ANNUAL REPORT 2011 - Instituto de Estructura de la Materia

screening of the interaction potential in the ladder series, and above a value α c ≈ 1.75 in the more sensible instanceof dynamical screening of the interaction.The main conclusion of our work is that the value α c ≈ 1.75, resulting from the most accurate treatment of the manybodycorrections, stillremains below the nominal value of the graphene fine structure constant in vacuum. Thismeans that an isolated free-standing layer of the material should be in the phase with dynamical gap generation, inagreement with most part of less precise theoretical studies on the subject, but apparently at odds with presentexperimental measures in suspended graphene samples. A key observation is however that, if chiral symmetrybreaking is to proceed in graphene accordingto our estimates, it should lead to a gap at least three orders ofmagnitudebelow the high-energy scale of the Dirac theory. This suggests then that thedynamical gap generationcannot be discarded in isolated free-standing graphene, though its experimental signature may be only found insuitable samples, for which the Fermi level can be tuned within an energy range belowthe milli-electron-Volt scaleabout the charge neutrality point.EXACTLY SOLVABLE MODELSThe Richardson-Gaudin (RG) integrable pairing models emerged from the combination of the exact solution of theBCS Hamiltonian obtained by Richardson in the sixties and the integrable model of quantum magnetism found byGaudin in the seventies. Formally, the RG models can be classified two families, the rational family and thehyperbolic family. The rational family contains the BCS Hamiltonian as well as many other exactly solvableHamiltonians studied in the last decade in the areas of nuclear structure, ultrasmall superconducting grains, quantumdots, trapped cold fermionic and bosonic gases, etc. In 2009 it has been shown that the px+ipy pairing Hamiltoniandescribing the interaction between spinless fermions in 2D lattices, is exactly solvable and it pertains to hyperbolicfamily. This Hamiltonian, studied by Read in connection to the fractionary quantum Hall effect, has a phasediagram with a topological phase transition (without order parameter). From de exact solution we studied the phasediagram of for one channel Hamiltonians (pairing interaction). The model displays a quantum phase transition of 3ºorder compatible with the lack of an order parameter. The exact wave function of the RG models is a product ofcorrelated pairs whose bound energies come from the solutions of the Richardson equations. The size of thecondensed wave function, which experimentally accessible in ultracold polarized gases, displays a divergence at thecritical point. Therefore, it can be used as an experimental signature. The structure of the Cooper pairs is moredifficult to be revealed experimentally. However, its size has been associated with threshold in energy of the radiofrequencyspectrum in trapped gases. The structure of p-wave pairs is different of the s-wave pairs. It shows anabrupt change between the weak pairing phase (extended pairs) and the strong pairing phase (bound pairs). Westudied the extension of the px+ipy model to a two channel model in which the interaction is mediated by a p-waveFeshbach resonance. The corresponding phase diagram also shows a 3º order quantum phase transition that wecharacterized by means of the quantum fidelity. The analysis of this magnitude led us to conclude that the phasetransition cannot be described by the Landau theory.The second realization of the hyperbolic family consists on an exactly solvable Hamiltonian with non-degeneratesingle particle energies and a separable pairing interaction with two free parameters. One of the parameters is thestrength of the pairing interaction and the other is an energy cutoff. We showed that these parameters can beconveniently chosen to reproduce the physical properties of the ground state of heavy nuclei described by the Gognyforce. In particular, we showed that the hyperbolic model describes with great precision the wave function of theGogny Hartre-Fock-Bogoliubov approximation. We also showed that the Gogny gaps display a behavior compatiblewith the square root in the form factors of the hyperbolic interaction. This new exactly solvable model opens theway for the construction of a selfconsistent algorithm of Hartree-Fock plus exact pairing for the description ofmasses and low energy properties of heavy nuclei.QUANTUM TRANSPORTDuring 2010 we have focused our efforts in the study of the transmission phase through quantum dots in theCoulomb blockade regime. Quantum mechanics fundamentally differs from classical mechanics in that timeevolutions are determined by complex probability amplitudes instead of real probabilities. Theassociated phase is akey element to understand mesoscopic transport experiments on Aharonov-Bohm (AB) conductance oscillations,weak localization, and conductance fluctuations. For a long time, the transmission amplitude phase through aquantum dot could not be measured directly. The pioneering experiments of Yacoby et al. opened a new field inmesoscopic physics at the end of the 90s when they embedded a quantum dot into one of the arms of an Aharanov-Bohm interferometer. However, these experiments have escaped a consistent and generic theoretical explanation.Specially surprising is the fact that a phase lapse of πalways appears between resonances in the transmission phasefor quantum dots with large number of electrons.51