Extreme scale simulations of high-temperature superconductivity. Thomas C. Schulthess

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1 Extreme scale simulations of high-temperature superconductivity Thomas C. Schulthess

2 T [K] Superconductivity: a state of matter with zero electrical resistivity Heike Kamerlingh Onnes ( ) Discovery 1911 Microscopic Theory 1957 Superconductivity in the cuprates HgBaCaCuO 1993 TIBaCaCuO 1988 BiSrCaCuO 1988 HgTlBaCuO 1995 YBa 2 Cu 3 O High temperature non-bcs Liquid He Nb Liquid N 2 Hg Pb NbC V 3 Si La 2-x Ba x CuO Nb=A1=Ge NbN Nb 3 Su Nb 3 Ge MgB BCS Theory Low temperature BCS Bednorz and Müller

3 From cuprate materials to the Hubbard model La2CuO4 CuO2 plane Holes form Zhang-Rice singlet states O-px O Sr doping introduces holes O-py Cu-dx 2 -y 2 La Cu Single band 2D Hubbard model

4 The challenge: a (quantum) multi-scale problem Antiferromagnetic correlations / nano-scale gap fluctuations Thurston et al. (1998) Superconductivity (macroscopic) On-site Coulomb repulsion (~A) N ~ complexity ~ 4 N Gomes et al. (2007)

5 Quantum cluster theories Maier et al., Rev. Mod. Phys. 05 Antiferromagnetic correlations / nano-scale gap fluctuations Thurston et al. (1998) Superconductivity (macroscopic) On-site Coulomb repulsion (~A) Explicitly treat correlations within a localized cluster Gomes et al. (2007) Coherently embed cluster into effective medium Treat macroscopic scales within meanfield

6 Systematic solution and analysis of the pairing mechanism in the 2D Hubbard Model First systematic solution demonstrates existence of a superconducting transition in 2D Hubbard model Maier,et al., Phys. Rev. Lett. 95, (2005) Study the mechanism responsible for pairing in the model - Analyze the particle-particle vertex - Pairing is mediated by spin fluctuations Maier, et al., Phys. Rev. Lett (2006) Spin fluctuation Glue

7 Moving toward a resolution of the debate over the pairing mechanism in the 2D Hubbard model We have a mammoth (U) and an elephant (J) in our refrigerator - do we care much if there is also a mouse? - P.W. Anderson, Science 316, 1705 (2007) - see also Scalapino is not a glue sniffer Relative importance of resonant valence bond and spin-fluctuation mechanisms - Maier et al., Phys. Rev. Lett (2008) Fraction of superconducting gap arising from frequencies Ω (a) I(k A, ) U=8 U=10 U=12 " d ( ) k A =(0, ); <n>= (b) U=10 Both retarded spin-fluctuations and nonretarded exchange interaction J contribute to the pairing interaction Dominant contribution comes from spin-fluctuations!

8 Significant increase of supercomputing transition temperature due to nanoscale stripe modulations Maier, Alvarez, Summers and Schulthess Phys. Rev. Lett. in press (2010) Monday, May 31, 2010 NVIDIA ISC Breakfast Briefing

9 Hirsch-Fye Quantum Monte Carole (HF-QMC) for the quantum cluster solver Hirsch & Fye, Phys. Rev. Lett. 56, 2521 (1988) Partition function & Metropolis Monte Carlo Z = e E[x]/kBT dx Acceptance criterion for M-MC move: min{1,e E[x k] E[x k+1 ] } Partition function & HF-QMC: Z s i,l det[g c (s i,l) 1 ] matrix of dimensions N t N t N c N l 10 2 N t = N c N l 2000 Acceptance: min{1, det[g c ({s i,l} k )]/ det[g c ({s i,l} k+1 )]} Update of accepted Green s function: G c ({s i,l} k+1 )=G c ({s i,l} k )+a k b k

10 HF-QMC with Delayed updates (or Ed updates) G c ({s i,l} k+1 )=G c ({s i,l} k )+a k b t k G c ({s i,l} k+1 )=G c ({s i,l} 0 )+[a 0 a 1... a k ] [b 0 b 1... b k ] t Complexity for k updates remains O(kN 2 t ) But we can replace k rank-1 updates with one matrix-matrix multiply plus some additional bookkeeping.

11 Performance improvement with delayed updates 6000 N c =16 N l =150 N t =2400 mixed precision double precision time to solution [sec] delay (k)

12 DCA++ speedup on GPU Meredith et al., Par. Comp. 35, 151 (2009) Speedup of HF-QMC updates (2GHz Opteron vs. NVIDIA 8800GTS GPU): - 9x for offloading BLAS to GPU & transferring all data (completely transparent to application code) - 13x for offloading BLAS to GPU & lazy data transfer - 19x for full offload HF-updates & full lazy data transfer CPU FSB GDDR3 DRAM at 2GHz (eff) PCIe x16 slot PCIe x16 slot North bridge PCI-Express bus DRAM GPU

13 DCA++ with mixed precision Run HF-QMC in single precision HF- QMC cluster solver DCA cluster mapping Keep the rest of the code, in particular cluster mapping in double precision

14 DCA++ with mixed precision Run HF-QMC in single precision HF- QMC cluster solver Multiple runs to compute Tc: Double Precision Double Precision CPU Mixed Single Precision GPU Mixed Single Precision Mean DCA cluster mapping Tc Keep the rest of the code, in particular cluster mapping in double precision

15 Performance improvement with delayed and mixed precision updates 6000 N c =16 N l =150 N t =2400 mixed precision double precision time to solution [sec] delay (k)

16 High-Tc superconductivity: DCA/QMC simulations of the 2D Hubbard model First simulation proving model describes superconducting transition High memory bandwidth helps performance of rank 1 update in DCA/QMC algortihm ~6 TF ~1 TF TF 18.5 TF Cray X1 / X1e Finally settled two-decade old debate over nature of pairing mechanism! ~26 TF <<5 TF 2006 ~60 TF TF 54 TF 119 TF 263 TF Cray XT3/4 Realistic models of disorder and nanoscale inhomogeneities >2 PF 1.34 PF ~260 TF 2009 ~140 TF 2008 Sustained performance of DCA/QMC Work with GPUs motivates change DCA/QMC algorithm to mixed precision Modify QMC algorithm: replaced (delayed) rank 1 updates by matrix multiply 1.4 PF 2.4 PF OLCF-3 Project: Cray XT5 In 5 years, factor 10 3 X compute 10-4 X time to solution First sustained petaflop/s under production condition PFlop/s Hybrid-Multi- Core systems based on NVIDIA Fermi (2011/12) Monday, May 31, 2010 NVIDIA ISC Breakfast Briefing

17 Collaborators (superconductivity / DCA++): Thomas Maier (ORNL) Gonzalo Alvarez (ORNL) QUESTIONS? Mike Summers (ORNL) Paul Kent (ORNL) Ed D Azevedo (ORNL -- Comp. Math.) Jeremy Meredith (ORNL -- future tech.) Jeff Vetter (ORNL -- future tech.) Trey White (ORNL -- NCCS) Markus Eisenbach (ORNL -- NCCS) Doug Scalapino (UCSB) Mark Jarrell (U. of Cincinnati)... many others Monday, May 31, 2010 NVIDIA ISC Breakfast Briefing