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A Duality-Odd Order Parameter as a Candidate Intrinsic Selection Mechanism for the Geometry-Matter Distinction

Qian Miao

Published Aug 24, 2026• 0 citations
Publisher

Abstract

We investigate whether an intrinsic duality can define a label-free distinction between two local sectors, without assuming that distinction from the outset. Given a system with an exact, order-2 duality automorphism D pairing two families of local operators, we show the geome-try/matter split can be defined purely by whether D is locally preserved or broken. We develop this construction on the {4, 4} (square) cellulation of T 2. This choice is not an incidental convenience: we derive an Euler-characteristic obstruction showing that no triangulated (simplicial) cellulation of non-trivial topology can support the construction, sharpening an earlier self-duality + non-trivial-topology + locality argument. Our main results are: (1) an exact duality auto- morphism of the minimal L = 2 toric code, with defects localizing exactly where D is broken, together with a continuum Bloch-vector analogue; (2) a generator-operator axiomatization and a Trichotomy Theorem classifying every local extension as defect-creating (a non-empty, even-cardinality defect set), free, or purely topological, verified by exact GF(2) computation on tori and a genus-2 surface, together with an exact discrete defect-pairing theorem and a numerically- supported continuum charge-conservation law; (3) a structural result excluding local sourcing between the two dual sectors by construction under the generator axioms, replaced by an exact topological braiding phase realizing Z2 mutual statistics; and (4) a proof that two natural lattice discretizations of a Gauss-map degree are each, separately, honest topological degrees under an explicit coherence bound, for odd lattice size, with their equality reduced to a single precisely-stated conjecture that we believe but do not prove and for which we identify the specific geometric obstruction a direct proof must overcome — even lattice size breaks the argument at an earlier, purely combinatorial step, which we leave open. All claims are either exact verified computations, proved propositions, or explicitly flagged as conjectural or open. Section 2 situates the construction relative to the pre-existing Kramers–Wannier/duality-defect and loop-quantum-gravity literature.

DOI
10.5281/zenodo.22075223

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We investigate whether an intrinsic duality can define a label-free distinction between two local sectors, without assuming that distinction from the outset. Given a system with an exact, order-2 duality automorphism D pairing two families of local operators, we show the geome-try/matter split can be defined purely by whether D is locally preserved or broken. We develop this construction on the {4, 4} (square) cellulation of T 2. This choice is not an incidental convenience: we derive an Euler-characteristic obstruction showing that no triangulated (simplicial) cellulation of non-trivial topology can support the construction, sharpening an earlier self-duality + non-trivial-topology + locality argument. Our main results are: (1) an exact duality auto- morphism of the minimal L = 2 toric code, with defects localizing exactly where D is broken, together with a continuum Bloch-vector analogue; (2) a generator-operator axiomatization and a Trichotomy Theorem classifying every local extension as defect-creating (a non-empty, even-cardinality defect set), free, or purely topological, verified by exact GF(2) computation on tori and a genus-2 surface, together with an exact discrete defect-pairing theorem and a numerically- supported continuum charge-conservation law; (3) a structural result excluding local sourcing between the two dual sectors by construction under the generator axioms, replaced by an exact topological braiding phase realizing Z2 mutual statistics; and (4) a proof that two natural lattice discretizations of a Gauss-map degree are each, separately, honest topological degrees under an explicit coherence bound, for odd lattice size, with their equality reduced to a single precisely-stated conjecture that we believe but do not prove and for which we identify the specific geometric obstruction a direct proof must overcome — even lattice size breaks the argument at an earlier, purely combinatorial step, which we leave open. All claims are either exact verified computations, proved propositions, or explicitly flagged as conjectural or open. Section 2 situates the construction relative to the pre-existing Kramers–Wannier/duality-defect and loop-quantum-gravity literature.
We extend non-invertible duality concepts from one-dimensional systems to two spatial dimensions by constructing a web of non-invertible dualities for lattice models with subsystem symmetries. For the Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ subsystem symmetry on the square lattice, we build two complementary dualities: a map that sends spontaneous subsystem symmetry-broken (SSSB) phases to the trivial phase (the analogue of the Kramers-Wannier (KW) duality in 1+1D), and a generalized subsystem Kennedy-Tasaki (KT) transformation that maps SSSB phases to subsystem symmetry-protected topological (SSPT) phases while leaving the trivial phase invariant. These dualities are boundary-sensitive. On open lattices, both subsystem KW and KT transformations act as unitary, invertible operators. In particular, the KT map not only matches the bulk Hamiltonians of the dual phases but also carries the spontaneous ground-state degeneracy of the SSSB phase onto the protected boundary degeneracy of the SSPT phase. On closed manifolds, however, both maps become intrinsically non-unitary and non-invertible when restricted to the original Hilbert space. We demonstrate this non-invertibility via ground-state degeneracy matching (in two copies of the Xu-Moore/Ising-plaquette model), analysis of symmetry-twist sectors mapping, and the fusion algebra of the duality operator. Enlarging the Hilbert space to include twisted sectors allows the subsystem KW map to be formulated as a projective unitary preserving quantum transition probabilities, consistent with generalized Wigner-theorem-based constructions. We also show that the KT map faithfully transmits the algebraic content of bulk and edge invariants diagnosing strong SSPT order: although strictly local SSPT repair operators map to highly nonlocal objects in the dual SSSB phase, the essential commutation algebra and the bulk-edge correspondence remain intact.
Electromagnetic duality in the toric code has order two at the level of anyon types and acts as a Hadamard-type logical transformation on the encoded quantum information. Here, we ask whether it can likewise be realized microscopically as an order-two Clifford operation. For the Z2\mathbb{Z}_2Z2​ toric code, we prove that any locality-preserving Clifford realization of electric-magnetic exchange cannot have order two. Our proof is fully general and requires neither translation symmetry with restricted families of system sizes nor a fixed pairing between vertex and plaquette stabilizers. Geometrically, the proof locally emulates the unavoidable crossing between electric and magnetic strings, similar to a cross-cap in a non-orientable manifold. Extending the problem to the ZN\mathbb{Z}_NZN​ toric code, we find that an order-two Clifford realization exists for odd NNN, whereas it is impossible for even NNN. Our results show that the group law of an emergent electromagnetic duality need not lift faithfully to its microscopic Clifford realization.
We construct an exactly solvable extension of the spin-1/21/21/2 Kitaev honeycomb model in which conserved Z2\mathbb{Z}_2Z2​ 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 2/32/32/3-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.
The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable inasmuch as it connects systems commonly associated to different areas of physics -- that of long range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically non-local, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this work, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs.
We propose and analyze a deformation of the 3+1d3+1\mathrm{d}3+1d lattice Z2{\mathbb{Z}}_{2}Z2​ gauge theory that preserves the noninvertible Wegner duality symmetry at the self-dual point. We identify a frustration-free point along this deformation where there are nine exactly degenerate ground states (on a periodic cubic lattice) even at finite volume. One of these ground states is a trivial product state and the rest are the topologically ordered ground states of the 3+1d3+1\mathrm{d}3+1d toric code, and they are exchanged by the Wegner duality. We also prove that the frustration-free point is gapped in the thermodynamic limit. Our model, therefore, realizes a gapped phase with spontaneously broken Wegner duality symmetry. Furthermore, by imposing the Gauss law constraints energetically, all the above features can be realized on a tensor product Hilbert space. Finally, we discuss a generalization of this deformation to the 3+1d3+1\mathrm{d}3+1d lattice ZN{\mathbb{Z}}_{N}ZN​ gauge theory and conjecture the possible phase diagram.
Parameterized Families of Toric Code Phase: ememem-duality family and higher-order anyon pumping
Within the toric-code phase, we study parameterized families of topologically ordered states. We construct 111- and 222-parameter families of local Hamiltonians and confirm their non-triviality via topological pumping. For the 111-parameter family, we show that the ememem-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the 222-parameter case, we construct a ``pump of a pump'' that transports an S1S^1S1-family of a system in one lower spatial dimension. Using similar methods, we also present a 111-parameter family with a higher-order anyon pump that produces corner-localized anyon modes. These constructions provide explicit lattice realizations and concrete diagnostics of family-level topology. We use recently developed boundary algebra methods to study the non-triviality of these families.