Diagrammatic Monte Carlo simulation of quantum impurity models

Diagrammatic Monte Carlo simulation of quantum impurity models Philipp Werner ETH Zurich IPAM, UCLA, Jan. 2009

Outline Continuous-time auxiliary field method (CT-AUX) Weak coupling expansion and auxiliary field decomposition Application: electron pockets in the 2D Hubbard model Hybridization expansion ``Strong coupling method for general classes of impurity models Application: spin freezing transition in a 3-orbital model Adaptation to non-equilibrium systems quantum dots / non-equilibrium DMFT Collaborators E. Gull, A. J. Millis, T. Oka, O. Parcollet, M. Troyer

Dynamical mean field theory Metzner & Vollhardt, PRL (1989) Georges & Kotliar, PRB (1992) Self-consistency loop lattice model impurity model t tk G latt dk 1 iω n +µ ɛ k Σ latt G latt G imp H imp impurity solver Σ latt Σ latt Σ imp G imp, Σ imp Computationally expensive step: solution of the impurity model

Diagrammatic QMC General recipe: Split Hamiltonian into two parts: H = H 1 + H 2 Use interaction representation in which O(τ) = e τh 1 Oe τh 1 Write partition function as time-ordered exponential, expand in powers of H 2 Z = T r [e βh 1 T e R ] β 0 dτh 2(τ) = k β 0 dτ 1... β 0 dτ k ( 1) k k! [ ] T r e βh 1 T H 2 (τ 1 )... H 2 (τ k ) Weak-coupling expansion: Rombouts et al., (1999), Rubtsov et al. (2005), Gull et al. (2008) expand in interactions, treat quadratic terms exactly Hybridization expansion: Werner et al., (2006), Werner & Millis (2006), Haule (2007) expand in hybridizations, treat local terms exactly

CT-auxiliary field QMC Rombouts et al., PRL (1999) Gull et al., EPL (2008) Impurity model given by H = H 0 + H U H 0 = K/β (µ U/2)(n + n ) + H hyb + H bath H U = U(n n (n + n )/2) K/β Expand partition function into powers of the interaction term Z = ( 1) k [ ] dτ 1... dτ k T r T e βh 0 H U (τ 1 )... H U (τ k ) k! k Decouple the interaction terms using Rombouts et al., PRL (1999) H U = K 2β s=±1 e γs(n n ), cosh(γ) = 1 + βu 2K

CT-auxiliary field QMC Rombouts et al., PRL (1999) Gull et al., EPL (2008) Configuration space: all possible time-ordered spin configurations Weight: w(τ 1, s 1 ;... ; τ k, s k ) = ( Kdτ 2β ) k N 1 σ = e Γ σ G 0σ (e Γ σ 1 σ det N 1 σ ({τ i, s i }) e Γ σ = diag(e γσs 1,..., e γσs k ) ) Monte Carlo updates: random insertion/removal of a spin Equivalent to Rubtsov et al., formally similar to Hirsch-Fye

M-I transition in the 2D Hubbard model Hubbard model with nn hopping t, nnn hopping t =0 (bandwidth 8t) H = p,α ɛ p c p,αc p,α + U i n i, n i, ɛ p = 2t(cos(p x ) + cos(p y )) DMFT: approximate momentum-dependence of the self-energy Σ(p, ω) = a φ a (p)σ a (ω) DCA: ``tiling of the Brillouin zone

M-I transition in the 2D Hubbard model 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 0.56 0.55 0.54!t=40, B!t=40, C!t=20, B!t=20, C n 0.53 0.52 0.51 B C B C 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4!/t

M-I transition in the 2D Hubbard model Doping the insulator produces electron/hole pockets Gull et al., arxiv (2008) 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 n 0.56 0.55 0.54 0.53!t=40, B!t=40, C!t=20, B!t=20, C 0.52 B 0.51 C 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4!/t

M-I transition in the 2D Hubbard model 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 n 0.56 0.55 0.54 0.53!t=40, B!t=40, C!t=20, B!t=20, C 0.52 B 0.51 C 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4!/t

M-I transition in the 2D Hubbard model 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 Assuming an ellipsoidal shape for the pocket, we can estimate the aspect ratio b a 1 a 10 b

Hybridization expansion Werner et al., PRL (2006) Werner & Millis, RPB (2006) Haule, PRB (2007) Impurity model given by H = H loc + H bath + H hyb H loc = Un n µ(n + n ) H hyb = t σ p c σa p,σ + h.c. p,σ Expand partition function into powers of the hybridization term Z = 1 [ ] dτ 1... dτ 2k T r T e β(h loc+h bath ) H hyb (τ 1 )... H hyb (τ 2k ) 2k! k Trace over bath degrees of freedom yields determinant of hybridization functions F T r bath [...] = σ det Mσ 1, Mσ 1 (i, j) = F σ (τ (c) i τ (c ) j ) t σ p 2 F σ ( iω n ) = p iω n ɛ p

Hybridization expansion Werner et al., PRL (2006) Werner & Millis, PRB (2006) Haule, PRB (2007) 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: w ( τ σ(c) 1, τ σ(c ) 1 ;... ; τ σ(c) k σ, τ σ(c )) k σ = e Ul overlap +µ(l +l ) }{{} T r imp [...] Determinant of a k x k matrix resums k! diagrams ( ) F det σ (τ (c) 1 τ (c ) 1 ) F σ (τ (c) 1 τ (c ) 2 ) F σ (τ (c) 2 τ (c ) 1 ) F σ (τ (c) 2 τ (c ) 2 ) σ det Mσ 1 }{{} T r bath [...] dτ 2k σ

Spin freezing transition in a 3-orbital model 1 site, 3 degenerate orbitals (semi-circular DOS, bandwidth 4t) H loc = µn α,σ + Un α, n α, α,σ α + α>β,σ U n α,σ n β, σ + (U J)n α,σ n β,σ Werner et al., PRL (2008) α β J(ψ α, ψ β, ψ β, ψ α, + ψ β, ψ β, ψ α, ψ α, + h.c.) Captures essential physics of SrRuO3 Similar models for other transition metal oxides, actinide compounds, Fe / Ni based superconductors,...

Spin freezing transition in a 3-orbital model Phase diagram for U = U = 2J, J/U = 1/6, βt = 50 Werner et al., PRL (2008) 16 14 12 10 0 1 2 3 16 14 12 10 U/t 8 U/t 8 6 6 4 2 4 2 Mott insulator (!t=50) 0-5 0 5 10 15 20 25 30 35 40 µ/t 0 0 0.5 1 1.5 2 2.5 3 n Mott insulating ``lobes with 1, 2, 3, (4, 5) electrons

Spin freezing transition in a 3-orbital model Phase diagram for U = U = 2J, J/U = 1/6, βt = 50 Werner et al., PRL (2008) 16 14 12 0 1 2 3 16 14 12 U/t 10 8 U/t 10 8 Fermi liquid frozen moment 6 6 4 2 glass transition 4 2 glass transition Mott insulator (!t=50) 0-5 0 5 10 15 20 25 30 35 40 µ/t 0 0 0.5 1 1.5 2 2.5 3 n Mott insulating ``lobes with 1, 2, 3, (4, 5) electrons In the metallic phase: transition from Fermi liquid to ``spin glass

Spin freezing transition in a 3-orbital model Phase diagram for U = U = 2J, J/U = 1/6, βt = 50 Werner et al., PRL (2008) 16 14 12 0 1 2 3 16 14 12 U/t 10 8 U/t 10 8 Fermi liquid frozen moment 6 6 4 2 glass transition 4 2 glass transition Mott insulator (!t=50) 0-5 0 5 10 15 20 25 30 35 40 µ/t 0 0 0.5 1 1.5 2 2.5 3 n Critical exponents associated with the transition can be seen in a wide quantum critical regime e. g. non Fermi-liquid self-energy ImΣ/t (iω n /t) α, α 0.5

Spin freezing transition in a 3-orbital model Werner et al., PRL (2008) A self-energy with frequency dependence Σ(ω) ω 1/2 implies an optical conductivity σ(ω) 1/ω 1/2

Real-time formalism Quantum dot coupled to two infinite leads Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) H = H dot + H leads + H mix H dot = ɛ d (n d + n d ) + Un d n d H leads = α=l,r H mix = pσ α=l,r p,σ ( ɛ α pσ µ α ) a α pσa α pσ ( V α p a α pσd σ + h.c. ) Goldhaber-Gordon (1998) Initial preparation of the dot: ρ 0,dot Non-interacting leads: ρ 0,leads (DOS, Fermi distribution function) Level broadening: Γ α (ω) = π Vp α 2 δ(ω ɛ α p ) p

Real-time formalism Quantum dot coupled to two infinite leads H = H dot + H leads + H mix H dot = ɛ d (n d + n d ) + Un d n d H leads = α=l,r H mix = pσ α=l,r p,σ ( ɛ α pσ µ α ) a α pσa α pσ ( V α p a α pσd σ + h.c. ) Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) µ L ɛ p V p U ɛ d V p Goldhaber-Gordon (1998) µ R µ R ɛ p Initial preparation of the dot: ρ 0,dot Non-interacting leads: ρ 0,leads (DOS, Fermi distribution function) Level broadening: Γ α (ω) = π Vp α 2 δ(ω ɛ α p ) p

Real-time formalism Interaction picture Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) H = H 1 + H 2, O(s) = e ish 1 Oe ish 1 O(t) = T r [ρ 0 e iht Oe iht] ( = T r [ρ 0 T e i R ) t 0 dsh 2(s) O(t) (T e i R )] t 0 dsh 2(s ) ``Keldysh contour ρ 0 e iht O 0 t e iht Expand time evolution operators in powers of H 2

Real-time formalism Interaction picture Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) H = H 1 + H 2, O(s) = e ish 1 Oe ish 1 O(t) = T r [ρ 0 e iht Oe iht] ( = T r [ρ 0 T e i R ) t 0 dsh 2(s) O(t) (T e i R )] t 0 dsh 2(s ) ``Keldysh contour ρ 0 ih 2 ih 2 ih 2 O 0 ih 2 ih 2 t Expand time evolution operators in powers of H 2

Weak-coupling expansion Werner, Oka & Millis, PRB (2009) Configuration space: all possible spin configurations on the Keldysh contour Weight: analogous to imaginary-time CT-AUX with { G G 0,σ (t K, t < K) = 0,σ (t, t ) t K < t K G > 0,σ (t, t ) t K t K ( G </> dω 0 (t, t ) = ±i 2π e iω(t t ) α=l,r Γα 1 tanh ( ω µ α )) 2T (ω ɛ d U/2) 2 + Γ 2

Weak-coupling expansion Werner, Oka & Millis, PRB (2009) Configuration space: all possible spin configurations on the Keldysh contour current Weight: analogous to imaginary-time CT-AUX with V p a pσd σ { G G 0,σ (t K, t < K) = 0,σ (t, t ) t K < t p K G > 0,σ (t, t ) t K t K ( G </> dω 0 (t, t ) = ±i 2π e iω(t t ) α=l,r Γα 1 tanh ( ω µ α )) 2T (ω ɛ d U/2) 2 + Γ 2

Weak-coupling expansion Werner, Oka & Millis, PRB (2009) Monte Carlo sampling: random insertion/removal of spins Current measurement: I = 2Im σ c w I σ c = 2Im σ [ w I σ c w c w c 1 φ c wc ]

Weak-coupling expansion Werner, Oka & Millis, PRB (2009) Interaction and voltage dependence of the current 0.74 0 I/! 0.72 0.7 0.68 0.66 0.64 0.62 0.6 0.58 U/!=2 U/!=3 (I(U)-I(0))/! -0.02-0.04-0.06-0.08-0.1-0.12 U/!=2 U/!=3 0.56 0 0.5 1 1.5 2 2.5 3 t! -0.14 0 2 4 6 8 10 V/! Interaction suppresses the current Correction largest for V U 4th order perturbation theory by Fujii & Ueda identical to MC for U = 2Γ

Weak-coupling expansion Werner, Oka & Millis, PRB (2009) Interaction and voltage dependence of the double occupancy double occupancy 0.25 0.24 0.23 0.22 0.21 0.2 U = 2Γ V/!=0 V/!=1 V/!=2 V/!=3 V/!=4 steady state double occupancy 0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 U/!=2 U/!=3 0.19 0 0.5 1 1.5 2 2.5 3 t! 0.17 0 2 4 6 8 10 V/! Convergence to steady state faster for larger voltage bias

Weak-coupling expansion Non-equilibrium DMFT Dynamics of the Hubbard model after a ``quantum quench Eckstein and Werner (work in progress) ReG(t, t ) 0.5 0.4 0.3 0.2 0.1-0.1 0-0.2-0.3-0.4-0.5 "G_u5t3_1" u 2:4:6 "G_u5t3_2" u 2:4:6 "G_u5t3_9" u 2:4:6 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 ImG(t, t ) 0.6 0.5 0.4 0.3 0.2 0.1-0.1 0-0.2-0.3 "G_u5t3_1" u 2:4:8 "G_u5t3_2" u 2:4:8 "G_u5t3_9" u 2:4:8 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3

Weak-coupling expansion Non-equilibrium DMFT Dynamics of the Hubbard model after a ``quantum quench Eckstein and Werner (work in progress) n k 1 0.8 eps k =-2... eps k =2 0.9 1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 t 1.2 1.4-2 -1.5-1 -0.5 0 0.5 eps k 1 1.5 2 n k 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t

Hybridization expansion Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) Configuration space: all possible segment configurations on the (doubled) Keldysh contour 0! 0! 2!+ U! 0! " 0 0 Hybridization matrix becomes Σ < (t 1 t 1) Σ < (t 2 t 1)... Σ > (t 1 t 2) Σ < (t 2 t 2)............ t Monte Carlo sampling: random insertions/removals of segments

Hybridization expansion Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) Configuration space: all possible segment configurations on the (doubled) Keldysh contour! 0 0 Hybridization matrix becomes Σ < (t 1 t 1) Σ < (t 2 t 1)... Σ > (t 1 t 2) Σ < (t 2 t 2)............ t Monte Carlo sampling: random insertions/removals of segments

Hybridization expansion Σ <,> depends on the DOS and voltage bias Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) "2# c! µ R # c µr! µ L µ L $# c soft cutoff: hard cutoff: Σ <,> soft (t) = Γ cos( V 2 ) β sinh( π β (t ± i/ω c)) Σ <,> hard (t) = Γ ( cos( V 2 t) β sinh( π β t) ) e±iωct ν sinh( π ν t)

Hybridization expansion Initial state: dot decoupled from the leads Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) Time evolution of the left, right and average current ( U/Γ = 8) 0.4 1 current/! 0.2 0-0.2-0.4-0.6-0.8-1 I L =-I R, V/!=0-1.2 I L, V/!=5-1.4 -I R, V/!=5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t! current/! 0.5 0-0.5-1 -1.5-2 dn/dt=i L -I R, V/!=0 dn/dt=i L -I L, V/!=5-2.5 2I=I L +I R, V/!=5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t! Initially, electrons rush to the dot from both leads; after tγ 2 a ``steady state is established with I L = I R

Hybridization expansion Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) Current-Voltage characteristic of a strongly interacting dot ( U/Γ = 8) measured at tγ = 1, 1.25 I/! 0.7 0.6 0.5 0.4 0.3 0.2 0.1 V/!=10 9 8 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t! I/! 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 U/!=0 4 6 0 0 2 4 6 8 10 12 14 16 8 V/! 10 12 14 Is tγ = 1, 1.25 close enough to steady state? Probably not for small voltage bias

Hybridization expansion Muehlbacher & Rabani (2008) Schmidt et al. (2008) Schiro & Fabrizio (2008) Werner et al. (2008) Comparison to ``fixed gap calculation and 4th order perturbation theory 1 1 0.9 0.8 U/!=0 0.9 0.8 U/!=0 0.7 0.7 4 6 I/! 0.6 0.5 8 12 I/! 0.6 0.5 8 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 2 4 6 8 10 12 14 16 V/! 0 0 2 4 6 8 10 12 14 16 V/!

(Dis)advantages of the two approaches Perturbation order: weak-coupling (left), hyb-expansion (right) 1 0.1 U/!=2 U/!=3 U/!=4 1 0.1 U/!=0 U/!=5 probability 0.01 0.001 probability 0.01 0.001 0.0001 0.0001 1e-05 0 2 4 6 8 10 perturbation order 1e-05 0 2 4 6 8 10 perturbation order Weak-coupling: restricted to small U, but can reach steady state Hybridization expansion: cannot quite reach steady state for U>0, requires finite bandwidth, but can treat strong interactions Both: sign problem which grows exponentially with time

Summary and Conclusions Diagrammatic QMC impurity solvers Enable efficient DMFT simulations of fermionic lattice models Weak-coupling solver scales favorably with number of sites/ orbitals: ideal for large impurity clusters Hybridization expansion allows to treat multi-orbital models with complicated interactions Keldysh implementation of diagrammatic QMC Enables the study of transport and relaxation dynamics Sign problem prevents the simulation of long time intervals Impurity solver for non-equilibrium DMFT