# Bulletin of the American Physical Society

# APS March Meeting 2022

## Volume 67, Number 3

## Monday–Friday, March 14–18, 2022; Chicago

### Session K65: Topological Superconductors: Theory II

3:00 PM–6:00 PM,
Tuesday, March 15, 2022

Room: Hyatt Regency Hotel -Grant Park C

Sponsoring
Unit:
DCMP

Chair: Victor Vakaryuk, Physical Review

### Abstract: K65.00010 : Desining $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ topological orders using networks of Majorana bound states*

4:48 PM–5:00 PM

#### Presenter:

Mehdi Kargarian

(Sharif University of Technology)

#### Authors:

Mehdi Kargarian

(Sharif University of Technology)

Fatemeh Mohammadi

(Sharif University of Technology)

In this work, we introduce several systems to simulate class of topologically ordered states with $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge symmetries. In the first system, we consider a network of Majorana fermions on a hexagonal lattice (square lattice) so that each vertex hosts six pairs of Majorana fermions. The latters are Kondo coupled to a hexagonal lattice of otherwise free magnetic ions, each carrying spin-$1/2$. We show that in the weak coupling limit, a $\mathbb{Z}_2 \times \mathbb{Z}_2$ ($\mathbb{Z}_2$) topological order is induced into the magnetic ions. Next, we introduce a scenario based on the Kramers pairs of Majorana fermions at the end of nanowires, which are experimentally more accessible than the one-dimensional Kitaev chains by a proper combination of heterostructures. We couple them to a hexagonal lattice of quantum dots, allowing electrons to tunnel to the dots. Using the perturbation theory, we show that the low-energy description of the whole systems is given by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ model of local moments. Majorana surface codes also carry topological order without coupling to free magnetic ions used above. We show that a $\mathbb{Z}_2 \times \mathbb{Z}_2$ topological order is realized in a network of Majorana fermions described above, paving a way to simulate Majorana color codes.

*The authors would like to acknowledge the support from Sharif University of Technology.

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