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

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1 1 22

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New scientific insights and new opportunities

Electric Conduction in Normal Metals

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In a perfect crystal

perfect conductor at T=0K

Real materials have defects

resistance finite at T=0K

at very low temperature metalscould 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 produceliquid helium (He) in 1908(Nobel prize in 1913)

Superconductor repels magnetic fieldMeissner 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 1972

1950s: Bardeen Cooper and Schrieffer develop the theory of (conventional) superconductors

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Phonon mediated attractive interaction: formation of Cooper Pairs

Coherence length of Cooper Pairs is ~ 10-4 cm

Superconducting state: Cooper Pairs condense into macroscopic quantum state~ 1023 particles are coherent!

But, at T>25K, lattice vibration destroy Cooper Pairs: fundamental upper limit for TTcc

In 1986, Bednorz and Müller discover superconductivity in La5-xBaxCu5O5(3-y)

La5-xBaxCu5O5(3-y) with x=.75 has Tc~30K, normal state is poor conductorParent compound, LaCuO2, 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

Cu O

O

Cu

O

Cu O Cu

O O

O

O

O Cu

O

O Cu

O

O O O

Cu O CuO O Cu

O O O

From experiment: superconductivity originates from 2-D CuO2 planes

Heavy ion (La, Y, Ba, Hg, ...) doping add / remove electrons to CuO2 planes

Doping with holes (electrons) leads to formation so-called Zhang-Rice singlet (Phys. Rev. B, 1988)

tU

Cu O

O

Cu

O

Cu O Cu

O O

O

O

O Cu

O

O Cu

O

O O O

Cu O CuO O Cu

O O O

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

superconductingAF

t

USimple 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 lattice Cluster coupled tomean-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 1998 ...)

Kinetic energy: Interaction energy:

Treated exactly in infinite system

Cut off correlations beyond cluster

K

k~

kx

2πL

ky

L

First Brillouin Zonecoarse 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 P4 @ CCS and Compaq @ PSC)

d-wavesuperconductingAF

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 up sample QMC time

(dger)

warm up

G G

warm up

G

warm up

Workhorses of the QMC-DCA code are DGER and DGEMM, hence, we analyze DGER

DGER Performance (N=4480)

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500

1000

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2500

3000

Cray X1 SGI Altix IBM p690

Performance of Concurrent DGERs

(N=4480)

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100

1000

10000

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Processors (MSPs)

Cray X1

SGI Altix

IBM p690

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

DCA-QMC Runtime

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10000

20000

30000

40000

50000

Problem Size

IBM p690, 8 PEs IBM p690, 32 PEs

Cray X1, 8 MSPs Cray X1, 32 MSPs

Code runs at 30-60% efficiency and scales to > 500 MSPs on the Cray X1

On the Cray X1 @ CCS we can simulate large enough clusters to validate the DCA algorithm

No antiferromagnetic order in 2D (Mermin Wagner Theorem)

Neel temperature (TN) indeed vanishes logarithmically

1 2

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1 1 22

2

2 1

1 2

34

1

1

2

2

3

3

4

4

Nc=2:1

neighbor

Nc=4:2

neighbors

Take a closer look at the Nc=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

+-

+-

8A

16B

16A

Zd=1 Zd=2 Zd=3

Number of independent neighboring d-wave plaquettes:

Cluster Zd

4A 0(MF)

Superconducting transition is where the d-wave pair-field susceptibility (Pd) diverges

8A 112A 216B

216A 320A 4

24A 426A 4

Tc ≈ 0.025tSecond 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 Tc 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 Nc>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 X1 @ 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-AC05-00OR22725. Work at Cincinnati was supported by the NSF Grant No. DMR-0113574.

Acknowledgement