Local frustration-free constraints can transport amplitudes consistently on every bond yet fail around a ring. We solve this obstruction for an interacting periodic spin-one Motzkin Hamiltonian with arbitrary nonzero complex ordered-bond parameters. The complete finite-size ground-space degeneracy and surviving-sector pattern depend only on the complex product of the bond parameters: for every ring size N ≥ 2, the dimension is 2N + 1 at trivial holonomy and exactly 3 otherwise. On the unit circle, the principal total boundary flux removes every nonextreme charged exact ground state and shifts the two charge-conjugate one-flat bands oppositely. Because those bands are invariant sectors of the full Hamiltonian, a fixed nonzero total flux gives a rigorous many-body gap upper bound of order N⁻². Moreover, every lifted charged sector contains a positive level at or below the squared principal flux divided by 4N, yielding at least 2N − 2 soft eigenvalues, counting multiplicity. Boundary flux therefore collapses the growing exact ground manifold without opening a thermodynamic gap. No gap lower bound, bulk transition, topology, or quantized transport is claimed. This v1 record contains the eight-page combined preprint and its minimal four-file LaTeX source archive. This is a preprint and has not undergone journal peer review.
Local frustration-free constraints can transport amplitudes consistently on every bond yet fail around a ring. We solve this obstruction for an interacting periodic spin-one Motzkin Hamiltonian with arbitrary nonzero complex ordered-bond parameters. The complete finite-size ground-space degeneracy and surviving-sector pattern depend only on the complex product of the bond parameters: for every ring size N ≥ 2, the dimension is 2N + 1 at trivial holonomy and exactly 3 otherwise. On the unit circle, the principal total boundary flux removes every nonextreme charged exact ground state and shifts the two charge-conjugate one-flat bands oppositely. Because those bands are invariant sectors of the full Hamiltonian, a fixed nonzero total flux gives a rigorous many-body gap upper bound of order N⁻². Moreover, every lifted charged sector contains a positive level at or below the squared principal flux divided by 4N, yielding at least 2N − 2 soft eigenvalues, counting multiplicity. Boundary flux therefore collapses the growing exact ground manifold without opening a thermodynamic gap. No gap lower bound, bulk transition, topology, or quantized transport is claimed. This v1 record contains the eight-page combined preprint and its minimal four-file LaTeX source archive. This is a preprint and has not undergone journal peer review.
We construct an exactly solvable extension of the spin-1/2 Kitaev honeycomb model in which conserved Z2 fluxes determine not only the signs but also the connectivity of nearest-neighbor Majorana hopping. At a tuned loop point, hopping survives only across opposite-flux plaquettes, so every vertex has active degree zero or two and the matter Hamiltonian fragments in each flux sector into independent Majorana rings and isolated zero modes. Exact ring spectra and bond counting then bound the matter energy over all local flux configurations and all four Wilson-loop sectors, and split it exactly into a frustrated triangular-lattice Ising term, whose extensively degenerate ground states are the fully packed loop coverings, and a non-negative Majorana residual. On admissible commensurate tori, the residual selects precisely the three translation-related 2/3-vison crystals, which saturate the bound, and an exact fermion-parity identity shows that for antiferromagnetic coupling their vacua also survive projection, making them rigorous ground states of the spin model. Within a single crystal, a depth-one local unitary then maps the ground space onto that of a sheared square-lattice toric code, giving fourfold topological degeneracy. The model thus realizes exact nonperturbative gauge-matter feedback: the flux fixes where the Majorana fermions may move, and their zero-point energy selects in return a topologically ordered vison crystal with spontaneously broken translation symmetry.
This thesis studies the mathematical structure of quantum Hall transport in lattice systems, with an emphasis on relating mutiple definitions of Hall conductance in different mathematical branches: Geometrically as a bulk Chern number for a Bloch bundle; analytically as an edge spectral flow for a half-space Hamiltonian; and in operator algebra language as a many body charge transport index generated by magnetic flux insertion. The main results of the thesis reveal restrictions imposed on these indices by frustration-free structure, and improved locality estimates for interacting many body systems.The first result concerns frustration free free-Fermion (F 4 ) Hamiltonians. For translation-invariant, strictly local, gapped F4 Hamiltonians on Z 2 , we prove that the bulk index vanishes. More precisely, the two Bloch bundle associated with the image/kernel of the momentum space Hamiltonian is shown to be trivial by an algebraic argument using polynomial modules and a free resolution. Thus a frustration-free free-Fermion model cannot realize a nontrivial integer quantum Hall phase at the level of the bulk Chern index.The second new result is the corresponding edge statement. For a translation-invariant F4 Hamiltonian restricted to a half-plane, we prove that the edge index vanishes. We first identified that frustration free is equivalent to flat ground state band. Detailed analysis of spectral flow of the edge Hamiltonian shows that the edge mode is an analytic function and can be absorbed by the flat band. Consequently, the edge spectral flow across a regular energy in the bulk gap is constant and must equal its trivial value below the spectrum. This establishes, in this setting, that the frustration-free structure rules out nonzero edge transport as well as nonzero bulk transport. The third new result is an exponential clustering theorem for finite-volume quantum spin systems with a possibly degenerate and split ground-state sector. Assuming only a uniform spectral gap above a low-energy spectral patch of sufficiently small width, we prove exponential decay of correlations in the form|⟨Ω, ABΩ⟩ − ⟨Ω, AP BΩ⟩| ≤ C∥P∥1∥A∥∥B∥e −cd(X,Y ),for local observables A and B with disjoint supports. This improves the locality control needed in many body Hall conductance arguments, where previous estimates under comparable hypotheses were superpolynomial. Together, these results clarify how locality and spectral gaps constrain both free-Fermion and interacting formulations of topological transport.
We study a frustrated seam produced by literal reflection gluing of Wen's plaquette model on a closed lattice Klein bottle. The seam operators anticommute along two cycles for even circumference and one cycle of twice the length for odd circumference. Their algebra reduces to free Majorana rings, while the commuting bulk stabilizers constrain the allowed ring sectors. This bulk locking yields exact ground energies, excitation gaps and degeneracies, checked by phase-aware Pauli traces and finite-size diagonalization, including two 24-qubit energy predictions. The seam degeneracy is not topological: the single-site seam fields tested split it at second order. We contrast this obstructed gluing with consistent toric-code realizations. An edge register preserves the anyon labels; a Hadamard-dressed site register has colour-dependent seam transport, with ground-state degeneracy 2 or 4 for even circumference and a sign-frustrated exception at odd circumference. For the enumerated lattice isometries and sizes, we compute logical gates and distances, distinguishing abstract qubit relabelling from local hardware depth. Finally, on three 16- and 24-qubit lattices, chiral matter selects a holonomy doublet whose electric-field splitting follows two competing tunnelling routes. These matter results remain finite-size statements. The main result is the exact obstructed-seam solution with bulk sector locking; the register, gate and matter calculations delimit its relation to topological codes.
In one fully specified, machine-checked special case — the charge-k magnetic flux family on an m × m torus of lossy (rank-deficient) links — we disambiguate three distinct quantities that the almost-commuting-operator and topological-protection literatures all call “the gap,” and determine which survive uniformly in the system size m. The local gaps — the commutator norm ‖[U,V]‖, the magnetic Hamiltonian's spectral gap, and the spectral-localizer gap they control — all vanish like the flux-per-plaquette 2πk/m² → 0. Two other gaps are m-uniform: the core gap (the rank/isometry structure of the links, independent of the magnetic data) and the Bott branch gap (well-definedness of the index). The consequence is a sharp negative with a constructive moral: the m-uniform operator-norm separation c(k) between charge and flatness — the Plateau quantity — cannot be sourced from any local spectral gap, and is therefore irreducibly topological (a Voiculescu distance-to-commuting that stays ≈ √2 while every local gap collapses), transported into the non-invertible setting by the rank-structure core gap, not by a spectral gap. This locates exactly where a proof of the Plateau Conjecture must, and must not, look. We then make the positive locale concrete: reading the rank loss as the ideal of an extension reduces the remaining unitary-to-contraction passage to a single off-diagonal Schur block, which numerics (Exp. E, and an adversarial worst-case Exp. F) show is core-gap-controlled and m-uniform — recasting “extend a dimension-free theorem to contractions” as “bound one block.” A direct minimization of the distance to flatness itself (Exp. G) independently fails to collapse the distance, converging up to the same √2 as the index-jump radius — the conjecture's most direct numerical test, passed. The √2 above is the codimension-1 value (d = r+1; the tested case d = 3, r = 2). For codim = d−r ≥ 2, a trivial flat competitor supported on an r-plane orthogonal to the rotating core sits at operator distance exactly 1 from the flux family at every m, so distOpFlat ≤ 1 < √2 there. The separation is therefore codimension-dependent: c(k) = √2 at codimension 1, c(k) = 1 at codimension ≥ 2 — both m- and k-uniform. The Plateau Conjecture (c(k) > 0) holds across 2 ≤ r ≤ d, with uniform floor 1; what is codimension-1-specific is the mechanism — the dimension-free unitary (Voiculescu/Exel–Loring) bound transports only at codimension 1, while for codimension ≥ 2 the binding competitor is the orthogonal one the unitary bound cannot see. House style. Every claim in this note compiles in Lean ([THEOREM]), cites the literature ([CITED]), carries numerical evidence with its script ([EVIDENCE]), or wears a [CONJECTURE] tag. Numerics (experiments A–G, full detail in §9) are the evidence base and are referenced throughout by letter.
Crystalline defects can permute topological charges, but the microscopic information that converts a geometric defect into a particular anyonic action is generally model dependent. We study this issue in the exactly solvable spin-3/2 kagome Γ model on closed 5–7 geometries. Odd faces obstruct a global trivialization of the copy-resolved Clifford frame. For connected even-site geometries, physicality of the generic solvable bond algebra leaves two projective implementations of this obstruction: a charge-preserving lift and a charge-conjugating lift. We construct both lifts, prove that their projected action is independent of the auxiliary odd-subgraph representative, and formulate a local flux-selected protocol for transporting an intrinsic vortex. After assembling the complete gauge orbit and projecting to the spin Hilbert space, the charge-conjugating cycle reverses the moving-core parity while preserving total matter parity. A gauge-invariant fermion-transfer line identifies the action as v ↦ v × ε; an independent |ν| = 2 Majorana Chern certificate supplies the conventional U(1)₄ identification. The bare lift and controlled deformations provide direct comparisons, while exact and numerical gap bounds delimit the range in which the construction is valid.
This preprint determines the complex Lie algebra generated by the original nonlocal ladder charges of the undeformed periodic spin-one Motzkin chain for every length N >= 2. A rank-one factorization shows that the charges act on the known (2N+1)-dimensional ground space and annihilate its orthogonal complement. Their algebra is so(2N+1,C) in its defining representation, establishing the conjectured rank and displayed Cartan matrix while correcting its type designation to B_N. An auxiliary sl(2) triple and a conditional-variance estimate yield a sufficient moment criterion for positive reflection-symmetric weighted ladders of either parity. For the periodic spin-1/2 Fredkin chain with N >= 3, the same criterion gives so(N+1,C) for even N and sp(N+1,C) for odd N. Central magnetization components and simple-coroot coefficients are obtained explicitly for both chains. Under independent signed control by the two Hermitian Motzkin charge components, the reachable group is SO(2N+1); the exact optimal transfer fidelity between pure ground states is derived. Adding quadratic collective magnetization gives full unitary control, and degree two is minimal among polynomial magnetization additions.