Numerical methods out of equilibrium -Threelectures-

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1 Numerical methods out of equilibrium -Threelectures- Michael Thorwart Freiburg Institute for Advanced Studies (FRIAS) Albert-Ludwigs-Universität Freiburg funded by the Excellence Initiative of the German Federal and State Governments

2 Transport through a nanoconductor V Source QD G V G Drain conductance G max Gate voltage µ(n + 1) µ(n) e2 2 + ε k T,eV addition energy Γ L Γ R a) Linear transport b) ev/2 ε U ev/2 Nonlinear transport

3 Motivation Quantum transport still challenging for theory: Transport through single molecules Many-body effects, nonequilibrium effects Theoretical methods at nonequilibrium: currently hot field of research Nonperturbative approaches Perturbative solutions very recently: Numerical techniques each has pro s and con s

4 Numerical methods out of equilibrium Real-time path-integral methods: Iterative summation of path integrals ISPI (MT, Egger) Real-time Quantum Monte arlo (Mühlbacher, Werner, Millis) Renormalization group methods: Numerical RG (Wilson, Anders, Schiller) Perturbative RG (Schoeller, König) Generalization of DMRG (Schmitteckert) Functional RG (Meden, Schoeller) Flow equation method (Kehrein)

5 Numerical methods out of equilibrium Real-time path-integral methods: Iterative summation of path integrals ISPI Lecture I Real-time Quantum Monte arlo Lecture II Renormalization group methods: Numerical RG Lecture III Perturbative RG Generalization of DMRG Functional RG Flow equation method

6 Lecture I: Iterative summation of real-time path integrals - ISPI - S Weiss, J Eckel, M Thorwart, R Egger, Phys Rev 77, (28)

7 Outline Anderson model as prototype example Nonequilibrium transport: Keldysh path-integral for current generating function Hubbard-Stratonovich transformation = path summation The ISPI scheme: numerically exact results Validation: exact analytical and perturbative results Regime of small transport voltage Regime of large transport voltage Recent results Summary S Weiss, J Eckel, M Thorwart, R Egger, Phys Rev 77, (28)

8 Anderson model as prototype ˆn σ = d σd σ dot operators, c kpσ lead operators ev/2 Γ L ε U Γ R ev/2 single particle energies E σ on-dot interaction U many-body effects finite bias voltage µ p=l/r = ±ev/2 noneq transport single particle energies of leads ǫ kp hybridization Γ p t p 2 H = H dot + H leads + H T = σ E σˆn σ + Uˆn ˆn + kpσ(ǫ kp µ p )c kpσ c kpσ kpσ t p c kpσ d σ + Hc

9 Nonequilibrium current Observable: symmetrized current I = (I L I R )/2 with I p (t) = eṅp(t) = ei[h,n p (t)]/ I(t) = ie 2 [pt p c kpσ d σ t pt p d σc ] kpσ t kpσ Nonequilibrium current: Keldysh formalism: _ 8 t 1 t 2 + t m t N _ 8 t 2N t 2N 1 _ extend real time axis Green function: Keldysh matrix t N+1 G αβ ij (t α,t β) = i T [ψ i (t α )ψ j (t β)] t 8

10 Keldysh path integral & generating function calculate expectation value I(t m ) via I(t m ) = i η ln Z[η] η= from a generating function ln Z[η] onstruct path integral for [ ] Z[η] = D d σ,d σ, c kpσ,c kpσ e is[ d σ,d σ, c kpσ,c kpσ ] σ with Grassmann fields d, d,c, c and action S = S dot + S leads + S T + S η

11 Keldysh path integral & generating function Action with S = S dot + S leads + S T + S η S dot = S dot, + S U [ = dt σ d σ (i t ǫ σ )d σ + U 2 (n n ) 2 ] S leads = dt c kpσ (i t ǫ kp + µ p )c kpσ kpσ S T = dt t p c kpσ d σ + hc kpσ S η = ieη p(t p c kpσ d σ t p 2 d σ c kpσ )(t m ) kpσ

12 For pedagocial reasons: noninteracting case U = Hamiltonian is quadratic Gaussian integrals Z ni [η] = σ det [ ig 1 σ (t,t ) + ησ J (t,t ) ] with dot GF in presence of the leads G σ (ω) = [(ω ǫ σ )τ z γ L (ω) γ R (ω)] 1 ( 1 = Γ 2 + (ω ǫ σ ) 2 ω ǫ σ + iγ(f 1) Γ L = Γ R = Γ/2 iγf iγ(f 2) ω + ǫ σ + iγ(f 1) ) and source term F = f(ω + ev/2) + f(ω ev/2) Σ J (t,t ) = e 2 [γ L(t,t ) γ R (t,t )] [δ(t t m ) δ(t t m )]

13 Interacting case U : Hubbard-Stratonovich Next: decouple quartic term: Hubbard-Stratonovich trafo e ±iδ tu(ˆn ˆn ) 2 /2 = 1 2 observe that n n =, ±1 s ± =± e δ tλ ± s ± (ˆn ˆn ) solution to this equation (in general, not unique): λ ± = λ ± iλ with δ t λ = sinh 1 sin(δ t U/2), δ t λ = sin 1 sin (δ t U/2) Integrate out all fermionic degrees of freedom

14 Interacting case U : Hubbard-Stratonovich Integrate out all fermions, price: introduce HS Ising fields s ± k : Z[η] = {s} σ det G 1 σ [{s},η] where now the path summation extends over all possible Ising fields {s} = (s 1 = ±1,s 2 = ±1,,s N = ±1,,s 2N = ±1) Time-discrete full GF takes the form ( G 1 σ ) αβ [{s},η] = ( G 1 kl σ ) αβ + iησ J,αβ kl kl iδ t δ kl λ α s α kδ αβ ISPI now computes this path sum in an exact way!

15 Iterative Summation of real-time Path Integrals ISPI G ++/-- (t-t ) (a) ev = ev = 2Γ T = Γ (b) ev = ev = 2Γ T = 1 Γ G ++/-- (t-t ) 2-2 (c) = = 3Γ ev = Γ (d) ε = Γ ε = 5Γ ev = Γ (t-t ) Γ (t-t ) Γ central: each Keldysh component of G σ,kl decays exponentially at finite T for k l time scale τ c of correlations exists: τ 1 c max(k T,eV ) D kl for k l > τ c Kδ t (convenient: D σ = G σ G 1 σ ) Discrete GF assume band matrix structure width of band memory K

16 Iterative Summation of real-time Path Integrals ISPI and structure of discrete GF (again, D σ = G σ G 1 σ ): D D (1,NK ) D 11 D 12 D 21 D 22 D 23 D 32 D 33 D 34 D 43 D 44 D N K 1N K D N KN K 1 D N KN K 1 A All blocks have size K K;N K = N/K

17 Iterative Summation of real-time Path Integrals ISPI We have to calculate the generating function Z[η] = {s} σ detg 1 σ [{s},η] Goal: alculate determinant iteratively, using Schur form: general matrix (a,d: square blocks, b,c: rect blocks) ( ) a b D = c d Gaussian elimination: ( I n D = LD = ca 1 I m ) ( a b c d ) = ( a b d ca 1 b ) det(d) = det(a) det(d ca 1 b)

18 Iterative Summation of real-time Path Integrals ISPI Let s apply this to the full GF: D D (1,NK ) = D 11 D 12 D 21 D 22 D 23 D 32 D 33 D 34 After the first step of the iteration: Z[η] = det { D 11 [s ± 1,,s ± K ]} s ± 1,,s± N det { D (2,NK )[s ± K+1,,s± N ] D21 [s ± K+1,,s± 2K ] [ D 11 [s ± 1,,s± K ]] 1 D 12 [s ± 1,,s± K ] }

19 Iterative Summation of real-time Path Integrals ISPI repeat this, finally: Z[η] = X s ± 1,,s± N det nd 11 [s ± 1,, s± K ] o N K 1 Y l=1 det n D l+1,l+1 [s ± lk+1,, s± (l+1)k ] D l+1,l [s ± lk+1,, s± (l+1)k ] h D l,l [s ± (l 1)K+1,, s± lk ] i 1 D l,l+1 [s ± (l 1)K+1,, s± lk ] o reorder and disentangle iterative scheme: Z l+1 [s ± lk+1,, s± (l+1)k ] = X Λ l [s ± (l 1)K+1,, s± lk, s± lk+1,, s± (l+1)k ] s ± (l 1)K+1,,s± lk final step: Z[η] = s ± N K+1,,s± N Z l [s ± (l 1)K+1,, s± lk ] Z NK [s ± N K+1,,s± N ]

20 ISPI: convergence & extrapolation still two numerical errors: 1 finite time discretization t = Nδ t : Trotter O(δ 2 t ) 2 finite memory τ c = Kδ t naive convergence δ t and K not possible! ut extrapolation: 2 δ t : ± 3 2 τ c : di/dv / (e 2 /h) δ t : ± 2 δ t : ± 2 δ t : ± 1 U = ; ev = 5Γ U = 5Γ, ev = 1Γ U = Γ; ev = 5Γ U = 2Γ; ev = 5Γ ( δ t Γ) 2 di/dv / (e 2 /h) τ c : τ c : 197 τ c : 1856 U =, ev = 5Γ U = 5Γ, ev = 1Γ U = Γ, ev = 5Γ U = 2Γ, ev = 5Γ / (τ c Γ) gives exact result I = I(δ t,τ c )

21 Results: time-dependent current 25 2 I / (eγ / h) 15 1 U=4Γ, ev=2γ, T=1Γ U=5Γ, ev=6γ, T=Γ analyt tγ after initial transient behavior, current saturates ǫ = = agreement with nonequilibrium Kondo theory of Rosch et al which is valid for ev T K (= 29Γ)

22 ISPI vsanalytics: U = I / (eγ/h) (a) ISPI analyt (b) I / (eγ/h) ε / Γ ev / Γ -6 warm-up check: U = complete agreement current conservation also checked! T = 1Γ, =, (a) ev = 2Γ, (b) ǫ =

23 ISPI vs analytics: small U interaction corrections δa A(U) A(U = ) δ I / (eγ/h) -1-2 ISPI U = 1Γ -2-6 ISPI U = 3Γ perturb ev / Γ ev / Γ δ I / (eγ/h) -2-4 δ (di/dv) (e 2 /h) ev / Γ U = 1Γ,3Γ U = Γ T = 1Γ perturbation theory in U: interaction self-energy up to O(U 2 ) possible only at the e-h-symmetric point ǫ = =

24 ISPI vs analytics: sequential regime T Γ δ (di/dv) / (e 2 /h) -1-2 δ G/(e 2 /h) -2-4 ISPI master eq T/ Γ T / Γ U = Γ, ev = 3Γ, ǫ = = comparison with simple rate equation valid at T Γ agreement for T 4Γ coherence effects important for T < Γ

25 Small biasregime ev Γ 2 15 = ; L = Γ ; L = 2Γ ; L = ; ISPI = Γ ; ISPI = 2Γ : ISPI G/(e 2 /h) ε /Γ -field splits conductance peak: ǫ 2 + U U = Γ, T = 1Γ, inset: = interaction corrections most pronounced on resonance

26 Linear conductance peak 2 G(e 2 /h) 15 1 U = U = 12Γ ISPI U = 3Γ ISPI U = 3Γ Hamann U = 4Γ ISPI U = 4Γ Hamann T/Γ temperature dependence for T K T ǫ = = Γ agreement with formula of Hamann, PR (1967) which agrees with NRG results of osti et al (1994) until now: no convergent results for T,V!

27 Large biasregime ev > Γ 1 (di/dv)/(e 2 /h) 5 = ; L = Γ ; L = 2Γ ; L = ; ISPI = Γ : ISPI =2Γ ; ISPI ε /Γ U = Γ, ev = 3Γ, T = 1Γ, inset: = Splitting of conductance peak at large V for = : ǫ ev/2 four peak structure at finite -field interaction corrections most pronounced on resonance

28 Result: nonlinear differential conductance di/dv / (e 2 /h) U = U = 12Γ U = 3Γ δ(di/dv)/(e 2 /h) T/Γ T / Γ ev = 2Γ, ǫ = = increase with lowering T increasing interaction suppresses conductance conductance decreases when bias voltage is raised

29 urrent studies: preliminary results nonequilibrium Kondo regime: ev T K influence of magnetic field: T K calculate spectral function out of equilibrium: splitting of Kondo resonance 8 1 di/dv (e 2 /h) U = 4Γ = 25Γ = 5Γ = 75Γ = Γ = 125Γ = 15Γ di/dv / (e 2 /h) T= 1Γ < Τ Κ U= 4Γ U = 5Γ ln -2 fit ln -2 fit ev / Γ ev / Γ T = 1Γ, ǫ = ǫ = =

30 Relation tobosonic QUAPI similar: idea of iterative evaluation of real-time path sum major difference: fermions Grassmann fields in the functional integrals QUAPI: dipole coupling H S = X j c jx j use eigenvalues of X to evaluate Feynman-Vernon influence functional while here: tunnel coupling H T = j (c jd + c j d) no simultaneous eigenbasis of d and d exists! QUAPI scheme not applicable! Makri, Makarov (1995), MT et al (1995-today)

31 onclusions Iterative Summation of real-time Path Integrals ISPI novel method to study quantum transport at nonequilibrium based on exact evaluation of real-time path integrals numerically exact, generic, no sign problem example of Anderson model thorough checks ISPI works in wide regime of parameters! does not yet converge for ev,t fancy property: the more nonequilibrium, the better ISPI works Phys Rev 77, (28)

32 Thanks to : Reinhold Egger Stephan Weiss Jens Eckel

33 Next lecture Lecture II: Real-time path-integral quantum Monte arlo approach to nonequilibrium transport