Superconductivity, the pseudogap, and the pairing glue in the 2d - PowerPoint PPT Presentation

Superconductivity, the pseudogap, and the pairing glue in the 2d Hubbard model Emanuel Gull � � Olivier Parcollet, A.J. Millis � Trieste, Italy Emanuel Gull and Andrew J. Millis, arXiv:1407.0704 Emanuel Gull, Andrew J. Millis, Olivier Parcollet, Phys. Rev. Lett. 110, 216405 (2013) Emanuel Gull and Andrew J. Millis, Phys. Rev. B 88, 075127 (2013)

Experiments: Pseudogap M e π φ Node in high-Tc materials: Electronic spectral function is Antinode suppressed along the BZ face, but not along zone 0 Γ diagonal. � Straight ARPES: Kanigel et al. , Nature Physics 2, 447 - 451 (2006). section − π π − 0 π Key physics dependence on momentum around Fermi ( π ,0) a b c surface, Difference of spectral function around Fermi Antinode 1 Bi2212 sample with Tc=90K, measured at 140K surface. � 2 3 Doping dependence of region with quasiparticles 4 5 6 7 ARPES: Shen et al. , Science 307, 901 (2005) ( π ,0) 8 A B C ( π , π ) 9 10 11 for 12 13 ( π /2, π /2) 14 ( π /2, π /2) 15 –0.2 0 0.2 Node (0,0) x = 0.05 x = 0.10 x = 0.12 Energy (eV)

Experiments: d-wave superconductivity Damascelli et al. , Rev. Mod. Phys 75, 2 (2003) He et al. , Science 331, 1579 (2011) C1 172 K (NS) 40 K (PG) 10 K (SC) 40 K A B C H A Arpes Intensity (arbitrary units) k F1 k F1 M 172 K ) T* ) SC k F2 k F2 k G2 FIG. 1. Phase diagram of n - and p -type superconductors, showing superconductivity (SC), antiferromagnetic (AF), pseudogap, and normal-metal regions. -0.1 0.0 H I J -0.1 0.0 190 J.G. Bednorz and K.A. Miiller: Ba-La-Cu-O System E - E F (eV) E - E F (eV) show strong Jahn-Teller (J.T.) effects [13]. While 0.06 oO 0.020 SrFe(VI)O3 is distorted perovskite insulator, o 9 ~ ~ o x* 9 LaNi(III)O3 is a J.T. undistorted metal in which the og transfer energy b~ of the J.T. eg electrons is sufficiently 0.05 t~ ~ a 9 ~176176 e~x "!2" 0.016 large [14] to quench the J.T. distortion. In analogy 9 o 9 x ( π , π ) Bednorz and Müller, Z. Phys. B 64, to Chakraverty's phase diagram, a J.T.-type polaron 9 9 o ~176 eex x Arpes EDC for cuts along Brillouin- 9 o ~ 9 x ~ o e 9 x formation may therefore be expected at the border- 9 o o x 0.04 9 % o o ~ x line of the metal-insulator transition in mixed perovs- 9 0%o o e~ 9149149 x 9 9 ooo~ o 9149 x x 9 9 . 9 ,x 0.012 189 (1986) kites, a subject on which we have recently carried zone boundary (near ( π ,0)), almost out a series of investigations [15]. Here, we report o x x o 0.03 x x on the synthesis and electrical measurements of com- Q. pounds within the Ba-La-Cu-O system. This sys- o x ~'~Kxxxxxxxxxxxxx~'~ 0.008 o.. (0 , 0) optimally doped Pb-Bi2201 with T c tem exhibits a number of oxygen-deficient phases x with mixed-valent copper constituents [16], i.e., with x o 0.25 A/cm 2 0.02 Ox ~,~ 9 0.50 A/cm 2 itinerant electronic states between the non-J.T. Cu a + ,x x 0.50 A/cm 2 of 38K, T* of 132K and the J.T. Cu z+ ions, and thus was expected to OOl -~: ~. 0.004 have considerable electron-phonon coupling and me- tallic conductivity. t I I I o 0 100 200 30( T (K) lI. Experimental Fig. 1. Temperature dependence ofresistivityin Ba~Las _=Cu505 (a y) for samples with x(Ba)= 1 (upper curves, left scale) and x(Ba)= 1. Sample Preparation and Characterization 0.75 (lower curve, right scale). The first two cases also show the influence of current density Samples were prepared by a coprecipitation method from aqueous solutions [17] of Ba-, La- and Cu-ni- trate (SPECPURE JMC) in their appropriate ratios. 3. Conductivity Measurements When added to an aqueous solution of oxalic acid as the precipitant, an intimate mixture of the corre- The dc conductivity was measured by the four-point sponding oxalates was formed. The decomposition method. Rectangular-shaped samples, cut from the of the precipitate and the solid-state reaction were sintered pellets, were provided with gold electrodes performed by heating at 900 ~ for 5 h. The product and contacted by In wires. Our measurements be- was pressed into pellets at 4 kbar, and reheated to tween 300 and 4.2 K were performed in a continuous- 900 ~ for sintering. flow cryostat (Leybold-Hereaus) incorporated in a computer-controlled (IBM-PC) fully-automatic sys- tem for temperature variation, data acquisition and 2. X-Ray Analysis processing. For samples with x(Ba)_<l.0, the conductivity X-ray powder diffract 9 (System D 500 SIE- measurements, involving typical current densities of 0.5 A/cm 2, generally exhibit a high-temperature me- MENS) revealed three individual crystallographic phases. Within a range of 10 ~ to 80 ~ (20), 17 lines tallic behaviour with an increase in resistivity at low could be identified to correspond to a layer-type per- temperatures (Fig. 1). At still lower temperatures, a sharp drop in resistivity (>90%) occurs, which for ovskite-like phase, related to the K2NiF, structure (a=3.79~ and c=13.21 ~) [16]. The second phase higher currents becomes partially suppressed (Fig. 1 : is most probably a cubic one, whose presence depends upper curves, left scale), This characteristic drop has on the Ba concentration, as the line intensity de- been studied as a function of annealing conditions, i.e., temperature and 02 partial pressure (Fig. 2). For creases for smaller x(Ba). The amount of the third phase (volume fraction > 30% from the x-ray intensi- samples annealed in air, the transition from itinerant ties) seems to be independent of the starting composi- to localized behaviour, as indicated by the minimum tion, and shows thermal stability up to 1,000 ~ For in resistivity in the 80 K range, is not very pro- nounced. Annealing in a slightly reducing atmo- higher temperatures, this phase disappears progres- sively, giving rise to the formation of an oxygen-defi- sphere, however, leads to an increase in resistivity cient perovskite (La3Ba3Cu601,) as described by Mi- and a more pronounced localization effect. At the chel and Raveau [16]. same time, the onset of the resistivity drop is shifted

Questions to theory Superconductivity at Pseudogap at intermediate interaction intermediate interaction strengths strengths Coexistence, precursor, competition, ? superconducting self energies? what is the pairing glue? What is the gap function? Contained within a well-defined model & systematic and controllable approximation? …………we will present a potential answer in this talk………

Theory: Hubbard model Restrict to simple minimal model with kinetic and potential energy terms: Hubbard model: t ij ( c † i σ c j σ + c † X X H = − j σ c i σ ) + U n i " n i # . h ij i , σ i Open theoretical question: how much of the physics on the last pages is contained in this model? Even for the most simple model, when kinetic energy ~ potential energy we have no working theoretical tools: quantum many-body theory needs numerical methods ! Here: Cluster DMFT : diagrammatic approximation based on mapping of the system onto a self-consistently adjusted multi-orbital quantum impurity model, solved by numerically exact ‘continuous-time’ QMC . Simulations of wide parameter regimes, for a range of cluster sizes/geometries. 
 Determine which features are robust, which may be artifacts of the model

Emanuel Gull and Andrew J. Millis, arXiv:1407.0704 Cluster DMFT Cluster DMFT: 
 Approximation to self energy: controlled approximation, Systematic truncation N c exact for N c → ∞ ; ‘single with cluster size N c X X Σ ( k, ω ) = Σ n ( ω ) φ n ( k ) ≈ Σ n ( ω ) φ n ( k ) site’ DMFT for N c =1. 
 Small parameter 1/N c n n Basis functions Example tiling of the BZ: 2d, N c = 16 ( π , π ) Cluster scheme: ‘Dynamical Cluster Approximation’ 
 (DCA), basis functions ϕ constant on patches in BZ Example tiling of the BZ: 2d, N c = 2, 4, 4, 8 (0 , 0) ( π , π ) (0 , π ) ( π , π ) ( π , 0) ( π , π ) (0 , π ) ( π , π ) ( π / 2 , π / 2) (0 , 0) (0 , 0) ( π , 0) (0 , 0) (0 , 0) ( π , 0) Resulting lattice system mapped onto impurity model & self-consistency DCA: Hettler et al., Phys. Rev. B 58, R 7475 (1998), 
 DMFT: Metzner, Vollhardt, Phys. Rev. Lett. 62, 324 (1989), 
 Lichtenstein, Katsnelson, Phys. Rev. B 62, R9283 (2000), 
 Georges, Kotliar, Phys. Rev. B 45, 6479 (1992), 
 CDMFT: Kotliar et al., Phys. Rev. Lett. 87, 186401 (2001), 
 Jarrell, Phys. Rev. Lett. 69, 168 (1992), 
 Review: T. Maier, et al., Rev. Mod. Phys. 77, 1027 (2005). Georges et al., Rev. Mod. Phys. 68, 13 (1996)

Phys. Rev. Lett. 106, 030401 (2011) Intermezzo: 3D Hubbard Model ‘Optical Lattice Emulator’: Goal is to experimentally simulate simple model Hamiltonians using cold atomic (fermionic) gases. T. Esslinger, Annu. Rev. Condens. Matter Phys. 1, 129-152 (2010) Test model: 3D Hubbard � t ij ( c † i σ c j σ + c † � H = − j σ c i σ ) + U n i � n i ⇥ . ⇤ ij ⌅ , σ i Temperatures in experiment are high (far above AFM phase). Can we emulate the optical lattice emulator? Numerically exact results needed!