1. 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” *e-mail: lara.benfatto@roma1.infn.itlara.benfatto@roma1.infn.it www: http://www.roma1.infn.it/~lbenfat/ 29-30 Novembre Conferenza di Progetto

2. 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

3. 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

4. Effect of the lattice Allowed electronic states forms energy bands

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

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

7. 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

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

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

10. 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

11. 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:0707.2487 (2007) Ohta et al. Science 313, 951 (2006) ≈

12. 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:0707.2487 (2007) Ohta et al. Science 313, 951 (2006)

13. “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) ≈

14. 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

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

16. 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

17. 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

18. 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

19. 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)

20. Understanding Kosterlitz-Thouless physics

21. 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, 117008 (07) L.Benfatto, C.Castellani and T.Giamarchi, in preparation

22. 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, 117008 (07) L.Benfatto, C.Castellani and T.Giamarchi, in preparation  Non-linear field-induced magnetization L.Benfatto, C.Castellani and T.Giamarchi, PRL 99, 207002 (07)

23. 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)

24. 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/0509223

25. The absence of the superfluid-density jump L.Benfatto, C. Castellani and T. Giamarchi, PRL 98, 117008 (07)

26. 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 )

27. 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

28. 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)

29. 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!!