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.
AI Summary
The paper addresses how to encode fusion, restricted mobility, and statistics in abelian fracton orders within a common algebraic framework. It considers translation-invariant bosonic systems with finite-order excitations, including extended excitations. Fusion complexes are chain complexes over translation Laurent-polynomial rings; a braiding chain map to a dual complex supplies statistical data. Equivalence under quasi-isometries defines braided fusion theories (BFTs).
The framework includes arbitrary translation-invariant Pauli codes and constructs nondegenerate descriptions of every two-dimensional abelian anyon theory, including chiral ones. Using derived duality, Ext modules, and commutative algebra, it derives constraints on excitation content. For nondegenerate theories containing only pointlike excitations, a nonzero fusion module $S$ must be pure $(d-2)$-dimensional Cohen–Macaulay and admit an antisymmetric isomorphism $b:S\to S^\vee$. At odd fusion order these conditions are sufficient, and $(S,b)$ completely determines the BFT. Statistical duality excludes the separate $X$ and $Z$ sectors of X-cube and anisotropic models as standalone pointlike nondegenerate BFTs, even after translation coarsening. Assuming a unique infinite-space ground state, nondegeneracy is proved equivalent to two-sided remote detectability of excitations and detectors.
Physical realizability of every nondegenerate BFT, and phase invariance of its assignment, remain conjectural; the algebraic obstructions become physical exclusions only under these assumptions. A representative-independent abstract formulation at even fusion order and simplified self-statistics data remain unresolved. Infinite fusion order and nonabelian systems are outside scope, and defining abelianity for extended excitations remains open.
No sign of AI-generated textAI use disclosed
The authors used ChatGPT, Claude, and Gemini, with various models
and versions available over the course of this work, to assist with drafting and revising exposition;
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
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.
The paper addresses how to encode fusion, restricted mobility, and statistics in abelian fracton orders within a common algebraic framework. It considers translation-invariant bosonic systems with finite-order excitations, including extended excitations. Fusion complexes are chain complexes over translation Laurent-polynomial rings; a braiding chain map to a dual complex supplies statistical data. Equivalence under quasi-isometries defines braided fusion theories (BFTs).
The framework includes arbitrary translation-invariant Pauli codes and constructs nondegenerate descriptions of every two-dimensional abelian anyon theory, including chiral ones. Using derived duality, Ext modules, and commutative algebra, it derives constraints on excitation content. For nondegenerate theories containing only pointlike excitations, a nonzero fusion module S must be pure (d−2)-dimensional Cohen–Macaulay and admit an antisymmetric isomorphism b:S→S∨. At odd fusion order these conditions are sufficient, and (S,b) completely determines the BFT. Statistical duality excludes the separate X and Z sectors of X-cube and anisotropic models as standalone pointlike nondegenerate BFTs, even after translation coarsening. Assuming a unique infinite-space ground state, nondegeneracy is proved equivalent to two-sided remote detectability of excitations and detectors.
Physical realizability of every nondegenerate BFT, and phase invariance of its assignment, remain conjectural; the algebraic obstructions become physical exclusions only under these assumptions. A representative-independent abstract formulation at even fusion order and simplified self-statistics data remain unresolved. Infinite fusion order and nonabelian systems are outside scope, and defining abelianity for extended excitations remains open.
We study Abelian planon-only fracton orders: a class of three-dimensional (3d) gapped quantum phases in which all fractional excitations are Abelian particles restricted to move in planes with a common normal direction. In such systems, the mathematical data encoding fusion and statistics comprises a finitely generated module over a Laurent polynomial ring \mathbb{Z}[t^±] ℤ [ t ± ] equipped with a quadratic form giving the topological spin. The principle of remote detectability requires that every planon braids nontrivially with another planon. While this is a necessary condition for physical realizability, we observe – via a simple example – that it is not sufficient. This leads us to propose the excitation-detector principle as a general feature of gapped quantum matter. For planon-only fracton orders, the principle requires that every detector – defined as a string of planons extending infinitely in the normal direction – braids nontrivially with some finite excitation. We prove this additional constraint is satisfied precisely by perfect theories of excitations – those whose quadratic form induces a perfect Hermitian form. To justify the excitation-detector principle, we consider the 2d Abelian anyon theory obtained by spatially compactifying a planon-only fracton order in a transverse direction. We prove the compactified 2d theory is modular if and only if the original 3d theory is perfect, showing that the excitation-detector principle gives a necessary condition for physical realizability that we conjecture is also sufficient. A key ingredient is a structure theorem for finitely generated torsion-free modules over \mathbb{Z}_{p^k} [t^±] ℤ p k [ t ± ] , where p p is prime and k k a natural number. Finally, as a first step towards classifying perfect theories of excitations, we prove that every theory of prime fusion order is equivalent to decoupled layers of 2d Abelian anyon theories.
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.
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) pointlike topological excitations with restricted mobility, with nontrivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore they serve as parent orders for both long-range-entangled quantum liquid and nonliquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional Z2 topological order with (i) the X-cube fracton order or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry G and subsystem symmetries of a normal subgroup N along lower-dimensional subregions.
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.
Generally, ``fracton'' topological orders are referred to as gapped phases that support pointlike topological excitations whose mobility is, to some extent, restricted. In our previous work [Phys. Rev. B 101, 245134 (2020)], a large class of exactly solvable models on hypercubic lattices are constructed. In these models, spatially extended excitations possess generalized fractonlike properties: not only mobility but also deformability is restricted. As a series work, in this paper, we proceed further to compute ground state degeneracy (GSD) in both isotropic and anisotropic lattices. We decompose and reconstruct ground states through a consistent collection of subsystem ground state sectors, in which mathematical game ``coloring method'' is applied. Finally, we are able to systematically obtain GSD formulas (expressed as log2GSD) which exhibit diverse kinds of polynomial dependence on system sizes. For example, the GSD of the model labeled as [0,1,2,4] in four-dimensional isotropic hypercubic lattice shows $12{L}^{2}\ensuremath{-}12L+4$ dependence on the linear size L of the lattice. Inspired by existing results [W. Shirley et al., Phys. Rev. X 8, 031051 (2018)], we expect that the polynomial formulas encode geometrical and topological fingerprints of higher-dimensional manifolds beyond toric manifolds used in this work. This is left to future investigation.
The statistics of particles and extended excitations, such as loops and membranes, are fundamental to modern condensed matter physics, high-energy physics, and quantum information science, yet a comprehensive lattice-level framework for computing them remains elusive. In this work, we develop a universal microscopic method to determine the generalized statistics of Abelian excitations on lattices of arbitrary dimension and demonstrate it by deriving the statistics of particles, loops, and membranes in up to three spatial dimensions. Our approach constructs a sequence of local unitary operators whose many-body Berry phase encodes the desired statistical invariant. The required sequence is generated automatically from the Smith normal form of locality constraints and therefore needs no extra physical input. We prove that the resulting invariants are quantized, provide an algorithm that computes them efficiently, and show how they unify familiar braiding and fusion data of particles while also uncovering new self- and mutual statistics of loop and membrane excitations. We further demonstrate that each statistical invariant corresponds to an ’t Hooft anomaly of a generalized symmetry; we show that a nontrivial invariant both (i) obstructs gauging that symmetry and (ii) forbids any short-range-entangled (symmetry-preserving) ground state. This establishes a precise connection between microscopic lattice anomalies and many-body dynamics, providing a generalization of the Lieb-Schultz-Mattis theorem that constrains a wide class of quantum lattice systems.