String operators for Cheshire strings in topological phases

TL;DR

This paper investigates the creation, deformation, and fusion of Cheshire strings in 3+1D topological phases, revealing that their generation requires linear depth circuits, unlike trivial excitations.

Contribution

It demonstrates that creating Cheshire strings necessitates linear depth circuits, distinguishing them from trivial excitations and providing insights into their manipulation in topological phases.

Findings

Cheshire strings can be created with linear depth circuits.

Deformation and fusion of Cheshire strings are achievable with finite depth circuits.

Creation of Cheshire strings is fundamentally different from trivial excitations.

Abstract

Elementary point charge excitations in 3+1D topological phases can condense along a line and form a descendant excitation called the Cheshire string. Unlike the elementary flux loop excitations in the system, Cheshire strings do not have to appear as the boundary of a 2d disc and can exist on open line segments. On the other hand, Cheshire strings are different from trivial excitations that can be created with local unitaries in 0d and finite depth quantum circuits in 1d and higher. In this paper, we show that to create a Cheshire string, one needs a linear depth circuit that acts sequentially along the length of the string. Once a Cheshire string is created, its deformation, movement and fusion can be realized by finite depths circuits. This circuit depth requirement applies to all nontrivial descendant excitations including symmetry-protected topological chains and the Majorana chain.