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.
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.
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.
Non-invertible symmetries in quantum many-body systems generally give rise to sequential unitary circuits that move symmetry defects. In this paper, we investigate invariants defined by sequences of such circuits, which move non-invertible defects and generate a Berry phase evaluated on quantum states with defects. We show that this Berry phase generally defines an invariant under local deformations, provided that the sequential circuits preserve the locality of those deformations. This invariant also rules out a short-range-entangled state that preserves the non-invertible symmetry, thereby signaling the 't Hooft anomaly of a non-invertible symmetry purely in terms of unitary operators acting on a state. We then apply this framework to loop excitations in three spatial dimensions and identify a new loop excitation in the (3+1)D D4 topological order, which we dub a non-Abelian fermionic loop. Using the invariant of sequential circuits, we characterize the statistics of non-Abelian fermionic loops. In addition, we find a new (3+1)D mixed topological order with a single non-Abelian fermionic loop, whose long-range entanglement is protected by an invariant of sequential circuits.
Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the e2m2-loop in the (4+1)D Z4 toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of −1. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by H5(B2G,U(1)). Beyond (4+1)D, the fermionic-loop toric codes form a family of Z2 topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using Z4 qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of Z2 topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that p-dimensional excitations in 2p+2 spatial dimensions can support anyonic p-brane statistics for only even p.
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.
Symmetries rigidly delimit the landscape of quantum matter. Recently uncovered spatially modulated symmetries, whose actions vary with position, enable excitations with restricted mobility, while Lieb-Schultz-Mattis (LSM) type anomalies impose sharp constraints on which lattice phases are realizable. In a one-dimensional spin chain, gauging procedures have linked modulated symmetry to LSM type anomaly, but a general understanding beyond 1D remains incomplete. We show that generalized LSM anomalies force the emergence of the modulated symmetries upon gauging. We construct explicit lattice models in two and three spatial dimensions and develop complementary field-theoretic descriptions in arbitrary spatial dimensions that connect LSM anomaly inflow to higher-group symmetry structures governing the modulated symmetries. Our results provide a unified, nonperturbative framework that ties together LSM constraints and spatially modulated symmetries across dimensions.
An ultimate theory of a phase of matter should describe all its universal properties via quantities that are measurable numerically and experimentally. In this work, we present a microscopic universal theory of symmetry-enriched topological quantum spin liquids (TQSLs) in two spatial dimensions, which directly utilizes microscopically measurable quantities to describe the universal properties. This theory applies to generic TQSLs, which can be Abelian or non-Abelian, chiral or non-chiral. The symmetries are also general, which can include both internal and lattice symmetries, unitary and anti-unitary symmetries, and discrete and continuous symmetries. There can be spin-orbit coupling, the microscopic degrees of freedom may transform linearly or projectively under the symmetries, and the symmetries can permute anyons. The input of the theory is some microscopic states with anyons, operators that control the dynamics of anyons, and symmetry actions in the TQSL, and its output is a set of data characterizing the universal properties, whose underlying mathematical structure is a generalization of category theory. Based on this theory, we find an explicit bijective map between the universal data characterizing a TQSL with a symmetry described by a group G, where the symmetry actions may include both lattice and internal symmetries, and the corresponding universal data for a TQSL with only an internal symmetry group G, and thus establish a precise crystalline equivalence principle. We demonstrate our theory in symmetry-enriched TQSLs realized on quantum processors based on superconducting qubits, trapped ions, and Rydberg atoms, and in each example we verify the Lieb-Schultz-Mattis anomaly matching condition. Our theory provides a solid basis for identifying and manipulating symmetry-enriched TQSLs, which further paves the way for fault-tolerant quantum computation based on these systems.