BCS Theory or

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1950s Bardeen Cooper and Schrieffer develop the
theory of (conventional) superconductors
BCS Theory or normal superconductors Physical
Review (1957), awarded Novel Prize in 1972
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 Tgt25K, lattice vibration destroy Cooper
Pairs fundamental upper limit for Tc
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In 1986, Bednorz and Müller discover
superconductivity in La5-xBaxCu5O5(3-y)
La5-xBaxCu5O5(3-y) with x.75 has Tc30K, normal
state is poor conductor Parent 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
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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 Greens
  function (electron density and density functional
  theory not adequate)

The plan is to create a model that can be solved
computationally
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The complex structure of high temperature
superconductors and where things happen
From experiment superconductivity originates
from 2-D CuO2 planes
Heavy ion (La, Y, Ba, Hg, ...) doping add /
remove electrons to CuO2 planes
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The one-band 2D Hubbard model may be simple, but
no simple solution is known for superconductivity!
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Increasing the cluster size leads to performance
problems on scalar processors
G
(dger)
warm up
sample
QMC time
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Workhorses of the QMC-DCA code are DGER and
DGEMM, hence, we analyze DGER
N4480 is a typical problem size for 20 site DCA
cluster
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This translates into about an order of magnitude
increase in performance on the Cray
Code runs at 30-60 efficiency and scales to gt
500 MSPs on the Cray X1
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On the Cray X1 _at_ 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
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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

Number of independent neighboring d-wave
plaquettes
Zd1
Zd2
Zd3
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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
  LDAU or SIC-LSD
• Analyze and understand the pairing mechanism
• Analyze convergence of DCA algorithm
• central problem in order to obtain analytic
  Greens functions!
• uniform convergence has been proved for cluster
  size 1, what about Ncgt1?
• 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

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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 _at_ 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
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Acknowledgement

• 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.

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