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arXiv:1107.2731 · Physical Review B

Weakly coupled non-Abelian anyons in three dimensions

Michael Hartley Freedman, Matthew B. Hastings, Chetan Nayak, Xiao-Liang Qi

Published Dec 15, 2011• 41 citations
arXivHTMLPDF

Abstract

We introduce a Hamiltonian coupling Majorana fermion degrees of freedom to a quantum dimer model. We argue that, in three dimensions, this model has deconfined quasiparticles supporting Majorana zero modes obeying nontrivial statistics. We introduce two effective field theory descriptions of this deconfined phase in which the excitations have Coulomb interactions. A key feature of this system is the existence of topologically nontrivial fermionic excitations, called ``Hopfions'' because, in a suitable continuum limit of the dimer model, such excitations correspond to the Hopf map and are related to excitations identified by Ran et al. [Phys. Rev. B 84, 184501 (2011)]. We identify corresponding topological invariants of the quantum dimer model (with or without fermions) that are present even on lattices with trivial topology. The Hopfion energy gap depends upon the phase of the model. We briefly comment on the possibility of a phase with a gapped, deconfined ${\mathbb{Z}}_{2}$ gauge field, as may arise on the stacked triangular lattice.

DOI
10.1103/physrevb.84.245119

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We introduce a Hamiltonian coupling Majorana fermion degrees of freedom to a quantum dimer model. We argue that, in three dimensions, this model has deconfined quasiparticles supporting Majorana zero modes obeying nontrivial statistics. We introduce two effective field theory descriptions of this deconfined phase in which the excitations have Coulomb interactions. A key feature of this system is the existence of topologically nontrivial fermionic excitations, called ``Hopfions'' because, in a suitable continuum limit of the dimer model, such excitations correspond to the Hopf map and are related to excitations identified by Ran et al. [Phys. Rev. B 84, 184501 (2011)]. We identify corresponding topological invariants of the quantum dimer model (with or without fermions) that are present even on lattices with trivial topology. The Hopfion energy gap depends upon the phase of the model. We briefly comment on the possibility of a phase with a gapped, deconfined Z2{\mathbb{Z}}_{2}Z2​ gauge field, as may arise on the stacked triangular lattice.
Dimer models have long been a fruitful playground for understanding topological physics. Here, we introduce a class, termed Majorana-dimer models, wherein bosonic dimers are decorated with pairs of Majorana modes. We find that the simplest examples of such systems realize an intriguing, intrinsically fermionic phase of matter that can be viewed as the product of a chiral Ising theory, which hosts deconfined non-Abelian quasiparticles, and a topological ${p}_{x}\ensuremath{-}i{p}_{y}$ superconductor. While the bulk anyons are described by a single copy of the Ising theory, the edge remains fully gapped. Consequently, this phase can arise in exactly solvable, frustration-free models. We describe two parent Hamiltonians: one generalizes the well-known dimer model on the triangular lattice, while the other is most naturally understood as a model of decorated fluctuating loops on a honeycomb lattice. Using modular transformations, we show that the ground-state manifold of the latter model unambiguously exhibits all properties of the $\text{Ising}\ifmmode\times\else\texttimes\fi{}({p}_{x}\ensuremath{-}i{p}_{y})$ theory. We also discuss generalizations with more than one Majorana mode per site, which realize phases related to Kitaev's 16-fold way in a similar fashion.
We show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of U(1)U(1)U(1) and Z2{Z}_{2}Z2​ gauge theories in d=3+1.d=3+1.d=3+1. The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the U(1)U(1)U(1) case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the Z2{Z}_{2}Z2​ case all collective excitations are gapped and, as in d=2,d=2,d=2, the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.
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We introduce a two-body quantum Hamiltonian model of spin 12\frac{1}{2}21​ on a two-dimensional spatial lattice with exact topological degeneracy in all coupling regimes. There exists a gapped phase in which the low-energy sector reproduces an effective color code model. High-energy excitations fall into three families of anyonic fermions that turn out to be strongly interacting. The model exhibits a ${\mathbf{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbf{Z}}_{2}$ gauge-group symmetry and string-net integrals of motion, which are related to the existence of topological charges that are invisible to moving high-energy fermions.
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 Dk 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 C/G twist liquids, where G=Z2 for C=U(1)l , SU(n)1 , and G=Z2 , Zk , Dk for 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.
Majorana modes can arise as zero energy bound states in a variety of solid state systems. A lattice of these quasiparticles, for instance, emerges on the surface of a topological superconductor with the zero modes localized at the cores of vortices. This thesis is devoted to the study of interactions between the Majorana zero modes when such a lattice has a triangular geometry. We map the phase diagram of this model using a combination of mean field theory and numerical simulation of ladders through the density-matrix renormalization-group technique. Our analysis suggests that interactions drive a topologically non-trivial gapped phase into a gapless Luttinger liquid.