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Quantum Cluster Theorynon-local corrections to DMFMark JarrellUniversity of Cincinnati
● DCA● Cluster Solvers● Convergence● Outlook
Collaborators and References
● K. Aryanpour● J. Deisz● O. Gonzalez● J. Hague● M. Hettler● C. Huscroft● H.R. Krishnamurthy● A. Macridin● Th. Maier● Th. Pruschke● Th. Schulthess● A.N. Tavilderzahdeh● F.C. Zhang
● Papers and talks (DCA):� www.physics.uc.edu/~jarrell/� www.physics.uc.edu/~jarrell/TALKS/� xxx.lanl.gov
● Figures:� www.lps.u-
psud.fr/Activites/ThemeA.asp● Further reading and Citations
� CDMF Kotliar et al., PRL 2001� MCPA F. Ducastelle, J. Phys. C. 7,
1795 (1974).
Local Approximations
Field
Mean
D=1 D=2 D=3 D=∞
CPADMF
Curie-WeissMigdal-Eliash.
1/D corrections?PvD 1995
The central site has 2D nearest neighbors
...
...
Two Causal Cluster ApproachesDynamical Cluster Approximation Cellular Dynamical Mean Field
Molecular CPA
Effective medium
Cluster
Effective medium
Cluster
L
�x
X
- k= 2Æ/L
�k
First Brillouin zone
K
Ducastelle 74 Kotliar 01
DCA Mapping to Cluster: Coarse Graining
kx
ky
Kx
Ky
M k = Kk
Kk1
k3
k2
- = N º k1� k2 ,k3
Ncº M k1 � M k2 ,M k3
DCA vs. DMFA
k1k3
k5
k2- =1
G k �G r= 0
r=0 r=0
- = N cº M k 1 � M k 2 , M k 3
k1k3
k5
k2G k �G K
Nc=1 DMFA
Nc >1 DCA
K
K' � Q
K� Q
K'
Mueller Hartmann (89)Metzner Vollhardt (89)
V k �V r= 0
V k �V K
V
G
k4k6
k6
k4 Q
Dynamical Cluster Approximation
´ G k ,V k H´ G K ,V k
¬ G k H¬ G,V
¶ = ´ � Tr ² G �Trln �Gº ¶º G
= 0�² G k H² G,V
mapping from the cluster back to the lattice
DCA Algorithm
ClusterSolver
G K
² K = 1G0 K
� 1G K
1G0 K
= ² K � 1G K
G0 K
G G k
Dynamical Cluster Approimation
Effective medium
Cluster
●fully causal●maintains lattice point group symmetries●maintains translational invariance●systematic (DMFA → Nc=1)●converges quickly Γ∝ 1/L2
¬
Cluster Solvers
ClusterSolver
1/G 0 K = ² K � 1/G K
G G k
² K = 1 /G0 K � 1/G K
Quantum Monte CarloFLEX
Non-Crossing ApproximationExact Enumeration
Average over Disorder
Quantum Monte Carlo Cluster Solver
G0QMC ClusterSolver on oneprocessor
G
QMCtimewarmup sample
QMC ClusterSolver on oneprocessor
QMC ClusterSolver on oneprocessor
QMC ClusterSolver on oneprocessor
G0
G
G
G
warmup sampleQMCtime
Serial
Perfectly Parallel
Hybrid Parallel QMC
QMC ClusterSolver on manyprocessors
QMC ClusterSolver on manyprocessors
G0
G
G
warmup sampleQMCtime
G GGG
G GGG
G GGG
G GGG
Perfectly parallel array of cpu's
GHybrid parallel array of cpu's
GG
GOpenMP
orPBLAS
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
Effective medium
Cluster
Effective medium
Cluster¬ ¬
¬� 1L2
L
¬�2D LD�1
LD = 2DL
●maintains point group symmetries●fully causal●violates translational invariance●converges slowly with corrections
●maintains point group symmetries●fully causal●maintains translational invariance●converges quickly with corrections
MCPA/CMDFDCA
USimplified Hubbard Model
Compare Cluster Approaches
UtÜ= t tÝ=0Simplified 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/L2 ))•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
X
x
Lx =
�x � X
Correlations within themolecules are treatedexplicitly; while thosebetween molecules areignored
G X 1 , X 2 ,�x Translational invariance
is violated
MCA vs. DMFA
x1 x2G x 1 , x 2 � G
�x = 0
X= 0 X= 0
G x 1 , x 2 � G X 1 , X 2 ,�x = 0
Nc=1 DMFA
Nc >1 MCA
Molecule Nc >1
Molecule Nc=1
x1 x2
X1 X2
Cellular Dynamical Mean FieldMolecular CPA
´ G x 1 � x 2 H ´ G X 1 , X 2 ,�x = 0
¶ = ´ � Tr ² G �Trln �Gº ¶º G
= 0�² x1� x2 H ² X 1 , X 2 ,�x= 0 º �
x1 ,�
x2
MCA Algorithm
ClusterSolver
G
² =G0�1�G�1G0
�1= ² �G�1
G0
G G
G0 ,² ,G... Are matrices in the cluster coordinates