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
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 www.physics.uc.edu/~jarrell/ www.physics.uc.edu/~jarrell/talks/ xxx.lanl.gov
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
Periodic Lattice Dynamical Cluster Approximation Effective Medium
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
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
Problems Simulating 1D Systems QMC requires significant computer power Correlations over many length scales QMC minus sign problem spectra
Problems Simulating 1D Systems QMC requires significant computer power Correlations of many length scales QMC minus sign problem spectra
We Solve The Cluster Problem with QMC ORNL/CES and OSC CRAY X1 ORNL IBM p690 (cheetah)
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
DCA-QMC Runtime 8 32 IBM 8 CRAY 32 T. Maier, www.cray.com
Performance of Concurrent DERs note the log scale ' = a bt N=4480
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
Performance of threaded DERs X1 eliminates the need for hybrid parallelization 1 DER 1Proc 1 DER 32 procs 32 DER 32 procs
Model of Spin-Charge Separation in 1D C. Slezak Velocities fit to Luttinger Liquid form (Zacher, PRB 57, 6370)
Problems Simulating 1D Systems QMC requires significant computer power Correlations over many length scales QMC minus sign problem spectra
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
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
The FluctuationExchange Approximation An infinite geometric resummation of certain classes of pp and ph graphs. N.E. Bickers, 1989
1D Hubbard Model N=1 T/W=0.04 J. Hague, PRB 69, 165113 Lieb and Wu, PRL 1968
Spin-Charge Separation with Hybrid FLEX C. Slezak Nc=8 Hybrid result, roughly = Nc=20 QMC Result, saving a factor of 16
Problems Simulating 1D Systems QMC requires significant computer power Correlations over many length scales QMC minus sign problem spectra
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
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
S. Doluweera Endres PRB 53 5530
Two-Chain Model Spectra
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
Configurational Correlations in Binary Alloys A B X 1 X X X + 1 + X 2 + + X X X X X + 1 X 1 2 + 1,2,...=(k,iωn) + 3 + X + + 1 2 M. Jarrell, D. Johnson,... preprint 1 2 X 3 X scattering potential V(r-r') Madelung PAB PA PB Neutron scattering
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
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