[Paper Review] State Space for Planar Majorana Zero Modes

R. Jackiw, So-Young Pi|Sep 21, 2011

Nonlinear Photonic Systems3 citations

TL;DR

This paper derives the state space dimensionality for N planar Majorana zero modes, showing it is $\mathcal{N} = 2^{N/2}$ for even $N$ and $\mathcal{N} = 2^{(N+1)/2}$ for odd $N$, using a fermion parity-preserving Clifford algebra realization. The construction uses Dirac matrices to realize the mode operators in a unitary, orthogonal representation that maintains topological invariance and non-Abelian statistics.

ABSTRACT

Zero modes arising from a planar Majorana equation in the presence of $N$ vortices require an $\mathcal{N}$-dimensional state-space, where $\mathcal{N} = 2^{N/2}$ for $N$ even and $\mathcal{N} = 2^{(N + 1)/2}$ for $N$ odd. The mode operators form a restricted $\mathcal{N}$-dimensional Clifford algebra.

Motivation & Objective

• To provide a unified, mathematically consistent description of the state space for N Majorana zero modes in planar geometry, including both even and odd N.
• To resolve the ambiguity in the state space structure for odd N, which is often overlooked in condensed matter physics due to focus on paired modes.
• To establish a fermion parity-preserving realization of the zero-mode operators using Clifford algebra, ensuring consistency with topological quantum field theory and non-Abelian statistics.
• To extend the known even-N result to odd N by introducing a phantom vortex at infinity, leading to an even-numbered system governed by the standard formula.
• To demonstrate that a two-dimensional representation fails for odd N, necessitating a higher-dimensional Dirac matrix realization to preserve fermion parity.

Proposed method

• Constructs the zero-mode operator algebra using anti-commutation relations $\{a_i, a_j^\dagger\} = \delta_{ij}$ and $a_i = a_i^\dagger$, ensuring Majorana statistics.
• Applies a fermion parity-preserving representation by requiring the mode operators to act on a Hilbert space where their action connects orthogonal states, avoiding diagonalization.
• Uses explicit matrix realizations via Pauli and Dirac matrices: $a = \sigma_1/\sqrt{2}$, $b = \sigma_2/\sqrt{2}$ for $N=2$, and $\boldsymbol{\alpha}, \beta$ matrices for higher $N$.
• For odd $N$, extends the even-$N$ formula by introducing a phantom vortex at infinity, transforming $N$ odd into $N+1$ even, and applying the standard even-$N$ counting.
• Verifies anti-commutation relations through the algebra of $\mathcal{N} \times \mathcal{N}$ Dirac matrices, ensuring orthogonal state evolution and fermion parity conservation.
• Rejects diagonal realizations (e.g., $\sigma_3/\sqrt{2}$) that would break fermion parity by making states eigenstates of the operator.

Experimental results

Research questions

• RQ1 What is the correct dimensionality of the state space for N planar Majorana zero modes when N is odd, given that the standard even-N formula does not apply?
• RQ2 Can a fermion parity-preserving representation of the zero-mode operators be consistently constructed for odd N, and if so, what is its minimal dimensionality?
• RQ3 How does the inclusion of a phantom vortex at infinity unify the treatment of odd and even N in the state space counting?
• RQ4 Why is a two-dimensional representation insufficient for three Majorana zero modes, and what algebraic obstruction prevents it?
• RQ5 What role does the Clifford algebra structure play in ensuring non-Abelian statistics and topological invariance in the state space?

Key findings

• For even $N$, the state space dimension is $\mathcal{N} = 2^{N/2}$, consistent with prior results and confirmed via Pauli matrix realizations for $N=2$.
• For odd $N$, the state space dimension is $\mathcal{N} = 2^{(N+1)/2}$, which is larger than the even-$N$ formula would suggest for the same $N$, due to the need for fermion parity conservation.
• A two-dimensional representation fails for $N=3$ because the anti-commutation relations between three operators cannot be simultaneously satisfied without breaking fermion parity.
• A four-dimensional representation is required for $N=3$, realized using $\mathcal{N} \times \mathcal{N}$ Dirac matrices $\boldsymbol{\alpha}$ and $\beta$, ensuring orthogonal state evolution and parity conservation.
• The state space for $N=4$ is $\mathcal{N} = 4$, verified by extending the $N=3$ Dirac matrix realization to four operators, confirming the general formula.
• The phantom vortex construction at infinity transforms an odd $N$ system into an even $N+1$ system, allowing the use of the standard even-$N$ formula and unifying the description across all $N$.