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

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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

SLIDE 1

Quantum Cluster Theory non-local corrections to DMF Mark Jarrell University of Cincinnati

DCA
Cluster Solvers
Convergence
Outlook

SLIDE 2

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).

SLIDE 3

Local Approximations

Field Mean

D=1 D=2 D=3 D=∞

CPA DMF Curie-Weiss Migdal-Eliash. 1/D corrections? PvD 1995

The central site has 2D nearest neighbors

... ...

SLIDE 4

Two Causal Cluster Approaches

Dynamical 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

SLIDE 5

DCA Mapping to Cluster: Coarse Graining

kx ky

Kx Ky

M k = K

k

K

k

k3 k2

= N º k1ƒ k2 ,k3

Ncº M k1 ƒ M k2 ,M k3

SLIDE 6

DCA vs. DMFA

k

k3 k5 k2

G k Œ G r= 0

r=0 r=0

= N cº M k1 ƒ M k2 , M k3

k

k3 k5 k2

G k Œ G K

Nc=1 DMFA Nc >1 DCA

K

K'ƒQ Kƒ Q

K'

Mueller Hartmann (89) Metzner Vollhardt (89)

V k ŒV r= 0 V k ŒV K

V G

k4 k6 k6 k4

Q

SLIDE 7

Dynamical Cluster Approximation

´ G k ,V k H´ G K ,V k ¬ G k H¬ G,V

¶ =´ ƒ Tr ² G ƒTrln G

º ¶ º G= 0Œ² G k H² G,V

mapping from the cluster back to the lattice

SLIDE 8

DCA Algorithm

Cluster Solver

G K

² K = 1 G0 K 1 G K 1 G0 K = ² K ƒ 1 G K

G0 K

G G k

SLIDE 9

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

¬

SLIDE 10

Cluster Solvers

Cluster Solver

1/G 0 K = ² K ƒ 1/G K

G G k

² K = 1/G0 K 1/G K

Quantum Monte Carlo FLEX Non-Crossing Approximation Exact Enumeration Average over Disorder

SLIDE 11

Quantum Monte Carlo Cluster Solver

G

QMC Cluster Solver on one processor

G

QMC time warmup sample

QMC Cluster Solver on one processor QMC Cluster Solver on one processor QMC Cluster Solver on one processor

G

G G G

warmup sample QMC time

Serial Perfectly Parallel

SLIDE 12

Hybrid Parallel QMC

QMC Cluster Solver on many processors QMC Cluster Solver on many processors

G

G G

warmup sample QMC time

G G G G G G G G G G G G G G G G

Perfectly parallel array of cpu's

G

Hybrid parallel array of cpu's

G G G

OpenMP

PBLAS

SLIDE 13

Sign Problem

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

SLIDE 14

Phase Diagram for 2DHubbard

SLIDE 15

FLEX as a Cluster Solver (2D Hubbard)

See poster by Karan Aryanpour

SLIDE 16

Compare Cluster Approaches

Effective medium Cluster Effective medium Cluster

¬ ¬

¬ 1 L

L

¬2D L

L

D = 2D

L

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/CMDF DCA U Simplified Hubbard Model

SLIDE 17

Compare Cluster Approaches

U

tÜ=t tÝ=0

Simplified Hubbard Model

SLIDE 18

Compare Cluster Approaches

SLIDE 19

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).

SLIDE 20

MCA Mapping to Cluster: Molecules

– x

X

x

L

x = – x ƒ X

Correlations within the molecules are treated explicitly; while those between molecules are ignored

G X 1 , X 2 , – x

Translational invariance is violated

SLIDE 21

MCA vs. DMFA

x

G x 1 , x 2 Œ G – x = 0

X=0 X=0

G x 1 , x 2 Œ G X 1 , X 2 , – x = 0

Nc=1 DMFA Nc >1 MCA

Molecule Nc >1 Molecule Nc=1

x

X1 X2

SLIDE 22

Cellular Dynamical Mean Field Molecular CPA

´ G x 1 x 2 H ´ G X 1 , X 2 , – x = 0

¶ =´ ƒ Tr ² G ƒTrln G

º ¶ º G = 0Œ² x1 x2 H ² X 1 , X 2 , – x= 0 º –

x1, – x2