Topological Orders from Reflection Positive Frustration-free Hamiltonians

TL;DR

This work develops spatial reflection positivity as a foundational principle for topological order, linking bulk ground-state degeneracy to boundary data through a Perron-Frobenius analysis of completely positive maps. It then constructs a boundary net via Osterwalder-Schrader reconstruction from bulk ground states, showing that the boundary algebra is generated by local symmetric operators and reduces to a boundary theory for finite-range interactions. The authors compute explicit boundary algebras for the toric code and Levin-Wen string-net models, illustrating how bulk topological order is encoded in boundary data. Overall, the RP framework provides a general, non-commuting-projector route to bulk-boundary correspondence and suggests connections to DHR theory and Haag duality in 2+1D topological phases.

Abstract

We establish a framework with reflection positivity as the first principle for establishing the boundary theory of topologically ordered quantum spin systems. For any reflection positive frustration-free Hamiltonian, We proved that the local topological quantum order (LTQO) condition of ground states on a disk holds if and only if the ground state on the sphere is non-degenerate. Furthermore, we show that the Osterwalder-Schrader reconstruction produces the boundary local net of operator algebras from the local ground states.

Figures (5)

• Figure 1: Spatial reflection positivity with periodic boundary condition in $\tau$.
• Figure 2: The light-shaded box represents the region $X$, the dark box represents $D\times [-R,R]$; the boundary algebra $\mathcal{A}_+(H^X_0)\widehat{\Pi}(X)$ appears as a corner of $\mathcal{A}_+(H^X_0)$, which is localized in $D\times [0,R]$.
• Figure 3: The toric code model on a square lattice: the shaded region $\mathcal{I}$ is the interval along the boundary of $X_+$ on which $\mathcal{A}_{+}(H^X_0)$ is supported; in the picture we have $L = 10$; the operator $\sigma^z_4\sigma^z_5$ acts on the blue edges and the operator $\sigma^x_7\sigma^x_8$ acts on the red edges. The translations of these operators along $\mathcal{I}$ generate $\mathcal{A}_{+}(H^X_0)$.
• Figure 4: A plaquette $p$ crossing the reflection axis.
• Figure 5: A symmetric tubular neighborhood of the reflection hyperplane that contains $6$ plaquettes across the reflection axis (the dashed line).

Theorems & Definitions (91)

• Theorem A: informal version of Theorem \ref{['thm:: local-indistinguishability and ground state degeneracy']}
• Theorem B: Theorem \ref{['theorem:: ground state of H']}
• Theorem C: Theorem \ref{['thm:: local net of field algebras']}
• Theorem D: Theorem \ref{['thm:: boundary algebra of a local RP frustration-free Hamiltonian']}
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Theorem 2.4: cf. JL2017, Theorem 7.1
• proof