Topological electrostatics.

We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the GrassmannianGr(M,N)manifold. These textures describe skyrmion lattices ofN-component fermions in a quantising magnetic field, relevant to the physics of graphene, bilayer and other multicomponent quantum Hall systems near integer filling factorsν > 1. We derive analytically the optimality condition, minimizing topological charge density fluctuations, for a general Grassmannian sigma modelGr(M,N)on a sphere and a torus, together with counting arguments which show that for any filling factor and number of components there is a critical value of topological chargedcabove which there are no optimal textures. Belowdca solution of the optimality condition on a torus is unique, while in the case of a sphere one has, in general, a continuum of solutions corresponding to new non-Goldstone zero modes, whose degeneracy is not lifted (via a order from disorder mechanism) by any fermion interactions depending only on the distance on a sphere. We supplement our general theoretical considerations with the exact analytical results for the case ofGr(2,4), appropriate for recent experiments in graphene.

Physical implementation of protected qubits.

We review the general notion of topological protection of quantum states in spin models and its relation with the ideas of quantum error correction. We show that topological protection can be viewed as a Hamiltonian realization of error correction: for a quantum code for which the minimal number of errors that remain undetected is N, the corresponding Hamiltonian model of the effects of the environment noise appears only in the Nth order of the perturbation theory.We discuss the simplest model Hamiltonians that realize topological protection and their implementation in superconducting arrays. We focus on two dual realizations: in one the protected state is stored in the parity of the Cooper pair number, in the other, in the parity of the flux number. In both cases the superconducting arrays allow a number of fault-tolerant operations that should make the universal quantum computation possible.

Topologically decoherence-protected qubits with trapped ions.

We show that trapped ions can be used to simulate a highly symmetrical Hamiltonian with eigenstates naturally protected against local sources of decoherence. This Hamiltonian involves long-range coupling between particles and provides a more efficient protection than nearest neighbor models discussed in previous works. Our results open the perspective of experimentally realizing, in controlled atomic systems, complex entangled states with decoherence times up to 9 orders of magnitude longer than isolated quantum systems.

Fluctuations and vortex-pattern ordering in the fully frustrated XY model on a honeycomb lattice.

The accidental degeneracy of various ground states of a fully frustrated XY model with a honeycomb lattice is shown to survive even when the free energy of the harmonic fluctuations is taken into account. The reason for that consists in the existence of a hidden gauge symmetry between the Hamiltonians describing the harmonic fluctuations in all these ground states. A particular vortex pattern is selected only when anharmonic fluctuations are taken into account. However, the observation of vortex ordering requires relatively large system size L>>Lc > or approximately equal to 10(5).

Topological order in the insulating Josephson junction array.

We propose a Josephson junction array which can be tuned into an unconventional insulating state by varying external magnetic field. This insulating state retains a gap to half-vortices; as a consequence, such an array with nontrivial global geometry exhibits a ground state degeneracy. This degeneracy is protected from the effects of external noise. We compute the gaps, separating higher energy states from the degenerate ground state, and we discuss experiments probing the unusual properties of this insulator.

Nesting induced precursor effects: a renormalization group approach.

We develop a controlled weak coupling renormalization group (RG) approach to itinerant electrons. Within this formalism we rederive the phase diagram for two-dimensional non-nested systems. We then study how nesting modifies this phase diagram. We show that competition between particle-particle and particle-hole channels leads to the manifestation of an unstable precursor fixed point in the RG flow. This effect should be experimentally measurable, and may be relevant for an explanation of pseudogaps in the high temperature superconductors, as a crossover phenomenon.

Interaction induced delocalization for two particles in a periodic potential

We consider two interacting particles evolving in a one-dimensional periodic structure embedded in a magnetic field. We show that the strong localization induced by the magnetic field for particular values of the flux per unit cell is destroyed as soon as the particles interact. We study the spectral and dynamical aspects of this transition.