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Ising–Majorana exchange interface

Definition

Let $m \geq 2$ and let $H = (C^{2})^{\otimes m}$ . A Majorana system of length $2 m$ is a family of operators $γ_{1}, \dots, γ_{2 m} \in End (H)$ satisfying $γ_{j}^{*} = γ_{j}, γ_{j}^{2} = I, γ_{j} γ_{k} = - γ_{k} γ_{j} (j \neq k) .$ Its parity operator is $Π = i^{m} γ_{1} \dots γ_{2 m}$ . For each adjacent pair, put $Q_{1} (j) = \frac{I - i γ_{j} γ_{j + 1}}{2}, Q_{ψ} (j) = \frac{I + i γ_{j} γ_{j + 1}}{2}, U_{j} = \exp ⁣ (- \frac{π}{4} γ_{j} γ_{j + 1}) .$

A Majorana–Ising exchange interface on a fixed parity sector consists of such a system, a choice $η \in {1, - 1}$ and the sector $H_{η} = \ker (Π - η I)$ , together with the following explicit model-identification hypotheses for every $1 \leq j < 2 m$ : the Ising pair-charge projections are identified with $Q_{1} (j), Q_{ψ} (j)$ , and the exchange is $R_{1} Q_{1} (j) + R_{ψ} Q_{ψ} (j)$ with the same relative phase $R_{ψ} / R_{1} = i$ . The last identity is required for all adjacent pairs as part of the interface; the three-anyon supplier below establishes it only in its stated sector and does not prove this all- $m$ identification.

Facts & Assumptions

Given: A Majorana system, a fixed parity sector, and the interface hypotheses in the Definition.

[F1]

In the three- $σ$ , total- $σ$ Ising sector, the supplier gives $R_{1} = e^{- i π / 8}$ and $R_{ψ} = e^{3 i π / 8}$ , hence $R_{ψ} / R_{1} = i$ . Ising fusion data and chiral convention

Proof

1.1givenalgebra

Let $A_{j} = i γ_{j} γ_{j + 1}$ . The Majorana relations give $A_{j}^{*} = A_{j}$ and $A_{j}^{2} = I$ . Therefore $Q_{1} (j) = (I - A_{j}) / 2$ and $Q_{ψ} (j) = (I + A_{j}) / 2$ are self-adjoint idempotents, $Q_{1} (j) Q_{ψ} (j) = 0$ , and $Q_{1} (j) + Q_{ψ} (j) = I$ . They are the orthogonal complementary pair-channel projections stipulated by the interface.

2.1givenF1algebra∎

Since $(γ_{j} γ_{j + 1})^{2} = - I$ , expansion of the exponential gives $U_{j} = 2^{- 1 / 2} (I - γ_{j} γ_{j + 1})$ . Its restrictions to $ran Q_{1} (j)$ and $ran Q_{ψ} (j)$ are respectively $e^{- i π / 4} I$ and $e^{i π / 4} I$ . Using the phase ratio in [F1] as imposed for every generator by the interface, the channel exchange equals $R_{1} Q_{1} (j) + R_{ψ} Q_{ψ} (j) = R_{1} e^{i π / 4} U_{j} .$ Thus it differs from $U_{j}$ by one common scalar.

Depends on

Ising fusion data and chiral convention

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)