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