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Reflection positivity and a refined index for 2d invertible phases

Nikita Sopenko

TL;DR

The paper develops a rigorous framework linking reflection positivity to the classification of 2d invertible quantum spin phases. It introduces a canonical purification that yields reflection-positive representatives for invertible states and shows that orientation-preserving deformations preserve the phase while orientation-reversing ones invert it. It then analyzes twist defects under discrete Z/N rotations, establishing a canonical lift of invertible 2d phases to Z/N-protected invertible phases and proving that RP selects a unique angular momentum, enabling a canonical Z/N-charge assignment. Building on these tools, the authors define a refined index θ_N whose N-th power recovers the prior invariant ω_N, and discuss its conjectured relation to the chiral central charge c_- in conformal boundary theories. The work thus provides a microscopic, RP-based approach to characterize 2d invertible phases beyond the previous mod-24/12 invariants, and suggests a concrete path toward fully capturing boundary chirality via a universal index.

Abstract

We analyze the validity of reflection positivity in the classification of invertible phases of quantum spin systems. We provide a mathematical model in which every 2d invertible state admits a reflection-positive representative. We prove that reflection positivity provides a canonical lift from the set of invertible phases to the set of invertible phases protected by a $\mathbb{Z}/N$-rotational symmetry. Using this, we define a refined version of the index recently introduced by the author. This refined version conjecturally provides a microscopic characterization of an invariant that coincides with the chiral central charge $c_-$ when conformal field theory effectively describes the boundary modes.

Reflection positivity and a refined index for 2d invertible phases

TL;DR

The paper develops a rigorous framework linking reflection positivity to the classification of 2d invertible quantum spin phases. It introduces a canonical purification that yields reflection-positive representatives for invertible states and shows that orientation-preserving deformations preserve the phase while orientation-reversing ones invert it. It then analyzes twist defects under discrete Z/N rotations, establishing a canonical lift of invertible 2d phases to Z/N-protected invertible phases and proving that RP selects a unique angular momentum, enabling a canonical Z/N-charge assignment. Building on these tools, the authors define a refined index θ_N whose N-th power recovers the prior invariant ω_N, and discuss its conjectured relation to the chiral central charge c_- in conformal boundary theories. The work thus provides a microscopic, RP-based approach to characterize 2d invertible phases beyond the previous mod-24/12 invariants, and suggests a concrete path toward fully capturing boundary chirality via a universal index.

Abstract

We analyze the validity of reflection positivity in the classification of invertible phases of quantum spin systems. We provide a mathematical model in which every 2d invertible state admits a reflection-positive representative. We prove that reflection positivity provides a canonical lift from the set of invertible phases to the set of invertible phases protected by a -rotational symmetry. Using this, we define a refined version of the index recently introduced by the author. This refined version conjecturally provides a microscopic characterization of an invariant that coincides with the chiral central charge when conformal field theory effectively describes the boundary modes.

Paper Structure

This paper contains 21 sections, 23 theorems, 10 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Canonical purification and locality
  3. Organization of the paper
  4. Invertible phases of 2d quantum spin systems
  5. Quantum spin systems
  6. Good conical covers
  7. Invertible phases
  8. Spatial deformations
  9. Reflection positivity and complex conjugation
  10. Twist defects and canonical angular momentum
  11. Twist defect states
  12. Rotational invariance
  13. Rotational invariance and CRT-symmetry
  14. A refined index for 2d invertible phases
  15. The index
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $\psi_1, \psi_2$ be invertible states on ${\mathbb R}^2$, and let $p \in {\mathbb R}^2$. Suppose for any good conical cover $C$ with an apex at $p$ there exist states $\tilde{\psi}_1$, $\tilde{\psi}_{2}$ on a spin system ${\mathcal{A}}$ such that $\tilde{\psi}_1$ is a trivial extension of $\psi_

Figures (7)

  • Figure 1: An observable ${\mathcal{O}}$ and its time reversed reflection $\overline{{\mathcal{O}}}'$.
  • Figure 2: An embedding of the Bravyi chain into a spin system on ${\mathbb R}^2$ separated by an orange line into two half-planes.
  • Figure 3: A twist defect for $N=3$.
  • Figure 4: A twist defect state on an $N$-sheeted branched cover of a complex plane can be transformed into a ${\mathbb Z}/N$-rotationally invariant state by a map $z \to z^{1/N}$, $z \in {\mathbb C}$.
  • Figure 5: An example of a good conical cover of ${\mathbb R}^2$ that consists of six cones.
  • ...and 2 more figures

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.1
  • Definition 2.4
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • ...and 44 more