Topological Orders from Reflection Positive Frustration-free Hamiltonians

TL;DR

This paper develops a framework based on reflection positivity to analyze the boundary theories of topologically ordered quantum spin systems, linking ground state properties to boundary operator algebras.

Contribution

It introduces a reflection positivity-based approach to connect bulk topological order with boundary theories in frustration-free Hamiltonians.

Findings

LTQO condition holds iff ground state on sphere is non-degenerate

Osterwalder-Schrader reconstruction yields boundary operator algebras

Framework applies to topologically ordered quantum spin systems

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.