Dirac fermions with zero effective mass in condensed matter: new perspectives Lara Benfatto* Centro Studi e Ricerche “Enrico Fermi” and University of Rome “La Sapienza” * www: Novembre Conferenza di Progetto

Outline Why Dirac fermions? Common denominator in emerging INTERESTING new materials Dirac fermions from lattice effect: the case of graphene  Bilayer graphene: “protected” optical sum rule Dirac fermions from interactions: d-wave superconductivity  Collective phase fluctuations: Kosterlitz-Thouless vortex physics Acknowledgments: C. Castellani, Rome, Italy T.Giamarchi, Geneva, Switzerland S. Sharapov, Macomb (Illinois), USA J. Carbotte, Hamilton (Ontario), Canada

Basic understanding of many electrons in a solid k values are quantized Pauli principle: N electrons cannot occupy the same quantum level Fermi-Dirac statistic: all level up to the Fermi level are occupied Excitations: unoccupied levels Quadratic energy-momentum dispersion

Effect of the lattice Allowed electronic states forms energy bands

Effect of the lattice Allowed electronic states forms energy bands and have an “effective mass” Quadratic energy-momentum dispersion Semiconductor physics!!

Dirac fermions from lattice effects: graphene One layer of Carbon atoms

Dirac fermions from lattice effects: graphene One layer of Carbon atoms Graphene: a 2D metal controlled by electric-field effect SiO 2 Si Au contacts GRAPHENE VgVg

Dirac fermions from lattice effects: graphene Carbon atoms: many allotropes Graphene: a 2D metal controlled by electric-field effect In momentum space

Dirac fermions from lattice effects: graphene Carbon atoms: many allotropes Graphene: a 2D metal controlled by electric-field effect In momentum space: Dirac cone

Universal conductivity Despite the fact that at the Dirac point there are no carriers the system has a finite and (almost) universal conductivity!! Dirac fermions are “protected” against disorder Deviations: charged impurities, self-doping, Coulomb interactions, vertex corrections

Bilayer graphene: tunable-gap semiconductor LARGE gap (a fraction of the Fermi energy) Does it affect the total spectral weight of the system? Oostinga et al. arXiv: (2007) Ohta et al. Science 313, 951 (2006) ≈

Bilayer graphene: tunable-gap semiconductor LARGE gap (a fraction of the Fermi energy) Does it affect the total spectral weight of the system? Analogous problem in oxides: electron correlations decrease considerably the carrier spectral weight Oostinga et al. arXiv: (2007) Ohta et al. Science 313, 951 (2006)

“Protected” optical sum rule The optical sum rule is almost constant despite the large gap opening: large redistribution of spectral weight is expected (a prediction to be tested experimentally) L.Benfatto, S.Sharapov and J. Carbotte, preprint (2007) ≈

Dirac fermions from interactions: d-wave superconductors Example of High-Tc superconductor La 1-x Sr x Cu 2 O 4 :  quasi two-dimensional in nature  CuO 2 layers are the key ingredient  La and Sr supply “doping” Superconductivity: formation of Cooper pairs which “Bose” condense High Tc: not explained within standard BCS theory for “conventional” low-Tc superconductors New quasiparticle excitations! New “collective” excitations! Cooper pair

Dirac fermions from interactions: d-wave superconductors s-wave Conventional s-wave SC: Δ=const over the Fermi surface Gapped excitations

Dirac fermions from interactions: d-wave superconductors massless massless Dirac Dirac fermions fermions vFvF vv s-wave  d-wave Conventional s-wave SC: Δ=const over the Fermi surface Gapped excitations High-Tc d-wave SC: Δ vanishes at nodal points Gapless Dirac excitations

Measuring Dirac excitations Dirac fermions are “protecetd” against disorder Low-energy part does not depend on the position High-energy part is affected by position, disorder, etc. Gomes et al. Nature 447, 569 (2007) vFvF vv

Collective phase fluctuations: vortices! In BCS superconductors superconductivity disappears when |Δ| 0 at Tc: standard paradigm applies In HTSC superconductivity is destroyed by phase fluctuations where |Δ| remains finite Crucial role of vortices water vortex

Collective phase fluctuations: vortices! In BCS superconductors superconductivity disappears when |Δ| ->0 at Tc: standard paradigm applies In HTSC superconductivity is destroyed by phase fluctuations where |Δ| remains finite Crucial role of vortices Kosterlitz-Thouless like physics Superconducting vortex is a topological defect in phase .  winds by 2π around the vortex core Superconducting hc/2e vortex J.M.K. and D.J.T. J. Phys. C (1973, 1974)

Understanding Kosterlitz-Thouless physics

Need of a new theoretical approach to the Kosterlitz-Thouless transition  Mapping to the sine-Gordon model Crucial role of the vortex-core energy  “Non-universal” jump of the superfluid density L.Benfatto, C.Castellani and T.Giamarchi, PRL 98, (07) L.Benfatto, C.Castellani and T.Giamarchi, in preparation

Understanding Kosterlitz-Thouless physics Need of a new theoretical approach to the Kosterlitz-Thouless transition  Mapping to the sine-Gordon model Crucial role of the vortex-core energy  “Non-universal” jump of the superfluid density L.Benfatto, C.Castellani and T.Giamarchi, PRL 98, (07) L.Benfatto, C.Castellani and T.Giamarchi, in preparation  Non-linear field-induced magnetization L.Benfatto, C.Castellani and T.Giamarchi, PRL 99, (07)

The absence of the superfluid-density jump In pure 2D superfluid/superconductors Js jumps discontinuously to zero, with an universal relation to T KT 4 He films McQueeney et al. PRL 52, 1325 (84)

The absence of the superfluid-density jump In pure 2D superfluid/superconductors Js jumps discontinuously to zero, with an universal relation to T KT YBCO D.Broun et al, cond-mat/

The absence of the superfluid-density jump L.Benfatto, C. Castellani and T. Giamarchi, PRL 98, (07)

Non-linear magnetization effects Field-induced magnetization is due to vortices but one does not recover the LINEAR regime as T approaches T c TcTc M=-a H L. Li et al, EPL 72, 451 (2005) Correlation length (diverges at T c )

Magnetization above T KT L.B. et al, PRL (2007) ξ diverges at T c ! No linear M in the range of fields accessible experimentally

Magnetization above T BKT ξ diverges at T c ! No linear M in the range of fields accessible experimentally L.B. et al, PRL (2007)

Conclusions New effects in emerging low-dimensional materials Need for new theoretical paradigms: quantum field theory for condensed matter borrows concepts and methods from high- energy physics Einstein cone Dirac cone!!