Universal coarse geometry of spin systems

TL;DR

The paper argues that quantum spin systems possess an intrinsic, large-scale geometry captured by a universal coarse structure $\\mathcal{E}_{\\phi}$ derived from correlation data $C_{\\phi}$. It develops both static and dynamical coarse structures, $\\mathcal{E}_{\\phi}$ and $\\mathcal{E}_{\\alpha}$, and proves stability under quasi-local perturbations, via connected completions $(\\mathcal{E}_{\\phi})_{con}$. It further shows that key order parameters, including topological order and correlation-decay classes, depend only on these coarse structures, and that locality-preserving circuits preserve or respect this coarse geometry, enabling coarse universality classifications. The framework unifies state- and dynamics-based locality, provides a principled route to coarse-grained universality classes, and points to practical ways to extract emergent geometry from finite quantum-device data.

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 associated coarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state $φ$ on an (abstract) spin system with an infinite collection of sites $X$, we define a universal coarse structure $\mathcal{E}_φ$ on the set $X$ with the property that a state has decay of correlations with respect to a coarse structure $\mathcal{E}$ on $X$ if and only if $\mathcal{E}_φ\subseteq \mathcal{E}$. We show that under mild assumptions, the coarsely connected completion $(\mathcal{E}_φ)_{con}$ is stable under quasi-local perturbations of the state $φ$. 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 to a quantum circuit $α$ and the coarse structure of the state $ψ\circ α$ where $ψ$ is any product state.

Theorems & Definitions (99)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 1.6
• Proposition 1.7
• Definition 2.1
• Definition 3.1
• Example 3.2
• Example 3.3