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Torus algebra and logical operators at low energy

Ying Chan, Tian Lan, Linqian Wu

Abstract

Given a modular tensor category $\mathscr{C}$, we construct an associative algebra $\mathrm{Tor({\mathscr{C}}})$, which we call the torus algebra. We prove that the torus algebra is semisimple by explicitly constructing all the simple modules. Suppose that a topological ordered phase described by $\mathscr{C}$ is put on a torus. Physically, each simple module over $\mathrm{Tor({\mathscr{C}}})$ consists of the low energy states on the torus with one anyon excitation, or equivalently, the ground states on a punctured torus where the anyon is enclosed by the puncture. Elements in $\mathrm{Tor({\mathscr{C}}})$ can be physically interpreted as anyon hopping processes on the torus. We give the precise formula how an arbitrary logical operator on the low energy states on a torus can be realized by moving anyons on the torus. Our work thus provides a theoretical proposal that the low energy states on a torus can serve as topological qudits and one can arbitrarily manipulate them by moving anyons around.

Torus algebra and logical operators at low energy

Abstract

Given a modular tensor category , we construct an associative algebra , which we call the torus algebra. We prove that the torus algebra is semisimple by explicitly constructing all the simple modules. Suppose that a topological ordered phase described by is put on a torus. Physically, each simple module over consists of the low energy states on the torus with one anyon excitation, or equivalently, the ground states on a punctured torus where the anyon is enclosed by the puncture. Elements in can be physically interpreted as anyon hopping processes on the torus. We give the precise formula how an arbitrary logical operator on the low energy states on a torus can be realized by moving anyons on the torus. Our work thus provides a theoretical proposal that the low energy states on a torus can serve as topological qudits and one can arbitrarily manipulate them by moving anyons around.
Paper Structure (10 sections, 14 theorems, 65 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Torus algebra
  4. Central idempotent decomposition
  5. Simple modules and punctured torus
  6. Unitary structures: involution, inner product on module, unitary module
  7. Punctured modular transformation
  8. Examples
  9. The Chiral Ising Anyon Model
  10. Fibonacci anyon model

Key Result

Lemma 2.1

Figures (1)

  • Figure 1: TQFT picture of punctured torus. The orange part denotes the elements of $\mathrm{Tor({ {\cal C} }})$. The blue part denotes modules over $\mathrm{Tor({ {\cal C} }})$. And the red circle is the puncture.

Theorems & Definitions (38)

  • Lemma 2.1
  • Definition 3.1
  • Remark 3.2
  • proof
  • Remark 3.3
  • Definition 4.1: Centrifugal loop
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 28 more