Continuous Time Monte Carlo methods for fermions

Transcription

1 Continuous Time Monte Carlo methods for fermions Alexander Lichtenstein University of Hamburg In collaboration with A. Rubtsov (Moscow University) P. Werner (ETH Zurich)

2 Outline Calculation of Path Integral Problems with Hirsch-Fay QMC scheme New fermionic solver - CT-QMC - weak coupling: CT-INT - strong coupling: CT-HYB Magnetic nanosystems Progress in DMFT Conclusions

3 Can we calculate a path integral? Interacting Fermions Partition Function Gaussian Integral

4 QMC for Fermions: Sign Problem 2 Приходится вычислять разность близких по величине членов, а это требует очень аккуратного вычисления каждого члена в отдельности Метод интегрирования по траекториям... фактически никогда не был полезен при рассмотрении вырожденных Ферми-систем 1 Р. Фейнман, А.Хиббс Квантовая механика и интегралы по траекториям

5 Path Integral for impurity problem Partition function: Bath Green-function Vk ε d Hybridization Local Interactions

6 Dynamical Mean Field Theory G 0 ( τ τ ) Σ Σ Σ Gˆ i BZ 1 Ω r ( ω ) = Gˆ ( k, i ) ˆ 1 1 G 0 n n ˆ k r ω ( iω ) = G ( iω ) + Σ( iω ) n ˆ n n Σ ΣU G 0 ( τ τ ) Σ Σ Σ Σ DMRG QMC ED Single Impurity Solver IPT FLEX W. Metzner and D. Vollhardt (1987) A. Georges and G. Kotliar (1992) Σˆ new ( ) 1( ) 1 iω = G iω G ( iω ) n ˆ 0 n ˆ n

7 Monte Carlo: basic M. Troyer (ETH) N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, "Equation of State Calculations by Fast Computing Machines" J. Chem. Phys. 21, 1087 (1953)

8 History of pre CT QMC

9 Continuous Time: World Lines

10 Quantum Monte Carlo

11 Discrete QMC: Hirsch Fye algorithm G b

12 Multi-band Hirsch-Fye QMC-scheme exp{ τu Discrete HS-transformation (Hirsch, 1983) mm' [ n n m m' cosh 1 ( n 2 m + n m' )]} = 1 2 S mm ' exp{ λ S Number of Ising fields: N = M( 2M 1), M = { m, σ} Green Functions: mm ' ( τ ) =± 1 ( λ ) exp τu mm ' 1 2 = mm ' 1 Gmm' ( τ, τ ') = Gmm '( τ, τ ', S) det G Z G ( τ, τ ', S) = G ( τ, τ ') + V ( τ ) δ δ V σ 1 1 mm' mm ' m mm ' ττ ' ( τ ) = λ S ( τ ) σ m mm ' mm ' mm ' m ' mm ' + 1, m < m ' = 1, m > m ' H = tc c + U nn S ij σ + i ij iσ jσ mm' mm' m m' 1 mm' τ mm' m ( n m n m' U )} mm m τ

13 Continuous Time Quantum Monte Carlo Partition function: Continuous Time Quantum Monte Carlo (CT-QMC) E. Gull, A. Millis, A.L., A. Rubtsov, M. Troyer, Ph. Werner, Rev. Mod. Phys. 83, 349 (2011)

14 CT QMC: configurations and weights

15 Continuous time QMC

16 Continuous Time QMC: CT-INT Partition function and action for fermionic system with pair interactions Z = Tr( Te S ) S = t c c drdr ' + w c c c c drdr ' dr dr ' r' + r r' r' + r + r r r' rr r' r' r = {,,} τ i s Splitting of the action into Gaussian part and interaction dr β = dτ 0 i s S = S0 + W ( r' r ( ) ) 2 r' 1 r' 2 r' 2 r' α 1 ' r S = t + w + w dr dr c c drdr ' + 0 r r' rr r r 2 2 r' ( + α )( + α ) W = w c c c c dr dr ' dr dr ' r' r' r r r r rr r' r' r' r' α r r ' - additional parameters - necessary to minimize a sign problem A. Rubtsov and A.L., JETP Lett. (2004)

17 CT-QMC formalism and Green function Perturbation-series expansion Ω = Z = dr dr '... dr dr ' Ω ( r, r ',..., r, r ' ) k= 0 ( 1) k! 1 1 2k 2k k 1 1 2k 2k r' 1r' 2 r' 2k 1r' 2k r1... r2k ( r1, r' 1,..., r2, r' 2 ) Z0 w... w D 12 2k 12k ' 1... ' 2k k k k rr r r r r k ( + r ) ( ) 1 r1 + α... α D = T c c c c r... r r r r'... r' r' r' r' r' 1 2k 2k 2k 1 2k 1 1 2k 2k Since S 0 is Gaussian one can apply the Wick theorem D can be presented as a determinant g0 The Green function can be calculated as follows g ( r ) ( ) 1 r1 r2k r2k α... α Tc c c c c c + r + + r r' r' r' r' r' r' ( k) = r r 1 1 2k 2k ( + r ) ( ) 1 r1 + 2k 2k r' αr'... r' αr' T c c c c 1 1 2k 2k In practice efficient calculation of a ratio is possible due to fast-update formulas ratio of determinants A. Rubtsov and A.L., JETP Lett. (2004)

18 Weak coupling QMC: CT-INT A. Rubtsov, 2004

19 CT INT: details Trivial sign problem: P-H transformation A. Rubtsov, cond-mat 2003 Possible updates:

20 CT INT: multiorbital scheme

21 CT-INT: random walks in the k space Z= Z k-1 + Z k + Z k+1 +. decrease k-1 k+1 Acceptance ratio increase Step k-1 k D w D k 1 k Distribution Step k+1 w k + 1 D D k+ 1 k Maximum at βun 2 k

22 Convergence with Temperature: CT-INT Maximum: βun 2

23 CT QMC Fast Update: k > k+1 Similar to QR-algorithm (K+1) 2 operations

24 Measurement of Green functions

25 Advantages of the CT QMC method non-local in time interactions: dynamical Coulomb screening non-local in space interactions: multi-band systems, E-DMFT Auxiliary field (Hirsch) algorithm is time-consuming since it s necessary to introduce large number of auxiliary fields, while CT-QMC scheme needs almost the same time as in local case Number of auxiliary spins in the Hirsch scheme Short-range interactions Local in time interactions Long-range interactions Non-local in time interactions

26 Complexity of the algorithm

27 Metal-insulator transition in the Hubbard model on Bethe lattice Initial Green function corresponds to semicircular density of state ( ( 2 )) 1 i Gi ( ω ) = µ ω+ ω + 1 We solve the effective one-site + problem by CTQMC method G( τ τ ') = Tc( τ) c ( τ ') S eff Equation of DMFT self-consistency G i i t G i 1 ( ω) ω µ 2 ( ω) 0 = + Self-energy Σ ( iω) = G ( iω) G ( iω) 1 1 0

28 Metal-insulator transition in the Hubbard model on Bethe lattice G(iω) U=3 U= Density of states for β=64: U=2; U=2.2; U=2.4; U= iω U=2 DOS Energy -1-2 U=3 4 3 coexistence of the metallic and insulating solutions: U=2.4, β=64, W=2 Σ(iω) G(iω) iω iω DMFT on Bethe lattice. Parameters: U=2, U=2.2, U=2.4, U=2.6, U=2.8, U=3 β=64, band width W=2 CTQMC scheme with β=64 V. Savkin et al PRB 2005

29 CT-QMC: Hybridization expansion (CT-HYB) Hamiltonian: Loc-d Hyb Bath-a Ph. Werner, et al PRL 97, (2006)

30 CT HYB: diagrammatics with Hubbard X (1,1) (0,1) (1,0) (0,0) 0 β Zk= exp{-u* + * )}*Δ* Δ

31 Strong-Coupling Expansion CT-HYB P. Werner, 2006

32 CT HYB

33 CT HYB: determinant weight

34 CT HYB: determinant weght

35 Diagrams vs. Determinats QMC Ph. Werner

36 CT HYB: Monte Carlo sampling

37 CT HYB: segment scheme +

38 CT HYB: multi orbital segment picture H int = ij U ij n i n j

39 CT QMC efficency

40 CT HYB: General multliorbital Interaction

41 CT HYB: matrix code

42 Use of Symmetry: CT HYB K. Haule PRB 75, (2007)

43 CT HYB: Krylov code

44 CT HYB: Krylov code

45 CT HYB: Krylov scaling

46 CT QMC Krylov: performance

47 ALPS project: CT QMC code CT-INT and CT-HYB

48 Continuous Time Monte Carlo methods for fermions Alexander Lichtenstein University of Hamburg In collaboration with A. Rubtsov (Moscow University) P. Werner, B. Surer (ETH Zurich) H. Hafermann (EPL Paris) T. Wehling (University of Bremen) A. Poteryaev (IMF Ekaterinburg)

49 Impurity solver: miracle of CT-QMC Interaction expansion CT-INT: A. Rubtsov et al, JETP Lett (2004) Hybridization expansion CT-HYB: P. Werner et al, PRL (2006) Efficient Krylov scheme: A. Läuchli and P. Werner, PRB (2009) E. Gull, et al, RMP 83, 349 (2011)

50 Comparisson of different CT QMC: U=W E. Gull et al cond-mat/060943

51 Comparison of different CT QMC Σ Σ Σ Σ Σ U Σ Σ Σ τ U G( τ τ ) τ Ch. Jung, unpublished CT-QMC review: E. Gull et al. RMP (2011)

52 Scaling of CT QMC Temperature Interactions

53 Benchmark for CT QMC

54 CT HYB: 1 band DMFT results Bethe lattice with W=4t

55 Kondo lattice model

56 KLM: MIT on Bethe lattice

57 CT HYB: 2 orbital model

58 CT HYB for 2 orbitals: OSMT

59 General Interaction: Multiorbital impurity with general U U = ij kl diσ d jσ ' 2 ijkl r12 σσ ' d lσ ' d kσ Krylov-CT-QMC A. Läuchli and Ph. Werner, et al PRB 80, (2009)

60 Anderson Impurity Model Hamiltonian of AIM: Hybridization function:

61 DFT+AIM using Projectors Projections of DFT basis Local Green function on local orbitals VASP PAW basis set G. Trimarchi, et al JPCM (2008), B. Amadon, et al., PRB (2008)

62 Hybridization function Co on/in Cu(111) Hybridization of Co in bulk twice stronger than on surface Hybridization in energy range of Cu d orbitals more anisotropic on surface Co d occupancy: n= 7 8 B. Surer, et al PRB 85, (2012)

63 Constrain GW calculations of U F. Aryasetiawanan et al PRB(2004)

64 Wannier GW and effective U(ω) T. Miyake and F. Aryasetiawan Phys. Rev. B 77, (2008) C-GW GW

65 Strength of Coulomb interactions: Graphene T. Wehling et al., PRL 106, (2011) C. Honerkamp, PRL 100, (2008) Z. Y. Meng et al., Nature 464, (2010)

66 Quasiparticle spectra: DFT vs. QMC Co in Cu: QMC and GGA agree qualitatively Quasiparticle peak twice narrower in QMC than in GGA Co on Cu QMC shows, both, quasiparticle peak and Hubbard like bands at higher energies Significantly reduced width of quasiparticle peak in QMC

67 Orbitally resolved Co DOS from QMC Orbitally resolved DOS of the Co impurities in bulk Cu and on Co (111) obtained from QMC simulations at temperature T = ev and chemical potential μ = 27 ev and μ = 28 ev, respectively. All Co d orbitals contribute to LDOS peak near EF=0

68 Self-energies: Local Fermi liquid Fermi liquid: Atomic Signatures limit: of low energy Fermi liquids in all orbitals!

69 Quasiparticle weight and Kondo temperature Quasiparticle weight QMC(Matsubara) Kondo temperature Exp:

70 Charge fluctuations: QMC results

71 Multi-orbital problems: general interaction New formalism allows one to consider the most general case of multi-orbital interactions Uˆ = U c c c c i, j, k, l; σσ, ' + + ijkl iσ jσ ' lσ ' kσ two band rotationally invariant impurity model three impurity atoms with Hubbard and exchange interaction 0.3 two bands U=4, J=1, β=4-0.2 U=2.4, J=-0.2 and J=0, β= DOS G(iω) Energy iω five bands U=2, J=0.2, β=4 0.6 U=2.4, J=-0.2 and J=0, β= DOS DOS Energy Energy

72 Cluster DMFT G 0 ( τ τ ) Σ Σ Σ Σ Σ ΣU Σ Σ V Σ ΣU Σ Σ Σ Σ Σ Σ M. Hettler et al, PRB 58, 7475 (1998) A. L. and M. Katsnelson, PRB 62, R9283 (2000) G. Kotliar, et al, PRL 87, (2001)

73 Double Bethe Lattice: exact C DMFT A. Ruckenstein PRB (1999)

74 Self consistent condition: C DMFT AF-between plane AF-plane

75 Finite temperature phase diagram order-disorder transition at tp / t=sqrt(2) MIT for intermediate U for large U H. Hafermann, et al. EPL, 85, (2009)

76 Density of States: large U

77 Spin correlations: large U

78 H. Park et al, PRL (2008) MIT in 2d: DMFT vs. C DMFT Uc=9.35t Uc=6.05t U=5.2t X= U=0 U=6t n=1

79 TM-Oxide VO 2 : singlet formation M. Marezio et al., (1972) Temperature (K) Metal Insulator Rutile structure Monoclinic distortion in the insulating phase G( ω) ij i U UH t ij j U ε i a b LH ε j Correlation vs. Bonding U/t

80 Cluster DMFT results for VO 2 Rutile 1.0 ρ(ω) LDA (dashed) DMFT (solid) U=4eV J=0.68eV VO 2 rutile U = 4 ev, J=0.68 ev β = 20 ev M ω[ev] Sharp peak below the gap DOS VO 2 M1 1.5 LDA (dashed) is NOT a Hubbard band! cluster DMFT 1.0 (solid) ω [ev] 2 4 New photoemission from Tjeng s group T. C. Koethe, et al. PRL (2006) S. Biermann, et al, PRL 94, (2005)

81 Conclusions Electronic Structure of correlated nano systems can be described in CT QMC scheme CT QMC is perfect for supercomputer applications

82 General Projection formalism for LDA+DMFT DELOCALIZED S,P-STATES L> G> CORRELATED D,F-STATES G. Trimarchi et al. JPCM 20, (2008) B. Amadon et al. PRB 77, (2008)

83 CT HYB example