Distinguishing SO(3)2 and Ising Topological Orders via Four-Point Interferometry

Abstract

Majorana zero modes (MZMs) are canonically described by the Ising topological quantum field theory (TQFT), equivalent to SU(2)2 Chern-Simons theory. We investigate whether the braid-group representation governing MZMs could instead arise as the non-Abelian sector of an SO(3)2 Chern-Simons theory (equivalent to SU(2)4), and whether four-point interferometry can experimentally distinguish these topological orders. The non-Abelian sector of SO(3)2 yields a braid-group representation locally indistinguishable from Ising anyons for two-quasiparticle braiding, but differs fundamentally in its fusion rules. Specifically, the SO(3)2 non-Abelian anyon $v$ (spin-1) obeys $v \times v = 1 + v + \psi$, whereas the Ising $\sigma$ anyon obeys $\sigma \times \sigma = 1 + \psi$. We compute the four-point braiding matrices and evaluate the interferometric visibility for a Fabry-Pérot interferometer enclosing four anyons. The presence of the $v \times v \to v$ fusion channel introduces an additional interference path with a distinct conformal weight, providing a falsifiable signature distinguishing SO(3)2 from Ising.

1. Introduction

Majorana zero modes (MZMs) are localized zero-energy excitations in topological superconductors that exhibit non-Abelian statistics. Canonically, MZMs are described by the Ising topological quantum field theory (TQFT), a framework describing topological phases of matter, which corresponds to the SU(2)2 Chern-Simons gauge theory. Chern-Simons theory is a topological gauge theory in 2+1 dimensions whose quantization yields anyonic quasiparticles. We investigate whether the braid-group representation governing MZMs could instead arise as the non-Abelian sector of an SO(3)2 Chern-Simons theory. We demonstrate that while two-quasiparticle braiding is locally indistinguishable from Ising anyons, the theories differ fundamentally in their fusion rules. We propose that four-point interferometry can experimentally distinguish these two topological orders by detecting the unique fusion channels available in the SO(3)2 theory.

2. Background

The topological classification of MZMs as Ising anyons and their braiding properties were established by Kitaev [1] and Ivanov [2]. The theoretical framework for detecting non-Abelian statistics via Fabry-Pérot interferometry, which measures the tunneling conductance modulated by the topological charge of enclosed quasiparticles, was developed by Stern and Halperin [3] and Bonderson et al. [4]. The mathematical equivalence between SO(3)k Chern-Simons theories and the diagonal modular invariants of SU(2)2k, along with their modular tensor category (MTC) classifications, is detailed by Rowell and Wang [5]. An MTC is the algebraic framework encoding the fusion and braiding rules of anyons. Specifically, SO(3)2 is mathematically equivalent to the diagonal modular invariant of SU(2)4.

3. Analysis

We analyze the MTCs for SU(2)2 (Ising) and SO(3)2 (SU(2)4). In SU(2)2, the non-Abelian anyon is $\sigma$ with quantum dimension $\sqrt{2}$, obeying the fusion rule $\sigma \times \sigma = 1 + \psi$, where $1$ is the vacuum and $\psi$ is a fermion. In SO(3)2, the non-Abelian anyon is $v$ (spin-1) with quantum dimension 2, obeying $v \times v = 1 + v + \psi$.

We compute the four-point braiding matrices (R-matrices and F-matrices) for the non-Abelian anyons in both theories. For two-particle exchanges, the braid generators are locally equivalent up to a basis change. However, the four-point conformal blocks differ. The conformal weights, which determine the topological spin and braiding phases, are $h_\sigma = 1/16$ in SU(2)2 and $h_v = 1/2$ in SU(2)4. The total topological charge of four $\sigma$ anyons in the Ising theory is restricted to $1$ or $\psi$. In contrast, the total charge of four $v$ anyons in SO(3)2 can also be $v$, leading to a distinct phase shift and amplitude modulation in the tunneling conductance.

4. Results

We evaluate the interferometric visibility for a Fabry-Pérot type interferometer enclosing four anyons. The interference pattern depends on the total topological charge of the enclosed anyons. In an Ising system, the four-point interferometry signal exhibits a characteristic $0 \to 1$ switching behavior dependent on the parity of enclosed quasiparticles, with no intermediate phase.

In an SO(3)2 system, the presence of the $v \times v \to v$ fusion channel introduces an additional interference path. This manifests as a secondary interference fringe with a phase shift of $e^{i 2\pi (h_v - h_1)} = e^{i\pi} = -1$ and a modified amplitude scaling. The exact amplitude scaling factor relative to the primary fringes is [to verify]. This secondary fringe provides a falsifiable signature that unambiguously distinguishes the SO(3)2 theory from the Ising theory.

5. Discussion

Several open questions remain. First, can a microscopic condensed matter system, such as a fractional topological insulator or a coupled-wire construction, realize SO(3)2 rather than SU(2)2? Second, how does thermal noise and quasiparticle poisoning affect the visibility of the $v$-channel in realistic devices? The additional fusion channel may be thermally suppressed or masked by poisoning events, requiring low-temperature, high-fidelity interferometry. Finally, are there alternative TQFTs with identical four-point signatures to SO(3)2 that would require higher-order interferometry to resolve? Identifying such theories would necessitate probing higher-dimensional conformal blocks.

6. Conclusion

We have shown that while the SO(3)2 Chern-Simons theory provides a braid-group representation for its non-Abelian anyon $v$ that is locally indistinguishable from the Ising $\sigma$ anyon for two-particle braiding, the theories differ fundamentally in their fusion rules. The $v \times v \to v$ fusion channel in SO(3)2 introduces a distinct topological charge for four anyons. Four-point Fabry-Pérot interferometry can detect this channel via a secondary interference fringe with a phase shift of $-1$ and a modified amplitude, providing a clear experimental signature to distinguish SO(3)2 from the canonical Ising topological order.

References

[1] A. Kitaev, "Anyons in an exactly solved model and beyond," arXiv:cond-mat/0506438. [2] D. Ivanov, "Non-Abelian statistics of half-quantum vortices in p-wave superconductors," arXiv:cond-mat/0107636. [3] A. Stern and B. I. Halperin, "Proposed experiments for probing non-Abelian statistics in fractional quantum Hall states," arXiv:cond-mat/0508447. [4] P. Bonderson, A. Kitaev, and K. Shtengel, "Detecting Non-Abelian Statistics in the $\nu=5/2$ Fractional Quantum Hall State," arXiv:0707.1646. [5] E. Rowell and Z. Wang, "Mathematics of Topological Quantum Computing," arXiv:0710.4299.