Isolated Flat Bands and Spin-1 Conical Bands in Two-Dimensional Lattices

Dispersionless bands, such as Landau levels, serve as a good starting point for obtaining interesting correlated states when interactions are added. With this motivation in mind, we study a variety of dispersionless ("flat") band structures that arise in tight-binding Hamiltonians defined on hexagonal and kagome lattices with staggered fluxes. The flat bands and their neighboring dispersing bands have several notable features: (a) Flat bands can be isolated from other bands by breaking time reversal symmetry, allowing for an extensive degeneracy when these bands are partially filled; (b) An isolated flat band corresponds to a critical point between regimes where the band is electron-like or hole-like, with an anomalous Hall conductance that changes sign across the transition; (c) When the gap between a flat band and two neighboring bands closes, the system is described by a single spin-1 conical-like spectrum, extending to higher angular momentum the spin-1/2 Dirac-like spectra in topological insulators and graphene; and (d) some configurations of parameters admit two isolated parallel flat bands, raising the possibility of exotic "heavy excitons"; (e) We find that the Chern number of the flat bands, in all instances that we study here, is zero.Comment: 7 pages. Sec. II slightly expanded. References adde

Alternative sets of hyperspherical harmonics: Satisfying cusp conditions through frame transformations

By extending the concept of Euler-angle rotations to more than three dimensions, we develop the systematics under rotations in higher-dimensional space for a novel set of hyperspherical harmonics. Applying this formalism, we determine all pairwise Coulomb interactions in a few-body system without recourse to multipole expansions. Our approach combines the advantages of relative coordinates with those of the hyperspherical description. In the present method, each Coulomb matrix element reduces to the ``1/r'' form familiar from the two-body problem. Consequently, our calculation accounts for all the cusps in the wave function whenever an interparticle separation vanishes. Unlike a truncated multipole expansion, the calculation presented here is exact. Following the systematic development of the procedure for an arbitrary number of particles, we demonstrate it explicitly with the simplest nontrivial example, the three-body system.Comment: 19 pages, no figure

Constructing Non-Abelian Quantum Spin Liquids Using Combinatorial Gauge Symmetry

We construct Hamiltonians with only 1- and 2-body interactions that exhibit an exact non-Abelian gauge symmetry (specifically, combinatiorial gauge symmetry). Our spin Hamiltonian realizes the quantum double associated to the group of quaternions. It contains only ferromagnetic and anti-ferromagnetic $ZZ$ interactions, plus longitudinal and transverse fields, and therefore is an explicit example of a spin Hamiltonian with no sign problem that realizes a non-Abelian topological phase. In addition to the spin model, we propose a superconducting quantum circuit version with the same symmetry.Comment: 9 pages, 4 figures. Minor edits to make exposition clearer in some place

A superconducting circuit realization of combinatorial gauge symmetry

We propose a superconducting quantum circuit based on a general symmetry principle -- combinatorial gauge symmetry -- designed to emulate topologically-ordered quantum liquids and serve as a foundation for the construction of topological qubits. The proposed circuit exhibits rich features: in the classical limit of large capacitances its ground state consists of two superimposed loop structures; one is a crystal of small loops containing disordered $U(1)$ degrees of freedom, and the other is a gas of loops of all sizes associated to $\mathbb{Z}_2$ topological order. We show that these classical results carry over to the quantum case, where phase fluctuations arise from the presence of finite capacitances, yielding ${\mathbb Z}_2$ quantum topological order. A key feature of the exact gauge symmetry is that amplitudes connecting different ${\mathbb Z}_2$ loop states arise from paths having zero classical energy cost. As a result, these amplitudes are controlled by dimensional confinement rather than tunneling through energy barriers. We argue that this effect may lead to larger energy gaps than previous proposals which are limited by such barriers, potentially making it more likely for a topological phase to be experimentally observable. Finally, we discuss how our superconducting circuit realization of combinatorial gauge symmetry can be implemented in practice.Comment: Joined by new author. Added section on experimental realization. Added analytical result

Adsorption on carbon nanotubes: quantum spin tubes, magnetization plateaus, and conformal symmetry

We formulate the problem of adsorption onto the surface of a carbon nanotube as a lattice gas on a triangular lattice wrapped around a cylinder. This model is equivalent to an XXZ Heisenberg quantum spin tube. The geometric frustration due to wrapping leads generically to four magnetization plateaus, in contrast to the two on a flat graphite sheet. We obtain analytical and numerical results for the magnetizations and transition fields for armchair, zig-zag and chiral nanotubes. The zig-zags are exceptional in that one of the plateaus has extensive zero temperature entropy in the classical limit. Quantum effects lift up the degeneracy, leaving gapless excitations which are described by a $c=1$ conformal field theory with compactification radius quantized by the tube circumference.Comment: 5 pages, 6 figure

U(1)U(1) symmetry-enriched toric code

We propose and study a generalization of Kitaev's $\mathbb Z_2$ toric code on a square lattice with an additional global $U(1)$ symmetry. Using Quantum Monte Carlo simulation, we find strong evidence for a topologically ordered ground state manifold with indications of UV/IR mixing, i.e., the topological degeneracy of the ground state depends on the microscopic details of the lattice. Specifically, the ground state degeneracy depends on the lattice tilt relative to the directions of the torus cycles. In particular, we observe that while the usual compactification along the vertical/horizontal lines of the square lattice shows a two-fold ground state degeneracy, compactifying the lattice at $45^\circ$ leads to a three-fold degeneracy. In addition to its unusual topological properties, this system also exhibits Hilbert space fragmentation. Finally, we propose a candidate experimental realization of the model in an array of superconducting quantum wires