Anomalies on the Lattice, Homotopy of Quantum Cellular Automata, and a Spectrum of Invertible States

TL;DR

The paper develops a topological framework for lattice anomalies using quantum cellular automata (QCA) and FDQC-invertible states, organizing their obstructions into an $\Omega$-spectrum structure and introducing blend and SRE anomalies that govern on-siteability and symmetric trivial states.Anomaly indices are formulated via a Whitehead tower to provide concrete cocycle data, with the Else– Nayak index capturing higher obstructions; a cofiber construction links the QCA spectrum to the FDQC-invertible spectrum, enabling a unified classification of anomalies and symmetry-protected topological (SPT) phases.The authors construct a Globular model of QCAs to compute homotopy groups as blend equivalence classes and demonstrate a detailed 1d fermionic ${\mathbb{Z}}_2$ case, revealing lattice-specific features and tensions with continuum cobordism pictures.The work establishes connections between bulk SPT data, lattice entanglers, and boundary anomalies, and outlines significant open problems, including extensions to approximate QCAs, nontrivial correlation-length invertible states, and deeper links to continuum topological field theories.

Abstract

We develop a rigorous topological theory of anomalies on the lattice, which are obstructions to gauging global symmetries and the existence of trivial symmetric states. We also construct $Ω$-spectra of a class of invertible states and quantum cellular automata, which allows us to classify both anomalies and symmetry protected topological phases up to blend equivalence.

Figures (15)

• Figure 1: This figure shows a 1-dimensional depth 2 circuit $C = \prod_i V_i\prod_i U_i$ composed of two-site unitaries $U_i$ and $V_i$, acting on a local two-site operator $a$ (green). Only a finite segment of an infinite circuit is shown. However, all the blue circuit elements combine with their inverses in the formal product $C a C^{-1}$, leaving a finite product involving only $a$ and the orange circuit elements (the "light cone"). Evaluated this way, $C a C^{-1}$ is a well-defined local operator. We see that the map $\alpha_C(a) = C a C^{-1}$ is a $\star$-algebra isomorphism with spread 2, and is a QCA.
• Figure 2: A depth 2 circuit acting on sites (black), consisting of swap gates (blue) performed in the order shown (first vertical, then diagonal), and realizing counter-propagating translations (red). By combining many of these circuits together in the vertical direction, adding more sites, we may obtain a depth 4 circuit which acts as the identity in the interior of a half-space, but realizes a translation on the boundary. Thus, QCA arise naturally at the boundaries of FDQC. With a little more work, one can show every QCA arises this way (see \ref{['propinvertibilityqcaentangler']}).
• Figure 3: This figure shows a blend between two depth two circuits $C = \prod_i V_i \prod_i U_i$ (blue) and $\prod_i V_i' \prod U_i'$ (orange), induces a blend of the corresponding QCA $\alpha_C$ and $\alpha_{C'}$, with blending interval $I = [-1,1]$ (magenta). For local operators supported in the region $H_{x_1<-1}$, only the blue circuit elements of $C$ will act (compare Fig. \ref{['figFDQCgivesQCA']}), while for $H_{x_1 > 1}$, only elements of $C'$ act. Any two finite depth circuits admit blends, including to the identity, by generalizing this construction.
• Figure 4: The domain $D(\alpha)$ of a stable $d$-QCA $\alpha$ is illustrated. $\alpha$ is required to preserve the subalgebra of operators supported in $D(\alpha)$ and act by the identity on all operators supported outside $D(\alpha)$. It is thus determined by a QCA on the "thickened" $d$-dimensional space $D(\alpha)$. This allows us to freely utilize ancillas and is crucial for obtaining an $\Omega$ spectrum of QCA.
• Figure 5: A "layer shifting" blend of $G$-representations along the first axis (other axes not shown), may be constructed by conjugating a QCA $G$-representation by layer swaps between layer $0$ and $1$ on a half-space of ${\mathbb{Z}}^d$. The result is shown. By construction, these blends satisfy the group law. By \ref{['lemmaGblend']}, this defines a homotopy of the corresponding maps $BG \to {\mathcal{Q}}^d_{\mathcal{H}}$.

Theorems & Definitions (60)

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