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Abstract The prospect of realizing highly entangled states on quantum processors with fundamentally different hardware geometries raises the question: to what extent does a state of a quantum spin system have an intrinsic geometry? In this paper, we propose that both states and dynamics of a spin system have a canonically associatedcoarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state$$\phi $$ $\varphi$on an (abstract) spin system with an infinite collection of sitesX, we define a universal coarse structure$$\mathcal {E}_{\phi }$$ $E_{\varphi }$on the setXwith the property that a state has decay of correlations with respect to a coarse structure$$\mathcal {E}$$ $E$onXif and only if$$\mathcal {E}_{\phi }\subseteq \mathcal {E}$$ $E_{\varphi }\subseteq E$. We show that under mild assumptions, the coarsely connected completion$$(\mathcal {E}_{\phi })_{con}$$ ${\left(E_{\varphi }\right)}_{\mathrm{con}}$is stable under quasi-local perturbations of the state$$\phi $$ $\varphi$. We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. We show that several order parameters of a state only depend on the coarse structure of an underlying spatial metric, and we establish a basic compatibility between the dynamical coarse structure associated with a quantum circuit$$\alpha $$ $\alpha$and the coarse structure of the state$$\psi \circ \alpha $$ $\psi \circ \alpha$where$$\psi $$ $\psi$is any product state. 

Abstract We consider the problem of building non-invertible quantum symmetries (as characterized by actions of unitary fusion categories) on noncommutative tori. We introduce a general method to construct actions of fusion categories on inductive limit C*-algberas using finite dimenionsal data, and then apply it to obtain AT-actions of arbitrary Haagerup-Izumi categories on noncommutative 2-tori, of the even part of the$$E_{8}$$ $E_8$subfactor on a noncommutative 3-torus, and of$$\text {PSU}(2)_{15}$$ $\text{PSU}{\left(2\right)}_{15}$on a noncommutative 4-torus. 

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a latticeL\subseteq \mathbb{R}^{n}satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebrasAof operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to\mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)), containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry\mathcal{D}, we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center\mathcal{Z}(\mathcal{D}). We use this to show that, for the double spin-flip action\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}, the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy ofS_{3}; hence, it is non-abelian, in contrast to the case with no symmetry. 

Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on$$\mathbb {Z}^{k}$$, we define a local net of boundary algebras on$$\mathbb {Z}^{k-1}$$, which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type$$\mathrm {II}$$, and we show that Levin-Wen models can have cone algebras of type$$\mathrm {III}$$. Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states. 

Abstract We show that the Levin-Wen model of a unitary fusion category$${\mathcal {C}}$$ $C$is a gauge theory with gauge symmetry given by the tube algebra$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$. In particular, we define a model corresponding to a$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case$${\mathcal {C}}=\textsf{Hilb}(G,\omega )$$ $C=\mathrm{Hilb}(G,\omega )$, we show how our procedure reduces to the twisted gauging of a trivalG-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter. 

Topological domain walls separating 2+1 dimensional topologically ordered phases can be understood in terms of Witt equivalences between the UMTCs describing anyons in the bulk topological orders. However, this picture does not provide a framework for decomposing stacks of multiple domain walls into superselection sectors — i.e., into fundamental domain wall types that cannot be mixed by any local operators. Such a decomposition can be understood using an alternate framework in the case that the topological order is anomaly-free, in the sense that it can be realized by a commuting projector lattice model. By placing these Witt equivalences in the context of a 3-category of potentially anomalous (2+1)D topological orders, we develop a framework for computing the decomposition of parallel topological domain walls into indecomposable superselection sectors, extending the previous understanding to topological orders with non-trivial anomaly. We characterize the superselection sectors in terms of domain wall particle mobility, which we formalize in terms of tunnelling operators. The mathematical model for the 3-category of topological orders is the 3-category of fusion categories enriched over a fixed unitary modular tensor category.