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Continuous Time Monte Carlo methods for fermions
Alexander LichtensteinUniversity of Hamburg
In collaboration withA. Rubtsov (Moscow University)
P. Werner (ETH Zurich)
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
Can we calculate a path integral?Interacting Fermions
Partition Function
Gaussian Integral
QMC for Fermions: Sign Problem
“Приходится вычислять разность близких по величинечленов, а это требует очень аккуратного вычислениякаждого члена в отдельности”
“Метод интегрирования по траекториям ... фактическиникогда не был полезен при рассмотрении вырожденныхФерми-систем”
Р. Фейнман, А.ХиббсКвантовая механика и интегралы по
траекториям
1
2
Path Integral for impurity problem
Partition function:
Bath Green-function
Hybridization
Local Interactions
d
εVk
Dynamical Mean Field Theory
( )ττ ′−0G
( ) ( )∑Ω=
BZ
knn ikGiG
r
rωω ,ˆ1ˆ
( ) ( ) ( )nnn iiGi ωωω Σ+= −− ˆˆˆ 110GΣ Σ Σ
Σ Σ
Σ Σ Σ
ΣU
QMC ED
DMRG IPTFLEX
( )ττ ′−0G
( ) ( ) ( )nnnnew iGii ωωω 110
ˆˆˆ −− −=Σ G
Single Impurity Solver
W. Metzner and D. Vollhardt (1987)A. Georges and G. Kotliar (1992)
Monte Carlo: basicM. 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)
History of pre‐CT‐QMC
Continuous Time: World Lines
Quantum Monte Carlo
Discrete QMC: Hirsch‐Fye algorithm
G
b
Multi-band Hirsch-Fye QMC-scheme
)(exp21)](
21[exp
''1
'''''
mmmmS
mmmmmmmmnnSnnnnU
mm
−=+−∆− ∑±=
λτ
Discrete HS-transformation (Hirsch, 1983)
Number of Ising fields: ,),12( σmMMMN =−=
( ) ⎟⎠⎞
⎜⎝⎛ ∆= '' 2
1expcosh mmmm Uτλ
Green Functions:
'
1' '
( )
1 1' ' ' '
' ' ''
'
1( , ') ( , ', ) det
( , ', ) ( , ') ( )( ) ( )
1, '1, '
m m
m m m mS
m m m m m m m
m m m m m m mm
m m
G G S GZ
G S VV S
m mm m
τ
ττ
τ τ τ τ
τ τ τ τ τ δ δ
τ λ τ σ
σ
−
− −
= ×
= +
=
+ <⎧= ⎨− >⎩
∑
∑G
' ''i
ij i j mm m mij mmH t c c U n nσ σσ
+= − +∑ ∑
U
m´
´mm´
τ
m
τ
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)
CT‐QMC: configurations and weights
Continuous time QMC
Continuous Time QMC: CT-INT
Partition function and action for fermionic system with pair interactions Tr( )SZ Te−=
1 2 1 2
1 2 1 2
' ''' ' ' 1 1 2 2' ' 'r r r rr r
r r r r r rS t c c drdr w c c c c drdr dr dr+ + += +∫ ∫ ∫ ∫ ∫ ∫
, , r i sτ=0 i s
dr dβ
τ= ∑∑∫ ∫Splitting of the action into
Gaussian part and interaction 0S S W= +
( )( )2 1 2 2 1
2 1 2 2 1
' ' ' ''0 ' 2 2 '' 'r r r r rr r
r r r r r r rS t w w dr dr c c drdrα += + +∫ ∫ ∫ ∫
( )( )1 2 1 1 2 2
1 2 1 1 2 2
' '' ' ' ' 1 1 2 2' 'r r r r r r
r r r r r rW w c c c c drdr dr drα α+ += − −∫ ∫ ∫ ∫
'rrα - additional parameters - necessary to minimize a sign problem
A. Rubtsov and A.L., JETP Lett. (2004)
CT-QMC formalism and Green function
Perturbation-series expansion1 1 2 2 1 1 2 2
0
' ... ' ( , ' ,..., , ' )k k k k kk
Z dr dr dr dr r r r r∞
=
= Ω∑∫ ∫ ∫ ∫
2 1 2 1 21 2
1 2 2 1 2 1 2
' ' ...' '1 1 2 2 0 ' ... '
( 1)( , ' ,..., , ' ) ...!
k k k
k k k
kr r r rr r
k k k r r r r r rr r r r Z w w Dk
−
−
−Ω =
( ) ( )1 2 2 21 1
1 2 1 1 2 2
...' ... ' ' ' ' '...k k k
k k k
r r r rr rr r r r r rD T c c c cα α+ += − −
Since S0 is Gaussian one can apply the Wick theorem
D can be presented as a determinant g0
( ) ( )( ) ( )
2 21 1
1 1 2 2
2 21 1
1 1 2 2
' ' ' ' ''
' ' ' '
...( )
...
k k
k k
k k
k k
r rr rrr r r r rr
r r rr rr r r r
Tc c c c c cg k
T c c c c
α α
α α
+ + +
+ +
− −=
− −The Green function can be
calculated as follows
ratio of determinantsIn practice efficient calculation
of a ratio is possible due to fast-update formulas
A. Rubtsov and A.L., JETP Lett. (2004)
Weak coupling QMC: CT-INT
A. Rubtsov, 2004
CT‐INT: detailsTrivial sign problem: P-H transformation
Possible updates:
A. Rubtsov, cond-mat 2003
CT‐INT: multiorbital scheme
CT-INT: random walks in the k space
Step k+1Step k-1
1
1
k
k
w Dk D
+
+
1k
k
k Dw D
−
Acceptance ratio
0 20 40 60
0
Dis
tribu
tion
k
decrease increase
Maximum at 2UNβ
k-1 k+1
Z=… Zk-1 + Zk + Zk+1+ ….
Convergence with Temperature: CT-INT
Maximum: 2UNβ
CT‐QMC Fast Update: k ‐> k+1
Similar to QR-algorithm
(K+1)2 operations
Measurement of Green functions
Advantages of the CT‐QMC method
Number of auxiliary spinsin the Hirsch scheme
Short-range interactions Long-range interactions
Local in time interactions Non-local in time interactions
• 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
Complexity of the algorithm
Metal-insulator transition in the Hubbard model on Bethe lattice
( )( ) 12( ) 0.5 1G i iω µ ω ω
−
= + + +
1 20 ( ) ( )G i i t G iω ω µ ω− = + −
1 10( ) ( ) ( )i G i G iω ω ω− −Σ = −
Initial Green function corresponds to semicircular density of state
Equation of DMFT self-consistency
Self-energy
We solve the effective one-site problem by CTQMC method ( ') ( ) ( ')
effSG Tc cτ τ τ τ+− = −
0.0 0.5 1.0 1.5 2.0 2.5 3.0-7
-6
-5
-4
-3
-2
-1
0
iω
iω
Σ(iω)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
U=2
U=3
U=2
G(iω
)
U=3
-4 -2 0 2 40.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
DO
S
Energy
Density of states for β=64:U=2; U=2.2; U=2.4; U=3
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
Metal-insulator transition in the Hubbard model on Bethe lattice
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-2
-1
0
1
2
3
4
coexistence of the metallic and insulating solutions: U=2.4, β=64, W=2
iω
G(iω
)
CTQMC scheme with β=64
V. Savkin et al PRB 2005
CT-QMC: Hybridization expansion (CT-HYB)
Hamiltonian:
Ph. Werner, et al PRL 97, 076405 (2006)
Bath-a
Loc-dHyb
CT‐HYB: diagrammatics with Hubbard‐X
(0,0)
(1,0)
(0,1)
(1,1)
β0
Zk= exp-U* + * )*Δ*… Δ
Strong-Coupling Expansion CT-HYB
P. Werner, 2006
CT‐HYB
CT‐HYB: determinant weight
CT‐HYB: determinant weght
Diagrams vs. Determinats QMCPh. Werner
CT‐HYB: Monte Carlo sampling
CT‐HYB: segment scheme
+
CT‐HYB: multi‐orbital segment picture
∑=ij
jiij nnUH int
CT‐QMC efficency
CT‐HYB: General multliorbital Interaction
CT‐HYB: matrix code
Use of Symmetry: CT‐HYB
K. Haule PRB 75, 155113 (2007)
CT‐HYB: Krylov code
CT‐HYB: Krylov code
CT‐HYB: Krylov – scaling
CT‐QMC‐Krylov: performance
ALPS‐project: CT‐QMC code
http://alps.comp-phys.org
CT-INT and CT-HYB
Continuous Time Monte Carlo methods for fermions
Alexander LichtensteinUniversity of Hamburg
In collaboration withA. Rubtsov (Moscow University)P. Werner, B. Surer (ETH Zurich)
H. Hafermann (EPL Paris)T. Wehling (University of Bremen)A. Poteryaev (IMF Ekaterinburg)
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)
E. Gull, et al, RMP 83, 349 (2011)
Efficient Krylov scheme: A. Läuchli and P. Werner, PRB (2009)
Comparisson of different CT‐QMC: U=W
E. Gull et al cond-mat/060943
Comparison of different CT‐QMC
Σ Σ Σ
Σ
Σ
Σ
ΣΣ
U
U
G( ’)τ−τ
ττ’
CT-QMC review: E. Gull et al. RMP (2011)
Ch. Jung, unpublished
Scaling of CT‐QMC
Temperature Interactions
Benchmark for CT‐QMC
CT‐HYB: 1‐band DMFT results
Bethe lattice with W=4t
Kondo‐lattice model
KLM: MIT on Bethe lattice
CT‐HYB: 2 orbital model
CT‐HYB for 2‐orbitals: OSMT
Multiorbital impurity with general U
General Interaction:
Krylov-CT-QMC
A. Läuchli and Ph. Werner, et al PRB 80, 235117 (2009)
σσσσ
σσ
kljiijkl
ddddklr
ijU ''
'12
121 ++∑=
Anderson Impurity Model
Hamiltonian of AIM:
Hybridization function:
DFT+AIM using Projectors
• Projections of DFT basis on local orbitals
• Local Green function
• VASP‐PAW basis set
G. Trimarchi, et al JPCM (2008), B. Amadon, et al., PRB (2008)
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= 7‐8B. Surer, et al PRB 85, 085114 (2012)
Constrain GW calculations of U
F. Aryasetiawanan et alPRB(2004)
Wannier ‐ GW and effective U(ω)
T. Miyake and F. Aryasetiawan Phys. Rev. B 77, 085122 (2008)
C-GW
GW
Strength of Coulomb interactions: Graphene
T. Wehling et al., PRL 106, 236805 (2011)
Z. Y. Meng et al., Nature 464, 847-851 (2010) C. Honerkamp, PRL 100, 146404 (2008)
• 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
Quasiparticle spectra: DFT vs. QMC
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 = 0.025 eV and chemical potential μ = 27 eV and μ = 28 eV, respectively.
All Co d‐orbitals contribute to LDOS peak near EF=0
Self-energies: Local Fermi liquid
• Fermi liquid:
• Atomic limit:Signatures of low energy Fermi liquids in all orbitals !Signatures of low energy Fermi liquids in all orbitals !
Quasiparticle weight and Kondo temperature
• Quasiparticle weight
– QMC (Matsubara)
– Kondo temperature Exp:
0.06
0.005
Charge fluctuations: QMC results
-4 -2 0 2 4
0.0
0.2
0.4
0.6
DO
S
Energy
U=2.4, J=-0.2 and J=0, β=64
-4 -2 0 2 40.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8five bands U=2, J=0.2, β=4
DO
S
Energy
0.0 0.5 1.0 1.5-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2 U=2.4, J=-0.2 and J=0, β=64
G(iω
)
iω
three impurity atoms with Hubbard and exchange interactiontwo band rotationally invariant impurity model
Multi-orbital problems: general interaction' '
, , , ; , '
ˆijkl i j l k
i j k l
U U c c c cσ σ σ σσ σ
+ += ∑New formalism allows one to consider the most general case of multi-orbital interactions
-4 -2 0 2 4
0.0
0.1
0.2
0.3 two bands U=4, J=1, β=4
DO
S
Energy
Σ Σ Σ
Σ Σ
Σ Σ
Σ
Σ
Σ
Σ Σ Σ Σ
Cluster DMFT
ΣU
( )ττ ′−0G
ΣU
V
M. Hettler et al, PRB 58, 7475 (1998)A. L. and M. Katsnelson, PRB 62, R9283 (2000)G. Kotliar, et al, PRL 87, 186401 (2001)
Double‐Bethe Lattice: exact C‐DMFT
A. RuckensteinPRB (1999)
Self‐consistent condition: C‐DMFT
AF-between plane AF-plane
Finite temperature phase diagram
• order-disorder transition at tp / t=Sqrt(2) for large U• MIT for intermediate U
H. Hafermann, et al. EPL, 85, 37006 (2009)
Density of States: large U
Spin‐correlations: large U
MIT in 2d: DMFT vs. C‐DMFT
n=1X=0.04
0.080.15
U=0U=5.2tU=6t
Uc=6.05tUc=9.35t
H. Park et al, PRL (2008)
M. Marezio et al., (1972)
TMTM--Oxide VOOxide VO22: singlet formation: singlet formation
Metal
Tem
pera
ture
(K)
Insulator
Rutile structure Monoclinic distortion inthe insulating phase
j
i
Gω( )ij
U
U
tij
U/t
ε εi jb
a
LH
UH
Correlation vs. Bonding
Cluster‐DMFT results for VO2
0
0.2
0.4
0.6
0.8
1.0
−2 0 2 4
U=4eV J=0.68eV
ρ(ω)
ω[eV]
LDA VO2
rutileDMFT
(dashed)(solid)
0
0.5
1.0
1.5
−4 −2 0 2 4
DOS VO2−M1
LDA
ω [eV]
cluster DMFT
(dashed)
(solid)
U = 4 eV, J=0.68 eV β = 20 eV-1Rutile
M1
New photoemission from Tjeng’s groupT. C. Koethe, et al. PRL (2006)
Sharp peak below the gap is NOT a Hubbard band !
S. Biermann, et al, PRL 94, 026404 (2005)
Conclusions
• Electronic Structure of correlated nano‐systems can be described in CT‐QMC scheme
• CT‐QMC is perfect for supercomputer applications
General Projection formalism for LDA+DMFT
DELOCALIZED S,P-STATES
CORRELATED D,F-STATES
G. Trimarchi et al. JPCM 20,135227 (2008)B. Amadon et al. PRB 77, 205112 (2008)
|L>
|G>
CT‐HYB example
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