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Functional renormalization group for nonequilibrium quantum many-body problems

R. Gezzi, Th. Pruschke, and V. Meden

Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen, Friedrich-Hund-Platz 1, D-37077 G¨ ottingen, Germany (Dated: February 6, 2008)

arXiv:cond-mat/0609457v2 [cond-mat.str-el] 23 Nov 2006

We extend the concept of the functional renormalization for quantum many-body problems to nonequilibrium situations. Using a suitable generating functional based on the Keldysh approach, we derive a system of coupled di?erential equations for the m-particle vertex functions. The approach is completely general and allows calculations for both stationary and time-dependent situations. As a speci?c example we study the stationary state transport through a quantum dot with local Coulomb correlations at ?nite bias voltage employing two di?erent truncation schemes for the in?nite hierarchy of equations arising in the functional renormalization group scheme.

PACS numbers: 71.27.+a,73.21.La,73.23.-b

I.

INTRODUCTION

The reliable calculation of physical properties of interacting quantum mechanical systems presents a formidable task. Typically, one has to cope with the interplay of di?erent energy-scales possibly covering several orders of magnitude even for simple situations. Approximate tools like perturbation theory, but even numerically exact techniques can usually handle only a restricted window of energy scales and are furthermore limited in their applicability by the approximations involved or the computational resources available. In addition, due to the divergence of certain classes of Feynman diagrams, some of the interesting many-particle problems cannot be tackled by straight forward perturbation theory. The situation becomes even more involved if one is interested in properties o? equilibrium, in particular timedependent situations. A standard approach for such cases is based on the Keldysh formalism1 for the time evolution of Green functions, resulting in a matrix structure of propagators and self-energies. This structure is a direct consequence of the fact that in nonequilibrium we have to calculate averages of operators taken not with respect to the ground state but with respect to an arbitrary state. Therefore, the Gell-Mann and Low theorem2 is not valid any more. Other approaches attempt to treat the time evolution of the nonequilibrium system numerically, for example using the density matrix renormalization group3 , the numerical renormalization group (NRG)4,5 or the Hamiltonian based ?ow-equation method.6 The Keldysh technique shows a big ?exibility and one therefore can ?nd a wide range of applications such as transport through atomic, molecular and nano devices under various conditions,7 systems of atoms interacting with a radiation ?eld in contact with a bath,8,9 or electron-electron interaction in a weakly ionized plasma.10 One powerful concept to study interacting manyparticle systems is the rather general idea of the renormalization group11 (RG), which has also been applied to time-dependent and stationary nonequilibrium situations recently.12,13,14,15,16 In the RG approach one usu-

ally starts from high energy scales, leaving out possible infrared divergences and works ones way down to the desired low-energy region in a systematic way. However, the precise de?nition of “systematic way” does in general depend on the problem studied. In order to resolve this ambiguity for interacting quantum mechanical many-particle systems in equilibrium, two di?erent schemes attempting a unique, problem independent prescription have emerged during the past decade. One is Wegner’s Hamiltonian based ?owequation technique,17,18 the second a ?eld theoretical approach, which we want to focus on in the following. This approach is based on a functional representation of the partition function of the system and has become known as functional renormalization group (fRG).19,20,21,22 . A detailed description of the various possible implementations of the fRG and its previous applications in equilibrium can be found e.g. in Refs. 23 and 24. In the present work we extend the fRG to nonequilibrium by formulating the problem on the real instead of the imaginary time axis using the functional integral representation of the action on the Keldysh contour. Within a diagrammatic approach a similar set of fRG ?ow equations has already been derived by Jakobs and Schoeller and applied to study nonlinear transport through one-dimensional correlated electron systems.25,26 We believe that this method will enable us to treat a variety of nonequilibrium problems within a scheme which is well established in equilibrium and in contrast to other approaches is comparatively modest with respect to the computer resources required. Our framework for nonequilibrium will turn out to be su?ciently general to allow for a treatment of systems disturbed by arbitrary external ?elds (bias voltage, laser ?eld, etc.), which can be constant or timedependent. For classical many-body problems the fRG (also called nonperturbative renormalization group) was used to study nonequilibrium transitions between stationary states.27 As a simple but nontrivial application to test the potential and weakness of our implementation of a nonequilibrium fRG we choose the single impurity Anderson

2

CK? ?∞ CK+ t ∞

where t, t′ ∈ CK? , the anti time-ordered Green function ? ψ (ξ ′ )ψ ? (ξ ) G++ (ξ ′ , ξ ) = ?i T = ?iθ(t ? t′ ) ψ (ξ ′ )ψ ? (ξ ) ? ζiθ(t′ ? t) ψ ? (ξ )ψ (ξ ′ ) , (5) with t, t′ ∈ CK+ and G?+ (ξ ′ , ξ ) = ?ζi ψ ? (ξ )ψ (ξ ′ ) , t′ ∈ CK? , t ∈ CK+ (7) G+? (ξ ′ , ξ ) = ?i ψ (ξ ′ )ψ ? (ξ ) , t′ ∈ CK+ , t ∈ CK? (6)

FIG. 1: (color online) Keldysh contour

model (SIAM).28 This model represents the paradigm for correlation e?ects in condensed matter physics and is at the heart of a large range of experimental29,30,31,32,33,34,35 and theoretical investigations.36,37 It furthermore is the standard model for the description of the transport properties of interacting single-level quantum dots. The paper is organized as follows. In Sec. II and the appendices we extend the equilibrium fRG to treat general nonequilibrium quantum many-body problems. As an example we study the ?nite bias, stationary transport through an Anderson impurity in Sec. III. Within the nonequilibrium extension we obtain a system of coupled tensor equations, which represent the ?ow of the di?erent components (in the Keldysh space) of the self-energy and the vertex function. We discuss two approximations derived from the fRG scheme that have been successfully applied to the equilibrium situation.38,39 A summary and outlook in Sec. IV concludes the paper.

II. FUNCTIONAL RG IN NONEQUILIBRIUM

where ζ = +1 for bosons and ζ = ?1 for fermions. Concerning the Keldysh indices we here follow the notation of Ref. 40. G?? (ξ ′ , ξ ) and G++ (ξ ′ , ξ ) take into account the excitation spectrum while G+? (ξ ′ , ξ ) and G?+ (ξ ′ , ξ ) describe the thermodynamic state of the system. Only three of the Green functions are independent and one commonly introduces the linear combinations GR (ξ ′ , ξ ) = θ(t′ ? t) G+? (ξ ′ , ξ ) + ζG?+ (ξ ′ , ξ ) (8)

GK (ξ ′ , ξ ) = G?+ (ξ ′ , ξ ) + G+? (ξ ′ , ξ ) ,

GA (ξ ′ , ξ ) = θ(t ? t′ ) G?+ (ξ ′ , ξ ) + ζG+? (ξ ′ , ξ ) (9)

(10)

We start from the standard de?nition of the two-time Green function in the interaction picture, where ξ comprises a set of single-particle quantum numbers and time t. The time evolution operators are given as ? ? ∞ ? ? S = T exp ?i VI (t)dt (2) ? ?

?∞ ? G(ξ ′ , ξ ) = ?i S ?1 ψI (ξ ′ )ψI (ξ )S

,

(1)

named retarded, advanced and Keldysh component, respectively. To derive a nonequilibrium fRG scheme for an interacting many-body problem along the lines of Refs. 23 and 24 one needs a formulation that allows to express the mparticle vertices as functional derivatives of a generating functional. Here we use the approach by Kameneev.43 To set up a functional integral representation of the generating functional we de?ne the matrix ?= G G?? G?+ G+? G++

and the short hand notation

∞

? Oψ ? ψ, =i

?∞

?(ξ )O ?ξ,ξ′ ψ (ξ ′ ) , dξdξ ′ ψ

? exp S ?1 = T

? ? ?

∞

i

?∞

VI (t)dt

? ? ?

,

(3)

where ψ= ψ? ψ+

? the anti timewith T the usual time ordering and T ordering operator. The interaction term VI (t) is arbitrary, including possible explicit time dependence. In a nonequilibrium situation, the propagation of the system from ?∞ → ∞ is not any more equivalent to the propagation from ∞ → ?∞, i.e. one has to distinguish whether the time arguments in Eq. (1) belong to the former or latter.40,41,42 This scheme is usually depicted by the Keldysh double-time contour shown in Fig. 1, where CK? represents the propagation ?∞ → ∞ (upper branch of the Keldysh contour) and CK+ the propagation ∞ → ?∞ (lower branch of the Keldysh contour). Consequently, one has to introduce four distinct propagators, namely the time-ordered Green function G?? (ξ ′ , ξ ) = ?i Tψ (ξ ′ )ψ ? (ξ ) = ?iθ(t′ ? t) ψ (ξ ′ )ψ ? (ξ ) ? ζiθ(t ? t′ ) ψ ? (ξ )ψ (ξ ′ ) , (4)

is a spinor of ?elds (Grassmann for fermions or complex for bosons) with ψ? having a time from the upper branch of the Keldysh contour and ψ+ a time from the lower. Later it will also prove useful to Fourier transform from time t to frequency ω . One then has to replace t in ξ by ω . The integrals over ξ and ξ ′ stand for summations over the quantum numbers and integrations over time or frequency. The following steps can be performed with ξ either containing time or frequency. The generalization of the functional integral representation of the partition function to nonequilibrium is43

3

Ξ=

1 Ξ0

? exp Dψψ

? G ?0 ψ,

?1

?}, {ψ } ψ ? iSint {ψ

, Ξ0 =

? exp Dψψ

? G ?0 ψ,

?1

ψ

.

(11)

? 0 denotes the propagator of the noninteracting part of the many-body problem and Sint is the (arbitrary) The matrix G interaction term. To construct a generating functional for m-particle Green functions, one introduces external source ?elds η and η ? W ({η ?}, {η }) = 1 Ξ0 ? exp Dψψ

(c )

? G ?0 ψ,

?1

? η ? (? ?}, {ψ }) ? ψ, η, ψ) ψ ? iSint ({ψ

.

(12)

The (connected) m-particle Green function Gm can then be obtained by taking functional derivatives

c) G( m ′ ; {ξj } = (ζi)m ξj

δm δm W (c) ({η ?}, {η }) ′ . . . δη ′ δηξ δη ?ξ1 ?ξm m . . . δηξ1

(13)

η =? η =0

with W c ( {η ?}, {η }) = ln [W ({η ?}, {η })] .

(c )

(14)

The derivatives in Eq. (13) are taken with respect to the spinors η and η ?, i.e. the resulting Green function Gm is a tensor of rank 2m in the Keldysh indices. In the following we adopt the notation that quantities with explicit index m are tensors of rank 2m; without index m we denote the propagator and self-energy, both carrying a hat to point out their matrix structure. Introducing the ?elds φξ = i δ W c ({η ?}, {η }) , δη ?ξ ?ξ = ζi δ W c ({η φ ?}, {η }) δηξ (15)

we can perform a Legendre transformation ?}, {φ} = ?W c ({η ? η ? (? ? G ?0 Γ {φ ?}, {η }) ? φ, η , φ) + φ,

?1

φ

,

(16)

to the generating functional of the one-particle irreducible vertex functions γm (tensors of rank 2m in the Keldysh indices) γm δm δm ′ ?}, {φ} ; {ξj } = im ? ξj Γ {φ ? ′ . . . δ φξ ′ δφξ δ φξ1 m . . . δφξ1 m .

?=0 φ =φ

(17)

In contrast to the usual de?nition44 of Γ for convenience ?1 ? G ?0 (see Appendix A) we added a term φ, φ in Eq. (16). The relations between the Gm and γm in imaginary time can be found in text books44 and can straightforwardly be extended to real times on the Keldysh contour. For example, for the one-particle Green function one obtains

′ G1 (ξ ′ ; ξ ) = Gc 1 (ξ ; ξ ) δ δ ? ξ′ ,ξ , W c = ?ζ G = i ′ δηξ δη ?ξ1 1 (c )

where ? ξ′ ,ξ = G ?0 G

?1 ?1

? ?Σ

,

ξ ′ ,ξ

? This implies with the proper one-particle self-energy Σ. ? the relation Σ = ζγ1 . We have a matrix structure not only with respect to ξ and ξ ′ (as in equilibrium), but also with respect to the Keldysh indices. How does this additional structure manifests itself in the Green functions (13)? For m = 1 we have two derivatives with respect to spinor ?elds, i.e. the structure of a tensor product, which can be made explicit by using a tensor product notation

′ Gc 1 (ξ , ξ ) = (ζi)

=

?0 γ1 ? ζ G

?1 ?1 ξ ′ ,ξ

δ δ ? Wc δη ?ξ′ δηξ

,

η =? η =0

4 leading to the matrix ?

′ ? Gc 1 (ξ , ξ ) = (ζi)

δ2 W c δη ?? (ξ ′ )δη? (ξ ) δ W δη ?+ (ξ ′ )δη? (ξ )

2 c

δ2 W c δη ?? (ξ ′ )δη+ (ξ ) δ W δη ?+ (ξ ′ )δη+ (ξ )

2 c

? ? (18)

ξ

ξ′

ξ

ξ′

=

G?? (ξ ′ , ξ ) G?+ (ξ ′ , ξ )

G+? (ξ ′ , ξ ) G++ (ξ ′ , ξ )

.

Λ FIG. 2: Diagrammatic form of the ?ow equation for γ1 . The ?Λ . slashed line stands for the single scale propagator S

ξ2 ξ1 ξ ′2

ξ ′2 ξ ′1 ξ2

ξ2 ξ1 ξ ′1

ξ ′2 ξ ′1

Up to now we extended standard text book manipulations44 from imaginary time to the real-time Keldysh contour. We emphasize that no translational invariance in time is assumed. To derive the fRG ?ow equations in nonequilibrium we can follow the steps of Ref. 24. In Eqs. (11) and (12) we replace the noninter?Λ acting propagator by a propagator G 0 depending on a parameter Λ ∈ [Λ0 , 0] and require ? Λ0 = 0 , G ? Λ=0 ?0 , G =G 0 0 (19)

ζ

ξ1

ξ2 ξ1 ξ ′1 ξ2

ξ ′2 ξ ′1 ξ ′2 ξ1

ζ

ξ1 ξ ′1

ζ

ξ2 ξ ′2 ξ1 ξ ′2 ξ2 ξ ′1

i.e. at the starting point Λ = Λ0 no degrees of freedom are “turned on” while at Λ = 0 the Λ free problem is recovered. In models with infrared divergences Λ can be used to regularize the problem. In equilibrium this is often be achieved by implementing Λ as an infrared cuto? in momentum or energy. One of the advantages of the fRG approach over other RG schemes is that one is not restricted to these choices and other ways of introducing the parameter Λ have turned out to be useful for equilibrium problems.45,46 All that is required to derive the fundamental ?ow equations (23) and (B1) (see below) are the conditions Eq. (19). In our application of the nonequilibrium fRG to the steady state transport through an interacting quantum dot it is natural to implement Λ as an energy cuto?. However, such a choice must not be the natural one in cases where one is interested in studying transient behavior. In this situation the propagator and the vertex functions in general depend on several spatial and time variables and there is no obvious momentum or energy cuto? scheme. Within the fRG several ways of introducing Λ can be worked out, compared and the one best suited for the problem under investigation can be identi?ed. ? Λ the quantities de?ned in Eqs. (11) to (17) Through G 0 acquire a Λ-dependence. Taking the derivative with respect to Λ one now derives a functional di?erential equation for ΓΛ . From this, by expanding in powers of the external sources, an in?nite hierarchy of coupled di?erΛ ential equations for the γm is obtained. Although the steps in the derivation are formally equivalent to Ref. 24, because of the real-time formulation additional factors i and signs appear in several places. We thus believe that it is helpful to present the details of the derivation for the present approach, which is done in Appendix A. In particular, for the ?ow of the self-energy one ?nds the expression d ?Λ d Λ ′ Σ′ γ1 (ξ ; ξ ) = ζ dΛ dΛ ξ ,ξ

Λ FIG. 3: Diagrammatic form of the ?ow equation for γ2 . The ? slashed line stands for the single scale propagator S Λ , the ?Λ. unslashed line for G

?Λ γ Λ (ξ ′ , ·; ξ, ·) , = Tr S 2

(20)

which can be visualized by the diagram in Fig. 2. The trace in Eq. (20) is meant to run over all quantum numbers and time respectively frequency. In Eq. (20) appears the so-called single scale propagator (the slashed line in Fig. 2) ?Λ = G ?ΛQ ?ΛG ?Λ , S ?1 ?Λ = d G ?Λ Q . 0 dΛ (21) (22)

Λ ′ and the quantity γ2 (ξ , ·; ξ, ·) denotes the matrix obtained by keeping the indices ξ and ξ ′ ?xed. We thus arrive at an expression that is formally identical to Eq. (19) in Ref. 24. The di?erence appears in the matrix structure, which now also contains the index components for the branches of the Keldysh contour. To make this explicit, we write out Eq. (20) with respect to the Keldysh structure

ζ

d αβ,Λ Σ′ = Tr dΛ ξ ,ξ

?ν

ανβ?,Λ ′ S ?ν,Λ γ2 (ξ , ·; ξ, ·) ,

(23)

where α, β, ? and ν denote Keldysh indices and take the Λ values ?. Apparently, the derivative of γ1 is determined Λ Λ by γ1 and the two-particle vertex γ2 . Thus an equation Λ for γ2 is required. The structure of this equation turns out to be similar to Eq. (21) of Ref. 24. Here we only show the diagrams representing it in Fig. 3, the full expression is given in Eq. (B1). The di?erential equation Λ Λ for γ2 does not only contain γ1 – implicitly via the propΛ Λ agators – and γ2 , but also the three-particle vertex γ3 . The ?ow of the three-particle vertex depends on the fourparticle vertex etc. It is generically impossible to solve

5 the full set of in?nitely many coupled di?erential equations. In applications one has to truncate it, and this is usually done at order m = 2, i.e. one replaces all vertices with m > 2 by their initial values, which for problems Λ Λ0 = 0 for with a two-particle interaction means γm = γm m > 2. Even within this truncated system the remaining set of di?erential equations must typically further be approximated to allow for an analytical or a numerical solution.38,39,47 At the end of the fRG ?ow, that is for Λ=0 Λ = 0, the γm = γm present approximations to the many-body vertex functions from which observables can be computed (for examples see Sec. III). We again note that a similar set of ?ow equations was derived by Jakobs and Schoeller using a diagrammatic approach.25,26 simple we use units with e = h ? = 1. The Hamiltonian reads H =

kσl

εkσl c? c +

σ

kσl kσl

VG d? σ dσ + U

kσl

n↑ ?

1 2

n↓ ?

1 2 (24)

+

kσl

Vkσl c? dσ + H.c.

III.

APPLICATION TO QUANTUM DOTS

The derivation of the ?ow equations in the previous section and the appendices was completely general and they can be used to study time dependent or stationary properties of systems at ?nite temperature T or T = 0. A simple but nonetheless nontrivial application is nonequilibrium, stationary transport through an interacting single-level quantum dot at T = 0. For a stationary situation, time translational invariance holds and one can Fourier transform to frequency space. Moreover, because the quantum dot is a zero-dimensional structure, no additional degrees of freedom except spin have to be taken into account, thus considerably reducing the complexity of the problem. Nonequilibrium theory of single-level quantum dots described by the SIAM (see below) has been a major subject of research over the past years. Several techniques have been developed respectively applied, for example statistical48,49,50 , perturbative,51,52,53,54,55,56,57,58,59,60 RG based13,15,16 and numerical procedures.5,6 In particular the RG based calculations were up to now restricted to situations away from the Kondo regime, either by looking at features in the mixed-valence state13 or in strong magnetic ?elds.15,16 Despite this tremendous ongoing e?ort, relatively little is known about nonequilibrium properties, even in the stationary state, of the model. On the other hand, the very detailed knowledge of the equilibrium properties61 makes it possible to interpret certain features or even dismiss approximations on the basis of this fundamental understanding.

in standard second quantized notation (nσ = d? σ dσ ). As usual, k is the wave vector of the band states in the leads and σ the spin. In addition, the index l = L, R distinguishes the left and right reservoirs which can have di?erent chemical potentials ?l through an applied bias voltage VB = ?L ? ?R . Since we do not include a magnetic ?eld lifting the spin-degeneracy of the spin up and down level, the one-particle Green function and the selfenergy are spin independent and we suppress the spin index. We assume the dispersions εkσl and hybridizations Vkσl between dot and the left and right leads to be √ identical, and Vk ≡ V / 2 to be k -independent. Equation (24) is written such that for VG = 0 we have particle-hole symmetry. The interaction in the Hamiltonian is reduced to the dot site only. We can thus reduce the problem to a zerodimensional one by “integrating out” the leads. For the ? d,0 at U = 0. ?ow equations we need the propagator G To this end we use the Dyson equation to obtain1,40,42 ? d,0 (ω ) = G where ? d,0 = G

?+ G?? d,0 Gd,0 ? ++ G+ d,0 Gd,0 kl G+? kl

ω ? VG 0 0 ?ω + VG

?1

? lead ?Σ

,

(25)

,

kl G++ kl

2 ? lead = V Σ 2N

G?? G?+

kl

.

(26)

To further evaluate Eq. (26), we have to insert the Green functions of the free electron gas, given by G?? (ω ) = kl 1 ω ? εk + ?l + iδ +2iπf (εk )δ (ω ? εk + ?l ) ,

kl ?+ G (ω ) kl G+? (ω ) kl ?

(27)

A.

Single impurity Anderson model

G++ (ω ) = ? G?? (ω )

kl

,

(28) (29) (30)

The standard model to describe transport through interacting quantum dots is the SIAM.28 Experimentally,29,62 the dot region is attached to two external leads, and the current is driven by an applied bias voltage VB . Furthermore, the ?lling of the dot can be controlled by a gate voltage VG . To keep the notation

= 2iπf (εk )δ (ω ? εk + ?l ) , = ?2iπf (?εk )δ (ω ? εk + ?l ) ,

into Eq. (25). This leads to G?? d,0 (ω ) = ω ? VG ? iΓ [1 ? fL (ω ) ? fR (ω )] , (31) (ω ? VG )2 + Γ2

6

?? ? G++ d,0 (ω ) = ?[Gd,0 (ω )] ,

(32) (33) (34)

+ G? d,0 (ω ) = i

Γ [fL (ω ) + fR (ω )] , (ω ? VG )2 + Γ2 Γ [fL (?ω ) + fR (?ω )] ? , G+ d,0 (ω ) = ?i (ω ? VG )2 + Γ2

the leads and Γ = π |V |2 NF , with NF the density of states at the Fermi level of the semi-in?nite leads, represents the tunnel barrier between the leads and the impurity. Finally, the quantities we want to calculate are the current J through the dot and the di?erential conductance G = dJ/dVB . For the model (24) the current is given by7,55

where fl (±ω ) = f (±(ω ? ?l )) are the Fermi functions of

J=

1 iΓ (JL + JR ) = 2 2π

? ?+ d? [fL (?) ? fR (?)] G+ d (?) ? Gd (?) ,

(35)

where we already used that the left and right couplings are identical, i.e. ΓL = ΓR = Γ/2, and summed over both ? d and JL/R are the currents spin directions. The interacting one-particle Green function of the dot is denoted by G across the left and right dot-lead contacts respectively. Equation (35) is written in a somewhat unusual form, not ? ?+ R A employing the relation G+ d (?) ? Gd (?) = Gd (?) ? Gd (?) = ?2πiρd (?), where ρd denotes the dot’s one-particle spectral function. The reason for this will be explained below. Another quantity of interest is ?J = JL ? JR .52 Obviously, since no charge is produced on the quantum dot, ?J = 0 in the exact solution. Using again the results of Refs. 7 and 55, the expression for ?J becomes Γ ?J = ? π

∞

dω

?∞

F (ω ) [?mΣ?+ (ω ) ? ?mΣ+? (ω )] ? 2?mΣ?+ (ω ) ? ω) ?(

2

(36)

? ω ) = ω ? VG + iΓ [1 ? F (ω )] ? Σ?? (ω ) ?( F (ω ) = fL (ω ) + fR (ω )

+ ΓF (?ω ) + ?mΣ+? (ω ) ΓF (ω ) ? ?mΣ?+ (ω ) (37)

where we used that Σ?+ (ω ) and Σ+? (ω ) are purely imaginary. Depending on the type of approximation used ?J = 0 might either hold for all parameters55,60 or not.52 We note that ful?lling ?J = 0 is, however, not suf?cient for an approximation to provide reliable results. E.g. the self-consistent Hartree-Fock approximation ful?lls ?J = 0, but nonetheless does not capture the correct physics even in equilibrium.61

B. Lowest order approximation

Λ ? Λ and γ2 Before studying the coupled system for Σ we Λ begin with the simpler case where we replace γ2 on the right hand side of Eq. (23) by the antisymmetrized bare ? The only interaction and consider only the ?ow of Σ. nonzero components of the bare two-particle vertex are those with all four Keldysh indices being the same (α = ?)40

Within this approximation the self-energy is always time or frequency independent, and no terms o?-diagonal in the Keldysh contour indices are generated. It leads to at least qualitatively good results in equilibrium,38 and has the additional advantage that the ?ow equations can be solved analytically. As the last step we specify how the parameter Λ is introduced. Since we are interested in a stationary situation, i.e. the propagators only depend on the time difference t ? t′ , all equations can be transformed into frequency space and one natural choice is a frequency cuto? of the form ?Λ ? G d,0 (ω ) = Θ (|ω | ? Λ) Gd,0 (ω ) (40)

? by means of the Morris with Λ0 → ∞.24 Evaluating S 21,24 lemma results in ?Λ (ω ) → δ (|ω | ? Λ) S 1 ? d,0 (ω ) G

?1

αααα,Λ0 ′ ′ ′ δσ ,σ ′ ? δσ ,σ ′ δσ ,σ ′ γ2 (ξ1 , ξ2 ; ξ1 , ξ2 ) = αiU δσ1 ,σ1 (38) . 2 1 2 2 2 1

With this replacement Eq. (23) reduces to d ??,Λ Σ = ±iU dΛ dω ??,Λ S (ω ) . 2π (39)

?Λ ?Σ

.

(41)

A straightforward calculation permits us thus to rewrite Eq. (39) as

7

iU d ??,Λ Σ =± dΛ 2π

G?? (ω ) d,0 ?(ω ) ω =± Λ G++ (ω ) d,0 ?(ω )

? Σ±±,Λ ? Σ++,Λ ?

? Σ??,Λ

G?? (ω ) d,0 ?(ω )

+ ? G? (ω )G + (ω ) d,0 d,0 ?(ω )2

,

(42)

where

++ ?+ +? ?(ω ) = G?? d,0 (ω )Gd,0 (ω ) ? Gd,0 (ω )Gd,0 (ω ) 1 = ? . (ω ? VG )2 + Γ2

0 -0.05 Σ /Γ

??

-0.1

Finally, the initial condition for the self-energy is ? Λ0 = 0.24 lim Σ

Λ0 →∞

-0.15 -0.2 -3 10

1.

Equilibrium

10

-2

10

-1

10 Λ/Γ

0

10

1

10

2

10

3

We now focus on T = 0. In a ?rst step we discuss the equilibrium situation, that is VB = ?L ? ?R → 0. Then we obtain the decoupled system VG ± Σ??,Λ U d ??,Λ Σ =i , dΛ π [(Λ ± iΓ)2 ? (VG ± Σ??,Λ )2 ] (43)

FIG. 4: Flow of Σ??,Λ /Γ with Λ/Γ for U/Γ = 1, VG /Γ = 0.5, and VB = 0. The full curve shows the real part, the dashed the imaginary part of Σ??,Λ .

5 0 Σ /Γ -5

Λ=∞→Λ=0 Λ=0→Λ=∞ exact solution

??

which can be solved analytically. We ?rst note that with ? Σ++,Λ = ? Σ??,Λ both equations are equivalent. For Σ??,Λ we obtain with the de?nition σ Λ = VG + Σ??,Λ the solution

πσ iσ Λ J1 ( πσ U ) ? (Λ + iΓ)J0 ( U )

Λ Λ Λ Λ

πσ iσ Λ Y1 ( πσ U ) ? (Λ + iΓ)Y0 ( U )

=

G J0 ( πV U ) G Y0 ( πV U )

-10

,

(44)

-15 -20 -3 10

-2 -1 0 1 2 3

where Jn and Yn are the Bessel functions of ?rst and second kind. The desired solution of the cuto? free problem is obtained by setting Λ = 0, i.e.

πσ G σJ1 ( πσ J0 ( πV U ) ? ΓJ0 ( U ) U ) . = πσ πσ πV σY1 ( U ) ? ΓY0 ( U ) Y0 ( UG )

10

10

10 Λ/Γ

10

10

10

(45)

which is precisely the result Eq. (4) obtained by Andergassen et al. 38 It is, however, important to note that in this work, based on the imaginary-time formulation of the fRG, the di?erential equation has a di?erent structure. It is real and has a positive de?nite denominator. Thus, while the solutions at Λ = 0 are identical for the imaginary-time and real-time formulations, the ?ow towards Λ = 0 will show di?erences. As we will see next, the complex nature of the di?erential equation (43) can lead to problems connected to its analytical structure when attempting a numerical solution. For small U/Γ no particular problems arise. As an example the result for the ?ow of Σ??,Λ as function of Λ for U/Γ = 1 and VG /Γ = 0.5 obtained with a standard Runge-Kutta solver is shown in Fig. 4. Consistent with the analytical

FIG. 5: (color online) Flow of Σ??,Λ /Γ with Λ/Γ for U/Γ = 15, VG /Γ = 6, and VB = 0. The full and dashed curves show real and imaginary part obtained from the integration Λ = ∞ → Λ = 0, the dashed-dotted and dotted curves real and imaginary part obtained from an integration Λ = 0 → Λ = ∞, using the solution from (45) as initial value for Σ??,Λ . The crosses denote the analytical solution (44).

solution Eq. (45), the imaginary part (dashed line) goes to zero as Λ → 0, while the real part (solid line) rapidly approaches the value given by formula (45). However, for larger values of U/Γ the numerical solution becomes unstable in a certain regime of VG . A typical result in such a situation is shown in Fig. 5. The di?erent curves were obtained as follows: The full and dashed ones from the numerical solution starting with Σ??,Λ0 = 0 at Λ0 → ∞, the dash-dotted and dotted

8

VG

a

1 0

VG

b

0 -0.1 Σ /Γ -0.2 -0.3

U/Γ=1, VB/Γ=0 U/Γ=1, VB/Γ=1 U/Γ=5, VB/Γ=0 U/Γ=5, VB/Γ=1

?m Λp

-2 -3 -4 -5 0

U/Γ=1 U/Γ=10 U/Γ=15 U/Γ=20

??

-1

-0.4 -0.5 -3 10

-2 -1 0

10

10

5

VG/Γ

10

15

10 Λ/Γ

10

1

10

2

10

3

FIG. 6: (color online) Imaginary part of Λp determined from (44) and VG + Σ??,Λp = Λp + iΓ for di?erent values of U and VB = 0 as function of VG . For U/Γ = 15 and 20 there exist a b an interval [VG , VG ] where ?mΛp > 0, while for small U or a b VG ∈ [VG , VG ] we always have ?mΛp < 0.

FIG. 7: (color online) Flow of Σ??,Λ /Γ with Λ for U/Γ = 1 and 5 for VG /Γ = 0.5 at VB /Γ = 0 and 1 (thick curves: real part; thin curves: imaginary part). The curve for ?mΣ??,Λ /Γ at U/Γ = 1 and VB /Γ = 1 (thin dashed line) lies on top of the corresponding zero bias curve and is thus not visible.

by integrating the di?erential equation (43) backwards from Λ = 0 with the correct solution for Λ = 0 as given by formula (45) as initial value. The crosses ?nally are the results from the analytical solution Eq. (44). Evidently, there exists a crossing of di?erent branches of solutions to the di?erential equation for Λ/Γ ≈ 1 and the numerical solution with starting point Λ = ∞ picks the wrong one as Λ → 0. The reason for this behavior a is that for large U there exists a certain VG such that a ??,Λp = Λp + iΓ with real Λp , resulting in a pole VG + Σ a in the di?erential equation (43). For VG = VG this pole does not appear for real Λp , but as shown in Fig. 6 ?mΛp a changes sign at VG , which in turn induces a sign change on the right hand side of the di?erential equation, leading to the behavior observed in Fig. 5. There also exists b b a second critical value VG such that for VG > VG we ?nd ?mΛp < 0 and the instability has vanished again. Obviously, this instability limits the applicability of the present approximation to su?ciently small values of U . This is di?erent from the imaginary-time approach by Andergassen et al.,38 where this simple approximation leads to qualitative correct results even for values of U signi?cantly larger than Γ.

imaginary part of order U 2 is generated in the ?ow which does not vanish for Λ → 0 (see the thin dotted line). Causality requires that the relation must hold for the exact solution. Because of Σ?+ (ω ) = Σ+? (ω ) = 0, the ?nite imaginary part of Σαα leads to a breaking of the condition (46) to order U 2 at the end of the fRG ?ow. This is consistent with the fact that by neglecting the ?ow of the vertex terms of order U 2 are only partially kept in the present fRG truncation scheme. To avoid any confusion we emphasize that our RG method is di?erent from any low-order perturbation theory. The weak breaking of causality can also be understood as a consequence of our approximation leading to a complex, energy-independent self-energy: The o?-diagonal components, being related to the distribution functions for electrons and holes, respectively, in general have di?erent support on the energy axis. The energy independence makes it impossible to respect this structure here. For our further discussion the order U 2 violation of Eq. (46) means that we may not rely on relations like Eqs. (8)-(10) but have to work with Gαβ , thus the somewhat unusual formula (35). A naive application of ? + ΣR = Σ?? ? Σ?+ and use of G+ ? G? = 2i?mGR d d d would have led to unphysical results. That working with ? + G+ ? G? is still sensible can be seen from a straightd d forward evaluation leading to

? ?+ G+ d (ω ) ? Gd (ω ) =

Σ?? (ω ) + Σ++ (ω ) = ? Σ?+ (ω ) + Σ+? (ω )

(46)

2.

Nonequilibrium

We now turn to the case of ?nite bias voltage VB . As a typical example, the ?ow of Σ??,Λ for U/Γ = 1 (full and dashed curves) and 5 (dashed-dotted and dotted curves) for VG /Γ = 0.5 at VB /Γ = 0 (equilibrium) and VB /Γ = 1 is shown in Fig. 7. Since the results for Σ++,Λ are related ? to those for Σ??,Λ by Σ++,Λ = ? Σ??,Λ we do not show them here. The VB dependence of the curves for ?e Σ??,Λ (thick lines) looks sensible. For VB = 0 an

(47) Γ

?2i

|ω ? VG + iΓ [1 ? F (ω )] ? Σ?? |2 + Γ2 F (ω )F (?ω )

F (ω ) = fL (ω ) + fR (ω ) which is purely imaginary with a de?nite sign. Inserting the expression (47) into the formula (35), one can

9

1 0.8 G/G0 0.6 0.4

-0.15 0

VB/Γ=0 VB/Γ=0.5 VB/Γ=1 VB/Γ=5

??

0

-0.05 Σ /Γ -0.1

-2

-4

0.2 0 -10 0 VG/Γ 10

-0.2 10

-2

-6 10 Λ/Γ

0

10

2

10

-2

10 Λ/Γ

0

10

2

FIG. 8: (color online) Conductance normalized to G0 = 2e2 /h as function of VG for U/Γ = 2 and several values of the bias voltage VB .

FIG. 9: (color online) Flow of Σ??,Λ /Γ with Λ for U/Γ = 1 (left panel) and U/Γ = 15 (right panel) at VG = U/2 and VB = 0. The full curves show the real part, the dashed the imaginary part of Σ??,Λ . The stars at the vertical axis denote the values as obtained from the imaginary-time fRG.39

calculate the current and thus the conductance. Since we are at T = 0, an explicit expression for the current of the cuto? free problem (at Λ = 0) can be obtained by noting that with ?L = VB /2, ?R = ?VB /2 one has fL (ω ) ? fR (ω ) = Θ(VB /2 ? |ω |) and F (±ω ) = 1 for ω ∈ [?VB /2, VB /2], which leads to Γ2 J = π = Γ Γ π Γ?

VB /2

approximate inclusion of the vertex ?ow. In the present approximation the current conservation ?J = 0 holds for all parameters as Σ?+ = Σ+? = 0 [cf. Eq. (36)].

?VB /2

1 dω |ω ? VG ? Σ?? |2 + Γ2 s arctan

? B VG + s V2 Γ?

C.

Flowing vertex

(48)

s=±1

with the abbreviations

? VG = VG + ?eΣ?? ,

(49) (50)

Γ? =

Γ2 + (?mΣ?? )2 .

Equation (48) for the current is equivalent to the noninteracting expression but with renormalized parameters ? VG and Γ? , which depend on the interaction as well as the bias and gate voltage. An example for the di?erential conductance as function of VG obtained from Eq. (48) for U/Γ = 2 and several values of VB is shown in Fig. 8, where G0 = 2e2 /h (after reintroducing e and h ? ). Increasing VB leads, as expected, ?rst to a decrease of the conductance close to VG = 0 and later to a splitting of order VB . Since we will discuss a more re?ned scheme including parts of the ?ow of the two-particle vertex next, we do not intend to dwell too much on the results of this simplest approximation. We note in passing that for VG = 0 due to particle-hole symmetry we obtain from the di?erential equation (42) that Σ?? = 0 independent of U . Consequently the current J calculated via Eq. (48) and the conductance are independent of U , too, and given by the corresponding expressions for the noninteracting system. As we will see in the next section, this de?ciency will be cured by the

A more re?ned approximation is obtained when we inΛ sert the ?owing two-particle vertex γ2 as given by expression (B2) in the calculation of the self-energy Eq. (23). By this we introduce an energy-dependence of the self-energy.24 However, because the size of the resulting system of di?erential equations becomes extremely large if the full frequency dependence is kept (for a discussion on this in equilibrium see Ref. 24), we only keep the ?ow of the frequency independent part of the vertex, an additional approximation which has successfully been used in equilibrium.39 As a consequence we again end up with a ? Λ . The resulting expression for frequency independent Σ the self-energy (see Appendix B) is 1 d αβ,Λ Σ =? dΛ 2π

Λ Gγδ, (ω ) 2U αδβγ,Λ ? U δαβγ,Λ , d

ω =± Λ

(51) where U αδβγ,Λ is the ?owing interaction given by the expression (B3). As has been observed by Karrasch et al.,39 this approximation leads to a surprisingly accurate description of the transport properties in equilibrium. In particular it is superior to the lowest order approximation including only the bare vertex.

1.

Equilibrium

We again begin with the discussion of the solution to Eq. (51) in equilibrium. Results for the ?ow of Σ??,Λ are presented in Fig. 9 for U/Γ = 1 (left panel) and U/Γ = 15

10

0 -0.05 ×10 -0.1 -0.15 -6 -0.2 10

-2 2

0

Σ /Γ, Σ /Γ

-2

??

-4

10 Λ/Γ

0

10

2

10

-2

10 Λ/Γ

0

10

2

FIG. 10: (color online) Flow of Σ??,Λ /Γ and Σ?+,Λ /Γ with Λ for U/Γ = 1 (left panel) and U/Γ = 15 (right panel), VG /Γ = U/2 and VB /Γ = 1. The full curves show the real part, the dashed the imaginary part of Σ??,Λ , the dot-dashed the imaginary part of Σ?+,Λ . The real part for the latter is zero.

part (dashed lines) of Σ?? a third curve is displayed, the imaginary part of Σ?+ (dashed-dotted lines), which now is generated during the ?ow. Note that ?eΣ?+,Λ = 0 ? and Σ+?,Λ = Σ?+,Λ . Furthermore, we always ?nd ?mΣ?+,Λ < 0. For U/Γ = 15 (right panel in Fig. 10) we have rescaled ?mΣ?+,Λ by a factor 102 to make it visible on the scale of Σ??,Λ . Since Σαβ,Λ is a complex energy independent quantity Eq. (46) is again not ful?lled. We note that the error is still of order U 2 , but for ?xed VG and VB it is signi?cantly smaller than in the simplest truncation scheme discussed above. The energy-independence of the self-energy allows to derive an analytical expression for the current at T = 0 similar to Eq. (48), which due to the appearance of Σ?+ now becomes ? Γ Γ s arctan π Γ? s=±1 as in Eq. (49) and J=

? B VG + s V2 Γ?

?+

(52)

? with VG

(right panel) for VG = U/2. Since Σ++,Λ = ? Σ??,Λ only one component is shown. The stars in Fig. 9 denote the solutions of the imaginary-time equations taken from Ref. 39. Note that for U/Γ = 15 and VG /Γ > 6 the simple approximation Eq. (42) showed an instability, while with the ?owing vertex the system is stable even for these large values of U and reproduces the correct equilibrium solutions for Λ → 0.39 The reason for this is that the ?ow of the vertex reduces the resulting e?ective interaction below the critical value in the instability region.39

2. Nonequilibrium

?

? = Γ ? ?m Σ ? + > Γ Γ Γ? = ? 2 + (?mΣ?? )2 ; Γ

For the same parameters as in Fig. 9 we present the resulting ?ow with ?nite bias VB /Γ = 1 in Fig. 10. In addition to the curves for real (solid lines) and imaginary

where Σαβ is taken at Λ = 0. Thus, the only change to ? in J/Γ. the expression (48) is a formal replacement Γ → Γ Equation (52) is of the same structure as for the noninteracting case with VG and Γ replaced by renormalized parameters. However, the two self-energy contributions Σ?? and Σ?+ enter distinctively di?erent in the expression for the current. While ?mΣ?? solely plays the role of an additional life-time broadening, ?mΣ?+ directly modi?es the tunneling rate both in the prefactor of J and in the expression for the life-time broadening. A problem occurs when using the results of the present approximation in Eq. (36), leading to

Γ ?J = 2 π

∞

dω

?∞

|ω ? VG + iΓ [1 ? F (ω )] ? Σ?? | + [ΓF (?ω ) + ?mΣ+? ] [ΓF (ω ) ? ?mΣ?+ ]

?mΣ?+ [1 ? F (ω )]

2

.

(53)

The requirement ?J = 0 is only ful?lled for VG = 0, because then Σ?? = 0 and the integrand is asymmetric with respect to ω . Thus our approximation of an energyindependent ?owing vertex violates current conservation for VG = 0 in nonequilibrium. We veri?ed that ?J ? U 2 which is consistent with the fact that not all terms of order U 2 are kept in our truncated fRG procedure. How does ?J behave in the limit VB → 0? To see this we note that, because ?mΣ?+ does not depend on the sign of VB V →0 and furthermore goes to zero as VB → 0, ?mΣ?+ B ?

2 2 VB . Consequently, ?J B ? VB and hence the violation of current conservation vanishes in the linear response regime VB → 0.

V →0

In Fig. 11 we show the current at VG = 0 as function of VB for U/Γ = 1, 6 and 15. With increasing U the current for intermediate VB is strongly suppressed. In addition there occurs a structure at low VB , which turns into a region of negative di?erential conductance with increasing U . The appearance of such a shoulder in the current was observed in other calculations as well.51,52,56 However,

11

0.4 0.2 J/J0 0 -0.2 -0.4 -8 -6 -4 -2 0 2 VB/Γ 4 6 8

U/Γ=6 U/Γ=15

G/G0

U/Γ=1

1 0.5 0 1 G/G0 0.5 0 -10

VB/Γ=0 VB/Γ=1 VB/Γ=5

0 VG/Γ

10

FIG. 11: (color online) Current normalized to J0 = G0 Γ e (after reintroducing e and h ? ) as function of VB for U/Γ = 1, 6 and 15 and VG = 0. For U/Γ = 15 we ?nd a region of negative di?erential conductance in the region |VB /Γ| ≈ 0.5 (c.f. Fig. 12).

FIG. 13: (color online) Di?erential conductance G as function of VG for di?erent values of VB and U/Γ = 1 (upper panel) and U/Γ = 15 (lower panel). Note the extended plateau at VB = 0 for U/Γ = 15, which is a manifestation of the pinning of spectral weight at the Fermi level.

1 0.8 G/G0 0.6 0.4 0.2 0 0 2 4 VB/Γ 6

U/Γ=2 U/Γ=4 U/Γ=6 U/Γ=8

8

FIG. 12: (color online) Di?erential conductance G as function of VB for VG = 0 and various values of U . For U/Γ > 5 a distinct minimum around VB /Γ ≈ 0.5 appears.

whether the negative di?erential conductance we ?nd for still larger values of U (c.f. Fig. 12) is a true feature of the model or rather an artifact of the approximations used is presently not clear and should be clari?ed in further investigations. However, negative di?erential conductance has also been observed in a slave-boson treatment of the model.63 Keeping VG = 0 ?xed, we can calculate the conductance G = dJ/dVB as function of VB for di?erent values of U . The results are collected in Fig. 12. In contrast to the simple approximation without ?ow of the vertex, the conductance is now strongly dependent on U , except for VB = 0, where due to the unitary limit at T = 0 we always ?nd G = G0 . As already anticipated from the current in Fig. 11, a minimum in G starts to form around VB /Γ ≈ 0.5 for U/Γ > 5, which is accompanied by a peak at VB /Γ ≈ 2. A similar behavior in the conductance was observed in a perturbative treatment,56 which in contrast to our current approximation involves the full

energy-dependence in the self-energy. This at least qualitative agreement – we of course cannot resolve structures like the Hubbard bands with an energy independent selfenergy – again supports our claim that despite the violation of the relation (46) we can obtain reasonable results from Gαβ . We ?nally discuss the variation of the conductance with VG for ?xed U and VB . We again emphasize, that for VG = 0, ?J = 0 only holds to leading order in U . In Fig. 13 we present the curves for two di?erent values of U , namely U/Γ = 1 (upper panel in Fig. 13) and U/Γ = 15 (lower panel in Fig. 13). In the former case, the variation of G with VB is rather smooth, as is to be expected from the current in Fig. 11. For large U , we observe an extended plateau at zero bias, which is a manifestation of the fact that in the strong coupling regime a pinning of spectral weight at the Fermi energy occurs. This feature is also observed in the imaginary-time fRG as well as in NRG calculations.39 Increasing VB quickly leads to a similarly extended region of negative di?erential conductance, which, assuming that this result is a true feature of the model, therefore seems to be linked to the “Kondo” pinning. We note that it is unlikely that the appearance of the negative di?erential conductance is related to the breaking of current conservation at order U 2 as it also appears for VG = 0 where ?J = 0. For large VB multiple structures appear in G, which are related to the energy scales VB and U .

IV.

SUMMARY AND OUTLOOK

Starting from a generating functional proposed by Kameneev43 we have derived an in?nite hierarchy of differential equations for the vertex functions of an interacting quantum mechanical many-body system in nonequilibrium (see also Refs. 25 and 26). The major di?erence

12 to the imaginary-time formulation comes from the use of Keldysh Green functions. The indices + and ? referring to the upper and lower branch of the (real-)time contour lead to an additional matrix structure to the problem. Our formulation is su?ciently general that it allows to treat bosonic and fermionic models with or without explicit time dependence and at T ≥ 0. Since the fRG leads to an in?nite hierarchy of coupled di?erential equations, one has to introduce approximations, at least a truncation at a certain level. Typically one neglects the ?ow of the three-particle vertex. As has been demonstrated in Ref. 24 for the imaginary-time fRG one can solve the remaining system of ?ow equations for simple models like the SIAM numerically, thus keeping the full energy-dependence. Due to the fact that the vertex function carries three continuous frequency arguments in addition to the discrete quantum numbers of the system, such a calculation can become computationally quite expensive. To reduce the numerical e?ort, further additional approximations can be introduced. A particularly important and successful one is obtained by neglecting the energy dependence of the vertex functions,38,39 which already leads to a surprisingly accurate description of local and transport properties of interacting quantum dots in the linear response regime. We applied the nonequilibrium fRG to the SIAM with ?nite bias voltage in the stationary state. It turned out, that for the simplest approximation where only the ?ow of the self-energy is kept, the analytic structure of the di?erential equation leads to problems in the numerical solution. In addition, this approximation leads to a violation of the causality relation (46) to order U 2 . The ?rst problem was resolved by including the two-particle vertex in the ?ow at least up to the largest interaction considered here (U/Γ = 15). At the present stage this was for computational reasons done by assuming it to be energyindependent, yielding again an energy-independent selfenergy. Although this approximation also violates Eq. (46) to order U 2 for a ?xed VG and VB the error is signi?cantly smaller compared to the simplest scheme. We were able to obtain reasonable expressions and numerical results for the current and the conductance using the functions Gαβ (ω ) instead of GR (ω ) in the current formula. We reproduced nonequilibrium features of the current and di?erential conductance known from the application of other approximate methods to the SIAM. In the more advanced truncation scheme and for VG = 0, the current conservation ?J = 0 only holds to leading order in U . This defect can be traced back to the energy independence of the two-particle vertex, leading to ?nite, but energy-independent Σ?+ and Σ+? . By a close inspection of formula (36), however, one can see that this de?ciency can for example be cured by assuming a coarse-grained energy dependence of the form

ω<? ?

VB : F (ω ) = 2 ? ?mΣ+?,Λ = 0 , ?mΣ?+,Λ =const. 2

(54) (55) (56)

VB VB <ω< : F (ω ) = 1 ? ?mΣ+?,Λ = ??mΣ?+,Λ = const. 2 2 VB < ω : F (ω ) = 0 ? ?mΣ?+,Λ = 0 , ?mΣ+?,Λ =const. 2

and corresponding energy dependencies for Σ?? and Σ++ . Here the const. might depend on Λ but not on ω . Such an approximation will quite likely also reestablish causality Eq. (46). Work along this line is in progress. Including ?nite temperature, magnetic ?eld etc. in the calculations of transport is easily possible, as well as the extension to more complicated “impurity” structures (see e.g. Ref. 39). The latter aspect is typically a rather cumbersome step for other methods like perturbation theories or in particular numerical techniques. Our present stage of work should mainly be seen as a “proof of principle”. Already for the SIAM there remain a variety of fundamental things to do; like the implementation of the full energy-dependence in the calculations, which then should cure the violation of charge conservation and causality; or implementing a reasonable “cut-

o? scheme” for time-dependent problems o? equilibrium, e.g. to calculate current transients. Despite its current limitations we expect the real-time nonequilibrium formulation of the fRG to be a useful tool to understand nonequilibrium features of mesoscopic systems, just like the imaginary-time version in the calculation of equilibrium properties.

Acknowledgments

We acknowledge useful conversations with H. Schoeller, S. Jakobs, S. Kehrein, J. Kroha, H. Monien, A. Schiller, A. Dirks, J. Freericks, K. Ueda, T. Fujii, K. Thygesen, and F.B. Anders. We thank H. Schmidt for pointing out typos in an earlier version of our paper.

13 This work was supported by the DFG through the collaborative research center SFB 602. Computer support was provided through the Gesellschaft f¨ ur wissenschaftliche Datenverarbeitung in G¨ ottingen and the Norddeutsche Verbund f¨ ur Hoch- und H¨ ochstleistungsrechnen.

APPENDIX A: DERIVATION OF THE FLOW EQUATIONS

Although the following derivation is mainly equivalent to the one presented in Ref. 24, the appearance of the factor i in the real-time formulation of the generating functional leads to some changes in signs and prefactors i in the equations. It thus appears to be helpful for the reader to repeat the derivation. As a ?rst step we di?erentiate W c,Λ with respect to Λ, which after straightforward algebra leads to d ?ΛG ? 0,Λ + iζ Tr W c,Λ = ζ Tr Q dΛ

2 c,Λ ?Λ δ W Q δη ?δη

+

δ W c,Λ ? Λ δ W c,Λ ,Q δη δη ?

.

(A1)

? as the fundamental variables we obtain from Eq. (16) Considering φ and φ d Λ ? d ? d ηΛ Γ {φ}, {φ} = ? W c,Λ {η ?Λ }, {η Λ } ? φ, dΛ dΛ dΛ Applying the chain rule and using Eq. (A1) this leads to d Λ ?ΛG ? 0,Λ ? iζ Tr Γ = ?ζ Tr Q dΛ

2 c,Λ ?Λ δ W Q Λ δη ? δη Λ

?

d Λ ? Q ?Λφ η ? , φ + φ, dΛ

.

,

where the last term in Eq. (16) cancels a corresponding contribution arising in (A1) thus a posterior justifying the inclusion of this term. Extending the well known relation44 between the second functional derivatives of Γ and W c to nonequilibrium we obtain the functional di?erential equation d Λ ,1 Λ ? 0,Λ ?ΛG ? 0,Λ ? Tr Q ?ΛV 1 Γ = ?ζ Tr Q ) ? (Γ , G φ,φ dΛ

1,1 where Vφ,φ stands for the upper left block of the matrix ?

,

(A2)

Λ and the upper index t denotes the transposed matrix. To obtain di?erential equations for the γm which include Λ 0,Λ ? ? self-energy corrections we express Vφ,φ in terms of G instead of G . This is achieved by de?ning ?

? Λ ? 0,Λ Vφ,φ )=? ? (Γ , G ?

?

2 Λ Γ ? 0,Λ iδ ?ζ G ? δ φδφ

?1

Γ iδ ? φ ? δ φδ Γ ? iδ ? + δφδ φ

2 Λ

2

Λ

Γ iζ δ δφδφ

2

Λ

? 0,Λ G

?1

t

? ?1 ? ? ?

(A3)

δ 2 ΓΛ Λ Uφ,φ ? =i ? ? γ1 δ φδφ and using ?Λ = G which leads to d Λ ,1 Λ ?Λ ?ΛQ ?ΛV ?1 ?ΛG ? 0,Λ + ζ Tr G Γ = ?ζ Tr Q ? (Γ , G ) , φ,φ dΛ with ?φ,φ V Γ ,G ?

Λ ?1

? 0,Λ G

?1

?1 Λ ? ζγ1

,

(A4)

(A5)

?Λ

= 1?

?Λ ζG 0

0 ?Λ G

t

Γ t (iζ ) δ ? δφδφ ζ Uφ,φ

Uφ,φ ?

2

Λ

Γ iδ ? φ ? δ φδ

2

Λ

.

(A6)

14

δ Γ Γ It is important to note that Uφ,φ as well as δ ? ? φ ? and δφδφ are at least quadratic in the external sources. The initial δ φδ condition for the exact functional di?erential equation (A5) can either be obtained by lengthy but straightforward ? 0,Λ0 = 0 (no algebra, which we are not going to present here or by the following simple argument: at Λ = Λ0 , G Λ0 degrees of freedom are “turned on”) and in a perturbative expansion of the γm the only term which does not vanish is the bare two-particle vertex. We thus ?nd

2 Λ 2 Λ

?}, {φ} . ?}, {φ} = Sint {φ ΓΛ0 {φ ?φ,φ Expanding V Eq. (A6) in a geometric series ?

2 3 2 ?Λ δ Γ ?φ,φ ? Λ Uφ,φ ? Λ Uφ,φ ? Λ Uφ,φ ?Λ V = 1 + ζG ? ? +G ? G ? +ζ i G ? φ ? G δ φδ t

(A7)

δ2Γ + ... δφδφ

(A8)

and ΓΛ in a Taylor series with respect to the external sources ?}, {φ} = ΓΛ { φ (iζ )m (m!)2 m=0

∞ Λ ′ ′ ?ξ′ . . . φ ?ξ′ φξ . . . φξ . γm (ξ1 , . . . , ξm ; ξ1 , . . . , ξm ) φ 1 m m 1

′ ,...,ξ ′ ξ ,...,ξ ξ1 m m 1

(A9)

Λ one can derive an exact in?nite hierarchy of ?ow equations for the γm . For example, the ?ow equation for the singleparticle vertex γ1 (the self-energy) can be obtained by taking the expansion (A8) up to ?rst order in Uφ,φ ? , inserting it into (A5), replacing ΓΛ on both sides by the expansion (A9) and comparing the terms quadratic in the external ?elds. This procedure leads to the expression (20).

APPENDIX B: FLOW EQUATION FOR THE TWO-PARTICLE VERTEX

? Λ we have expanded Eq. (A6) in a geometric series up to ?rst order and ΓΛ (A9) To obtain the ?ow equation for Σ Λ in the external sources up to m = 2. To ?nd the ?ow equation for the vertex function, γ2 , we have to proceed one Λ step further, that is expand (A6) up to the second order and Γ up to m = 3. After comparison of the terms with the same power in the ?elds we obtain an expression similar to the equilibrium one (see Eq. (15) of Ref. 39) d αβγδ,Λ ′ ′ γ (ξ1 , ξ2 ; ξ1 , ξ2 ) = dΛ

Λ ν?,Λ αβρν,Λ ′ ′ ′ ′ Gρη, ; ξ3 , ξ4 )γ η?γδ,Λ (ξ3 , ξ4 ; ξ1 , ξ2 ) (ξ1 , ξ2 ξ ′ ,ξ3 Sξ4 ,ξ ′ γ

′ ξ ,ξ ′ ?,ν,ρ,η ξ3 ,ξ3 4 4 3 4

Λ ν?,Λ α?γη,Λ ′ ′ ′ ′ ? Gηρ, (ξ1 , ξ4 ; ξ1 , ξ3 )γ ρβνδ,Λ (ξ3 , ξ2 ; ξ4 , ξ2 ) ξ3 ,ξ ′ Sξ4 ,ξ ′ γ

3 4

′ ′ ′ ′ + γ αργν,Λ (ξ1 , ξ3 ; ξ1 , ξ4 )γ ?βηδ,Λ (ξ4 , ξ2 ; ξ3 , ξ2 ) β?γη,Λ ′ ′ ρανδ,Λ ′ ′ ? γ (ξ2 , ξ4 ; ξ1 , ξ3 )γ (ξ3 , ξ1 ; ξ4 , ξ2 ) ′ ′ ′ ′ ? γ βργν,Λ (ξ2 , ξ3 ; ξ1 , ξ4 )γ ?αηδ,Λ (ξ4 , ξ1 ; ξ3 , ξ2 )

,

(B1)

Λ where the contribution involving γ3 was replaced by its Λ0 Λ initial value, i.e. γ3 = γ3 = 0. The main di?erence to equilibrium is the presence of mixed contractions for ? Λ and S ?Λ and the the Keldysh indices of the matrices G Λ four-index tensors γ2 . Up to this point the derivation of the ?ow equation for the two-particle vertex was as general as the discussion in Sec. II. We now consider the application to the steady state current through a quantum dot described by the SIAM presented in Sec. III. We thus go over to frequency space and introduce the cuto? Λ as in Eq. (40). The single-particle quantum number in ξ is given by the spin index. We can take advantage of spin and energy ? conservation which implies that the lower indices of G

′ ′ ? are equal, that is ξ3 = ξ3 and S and ξ4 = ξ4 . This does not hold for the Keldysh indices. The integrals over ω3′ ′ ′ and σ4 can then be and ω4′ as well as the sums over σ3 performed trivially. Using the Morris lemma,21 we can ? and S ? as rewrite the matrix product of G

? Λ (ω )S ?Λ (ω ′ ) → 1 δ (|ω ′ | ? Λ)G ? Λ (ω )G ? Λ (ω ′ ) , G d d 2 with ? Λ (ω ) = G d 1 ? d,0 (ω ) G

?1

? Λ (ω ) ?Σ

.

The δ -function can be used to perform another of the

15 frequency integrals, and the remaining one can be evaluated because of energy conservation of the two-particle vertex. Using the spin independence of the Green function for zero magnetic ?eld, this leads to

1 d αβγδ,Λ γ ′ ′ = dΛ σ1 ,σ2 ;σ1 ,σ2 4π

ω =±Λ σ3 ,σ4 ?,ν,ρ,η

Λ Λ αβρν,Λ η?γδ,Λ Gρη, (?ω )Gν?, (ω ) γσ ′ ,σ ′ ;σ ,σ γσ3 ,σ4 ;σ1 ,σ2 d d 3 4

1 2

?βηδ,Λ αργν,Λ ρβνδ,Λ Λ Λ α?γη,Λ ?Gηρ, (ω )Gν?, (ω ) γσ ′ ,σ ;σ ,σ γσ ,σ ′ ;σ ,σ + γσ ′ ,σ ;σ ,σ γσ ,σ ′ ;σ ,σ d d 4 3 2 3 1 4 3 4 2 4 1 3

1 2 1 2

β?γη,Λ ρανδ,Λ βργν,Λ ?αηδ,Λ ?γσ ′ ,σ ;σ ,σ γσ ,σ ′ ;σ ,σ ? γσ ′ ,σ ;σ ,σ γσ ,σ ′ ;σ ,σ 4 1 3 3 4 2 3 1 4 4 3 2

2 1 2 1

(B2) .

Comparing (B2) with Eq. (20) in Ref. 39, were similar approximations were made in equilibrium, we see that we have Λ Λ two more terms because of the Keldysh indices. In the ?rst term appears Gρη, , while its transpose Gηρ, enters d d everywhere else. Λ Using the antisymmetry of γ2 in the spin indices and spin conservation we can further simplify Eq. (B2) by introducing the ?owing interaction U αβγδ,Λ de?ned as

αβγδ,Λ αβγδ,Λ βαγδ,Λ ′ ,σ δσ ′ ,σ U γσ ? δσ2 . ′ ,σ ′ ;σ ,σ = δσ ′ ,σ1 δσ ′ ,σ2 U 1 2 1 2 1 1 2

1 2

This leads to the ?ow equation d αβγδ,Λ 1 U = dΛ 4π

Λ Λ Gρη, (?ω )Gν?, (ω ) U αβρν,Λ U η?γδ,Λ + U βαρν,Λ U ?ηγδ,Λ d d Λ Λ ?Gηρ, (ω )Gν?, (ω ) 2U α?γη,Λ U ρβνδ,Λ ? U α?γη,Λ U βρνδ,Λ ? U ?αγη,Λ U ρβνδ,Λ d d

ω =±Λ ?,νρ,η

+2U αργν,Λ U ?βηδ,Λ ? U αργν,Λ U β?ηδ,Λ ? U ραγν,Λ U ?βηδ,Λ ?U ?βγη,Λ U αρνδ,Λ ? U ρβγν,Λ U α?ηδ,Λ . (B3)

Λ From the initial value of γ2 at Λ = Λ0 → ∞ given in Eq. (38) we can read o? as the initial value for U αβγδ,Λ0

while all other components are zero.

U αααα,Λ0 = αiU ,

1 2

3 4 5 6

7

8 9

10 11 12

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- Non-Perturbative Renormalization Group Analysis in Quantum Mechanics
- The Quantum-mechanical Many Body Problem The Bose Gas
- Renormalization group transformations on quantum states
- Quantum-mechanical tunnelling and the renormalization group
- The many body problem in relativistic quantum mechanics
- Density matrix renormalization group approach for many-body open quantum systems
- Density functional approach to the many-body problem_key concepts and exact functionals
- New Algebraic Quantum Many-body Problems
- The Kondo Effect in Non-Equilibrium Quantum Dots Perturbative Renormalization Group
- Spatial reflection and renormalization group flow of quantum many-body systems with matrix

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