Exchange statistics is classified by representations of the symmetric group in three dimensions and of the braid group in two. Both classifications presuppose that the exchanges they represent are physically realizable operations. In fracton phases this premise fails: fractons are immobile, lineons move only along lines, and planons only within planes. We show that the correct replacement is not a new algebraic structure but a computable one. For an excitation whose mobility support is a graph Γ, the group of realizable exchange processes is the graph braid group Bn(Γ)=π1(UCn(Γ)), and abelian statistics form Hom(H1(UCn(Γ)),U(1)). We give an algorithm that derives mobility supports directly from a Pauli stabilizer Hamiltonian by linear algebra over F2, and apply it to the X-cube model, recovering lineon, fracton and fracton-dipole mobility without assuming any of it. We then prove an obstruction: in any lattice model with translation symmetry, the mobility support of a superselection sector is a coset of a subgroup of the translation group, and is therefore a point, a line, a plane, or all of space. No junction can occur, and consequently no translation-invariant fracton phase supports exchange statistics beyond bosons, fermions and ordinary two-dimensional anyons. Breaking translation symmetry evades this: a mobility support with a vertex of degree e≥3 has rankH1(UC2(Γ))=(e−1)(e−2)/2≥1, giving free U(1) exchange phases in three spatial dimensions that are not quantized to ±1. Because π1 is then not the symmetric group, this lies outside the scope of recent three-dimensional no-go theorems, and because no internal index is introduced it is not reducible by the Doplicher–Roberts construction. We exhibit the gauge-invariant signature explicitly: for hard-core particles on a junction the many-body spectrum depends on a statistical parameter θ through a flat, locally undetectable connection on configuration space, whereas on an open line no such parameter exists. All computations are reproducible.
Exchange statistics is classified by representations of the symmetric group in three dimensions and of the braid group in two. Both classifications presuppose that the exchanges they represent are physically realizable operations. In fracton phases this premise fails: fractons are immobile, lineons move only along lines, and planons only within planes. We show that the correct replacement is not a new algebraic structure but a computable one. For an excitation whose mobility support is a graph Γ, the group of realizable exchange processes is the graph braid group Bn(Γ)=π1(UCn(Γ)), and abelian statistics form Hom(H1(UCn(Γ)),U(1)). We give an algorithm that derives mobility supports directly from a Pauli stabilizer Hamiltonian by linear algebra over F2, and apply it to the X-cube model, recovering lineon, fracton and fracton-dipole mobility without assuming any of it. We then prove an obstruction: in any lattice model with translation symmetry, the mobility support of a superselection sector is a coset of a subgroup of the translation group, and is therefore a point, a line, a plane, or all of space. No junction can occur, and consequently no translation-invariant fracton phase supports exchange statistics beyond bosons, fermions and ordinary two-dimensional anyons. Breaking translation symmetry evades this: a mobility support with a vertex of degree e≥3 has rankH1(UC2(Γ))=(e−1)(e−2)/2≥1, giving free U(1) exchange phases in three spatial dimensions that are not quantized to ±1. Because π1 is then not the symmetric group, this lies outside the scope of recent three-dimensional no-go theorems, and because no internal index is introduced it is not reducible by the Doplicher–Roberts construction. We exhibit the gauge-invariant signature explicitly: for hard-core particles on a junction the many-body spectrum depends on a statistical parameter θ through a flat, locally undetectable connection on configuration space, whereas on an open line no such parameter exists. All computations are reproducible.
We develop an algebraic theory of fusion, mobility, and statistics in abelian gapped fracton orders and topological phases. The framework generalizes the algebraic description of abelian anyons to systems with restricted-mobility and spatially extended excitations. We consider translation-invariant bosonic systems whose pointlike and extended excitations have finite fusion order. The basic objects are braided fusion complexes (BFCs): chain complexes encoding fusion and mobility, equipped with a braiding chain map encoding statistics. Their equivalence classes define braided fusion theories (BFTs). Our framework encompasses arbitrary translation-invariant Pauli codes, as well as all abelian anyon theories in two spatial dimensions, including chiral theories. We give evidence for the conjecture that every nondegenerate BFT can be realized as the excitation theory of a gapped phase and that the BFT describing such a phase is a phase invariant. We derive constraints on pointlike fusion and mobility, showing, for example, that neither the Pauli X nor Z sectors of the X-cube and anisotropic models can occur alone in a nondegenerate BFT. For nondegenerate theories with only pointlike excitations, we extract a simpler theory of mutual statistics, consisting of a fusion module and a map to its statistical dual. This theory of mutual statistics is an invariant of the BFT and, at odd fusion order, gives an equivalent description of the full braided fusion theory. We interpret remote detection as the generalization of mutual statistics to fracton orders. In the BFC setting, we give a precise formulation of the excitation-detector principle: every nontrivial excitation can be detected by an operator supported at spatial infinity, and every nontrivial detector detects some excitation. Assuming a unique infinite-space ground state, we prove that this principle is equivalent to nondegeneracy.
Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.
We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile ``fracton'' excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub ``twisted fracton models.'' A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a 3-torus and computing their ground-state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility.
Braiding phenomena, from the charge-flux Aharonov-Bohm effect to anyonic statistics in fractional quantum Hall systems, are paradigmatic manifestations of topology in quantum physics. Ordinary mutual braiding between p- and q-dimensional excitations occurs in d=p+q+2 spatial dimensions. In this work, we introduce a universal construction of mutual statistics in the adjacent dimension d=p+q+1, applicable to excitations obeying ZN fusion for arbitrary N and all excitation dimensions p and q. The corresponding invariant is the Berry phase accumulated in a simple 4N-step microscopic unitary process built from local excitation operators on lattices. This process measures the linking of one excitation with the N-fold fusion junction of the other, encompassing particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. We establish the quantization and bilinearity of the invariant and show that its field-theory response is governed by the Bockstein homomorphism, motivating the name Bockstein braiding statistics. Interpreting the excitation operators as open symmetry operators turns the same invariant into a direct microscopic diagnostic of mixed anomalies between symmetries. We demonstrate this diagnostic in a (1+1)D spin chain, where the nontrivial Bockstein braiding phase proves the mixed anomaly between the spin-flip symmetry ∏X and the nearest-neighbor controlled-Z symmetry ∏CZ. We construct explicit (2+1)D and (3+1)D lattice analogs, yielding new anomalous symmetry pairs, and apply the framework to strongly coupled (3+1)D continuum gauge theories. Nontrivial Bockstein braiding rules out a fully symmetric gapped phase, obstructs simultaneous condensation of the two excitations, and implies fractionalization of higher-form symmetries.
Configuration-space topology (CST) derives exchange statistics from the topology of the space in which identical particles move: the fundamental group pi_1(C_N(M)) is the symmetric group S_N in d >= 3 spatial dimensions (exactly two exchange phases: boson +1, fermion -1) and the braid group B_N in d = 2 (anyons, including the non-abelian braiding used in topological quantum computation). This paper draws the boundary of that framework. It names the five silent scaffolds CST imports from classical geometry -- point-like particles, a fixed background manifold, the deleted hard-core diagonal, externally imposed dimension, adiabatic exchange -- and engages the active counter-literature testing each one (orbifolds, traid groups, graph configuration spaces, extended objects, supersymmetric anyons). It concedes the framework's hard boundary: the spin-statistics connection requires Lorentz symmetry, microcausality, and positive energy, which no topological or logical argument supplies. Against this boundary it reports the status of the QNFO pre-registered derivation program (exchange phase as a logical scalar R = e^{2 pi i s} from the re-entrant mark): T2 now carries construction content -- the +/-1 dichotomy is the ribbon identity evaluated in the d >= 3 sector; T1 is partially constructed; T3 is restated with named boundary conditions. It then states a synthesis conjecture -- the Distinction Calculus lifted into homotopy type theory, with the Grothendieck-Teichmuller group acting on braided monoidal categories, renders boundary-drawing an arithmetic act -- with explicit falsification conditions F1-F3, and differentiates it from the cohesive-HoTT program of Sati and Schreiber. The paper proves no new theorem; its contribution is a verified boundary map of a mature framework and a falsifiable route toward the logical origin of quantum statistics.
As an unconventional realization of topological orders with an exotic interplay of topology and geometry, fracton (topological) orders feature subextensive topological ground-state degeneracy and subdimensional excitations that are movable only within a certain subspace. It has been known, in the exactly solvable three-dimensional X-cube model that universally represents the type-I fracton orders, that mobility constraints on subdimensional excitations originate from the absence of spatially deformable stringlike operators. To unveil the interplay of topology and geometry, in this paper, we study the dynamical signature in the X-cube model in the presence of external Zeeman fields via large-scale quantum Monte Carlo simulation and stochastic analytic continuation. We compute both real-space correlation functions and dynamic structure factors of subdimensional excitations (i.e., fractons, lineons, and planons) in the fracton phase and their evolution into the trivial paramagnetic phase by increasing external fields. We find, in the fracton phase, that the correlation functions and the spectral functions show clear anisotropy exactly caused by the underlying mobility constraints. On the other hand, the external fields successfully induce quantum fluctuations and offer mobility to excitations along the subspace allowed by mobility constraints. These numerical results provide the evolution of a dynamical signature of subdimensional particles in fracton orders, indicating that the mobility constraints on local dynamical properties of subdimensional excitations are deeply related to the existence of fracton topological order. The results will also be helpful in potential experimental identifications in spectroscopy measurements such as neutron scattering and nuclear magnetic resonance.