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Symmetric entanglers for non-invertible SPT phases

Minyoung You

TL;DR

The paper addresses the puzzle of symmetric entanglers for non-invertible SPT phases by leveraging topological holography (the symTFT framework). It proves a key theorem: if a bulk duality is a fixed-charge duality (FCD), it preserves boundary charges, thereby implying the existence of a symmetric entangler for the corresponding boundary SPT phases. As a concrete demonstration, it constructs a matrix-product-unitary (MPU) entangler that connects two Rep$(A_4)$ SPT phases, showing the entangler is a globally symmetric finite-depth circuit with index zero. This result decouples stacking structure from symmetric entanglers in the non-invertible setting and suggests a generalized notion of stacking applicable to SBP pairs linked by FCDs, with potential bulk-boundary realizations of the entangler from the symTFT data.

Abstract

It has been suggested that non-invertible symmetry protected topological phases (SPT), due to the lack of a stacking structure, do not have symmetric entanglers (globally symmetric finite-depth quantum circuits) connecting them. Using topological holography, we argue that a symmetric entangler should in fact exist for $1+1$d systems whenever the non-invertible symmetry has SPT phases connected by fixed-charge dualities (FCD). Moreover, we construct an explicit example of a symmetric entangler for the two SPT phases with $\mathrm{Rep}(A_4)$-symmetry, as a matrix product unitary (MPU).

Symmetric entanglers for non-invertible SPT phases

TL;DR

The paper addresses the puzzle of symmetric entanglers for non-invertible SPT phases by leveraging topological holography (the symTFT framework). It proves a key theorem: if a bulk duality is a fixed-charge duality (FCD), it preserves boundary charges, thereby implying the existence of a symmetric entangler for the corresponding boundary SPT phases. As a concrete demonstration, it constructs a matrix-product-unitary (MPU) entangler that connects two Rep SPT phases, showing the entangler is a globally symmetric finite-depth circuit with index zero. This result decouples stacking structure from symmetric entanglers in the non-invertible setting and suggests a generalized notion of stacking applicable to SBP pairs linked by FCDs, with potential bulk-boundary realizations of the entangler from the symTFT data.

Abstract

It has been suggested that non-invertible symmetry protected topological phases (SPT), due to the lack of a stacking structure, do not have symmetric entanglers (globally symmetric finite-depth quantum circuits) connecting them. Using topological holography, we argue that a symmetric entangler should in fact exist for d systems whenever the non-invertible symmetry has SPT phases connected by fixed-charge dualities (FCD). Moreover, we construct an explicit example of a symmetric entangler for the two SPT phases with -symmetry, as a matrix product unitary (MPU).

Paper Structure

This paper contains 11 sections, 68 equations, 1 figure.

Table of Contents

  1. Introduction and background
  2. Fixed-charge dualities preserve symmetries
  3. Examples
  4. SPT phases
  5. cluster states
  6. SPT phases
  7. SPT phase 1: product state
  8. SPT phase 2
  9. Symmetric entangler as an MPU
  10. Discussion
  11. -symbols and torus partition functions for SPT phases

Figures (1)

  • Figure 1: SymTFT setup on $\Sigma \times I$, where $\Sigma$ is a $2$-manifold and $I$ is the interval. $\mathsf{D}$ is the reference Dirichlet boundary condition, and $\mathbb{B}_Q$ is the physical boundary condition. $V_\mu$ is the space of local operators which tells us how the bulk anyons $\mu$ can end on the physical boundary. $W_a^\mu$ is the space of junctions between $\mu$ and the symmetry line $a$ which lives on the reference boundary. This tells us how $W$ can be transmuted into a boundary line $a$. Compactifying the interval leads to a ${\cal C}$-symmetric $1+1$d system $Q$, with the anyon $\mu$ turning into an $a$-twisted sector local operator $O$. Lin_2023