, les dégénérescences dépendent non seulement du centre de Drinfeld mais aussi de la catégorie d'entrée. Ensuite, nous calculons la fonction de partition, à partir de laquelle nous obtenons l'entropie, la chaleur spécifique, et montrons qu'il n'y a pas de transition de phase à température finie. Nous identifions un ensemble particulier d'objets du centre de Drinfeld, appelés fluxons purs, qui dominent le comportement de la fonction de partition dans la limite thermodynamique, et étudions leurs propriétés. Nous obtenons également les moyennes thermiques des opérateurs de cordes fermées et étudions l'information mutuelle. Enfin, nous appliquons notre approche aux modèles du double quantique, où nous dérivons également une formule générale pour les dégénérescences spectrales, la fonction de partition et l'entropie d'intrication, permettant une étude plus générale et détaillée des propriétés à température finie par rapport aux études précédentes.