Non-abelian anyons are highly desired for topological quantum computation purposes, with Majorana fermions providing a promising route, particularly zero modes with non-trivial mutual statistics. Yet realizing Majorana zero modes in matter is a challenge, with various proposals in chiral superconductors, nanowires, and spin liquids, but no clear experimental examples. Heavy fermion materials have long been known to host Majorana fermions at two-channel Kondo impurity sites, however, these impurities cannot be moved adiabatically and generically occur in metals, where the absence of a gap removes the topological protection. Here, we consider an ordered lattice of these two-channel Kondo impurities, which at quarter-filling form a Kondo insulator. We show that topological defects in this state will host Majorana zero modes, or possibly more complicated parafermions. These states are protected by the insulating gap and may be adiabatically braided, providing the novel possibility of realizing topological quantum computation in heavy fermion materials.Majorana fermions (γ) are real fermions that are their own antiparticles. These may be free particles, but are most interesting when bound to defects as a zero energy state in two-dimensions [1]. These defects occur in pairs, which may be spatially separated, with the two Majoranas encoding a single complex fermion, c † = γ 1 + iγ 2 . The resulting ground state degeneracy is associated with the complex fermion parity, c † c = 0, 1, and can encode qubits. If two defects can be braided adiabatically, the system evolves smoothly within the degenerate ground state manifold, encoding non-Abelian mutual statistics, even as the self-statistics remain fermionic. These defects are called Majorana zero modes (MZMs) to distinguish them from free Majoranas, and are Ising anyons. The encoded quantum information can be controlled by manipulating the defects and is topologically protected by the system gap. MZMs require only that the fermions are real, the defect state is localized, and the fermion parity is conserved. Many proposals use superconductivity to ensure parity conservation [2-6], but the emergent fermions in spin liquids, and in some heavy fermions, as we shall see, are also viable candidates [7][8][9].Kondo impurities are isolated local moments that interact antiferromagnetically with a metallic host, and are screened by the conduction electron spin density at low temperatures. Usually, this screening involves a sin-

No abstract

The two-channel Kondo lattice likely hosts a rich array of phases, including hastatic order, a channel-symmetry breaking heavy Fermi liquid. We revisit its one-dimensional phase diagram using density matrix renormalization group. In contrast to previous work, we find algebraic hastatic orders generically for strong coupling. These are heavy Tomonaga-Luttinger liquids with nonanalyticities at Fermi wave-vectors captured by hastatic density waves. Intriguingly, we find a recently predicted additional order parameter, not present at large-N , arising from RKKY mediated interference between hastatic spinors, and indications of residual repulsive interactions at strong coupling.