Dihedral twist liquid models from emergent Majorana fermions

Jeffrey C. Y. Teo1 and Yichen Hu2

1Department of Physics, University of Virginia, Charlottesville, VA22904, USA
2The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK

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We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the $SO(N)$ Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral $D_k$ symmetry of $SO(2n)_1$, and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the $\mathcal{C}/G$ twist liquids, where $G=\mathbb{Z}_2$ for $\mathcal{C}=U(1)_l$, $SU(n)_1$, and $G=\mathbb{Z}_2$, $\mathbb{Z}_k$, $D_k$ for $\mathcal{C}=SO(2n)_1$. We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in $SO(2n)_1/G$ by identifying the non-chiral components of the twist liquids with discrete gauge theories.

Strongly interacting electrons in two-dimensions can give rise to exotic quantum-entangled topological phases of matter. Fractional quantum Hall states with fractionally charged quasiparticles, among others, are well-known examples. Recently, substantial theoretical progress has been made in the classification of topological phases with symmetries, where symmetry fluxes can be promoted from classical extrinsic vortices to quantum dynamical excitations. In this work, using an exactly-solvable model, we provide new insight to the physical origin and its many-body microscopic dynamics of a prototypical family of such quantum phases.

We focus on electron-based bosonic topological phases supporting emergent Majorana fermions that are their own anti-particles and are fractions of electrons. The dihedral symmetry that “rotates” the fermion species is promoted to a local gauge invariance and flux-charge excitations are deconfined. We demonstrate how many-body interactions microscopically dictates locality properties of combinations of fermions and thereby governs the local and quantum properties of the symmetry. Flux excitations, such as the metaplectic anyons and the novel “Ising-fluxon”, have exotic properties and may enable quantum technologies protected from environmental decoherences. We further discover a periodic classification scheme for dihedral symmetry gauged bosonic topological phases.

The method employed in our work will be beneficial for future works exploring quantum vortex dynamics and subsequently their usefulness for quantum technologies. Our models will provide useful guidance for experimental search of the desired topological phases in real materials.

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Cited by

[1] Pak Kau Lim, Michael Mulligan, and Jeffrey C. Y. Teo, "Partial fillings of the bosonic $E_8$ quantum Hall state", arXiv:2212.14559, (2022).

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