Quantum Cluster Theory non-local corrections to DMF Mark Jarrell - PowerPoint PPT Presentation

Quantum Cluster Theory non-local corrections to DMF Mark Jarrell University of Cincinnati ● DCA ● Cluster Solvers ● Convergence ● Outlook

Collaborators and References ● Papers and talks (DCA): ● K. Aryanpour – www.physics.uc.edu/~jarrell/ ● J. Deisz – www.physics.uc.edu/~jarrell/TALKS/ ● O. Gonzalez – xxx.lanl.gov ● J. Hague ● Figures: ● M. Hettler ● C. Huscroft – www.lps.u- ● H.R. Krishnamurthy psud.fr/Activites/ThemeA.asp ● A. Macridin ● Further reading and Citations ● Th. Maier – CDMF Kotliar et al. , PRL 2001 ● Th. Pruschke – MCPA F. Ducastelle, J. Phys. C. 7 , ● Th. Schulthess 1795 (1974). ● A.N. Tavilderzahdeh ● F.C. Zhang

Local Approximations Mean CPA ... DMF Curie-Weiss Field Migdal-Eliash. D= ∞ D=1 D=2 D=3 The central site has 2D nearest neighbors 1/D corrections? ... PvD 1995

Two Causal Cluster Approaches Cellular Dynamical Mean Field Dynamical Cluster Approximation Molecular CPA Ducastelle 74 Kotliar 01 Cluster Cluster Effective medium Effective medium – X k K – x - k = 2 Æ / L L First Brillouin zone

DCA Mapping to Cluster: Coarse Graining Ky k 1 k 2 K Kx k 3 - = N º k 1 ƒ k 2 ,k 3 ky M k = K k Nc º M k 1 ƒ M k 2 ,M k 3 kx

Mueller Hartmann (89) DCA vs. DMFA Metzner Vollhardt (89) Nc=1 DMFA G r = 0 r = 0 k - = 1 k 3 1 k 4 k 2 V G k Œ G r = 0 k 6 k 5 V k Œ V r = 0 K ƒ Q Nc >1 DCA k 1 - = N c º M k 1 ƒ M k 2 , M k 3 k 3 K k 4 k 2 Q G k Œ G K K' k 6 k 5 V k Œ V K K' ƒ Q

Dynamical Cluster Approximation mapping from the cluster back to the lattice ´ G k ,V k H´ G K ,V k ¶ =´ ƒ Tr ² G ƒ Trln � G º ¶ º G = 0 Œ² G k H² G,V ¬ G k H¬ G,V

DCA Algorithm Cluster G K G 0 K Solver 1 1 1 1 ² K = G 0 K � G 0 K = ² K ƒ G K G K G G k

Dynamical Cluster Approimation ¬ Cluster Effective medium ● fully causal ● maintains lattice point group symmetries ● maintains translational invariance ● systematic (DMFA → Nc=1) ● converges quickly Γ∝ 1/L 2

Cluster Solvers Quantum Monte Carlo FLEX Non-Crossing Approximation Exact Enumeration Average over Disorder Cluster Solver 1 / G 0 K = ² K ƒ 1 / G K ² K = 1 / G 0 K � 1 / G K G G k

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

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

Sign Problem Finite-Size Simulations (FSS): White (1989)

Phase Diagram for 2DHubbard

FLEX as a Cluster Solver (2D Hubbard) See poster by Karan Aryanpour

Compare Cluster Approaches U Simplified Hubbard Model MCPA/CMDF DCA ¬ ¬ Cluster L Cluster Effective medium Effective medium ● maintains point group symmetries ● maintains point group symmetries ● fully causal ● fully causal ● maintains translational invariance ● violates translational invariance ● converges quickly with corrections ● converges slowly with corrections D � 1 ¬� 1 ¬� 2D L D = 2D 2 L L L

Compare Cluster Approaches t Ü = t t Ý = 0 U Simplified Hubbard Model

Compare Cluster Approaches

Conclusion • DCA: systematic non-local corrections to the DMFA • Preserves translational and point group symmetries • Converges quickly (correction O(1/L 2 )) • Converges quickly even in 1D. • Many cluster solvers may be used. • QMC: very mild minus sign problem • DCA complementary to FSS. • See http://www.physics.uc.edu/~jarrell for more info. Outlook • MFT for the cuprates (lanl.gov) • LDA+DCA (with Th. Schulthess, ORNL). • DCA for nanotubes (lanl.gov shortly). • QMC+MEM codes available for collaboration. • GPL codes within 2 years (some sooner).

MCA Mapping to Cluster: Molecules X Correlations within the molecules are treated – explicitly; while those x between molecules are ignored – L Translational invariance G X 1 , X 2 , x – is violated x = x ƒ X x

MCA vs. DMFA Nc=1 DMFA X = 0 X = 0 Molecule Nc=1 x x 1 2 РΠG x = 0 G x 1 , x 2 Nc >1 MCA X 1 Molecule Nc >1 x X 2 x 1 2 РΠG x = 0 G x 1 , x 2 X 1 , X 2 ,

Cellular Dynamical Mean Field Molecular CPA – ´ x 1 � x 2 H ´ x = 0 G G X 1 , X 2 , ¶ =´ ƒ Tr ² G ƒ Trln � G – º ¶ º G = 0 Œ² x 1 � x 2 H ² x = 0 º – X 1 , X 2 , x 1 , – x 2

MCA Algorithm Cluster G G 0 Solver � 1 � G � 1 � 1 =² � G � 1 ² = G 0 G 0 G G G 0 , ² ,G ... Are matrices in the cluster coordinates