Diagrammatic Monte Carlo simulation of quantum impurity modelshelper.ipam.ucla.edu/publications/qs2009/qs2009_8065.pdf · 2009. 2. 6. · Diagrammatic Monte Carlo simulation of quantum

Philipp Werner

ETH Zurich

Diagrammatic Monte Carlo simulation of quantum impurity models

IPAM, UCLA, Jan. 2009

Outline

Continuous-time auxiliary field method (CT-AUX)Weak coupling expansion and auxiliary field decompositionApplication: electron pockets in the 2D Hubbard model

Hybridization expansion``Strong coupling” method for general classes of impurity modelsApplication: spin freezing transition in a 3-orbital model

Adaptation to non-equilibrium systemsquantum dots / non-equilibrium DMFT

CollaboratorsE. Gull, A. J. Millis, T. Oka, O. Parcollet, M. Troyer

Dynamical mean field theory

Self-consistency loop

Computationally expensive step: solution of the impurity model

Metzner & Vollhardt, PRL (1989)Georges & Kotliar, PRB (1992)

tkt

lattice model impurity model

impurity solver

Σlatt

∫dk 1

iωn+µ−εk−Σlatt

Glatt Himp

Gimp, ΣimpΣlatt ≡ Σimp

Glatt ≡ Gimp

Diagrammatic QMC

General recipe:Split Hamiltonian into two parts:Use interaction representation in whichWrite partition function as time-ordered exponential, expand in powers of

Weak-coupling expansion: Rombouts et al., (1999), Rubtsov et al. (2005), Gull et al. (2008)

expand in interactions, treat quadratic terms exactlyHybridization expansion: Werner et al., (2006), Werner & Millis (2006), Haule (2007)

expand in hybridizations, treat local terms exactly

H = H1 + H2

O(τ) = eτH1Oe−τH1

H2

Z = Tr[e−βH1Te−

R β0 dτH2(τ)

]

=∑

k

∫ β

0dτ1 . . .

∫ β

0dτk

(−1)k

k!Tr

[e−βH1TH2(τ1) . . .H2(τk)

]

CT-auxiliary field QMC

Impurity model given by

Expand partition function into powers of the interaction term

Decouple the interaction terms using Rombouts et al., PRL (1999)

Rombouts et al., PRL (1999)Gull et al., EPL (2008)

H = H0 + HU

H0 = K/β − (µ− U/2)(n↑ + n↓) + Hhyb + Hbath

HU = U(n↑n↓ − (n↑ + n↓)/2)−K/β

−HU =K

2β

∑

s=±1

eγs(n↑−n↓), cosh(γ) = 1 +βU

2K

Z =∑

k

(−1)k

k!

∫dτ1 . . .

∫dτkTr

[Te−βH0HU (τ1) . . .HU (τk)

]


Configuration space: all possible time-ordered spin configurations

Weight:

Monte Carlo updates: random insertion/removal of a spin

Equivalent to Rubtsov et al., formally similar to Hirsch-Fye


w(τ1, s1; . . . ; τk, sk) =(Kdτ

2β

)k ∏

σ

det N−1σ ({τi, si})

N−1σ = eΓσ −G0σ

(eΓσ − 1

)

eΓσ = diag(eγσs1 , . . . , eγσsk)



Weight:




w(τ1, s1; . . . ; τk, sk) =(Kdτ

2β

)k ∏

σ



(eΓσ − 1

)




Weight:




w(τ1, s1; . . . ; τk, sk) =(Kdτ

2β

)k ∏

σ



(eΓσ − 1

)




Weight:




w(τ1, s1; . . . ; τk, sk) =(Kdτ

2β

)k ∏

σ



(eΓσ − 1

)




Weight:




w(τ1, s1; . . . ; τk, sk) =(Kdτ

2β

)k ∏

σ



(eΓσ − 1

)


M-I transition in the 2D Hubbard model

Hubbard model with nn hopping t, nnn hopping t’=0 (bandwidth 8t)

DMFT: approximate momentum-dependence of the self-energy

DCA: ``tiling” of the Brillouin zone

Σ(p, ω) =∑

a

φa(p)Σa(ω)

H =∑

p,α

εpc†p,αcp,α + U

∑

i

ni,↑ni,↓ εp = −2t(cos(px) + cos(py))


Doping the insulator produces electron/hole pockets

8-site cluster has a ``tile” at the expected position of the pockets

8-site DCA-result at U/t=7: first 8% of dopants go into the B sector

BC

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0 0.2 0.4 0.6 0.8 1 1.2 1.4

n

!/t

C

B

!t=40, B!t=40, C!t=20, B!t=20, C





Gull et al., arXiv (2008)

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0 0.2 0.4 0.6 0.8 1 1.2 1.4

n

!/t

C

B

!t=40, B!t=40, C!t=20, B!t=20, C





Gull et al., arXiv (2008)

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0 0.2 0.4 0.6 0.8 1 1.2 1.4

n

!/t

C

B

!t=40, B!t=40, C!t=20, B!t=20, C





0.5

0.51

0.52

0.53

0.54

0.55

0.56

0 0.2 0.4 0.6 0.8 1 1.2 1.4

n

!/t

C

B

!t=40, B!t=40, C!t=20, B!t=20, C





Assuming an ellipsoidal shape for the pocket, we can estimate the aspect ratio

b

a≈ 1

10b

a

Hybridization expansion

Impurity model given by

Expand partition function into powers of the hybridization term

Trace over bath degrees of freedom yields determinant of hybridization functions F

Werner et al., PRL (2006)Werner & Millis, RPB (2006)

Haule, PRB (2007)

Trbath[. . .] =∏

σ

det M−1σ , M−1

σ (i, j) = Fσ(τ (c)i − τ (c†)

j )

Z =∑

k

12k!

∫dτ1 . . .

∫dτ2kTr

[Te−β(Hloc+Hbath)Hhyb(τ1) . . .Hhyb(τ2k)

]

Fσ(−iωn) =∑

p

|tσp |2

iωn − εp

H = Hloc + Hbath + Hhyb

Hloc = Un↑n↓ − µ(n↑ + n↓)

Hhyb =∑

p,σ

tσpc†σap,σ + h.c.


Monte Carlo configurations consist of segments for spin up and down

Monte Carlo updates: random insertion/removal of (anti-)segments

Weight of a segment configuration:

Determinant of a k x k matrix resums k! diagrams

Werner et al., PRL (2006)Werner & Millis, PRB (2006)

Haule, PRB (2007)

det

(Fσ(τ (c)

1 − τ (c†)1 ) Fσ(τ (c)

1 − τ (c†)2 )

Fσ(τ (c)2 − τ (c†)

1 ) Fσ(τ (c)2 − τ (c†)

2 )

)

w(τσ(c)1 , τσ(c†)

1 ; . . . ; τσ(c)kσ

, τσ(c†)kσ

)= e−Uloverlap+µ(l↑+l↓)

︸︷︷︸Trimp[...]

∏

σ

det M−1σ︸︷︷︸

Trbath[...]

dτ2kσ







Haule, PRB (2007)

det

(Fσ(τ (c)

1 − τ (c†)1 ) Fσ(τ (c)

1 − τ (c†)2 )

Fσ(τ (c)2 − τ (c†)

1 ) Fσ(τ (c)2 − τ (c†)

2 )

)


1 ; . . . ; τσ(c)kσ

, τσ(c†)kσ


︸︷︷︸Trimp[...]

∏

σ


Trbath[...]

dτ2kσ







Haule, PRB (2007)

det

(Fσ(τ (c)

1 − τ (c†)1 ) Fσ(τ (c)

1 − τ (c†)2 )

Fσ(τ (c)2 − τ (c†)

1 ) Fσ(τ (c)2 − τ (c†)

2 )

)


1 ; . . . ; τσ(c)kσ

, τσ(c†)kσ


︸︷︷︸Trimp[...]

∏

σ


Trbath[...]

dτ2kσ







Haule, PRB (2007)

det

(Fσ(τ (c)

1 − τ (c†)1 ) Fσ(τ (c)

1 − τ (c†)2 )

Fσ(τ (c)2 − τ (c†)

1 ) Fσ(τ (c)2 − τ (c†)

2 )

)


1 ; . . . ; τσ(c)kσ

, τσ(c†)kσ


︸︷︷︸Trimp[...]

∏

σ


Trbath[...]

dτ2kσ







Haule, PRB (2007)

det

(Fσ(τ (c)

1 − τ (c†)1 ) Fσ(τ (c)

1 − τ (c†)2 )

Fσ(τ (c)2 − τ (c†)

1 ) Fσ(τ (c)2 − τ (c†)

2 )

)


1 ; . . . ; τσ(c)kσ

, τσ(c†)kσ


︸︷︷︸Trimp[...]

∏

σ


Trbath[...]

dτ2kσ

Spin freezing transition in a 3-orbital model

1 site, 3 degenerate orbitals (semi-circular DOS, bandwidth 4t)

Captures essential physics of SrRuO3

Similar models for other transition metal oxides, actinide compounds, Fe / Ni based superconductors, ...

Hloc = −∑

α,σ

µnα,σ +∑

α

Unα,↑nα,↓

+∑

α>β,σ

U ′nα,σnβ,−σ + (U ′ − J)nα,σnβ,σ

−∑

α%=β

J(ψ†α,↓ψ

†β,↑ψβ,↓ψα,↑ + ψ†

β,↑ψ†β,↓ψα,↑ψα,↓ + h.c.)

Werner et al., PRL (2008)


Phase diagram for

Mott insulating ``lobes” with 1, 2, 3, (4, 5) electrons


U ′ = U = 2J, J/U = 1/6,βt = 50

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5 3

U/t

n

Mott insulator (!t=50)

0

2

4

6

8

10

12

14

16

-5 0 5 10 15 20 25 30 35 40

U/t

µ/t

0 1 2 3


Phase diagram for

Mott insulating ``lobes” with 1, 2, 3, (4, 5) electrons

In the metallic phase: transition from Fermi liquid to ``spin glass”


U ′ = U = 2J, J/U = 1/6,βt = 50

0

2

4

6

8

10

12

14

16

-5 0 5 10 15 20 25 30 35 40

U/t

µ/t

0 1 2 3

glass transition

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5 3

U/t

n

Fermi liquidfrozenmoment

glass transitionMott insulator (!t=50)


Phase diagram for

Critical exponents associated with the transition can be seen in a wide quantum critical regime

e. g. non Fermi-liquid self-energy


U ′ = U = 2J, J/U = 1/6,βt = 50

0

2

4

6

8

10

12

14

16

-5 0 5 10 15 20 25 30 35 40

U/t

µ/t

0 1 2 3

glass transition

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5 3

U/t

n

Fermi liquidfrozenmoment

glass transitionMott insulator (!t=50)

ImΣ/t ∼ (iωn/t)α, α ≈ 0.5


A self-energy with frequency dependence implies an optical conductivity


Σ(ω) ∼ ω1/2

σ(ω) ∼ 1/ω1/2

Real-time formalism

Quantum dot coupled to two infinite leads

Initial preparation of the dot:Non-interacting leads: (DOS, Fermi distribution function)Level broadening:

Muehlbacher & Rabani (2008)Schmidt et al. (2008)

Schiro & Fabrizio (2008)Werner et al. (2008)

Goldhaber-Gordon (1998)

ρ0,dot

ρ0,leads

Γα(ω) = π∑

p

|V αp |2δ(ω − εα

p )

H = Hdot + Hleads + Hmix

Hdot = εd(nd↑ + nd↓) + Und↑nd↓

Hleads =∑

α=L,R

∑

pσ

(εαpσ − µα

)aα†

pσaαpσ

Hmix =∑

α=L,R

∑

p,σ

(V α

p aα†pσdσ + h.c.

)

Real-time formalism

Quantum dot coupled to two infinite leads

Initial preparation of the dot:Non-interacting leads: (DOS, Fermi distribution function)Level broadening:



Goldhaber-Gordon (1998)

ρ0,dot

ρ0,leads

µL

µR

µRεd

Vp Vpεp εp

Γα(ω) = π∑

p

|V αp |2δ(ω − εα

p )

H = Hdot + Hleads + Hmix

Hdot = εd(nd↑ + nd↓) + Und↑nd↓

Hleads =∑

α=L,R

∑

pσ

(εαpσ − µα

)aα†

pσaαpσ

Hmix =∑

α=L,R

∑

p,σ

(V α

p aα†pσdσ + h.c.

)

U

Real-time formalism

Interaction picture

``Keldysh contour”

Expand time evolution operators in powers of



ρ0 O

0 t

e−iHt

eiHt

H2

H = H1 + H2, O(s) = eisH1Oe−isH1

〈O(t)〉 = Tr[ρ0e

iHtOe−iHt]

= Tr[ρ0

(T̃ ei

R t0 dsH2(s)

)O(t)

(Te−i

R t0 dsH2(s

′))]

Real-time formalism

Interaction picture

``Keldysh contour”

Expand time evolution operators in powers of



ρ0 O

0 t

H2

H = H1 + H2, O(s) = eisH1Oe−isH1

〈O(t)〉 = Tr[ρ0e

iHtOe−iHt]

= Tr[ρ0

(T̃ ei

R t0 dsH2(s)

)O(t)

(Te−i

R t0 dsH2(s

′))]

iH2

−iH2

iH2

−iH2−iH2

Weak-coupling expansion

Configuration space: all possible spin configurations on the Keldysh contour

Weight: analogous to imaginary-time CT-AUX with

Werner, Oka & Millis, PRB (2009)

G0,σ(t′K , t′′K) ={

G<0,σ(t′, t′′) t′K < t′′K

G>0,σ(t′, t′′) t′K ≥ t′′K

G</>0 (t′, t′′) = ±i

∫dω

2πe−iω(t′−t′′)

∑α=L,R Γα

(1∓ tanh

(ω−µα

2T

))

(ω − εd − U/2)2 + Γ2


Configuration space: all possible spin configurations on the Keldysh contour

Weight: analogous to imaginary-time CT-AUX with

G0,σ(t′K , t′′K) ={

G<0,σ(t′, t′′) t′K < t′′K

G>0,σ(t′, t′′) t′K ≥ t′′K

G</>0 (t′, t′′) = ±i

∫dω

2πe−iω(t′−t′′)

∑α=L,R Γα

(1∓ tanh

(ω−µα

2T

))

(ω − εd − U/2)2 + Γ2

∑

p

Vpa†pσdσ

current



Monte Carlo sampling: random insertion/removal of spins

Current measurement:

I = −2Im∑

σ

∑

c

wIσc = −2Im

∑

σ

[⟨wIσc

|wc|

⟩

|wc|

1〈φc〉|wc|

]





I = −2Im∑

σ

∑

c

wIσc = −2Im

∑

σ

[⟨wIσc

|wc|

⟩

|wc|

1〈φc〉|wc|

]





I = −2Im∑

σ

∑

c

wIσc = −2Im

∑

σ

[⟨wIσc

|wc|

⟩

|wc|

1〈φc〉|wc|

]





I = −2Im∑

σ

∑

c

wIσc = −2Im

∑

σ

[⟨wIσc

|wc|

⟩

|wc|

1〈φc〉|wc|

]





I = −2Im∑

σ

∑

c

wIσc = −2Im

∑

σ

[⟨wIσc

|wc|

⟩

|wc|

1〈φc〉|wc|

]





I = −2Im∑

σ

∑

c

wIσc = −2Im

∑

σ

[⟨wIσc

|wc|

⟩

|wc|

1〈φc〉|wc|

]



Interaction and voltage dependence of the current

Interaction suppresses the currentCorrection largest for 4th order perturbation theory by Fujii & Ueda identical to MC for

V ≈ U

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0 0.5 1 1.5 2 2.5 3

I/!

t!

U/!=2

U/!=3

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0 2 4 6 8 10

(I(U

)-I(

0))

/!

V/!

U/!=2U/!=3

U = 2Γ



Interaction and voltage dependence of the double occupancy

Convergence to steady state faster for larger voltage bias

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0 0.5 1 1.5 2 2.5 3

double

occupancy

t!

V/!=0V/!=1V/!=2V/!=3V/!=4

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0 2 4 6 8 10

ste

ady s

tate

double

occupancy

V/!

U/!=2U/!=3U = 2Γ



Non-equilibrium DMFTDynamics of the Hubbard model after a ``quantum quench” Eckstein and Werner (work in progress)

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

-0.5-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5

"G_u5t3_1" u 2:4:6"G_u5t3_2" u 2:4:6"G_u5t3_9" u 2:4:6

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

"G_u5t3_1" u 2:4:8"G_u5t3_2" u 2:4:8"G_u5t3_9" u 2:4:8

ReG(t, t′)

ImG(t, t′)


Non-equilibrium DMFTDynamics of the Hubbard model after a ``quantum quench” Eckstein and Werner (work in progress)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

n k

t

epsk=-2 ...epsk=2

-2-1.5-1-0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

nk

epsk

t

nk

Configuration space: all possible segment configurations on the (doubled) Keldysh contour

Hybridization matrix becomes

Monte Carlo sampling: random insertions/removals of segments

Hybridization expansionMuehlbacher & Rabani (2008)

Schmidt et al. (2008)Schiro & Fabrizio (2008)

Werner et al. (2008)

Σ<(t1 − t′1) Σ<(t2 − t′1) . . .Σ>(t1 − t′2) Σ<(t2 − t′2) . . .. . . . . . . . .

U0 ! ! 02!+! !0

t0

"0

Configuration space: all possible segment configurations on the (doubled) Keldysh contour

Hybridization matrix becomes

Monte Carlo sampling: random insertions/removals of segments




Σ<(t1 − t′1) Σ<(t2 − t′1) . . .Σ>(t1 − t′2) Σ<(t2 − t′2) . . .. . . . . . . . .

0

t0

!

depends on the DOS and voltage bias

soft cutoff:

hard cutoff:




Σ<,>

!! µ

µL

R

"2#c

Lµ

$#c

#c

µR

Σ<,>soft (t) = Γ

cos(V2 )

β sinh(πβ (t± i/ωc))

Σ<,>hard(t) = Γ

(cos(V

2 t)β sinh(π

β t)− e±iωct

ν sinh(πν t)

)

Initial state: dot decoupled from the leads

Time evolution of the left, right and average current ( )

Initially, electrons rush to the dot from both leads; after a ``steady state” is established with




-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

curr

ent/!

t!

IL=-IR, V/!=0IL, V/!=5

-IR, V/!=5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

curr

ent/!

t!

dn/dt=IL-IR, V/!=0dn/dt=IL-IL, V/!=5

2I=IL+IR, V/!=5

U/Γ = 8

tΓ ! 2IL = −IR

Current-Voltage characteristic of a strongly interacting dot ( ) measured at

Is close enough to steady state ?Probably not for small voltage bias




U/Γ = 8

tΓ = 1, 1.25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4

I/!

t!

V/!=109

8

7

6

5

4

3

2

1

tΓ = 1, 1.25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16

I/!

V/!

U/!=0

4

6

8

10

12

14

Comparison to ``fixed gap calculation” and 4th order perturbation theory




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16

I/!

V/!

U/!=0

46

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16

I/!

V/!

U/!=0

8

12

Perturbation order: weak-coupling (left), hyb-expansion (right)

Weak-coupling: restricted to small U, but can reach steady stateHybridization expansion: cannot quite reach steady state for U>0, requires finite bandwidth, but can treat strong interactionsBoth: sign problem which grows exponentially with time

(Dis)advantages of the two approaches

1e-05

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10

pro

babili

ty

perturbation order

U/!=2U/!=3U/!=4

1e-05

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10

pro

babili

ty

perturbation order

U/!=0U/!=5

Summary and Conclusions

Diagrammatic QMC impurity solvers Enable efficient DMFT simulations of fermionic lattice modelsWeak-coupling solver scales favorably with number of sites/orbitals: ideal for large impurity clustersHybridization expansion allows to treat multi-orbital models with complicated interactions

Keldysh implementation of diagrammatic QMCEnables the study of transport and relaxation dynamicsSign problem prevents the simulation of long time intervalsImpurity solver for non-equilibrium DMFT

Diagrammatic Monte Carlo simulation of quantum impurity modelshelper.ipam.ucla.edu/publications/qs2009/qs2009_8065.pdf · 2009. 2. 6. · Diagrammatic Monte Carlo simulation of quantum

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