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arXiv:2607.08583

Holographic Theory of Mixed-Dimensional Statistics and Conservation-Encoding Hopping-Operator Algebras

Hanyu Xue, Xiao-Gang Wen

cond-mat.str-elhep-thquant-phSubmitted Jul 9, 2026• Announced Jul 10, 2026• 0 citations
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Abstract

We develop a general framework for the statistics of mixed-dimensional excitations subject to intertwined conservation laws, extending the familiar Fermi statistics with conserved particle number. We define statistics microscopically through a \emph{hopping-operator algebra}: a local operator subalgebra (LOsA) generated by operators that locally move or deform excitations while preserving the conservation law. Nontrivial statistics arise when this subalgebra is nontrivial. We first focus on LOsAs that encode \emph{pointed} conservation laws. These give rise to invertible excitations, whose fusion rules are exactly those of the symmetry defects of a higher group $\cG$. For such $\cG$-conserved excitations in $d$-dimensional space, we show that the corresponding LOsA -- and hence the statistics it defines -- is classified by a cohomology class $[ω] \in H^{d+2}(B\cG;\R/\Z)$, where changing $[ω]$ by a coboundary corresponds merely to a rephasing of the local operators. We further provide a holographic realization: excitations with this prescribed conservation law and statistics live on the boundary of a $\cG$ higher-group gauge theory in $(d+1)$-dimensional space, twisted by $[ω]$. More generally, non-pointed conservation laws and the associated statistics of non-invertible excitations are defined by a pair: a LOsA together with its excitation-complex representation. This is equivalent to the pair consisting of a LOsA and its Hilbert-space representation, which is the data defining a generalized symmetry. Consequently, non-pointed conservation laws and their statistics in $d$-dimensional space are classified by fusion $d$-categories, just as generalized symmetries are. The higher-group results above are the fully-pointed special cases of this more general classification.

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cond-mat.str-elCross-listedarXiv:2606.11527•Jun 11, 2026•0 citations
We develop a general framework for the statistics of mixed-dimensional excitations subject to intertwined conservation laws, extending the familiar Fermi statistics with conserved particle number. We define statistics microscopically through a \emph{hopping-operator algebra}: a local operator subalgebra (LOsA) generated by operators that locally move or deform excitations while preserving the conservation law. Nontrivial statistics arise when this subalgebra is nontrivial.
We first focus on LOsAs that encode \emph{pointed} conservation laws. These give rise to invertible excitations, whose fusion rules are exactly those of the symmetry defects of a higher group \cG\cG\cG. For such \cG\cG\cG-conserved excitations in ddd-dimensional space, we show that the corresponding LOsA -- and hence the statistics it defines -- is classified by a cohomology class [ω]∈Hd+2(B\cG;R/Z)[ω] \in H^{d+2}(B\cG;\R/\Z)[ω]∈Hd+2(B\cG;R/Z), where changing [ω][ω][ω] by a coboundary corresponds merely to a rephasing of the local operators. We further provide a holographic realization: excitations with this prescribed conservation law and statistics live on the boundary of a \cG\cG\cG higher-group gauge theory in (d+1)(d+1)(d+1)-dimensional space, twisted by [ω][ω][ω].
More generally, non-pointed conservation laws and the associated statistics of non-invertible excitations are defined by a pair: a LOsA together with its excitation-complex representation. This is equivalent to the pair consisting of a LOsA and its Hilbert-space representation, which is the data defining a generalized symmetry. Consequently, non-pointed conservation laws and their statistics in ddd-dimensional space are classified by fusion ddd-categories, just as generalized symmetries are. The higher-group results above are the fully-pointed special cases of this more general classification.
Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators---transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in nnn-dimensional space gives rise to a nondegenerate braided fusion nnn-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, noninvertible (higher) symmetries (also known as algebraic higher symmetries), and noninvertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to as symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: boundary uniquely determines the bulk, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry with mixed anomaly and Z4{\mathbb{Z}}_{4}Z4​ symmetry, as well as between many other symmetries, in 1-dimensional space.
We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in nnn-dimensional space is characterized and classified by a local fusion nnn-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also refer to non-invertible gravitational anomaly as categorical symmetry to stress its connection to symmetry. This provides a holographic and entanglement view of symmetries. For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low energy excitations, as long as they have the same categorical symmetry; (4) the classification of gapped liquid phases for bosonic/fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry). This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.
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 ppp- and qqq-dimensional excitations occurs in d=p+q+2d=p+q+2d=p+q+2 spatial dimensions. In this work, we introduce a universal construction of mutual statistics in the adjacent dimension d=p+q+1d=p+q+1d=p+q+1, applicable to excitations obeying ZN\mathbb Z_NZN​ fusion for arbitrary NNN and all excitation dimensions ppp and qqq. The corresponding invariant is the Berry phase accumulated in a simple 4N4N4N-step microscopic unitary process built from local excitation operators on lattices. This process measures the linking of one excitation with the NNN-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\prod X∏X and the nearest-neighbor controlled-ZZZ symmetry ∏CZ\prod CZ∏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.
A global symmetry (0-symmetry) in an n-dimensional space acts on the whole space.A higher symmetry acts on closed submanifolds (i.e., loops and membranes, etc.), and those transformations form a higher group.In this paper, we introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group.We show that an algebraic higher symmetry in a bosonic system in n-dimensional space is characterized and classified by a local fusion n-category.We find that, when restricted to its symmetric sub-Hilbert space, an algebraic higher symmetry can be fully characterized by a noninvertible gravitational anomaly (i.e., a topological order in one higher dimension).Thus, we also refer to a noninvertible gravitational anomaly as categorical symmetry to stress its connection to symmetry.This provides a holographic and entanglement view of symmetries.For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless.Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low-energy excitations, as long as they have the same categorical symmetry; and (4) the classification of gapped liquid phases for bosonic and/or fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry).This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.
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.
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\mathbb{D}_4D4​ 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.