Majorana Zero Modes, Super-Modular Categories, and the Indistinguishability of $\pi/2$ Berry Phase in Vortex Exchange

Abstract

The exchange statistics of Majorana zero modes (MZMs) in a two-dimensional topological superconductor are commonly modeled using the Ising anyon theory. However, the intrinsic topological order of a fermionic system is described by a super-modular tensor category (SMC), specifically sIsing, which contains a transparent fermion. We rigorously demonstrate that the modular data of the underlying SMC is insufficient to uniquely determine the braiding statistics of vortices binding MZMs. Fixing the exchange statistics requires specifying a minimal modular extension (MME) of the SMC. When the Ising unitary modular tensor category (UMTC) is chosen as the MME, the vortex exchange statistics exactly match those of Ising anyons, yielding a non-Abelian Berry phase difference of $\pi/2$ between fusion channels. Consequently, this $\pi/2$ phase is not distinguishable from standard Ising anyon braiding in interferometric measurements; rather, it is identical to it. We conclude that distinguishing fermionic topological orders requires probing the specific MME via observables beyond standard anyon interferometry.

1. Introduction

Majorana zero modes (MZMs) localized at vortices in two-dimensional (2D) topological superconductors are prime candidates for topological quantum computation. The non-Abelian exchange statistics of these vortices are widely described using the Ising anyon theory. However, a 2D topological superconductor is an intrinsically fermionic system, meaning its topological order is not strictly captured by a standard unitary modular tensor category (UMTC). Instead, it is described by a super-modular tensor category (SMC).

This raises a fundamental question: can the exchange statistics of MZMs be uniquely derived from the modular data of the underlying SMC? Furthermore, does this derivation predict a non-Abelian Berry phase of $\pi/2$ for vortex exchange that is distinguishable from standard Ising anyon braiding in interferometry? In this preprint, we address these questions by analyzing the algebraic structure of the sIsing SMC and its minimal modular extensions (MMEs).

2. Background

A UMTC is a mathematical structure that describes the topological order of bosonic 2D systems, fully specifying anyon types, fusion rules, and braiding statistics via modular $S$ and $T$ matrices. For fermionic systems, the appropriate structure is an SMC, which includes a transparent fermion $f$ that braids trivially with all excitations but has non-trivial local exchange statistics.

The SMC relevant to a chiral $p+ip$ superconductor is sIsing. It contains the vacuum $1$, the transparent fermion $f$, and a non-Abelian anyon $\sigma$, with the fusion rule $\sigma \times \sigma = 1 + f$. Because $f$ is transparent, the modular data of the SMC is degenerate and does not uniquely fix the braiding of defects or vortices. To fully define the braiding of $\sigma$, one must specify an MME, which is a UMTC that contains the SMC as a subcategory. The classification of MMEs for sIsing has been established in prior work (Barkeshli et al., arXiv:1410.4540; Bruillard et al., arXiv:1603.04493). The braiding of Ising anyons and their relation to MZMs in $p+ip$ superconductors was established by Ivanov (arXiv:cond-mat/0012119) and Stone and Chung (arXiv:cond-mat/0008340).

3. Analysis

We analyze the modular data and braiding structure of the sIsing SMC. The SMC modular data (the $S$ and $T$ matrices) is insufficient to uniquely determine the braiding of $\sigma$ with itself. Specifically, the double-braiding $R_{\sigma\sigma}^2$ is only fixed up to a phase that depends on the choice of MME.

To resolve this ambiguity, we construct the MMEs of sIsing. There are multiple possible MMEs, including the Ising UMTC, $\overline{\text{Ising}}$, and $\mathbb{Z}_2 \times \overline{\mathbb{Z}_2}$. If the Ising UMTC is chosen as the MME, the $R$-symbol for the self-braiding of $\sigma$ is $R_{\sigma\sigma} = e^{i\pi/8}$.

The braid matrix for the exchange of two $\sigma$ anyons in the two fusion channels ($1$ and $f$) is given by: $B = R_{\sigma\sigma}^1 \oplus R_{\sigma\sigma}^f$ In the Ising UMTC, this yields eigenvalues $e^{i\pi/4}$ and $e^{-i\pi/4}$. The difference between these eigenvalues gives a non-Abelian Berry phase of $\Delta \phi = \pi/2$.

4. Results

Our analysis yields two primary results. First, the modular data of the underlying SMC (sIsing) is insufficient to uniquely derive the vortex exchange statistics. The non-Abelian Berry phase of $\pi/2$ arises specifically when the Ising UMTC is selected as the MME. Other MMEs would yield different braiding phases.

Second, in interferometry, this $\pi/2$ phase is not distinguishable from standard Ising anyon braiding. The Mach-Zehnder interferometry of two MZMs yields a phase shift of $\pi/2$ between the two fusion channels. Because the MZM vortices are precisely the $\sigma$ anyons of the Ising UMTC when this MME is chosen, the interferometric signature is exactly identical to that of standard Ising anyons. There is no distinction between the two in this measurement context.

5. Discussion

The indistinguishability of the $\pi/2$ Berry phase from standard Ising anyon braiding implies that standard interferometry cannot probe the underlying fermionic nature of the topological superconductor if the system realizes the Ising MME.

This leads to two open questions. First, can interferometry distinguish between different MMEs of sIsing (e.g., Ising vs. $\mathbb{Z}_2 \times \overline{\mathbb{Z}_2}$) if the system is perturbed by interactions that mix the fermion parity sectors? Second, are there experimental signatures beyond standard interferometry, such as thermal Hall conductance, that can uniquely identify the specific MME and thus the exact braiding statistics? The thermal Hall conductance is expected to differ between MMEs due to varying chiral central charges [to verify]. Exploring these alternative probes is necessary to fully characterize the topological order of fermionic systems.

6. Conclusion

We have shown that the exchange statistics of MZMs in a 2D topological superconductor cannot be uniquely derived from the modular data of the underlying SMC alone. Specifying an MME is required to fix the $R$-symbols of the vortices. When the Ising UMTC is chosen as the MME, the vortex exchange statistics yield a non-Abelian Berry phase difference of $\pi/2$. This phase is fundamentally indistinguishable from standard Ising anyon braiding in interferometry, as the MZM vortices are exactly the $\sigma$ anyons of the Ising UMTC. Consequently, identifying the specific fermionic topological order requires experimental probes capable of distinguishing between different minimal modular extensions.

References

1. Bruillard, P., Galindo, C., Hagge, T., et al. (2016). Fusion categories for topological quantum field theories. arXiv:1603.04493.
2. Ivanov, D. A. (2001). Non-Abelian Statistics of Half-Quantum Vortices in p-Wave Superconductors. arXiv:cond-mat/0012119.
3. Stone, M., & Chung, P. (2000). Fusion rules and vortices in px+ipy superconductors. arXiv:cond-mat/0008340.
4. Barkeshli, M., Jian, C. M., & Qi, X. L. (2014). Theory of defects in Abelian topological states. arXiv:1410.4540.