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arXiv:2610.14165

Categorifying Quantum Spin Systems

Matthew Yu

math-phcond-mat.str-elhep-thmath.CTquant-phSubmitted Oct 9, 2026• Announced Oct 13, 2026
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Abstract

We develop a homotopical framework for understanding the space of quantum spin systems based on topological $n$-nets, which assign associative unital algebras to local regions in an $n$-dimensional space in a manner that captures the locality structure of the underlying lattice theories. The net-theoretic perspective admits a natural categorification, in which the category of topological $n$-nets carries the structure of a higher Morita category. Moreover, the nets of greatest physical interest form a Karoubi-complete subcategory that shares many of the characteristic structural features of the category of finite TQFTs. Our main result identifies the group of locality-preserving symmetries of (3+1)-dimensional quantum spin systems (i.e. quantum cellular automata modulo stabilization, finite-depth quantum circuits, and translations) with the full Witt group of nondegenerate braided fusion categories, thereby answering a question posed by Haah.

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38 pages, comments welcome

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We develop a homotopical framework for understanding the space of quantum spin systems based on topological nnn-nets, which assign associative unital algebras to local regions in an nnn-dimensional space in a manner that captures the locality structure of the underlying lattice theories. The net-theoretic perspective admits a natural categorification, in which the category of topological nnn-nets carries the structure of a higher Morita category. Moreover, the nets of greatest physical interest form a Karoubi-complete subcategory that shares many of the characteristic structural features of the category of finite TQFTs. Our main result identifies the group of locality-preserving symmetries of (3+1)-dimensional quantum spin systems (i.e. quantum cellular automata modulo stabilization, finite-depth quantum circuits, and translations) with the full Witt group of nondegenerate braided fusion categories, thereby answering a question posed by Haah.
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the Z8×Z2\mathbb{Z}_8\times\mathbb{Z}_2Z8​×Z2​ subgroup of the Witt group, including the U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ QCAs. The same algebraic structure yields new infinite families of generalized U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ non-Clifford QCAs in spatial dimensions d=4k−1d=4k-1d=4k−1. We also reformulate the 4-dimensional w2w3w_2w_3w2​w3​ QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families: w2nw3mw_2^nw_3^mw2n​w3m​ QCAs in dimension d=2n+3m−1d=2n+3m-1d=2n+3m−1 and w2w4k−1w_2w_{4k-1}w2​w4k−1​ QCAs in dimension d=4kd=4kd=4k. For the 5-dimensional w32w_3^2w32​ and w23w_2^3w23​ QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a Zp\mathbb{Z}_pZp​ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality SSS, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT TTT. In the continuum, they form a central extension of PSL(2,Z4)PSL(2,\mathbb{Z}_4)PSL(2,Z4​) for p=2p=2p=2, and of SL(2,Zp)SL(2,\mathbb{Z}_p)SL(2,Zp​) for odd primes ppp, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For p=2p=2p=2 the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a Zp\mathbb{Z}_pZp​ 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For p=2p=2p=2 they are generated by the non-trivial semion QCA, and for odd prime ppp they are generated by the non-trivial Zp\mathbb{Z}_pZp​ Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.
We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible symmetries. Our framework relies on an approach to dualities whereby dual quantum lattice models only differ in a choice of module 2-category over some input fusion 2-category. Given an arbitrary two-dimensional spin system with an ordinary symmetry, we explain how to perform the (twisted) gauging of any of its sub-symmetries. We then demonstrate that the resulting model has a symmetry structure encoded into the Morita dual of the input fusion 2-category with respect to the corresponding module 2-category. We exemplify this approach by specialising to certain finite group generalisations of the transverse-field Ising model, for which we explicitly define lattice symmetry operators organised into fusion 2-categories of higher representations of higher groups.
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 lattice L\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 algebras A of 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 of S_{3} ; hence, it is non-abelian, in contrast to the case with no symmetry.
We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra – the symmetric subalgebra under a finite Abelian group symmetry G . For systems where each site carries a regular representation of G , we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a ( 2 + 1 ) d G gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of G -symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a Z 2 symmetric subalgebra, demonstrating that it maps to the e - m permutation in the two-dimensional toric code and has an irrational index of 2 . Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.
Abstract Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work, we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators in the the unit matrix-product operator sector. For on-site representations of Hopf algebra symmetries, this larger algebra is the usual tensor product quasi-local algebra. We present a solution to the extension problem using the machinery of Doplicher–Haag–Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.