Exciting New Insights into Strongly Correlated Oxides with Advanced Computing: Solving a Microscopic Model for High Temperature Superconductivity T. Maier, J. B. White, T. C. Schulthess (ORNL) M. Jarrell (University of Cincinnati) P. Kent (UT/JICS & ORNL)

What is superconductivity Outline of this Talk A model for high temperature superconductors t U Algorithm and leadership computing New scientific insights and new opportunities

Electric Conduction in Normal Metals In a perfect crystal perfect conductor at T=0K Real materials have defects resistance finite at T=0K at very low temperature metals could become insulators (?) (proposal by Kelvin, 1902)

While verifying Kelvin’s theory, Kamerling Onnes discovers superconductivity in Hg at 4K Resistance in pure mercury (Hg) drops to zero at liquid He temp. Kamerling Onnes first to produce liquid helium (He) in 1908 (Nobel prize in 1913) Superconductor repels magnetic field Meissner and Ochsenfeld, Berlin 1933 Superconducting state is a new phase with zero resistance and perfect diamagnetism

BCS Theory or “normal” superconductors : Physical Review (1957), awarded Novel Prize in s: Bardeen Cooper and Schrieffer develop the theory of (conventional) superconductors Phonon mediated attractive interaction: formation of Cooper Pairs Coherence length of Cooper Pairs is ~ cm Superconducting state: Cooper Pairs condense into macroscopic quantum state ~ particles are coherent! T c But, at T>25K, lattice vibration destroy Cooper Pairs: fundamental upper limit for T c

In 1986, Bednorz and Müller discover superconductivity in La 5-x Ba x Cu 5 O 5(3-y) La 5- x Ba x Cu 5 O 5(3- y ) with x =.75 has T c ~30K, normal state is poor conductor Parent compound, LaCuO 2, is an insulator! (Bednorz and Müller, Z. für Physik 1986, Nobel Prize 1987) Something other than phonon mediate the formation of Cooper Pairs

Why modeling high temperature super- conductors is a challenge We have to account for a macroscopic number of particles The particles are correlated over several nanometers (from measured antiferromagnetic fluctuations) We need the many-body wave function or Green’s function (electron density and density functional theory not adequate) The plan is to create a model that can be solved computationally

The complex structure of high temperature superconductors and where things happen CuO O O O O O O O O O O O OOO O OO OOO From experiment: superconductivity originates from 2-D CuO 2 planes Heavy ion (La, Y, Ba, Hg,...) doping add / remove electrons to CuO 2 planes

Doping with holes (electrons) leads to formation so-called Zhang-Rice singlet (Phys. Rev. B, 1988) t U CuO O O O O O O O O O O O OOO O OO OOO Map onto a simple one-band and 2-D Hubbard model:

The cuprate high temperature superconductors are complex in a canonical way Complex crystal structure Canonical phase diagram of the cpurates superconducting AF t U Simple model can the simple 2D Hubbard model describe such rich physics?

The one-band 2D Hubbard model may be simple, but no simple solution is known for superconductivity!

Outline of the Dynamical Cluster Approximation (DCA) Infinite latticeCluster coupled to mean-field DCA  Short-ranged correlations within cluster treated explicitly  Longer length scales treated on mean-field level

The key idea of the DCA is to systematically coarse-grain the self-energy (Jarrell et al., Phys. Rev. B ) Kinetic energy:Interaction energy: Treated exactly in infinite system Cut off correlations beyond cluster K k ~ kxkx kyky First Brillouin Zone coarse grain self-energy:

What the DCA accomplishes in a nutshell: Non-local correlations Thermodynamic limit Cluster in reciprocal space Translational symmetry QMC We use Quantum Monte Carlo (QMC) to solve the many-body problem on the cluster

QMC-DCA generated phase diagram using a 2x2 cluster (calculations done on IBM CCS and PSC) d-wave superconducting AF Issue:no antiferromagnetic (AF) transition in a strictly 2D model Consequence of small (4-site) cluster: Need to study larger clusters!

Increasing the cluster size leads to performance problems on scalar processors G warm upsampleQMC time (dger) warm up G G G

Workhorses of the QMC-DCA code are DGER and DGEMM, hence, we analyze DGER N=4480 is a typical problem size for ~20 site DCA cluster

This translates into about an order of magnitude increase in performance on the Cray Code runs at 30-60% efficiency and scales to > 500 MSPs on the Cray X1

On the Cray CCS we can simulate large enough clusters to validate the DCA algorithm No antiferromagnetic order in 2D (Mermin Wagner Theorem) Neel temperature ( T N ) indeed vanishes logarithmically Nc=2: 1 neighbor Nc=4: 2 neighbors Take a closer look at the N c =2 and 4 cases

Pay attention when running larger clusters to study the superconducting transition Problem: - d-wave order parameter non-local (4 sites) - Expect large size and geometry effects in small clusters A 16B 16A Z d =1 Z d =2Z d =3 Number of independent neighboring d-wave plaquettes:

Cluster Z d - 4A 0(MF) Superconducting transition is where the d- wave pair-field susceptibility (P d ) diverges 8A 1 12A 2 16B 2 16A 3 20A 4 24A 4 26A 4 T c  0.025t Second neighbor shell difficult due to QMC sign problem Superconductivity can be a consequence of strong electron correlations

What next? Materials specific model: try to understand the differences in T c for different Cuprates (La vs. Hg based compounds) - use input band structure from density functional ground state calculations - explore better functionals than LDA, for example LDA+U or SIC-LSD Analyze and understand the pairing mechanism Analyze convergence of DCA algorithm - central problem in order to obtain analytic Green’s functions! - uniform convergence has been proved for cluster size 1, what about N c >1? Develop a multi-scale DCA approach - QMC sign problem WILL limit maximum cluster size and parameter range! - different approximations of the self-energy at different length and time scales

Summary / Conclusions Superconductivity, a macroscopic quantum effect 2-D Hubbard model for strongly correlated high temperature superconducting cuprates Dynamical Cluster Approximation, QMC-DCA code, and the impact of the Cray CCS to solve the 2-D Hubbard model We can model phase diagram of the cuprates microsopically Superconductivity can be a result of strong electron correlations

This research used resources of the Center for Computational Sciences and was sponsored in part by the offices Basic Energy Sciences and of Advance Scientific Computing Research, U.S. Department of Energy. The work was performed at Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR Work at Cincinnati was supported by the NSF Grant No. DMR Acknowledgement