Hyperbolic Lattices and Two-Dimensional Yang-Mills Theory.

Hyperbolic lattices are a new type of synthetic quantum matter emulated in circuit quantum electrodynamics and electric-circuit networks, where particles coherently hop on a discrete tessellation of two-dimensional negatively curved space. While real-space methods and a reciprocal-space hyperbolic band theory have been recently proposed to analyze the energy spectra of those systems, discrepancies between the two sets of approaches remain. In this work, we reconcile those approaches by first establishing an equivalence between hyperbolic band theory and U(N) topological Yang-Mills theory on higher-genus Riemann surfaces.

Anderson Localization Transition in Disordered Hyperbolic Lattices.

We study Anderson localization in disordered tight-binding models on hyperbolic lattices. Such lattices are geometries intermediate between ordinary two-dimensional crystalline lattices, which localize at infinitesimal disorder, and Bethe lattices, which localize at strong disorder. Using state-of-the-art computational group theory methods to create large systems, we approximate the thermodynamic limit through appropriate periodic boundary conditions and numerically demonstrate the existence of an Anderson localization transition on the {8,3} and {8,8} lattices.

Hyperbolic Non-Abelian Semimetal.

We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in a negatively curved plane. Because of their distinct translation group structure, such lattices are associated with a high-dimensional reciprocal space. In addition, they support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations.

Non-Abelian Hyperbolic Band Theory from Supercells.

Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states.

Chiral Ising Gross-Neveu Criticality of a Single Dirac Cone: A Quantum Monte Carlo Study.

We perform large-scale quantum Monte Carlo simulations of SLAC fermions on a two-dimensional square lattice at half filling with a single Dirac cone with N=2 spinor components and repulsive on-site interactions. Despite the presence of a sign problem, we accurately identify the critical interaction strength U_{c}=7.28±0.

Automorphic Bloch theorems for hyperbolic lattices.

Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (2D) hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface.

Triangular Pair Density Wave in Confined Superfluid ^{3}He.

Recent advances in experiment and theory suggest that superfluid ^{3}He under planar confinement may form a pair density wave (PDW) whereby superfluid and crystalline orders coexist. While a natural candidate for this phase is a unidirectional stripe phase predicted by Vorontsov and Sauls in 2007, recent nuclear magnetic resonance measurements of the superfluid order parameter rather suggest a two-dimensional PDW with noncollinear wave vectors, of possibly square or hexagonal symmetry. In this Letter, we present a general mechanism by which a PDW with the symmetry of a triangular lattice can be stabilized, based on a superfluid generalization of Landau's theory of the liquid-solid transition.

Hyperbolic band theory.

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit quantum electrodynamics, we exploit ideas from algebraic geometry to construct a hyperbolic generalization of Bloch theory, despite the absence of commutative translation symmetries. For a quantum particle propagating in a hyperbolic lattice potential, we construct a continuous family of eigenstates that acquire Bloch-like phase factors under a discrete but noncommutative group of hyperbolic translations, the Fuchsian group of the lattice.

Fractional Quantum Hall Phases of Bosons with Tunable Interactions: From the Laughlin Liquid to a Fractional Wigner Crystal.

Highly tunable platforms for realizing topological phases of matter are emerging from atomic and photonic systems and offer the prospect of designing interactions between particles. The shape of the potential, besides playing an important role in the competition between different fractional quantum Hall phases, can also trigger the transition to symmetry-broken phases, or even to phases where topological and symmetry-breaking order coexist. Here, we explore the phase diagram of an interacting bosonic model in the lowest Landau level at half filling as two-body interactions are tuned.

Emergence of Supersymmetric Quantum Electrodynamics.

Supersymmetric (SUSY) gauge theories such as the minimal supersymmetric standard model play a fundamental role in modern particle physics, but have not been verified so far in nature. Here, we show that a SUSY gauge theory with dynamical gauge bosons and fermionic gauginos emerges naturally at the pair-density-wave (PDW) quantum phase transition on the surface of a correlated topological insulator hosting three Dirac cones, such as the topological Kondo insulator SmB_{6}. At the quantum tricritical point between the surface Dirac semimetal and nematic PDW phases, three massless bosonic Cooper pair fields emerge as the superpartners of three massless surface Dirac fermions.

Time-Resolved Imaging of Negative Differential Resistance on the Atomic Scale.

Negative differential resistance remains an attractive but elusive functionality, so far only finding niche applications. Atom scale entities have shown promising properties, but the viability of device fabrication requires a fuller understanding of electron dynamics than has been possible to date. Using an all-electronic time-resolved scanning tunneling microscopy technique and a Green's function transport model, we study an isolated dangling bond on a hydrogen terminated silicon surface.

Publisher's Note: Optical Conductivity of Topological Surface States with Emergent Supersymmetry [Phys. Rev. Lett. 116, 100402 (2016)].

This corrects the article DOI: 10.1103/PhysRevLett.116.

Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model.

We study the two-dimensional (2D) Hubbard model using exact diagonalization for spin-1/2 fermions on the triangular and honeycomb lattices decorated with a single hexagon per site. In certain parameter ranges, the Hubbard model maps to a quantum compass model on those lattices. On the triangular lattice, the compass model exhibits collinear stripe antiferromagnetism, implying d-density wave charge order in the original Hubbard model.

Optical Conductivity of Topological Surface States with Emergent Supersymmetry.

Topological states of electrons present new avenues to explore the rich phenomenology of correlated quantum matter. Topological insulators (TIs) in particular offer an experimental setting to study novel quantum critical points (QCPs) of massless Dirac fermions, which exist on the sample's surface. Here, we obtain exact results for the zero- and finite-temperature optical conductivity at the semimetal-superconductor QCP for these topological surface states.

Landau Theory of Helical Fermi Liquids.

We construct a phenomenological Landau theory for the two-dimensional helical Fermi liquid found on the surface of a three-dimensional time-reversal invariant topological insulator. In the presence of rotation symmetry, interactions between quasiparticles are described by ten independent Landau parameters per angular momentum channel, by contrast with the two (symmetric and antisymmetric) Landau parameters for a conventional spin-degenerate Fermi liquid. We project quasiparticle states onto the Fermi surface and obtain an effectively spinless, projected Landau theory with a single projected Landau parameter per angular momentum channel that captures the spin-momentum locking or nontrivial Berry phase of the Fermi surface.

Topological order in a correlated three-dimensional topological insulator.

Motivated by experimental progress in the growth of heavy transition metal oxides, we theoretically study a class of lattice models of interacting fermions with strong spin-orbit coupling. Focusing on interactions of intermediate strength, we derive a low-energy effective field theory for a fully gapped, topologically ordered, fractionalized state with an eightfold ground-state degeneracy. This state is a fermionic symmetry-enriched topological phase with particle-number conservation and time-reversal symmetry.

Fractional topological insulators in three dimensions.

Topological insulators can be generally defined by a topological field theory with an axion angle θ of 0 or π. In this work, we introduce the concept of fractional topological insulator defined by a fractional axion angle and show that it can be consistent with time reversal T invariance if ground state degeneracies are present. The fractional axion angle can be measured experimentally by the quantized fractional bulk magnetoelectric polarization P₃, and a "halved" fractional quantum Hall effect on the surface with Hall conductance of the form σH=p/q e²/2h with p, q odd.

Topological quantization in units of the fine structure constant.

Fundamental topological phenomena in condensed matter physics are associated with a quantized electromagnetic response in units of fundamental constants. Recently, it has been predicted theoretically that the time-reversal invariant topological insulator in three dimensions exhibits a topological magnetoelectric effect quantized in units of the fine structure constant α=e²/ℏc. In this Letter, we propose an optical experiment to directly measure this topological quantization phenomenon, independent of material details.

Kondo effect in the helical edge liquid of the quantum spin Hall state.

Following the recent observation of the quantum spin Hall (QSH) effect in HgTe quantum wells, an important issue is to understand the effect of impurities on transport in the QSH regime. Using linear response and renormalization group methods, we calculate the edge conductance of a QSH insulator as a function of temperature in the presence of a magnetic impurity. At high temperatures, Kondo and/or two-particle scattering give rise to a logarithmic temperature dependence.

Nonlocal transport in the quantum spin Hall state.

Nonlocal transport through edge channels holds great promise for low-power information processing. However, edge channels have so far only been demonstrated to occur in the quantum Hall regime, at high magnetic fields. We found that mercury telluride quantum wells in the quantum spin Hall regime exhibit nonlocal edge channel transport at zero external magnetic field.