nLab Majorana dimer

Idea

In solid state physics and quantum many-body physics, and specifically in the context of topological phases of matter, by a Majorana dimer one means a dimer $a \to b$ (an edge in some graph) that is labeled by an entangled state $\psi_{a b}$ of two qbits (“spins”) characterized by the equation

$$\big( \gamma_a + i p_{a b} \gamma_b \big) \left\vert \psi \right\rangle \;=\; 0 \,,$$

where

• $\gamma_i = f^\dagger_i + f_i$ are Clifford algebra operators defined in terms of the canonical anti-commutation relations $\{f_j, f^\dagger_j\} = \delta_{i j}$ that define qbit spaces of quantum states $\simeq \mathbb{C}^2$ at each of any given lattice sites $i,j, \cdots$,

• $p_{a b} \in \{\pm 1\}$ is a sign (a direction/orientation of the dimer)

(e.g. JGPE 19, III.A)

References

The notion is due to:

• Brayden Ware, Jun Ho Son, Meng Cheng, Ryan V. Mishmash, Jason Alicea, and Bela Bauer, Ising anyons in frustration-free Majorana-dimer models, Phys. Rev. B 94, 115127 2016 (arXiv:10.1103/PhysRevB.94.115127)

  (in relation to Ising anyons)

• Nicolas Tarantino and Lukasz Fidkowski, Discrete spin structures and commuting projector models for two-dimensional fermionic symmetry-protected topological phases, Phys. Rev. B 94, 115115 2016 (doi:10.1103/PhysRevB.94.115115)

Majorana dimer states are used (not under that name though) for discussion of the SYK model and its AdS2/CFT1 dual in:

• Ioanna Kourkoulou, Juan Maldacena, around (2.2) of: Pure states in the SYK model and nearly-$AdS_2$ gravity (arXiv:1707.02325)

Application of Majorana dimers to quantum error correcting codes and holographic entanglement entropy in the Majorana dimer code:

• Alexander Jahn, Marek Gluza, Fernando Pastawski, Jens Eisert, Majorana dimers and holographic quantum error-correcting code, Phys. Rev. Research 1, 033079 (2019) (arXiv:1905.03268)