Simulating Dirac fermions with Abelian and non-Abelian gauge fields in optical lattices

In this work we present an optical lattice setup to realize a full Dirac Hamiltonian in 2+1 dimensions. We show how all possible external potentials coupled to the Dirac field can arise from perturbations of the existing couplings of the honeycomb lattice pattern. This greatly simplifies the proposed implementations, requiring only spatial modulations of the intensity of the laser beams to induce complex non-Abelian potentials. We finally suggest several experiments to observe the properties of the quantum field theory in the setup.     

Quantum information and triangular optical lattices

The regular structures obtained by optical lattice technology and their behavior are analyzed from the quantum information perspective. Initially, we demonstrate that a triangular optical lattice of two atomic species, bosonic or fermionic, can be employed to generate a variety of novel spin-1/2 models that include effective three-spin interactions. Such interactions can be employed to simulate specific one-or two-dimensional physical systems that are of particular interest for their condensed matter and entanglement properties. In particular, connections between the scaling behavior of entanglement and the entanglement properties of closely spaced spins are drawn. Moreover, three-spin interactions are well suited to support quantum computing without the need to manipulate individual qubits. By employing Raman transitions or the interaction of the atomic electric dipole moment with magnetic field gradients, one can generate Hamiltonians that can be used for the physical implementation of geometric or topological objects. This work serves as a review article that also includes many new results.     

Manifestations of topological effects in graphene

Graphene is a monoatomic layer of graphite with carbon atoms arranged in a two-dimensional honeycomb lattice configuration. It has been known for more than 60 years that the electronic structure of graphene can be modelled by two-dimensional massless relativistic fermions. This property gives rise to numerous applications, both in applied sciences and in theoretical physics. Electronic circuits made out of graphene could take advantage of its high electron mobility that is witnessed even at room temperature. In the theoretical domain the Dirac-like behaviour of graphene can simulate high energy effects, such as the relativistic Klein paradox. Even more surprisingly, topological effects can be encoded in graphene such as the generation of vortices, charge fractionalisation and the emergence of anyons. The impact of the topological effects on graphene's electronic properties can be elegantly described by the Atiyah–Singer index theorem. Here we present a pedagogical encounter of this theorem and review its various applications to graphene. A direct consequence of the index theorem is charge fractionalisation that is usually known from the fractional quantum Hall effect. The charge fractionalisation gives rise to the exciting possibility of realising graphene based anyons that unlike bosons or fermions exhibit fractional statistics. Besides being of theoretical interest, anyons are a strong candidate for performing error free quantum information processing.     

Thermal states of anyonic systems

A study of the thermal properties of two-dimensional topological lattice models is presented. This work is relevant to assess the usefulness of these systems as a quantum memory. For our purposes, we use the topological mutual information Itopo$I_{\mathrm{topo}}$ as a “topological order parameter”. For Abelian models, we show how Itopo$I_{\mathrm{topo}}$ depends on the thermal topological charge probability distribution. More generally, we present a conjecture that Itopo$I_{\mathrm{topo}}$ can (asymptotically) be written as a Kullback–Leitner distance between this probability distribution and that induced by the quantum dimensions of the model at hand. We also explain why Itopo$I_{\mathrm{topo}}$ is more suitable for our purposes than the more familiar entanglement entropy Stopo$S_{\mathrm{topo}}$. A scaling law, encoding the interplay of volume and temperature effects, as well as different limit procedures, are derived in detail. A non-Abelian model is next analyzed and similar results are found. Finally, we also consider, in the case of a one-plaquette toric code, an environment model giving rise to a simulation of thermal effects in time.     

Graphene with geometrically induced vorticity

At half filling, the electronic structure of graphene can be modeled by a pair of free two-dimensional Dirac fermions. We explicitly demonstrate that in the presence of a geometrically induced gauge field an everywhere-real Kekulé modulation of the hopping matrix elements can correspond to a nonreal Higgs field with nontrivial vorticity. This provides a natural setting for fractionally charged vortices with localized zero modes. For fullerenelike molecules we employ the index theorem to demonstrate the existence of six low-lying states that do not depend strongly on the Kekulé-induced mass gap.     

Holographic correspondence in topological superconductors

We analytically derive a compatible family of effective field theories that uniquely describe topological superconductors in 3D, their 2D boundary and their 1D defect lines. We start by deriving the topological field theory of a 3D topological superconductor in class DIII, which is consistent with its symmetries. Then we identify the effective theory of a 2D topological superconductor in class D living on the gapped boundary of the 3D system. By employing the holographic correspondence we derive the effective chiral conformal field theory that describes the gapless modes living on the defect lines or effective boundary of the class D topological superconductor. We demonstrate that the chiral central charge is given in terms of the 3D winding number of the bulk which by its turn is equal to the Chern number of its gapped boundary.     

Quantum computation with a one-dimensional optical lattice

We present an economical dynamical control scheme to perform quantum computation on a one-dimensional optical lattice, where each atom encodes one qubit. The model is based on atom tunneling transitions between neighboring sites of the lattice. They can be activated by external laser beams resulting in a two-qubit phase gate or in an exchange interaction. A realization of the Toffoli gate is presented, which requires only a single laser pulse and no individual atom addressing.     

Bringing order through disorder: localization of errors in topological quantum memories

Anderson localization emerges in quantum systems when randomized parameters cause the exponential suppression of motion. Here we consider this phenomenon in topological models and establish its usefulness for protecting topologically encoded quantum information. For concreteness we employ the toric code. It is known that in the absence of a magnetic field this can tolerate a finite initial density of anyonic errors, but in the presence of a field anyonic quantum walks are induced and the tolerable density becomes zero. However, if the disorder inherent in the code is taken into account, we demonstrate that the induced localization allows the topological quantum memory to regain a finite critical anyon density and the memory to remain stable for arbitrarily long times. We anticipate that disorder inherent in any physical realization of topological systems will help to strengthen the fault tolerance of quantum memories.     

Conformally flat Kaluza–Klein spaces, pseudo-/para-complex space forms and generalized gravitational kinks

The equations describing the Kaluza–Klein reduction of conformally flat spaces are investigated in arbitrary dimensions. Special classes of solution related to pseudo-Kähler and para-Kähler structures are constructed and classified according to spacetime dimension, signature and gauge field rank. Remarkably, rank two solutions include gravitational kinks together with their centripetal and centrifugal deformations.     

Cold atom simulation of interacting relativistic quantum field theories

We demonstrate that Dirac fermions self-interacting or coupled to dynamic scalar fields can emerge in the low energy sector of designed bosonic and fermionic cold atom systems. We illustrate this with two examples defined in two spacetime dimensions. The first one is the self-interacting Thirring model. The second one is a model of Dirac fermions coupled to a dynamic scalar field that gives rise to the Gross-Neveu model. The proposed cold atom experiments can be used to probe spectral or correlation properties of interacting quantum field theories thereby presenting an alternative to lattice gauge theory simulations.