Quantum Cluster Simulations of Low D Systems

Transcription

1 Quantum Cluster Simulations of Low D Systems Electronic Correlations on Many Length Scales M. Jarrell, University of Cincinnati High Perf. QMC Hybrid Method SP Sep. in 1D NEW MEM SP Sep. in 1D 2-Chain Spectra Other Projects

2 Collaborators and References J. Hague S. Doluweera O. onazalez A. Macridin Th. Maier Th. Pruschke C. Slezak Th. Schulthess D. Johnson f f Papers, talks, and example codes xxx.lanl.gov

3 Carbon Nanotubes Purpose NaV2O5 J. Mintmire PRL 68, 631 vertically coupled aas/alaas quantum wires H. Smolinski PRL 80, 5164 Non-perturbative physics correlations over all length scales very low temperatures M. Weckworth, Superlattices and Microstruct 20, 561

4 Periodic Lattice Dynamical Cluster Approximation Effective Medium

5 DCA Mapping to Cluster: Coarse raining Ky k1 K Kx ky k M k =K kx k2 k3 =N k k 1 2, k3 N c M k M k, M k 1 2 3

6 The Nature of Cluster Approximations screening cloud (r) (x) x=0 Self Energy DMFA DCA Local Short Ranged (k,w) (w) (k,w) (K,w) few K

7 Problems Simulating 1D Systems QMC requires significant computer power Correlations over many length scales QMC minus sign problem spectra

8 Problems Simulating 1D Systems QMC requires significant computer power Correlations of many length scales QMC minus sign problem spectra

9 We Solve The Cluster Problem with QMC ORNL/CES and OSC CRAY X1 ORNL IBM p690 (cheetah)

10 Quantum Monte Carlo Cluster Solver Serial 0 QMC Cluster Solver on one processor warmup Perfectly Parallel QMC Cluster Solver on one processor 0 sample QMC time QMC Cluster Solver on one processor QMC Cluster Solver on one processor warmup sample QMC time

11 DCA-QMC Runtime 8 32 IBM 8 CRAY 32 T. Maier,

12 Performance of Concurrent DERs note the log scale ' = a bt N=4480

13 Hybrid Parallel QMC Perfectly parallel array of cpu's 0 Hybrid parallel array of cpu's QMC Cluster Solver on many processors QMC Cluster Solver on many processors warmup sample OpenMP PBLAS QMC time

14 Performance of threaded DERs X1 eliminates the need for hybrid parallelization 1 DER 1Proc 1 DER 32 procs 32 DER 32 procs

15 Model of Spin-Charge Separation in 1D C. Slezak Velocities fit to Luttinger Liquid form (Zacher, PRB 57, 6370)

16 Problems Simulating 1D Systems QMC requires significant computer power Correlations over many length scales QMC minus sign problem spectra

17 Hybrid, Multiple Embedding Effective Medium Perturbation theory Length scales within the small cluster are treated explicitly Length scales between the large and small cluster are treated perturbatively Length scales beyond the large cluster are treated with a mean field Effective Medium K. Aryanpour, PRB, 2003

18 Ingredients of the Hybrid Approach Dynamical Cluster Approximation Quantum Monte Carlo glue small cluster FLEX perturbation theory large cluster N.E. Bickers, 1989 Effective Medium

19 The FluctuationExchange Approximation An infinite geometric resummation of certain classes of pp and ph graphs. N.E. Bickers, 1989

20 1D Hubbard Model N=1 T/W=0.04 J. Hague, PRB 69, Lieb and Wu, PRL 1968

21 Spin-Charge Separation with Hybrid FLEX C. Slezak Nc=8 Hybrid result, roughly = Nc=20 QMC Result, saving a factor of 16

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23 Problems Simulating 1D Systems QMC requires significant computer power Correlations over many length scales QMC minus sign problem spectra

24 MEM Analytic Continuation of bad data The minus sign problem happens when the QMC sampling weight is not positive definite. In this case, we associate the sign with the measurement, so s W = W s W MEM recasts the analytic continuation problem 2 P(A ) = P( A) P(A) where P( A)=exp(- /2) A. Macridin, preprint

25 Spectra of 2-chain model Carbon Nanotubes NaV2O5 J. Mintmire PRL 68, 631 vertically coupled aas/alaas quantum wires H. Smolinski PRL 80, 5164 M. Weckworth, Superlattices and Microstruct 20, 561

26 S. Doluweera Endres PRB

27 Two-Chain Model Spectra

28 Other Projects Spectra of 1D Hubbard model Thermodynamics of 2-chain model More Accurate Hybrid Method First-Principles simulations of disorder D. Johnson, W. Shelton

29 Configurational Correlations in Binary Alloys A B X 1 X X X X X X X X X + 1 X ,2,...=(k,iωn) X M. Jarrell, D. Johnson,... preprint 1 2 X 3 X scattering potential V(r-r') Madelung PAB PA PB Neutron scattering

30 More Accurate Hybrid Approach 0 F QMC Cluster QMC Cluster Solver on one Solver processor = Γ k, iωn, σ Σ Γ + F Γ F k, iωn, σ -σ k+q, iωn +iυ, σ

31 Conclusions QMC + MEM allow us to study 1D systems spin-charge separation coupled chains Improved efficiency with hybrid approach. New MEM much greater frequency resolution Improved formalism for alloys