Effects of Coulomb interaction on photon-assisted current noises through a quantum dot
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Effects of Coulomb interactionon photon-assisted current noises through a quantum dot
Takafumi J. Suzuki and Takeo Kato
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba, 277-8581, Japan (Dated: August 27, 2018)We study photon-assisted transport in a single-level quantum dot system under a periodicallyoscillating field. Photon-assisted current noises in the presence of the Coulomb interaction arecalculated based on a gauge-invariant formulation of time-dependent transport. We derive the ver-tex corrections within the self-consistent Hartree-Fock approximation in terms the Floquet-Green’sfunctions (Floquet-GFs), and examine the effects of the Coulomb interaction on the photon-assistedcurrent noises. Moreover, we introduce a concept of an effective temperature to characterize nonequi-librium properties under the influence of the AC field. The vertex corrections are suppressed by therise of the effective temperature, whereas characteristic resonant structures appear in the frequencyspectra of the vertex corrections. The present result provides a useful viewpoint for understandingphoton-assisted transport in interacting electron systems.
I. INTRODUCTION
Photon-assisted transport through mesoscopic conduc-tors has attracted much attention because the externalfields open up additional transport channels via pho-ton absorption and emission.
Nonperturbative effectsof the time-dependent fields significantly modify thequantum nature of transport processes, and these ef-fects have been studied from various viewpoints, such asthe coherent destruction of tunneling, the nonstation-ary Aharonov-Bohm effect, and so on. Photon-assistedtransport has also been utilized in various applicationsfor electronic devices, e.g., classical current sources, op-eration of solid state quantum bits, and on-demand gen-eration of quantum excitations. In recent years, much studies have revealed that cur-rent noises provide significant information about the mi-croscopic processes involved in photon-assisted trans-port.
Levitov and Lesovik pointed out that photon-assisted current noises can be used to detect the phase ofthe transmission amplitudes induced by the external ACfield. In subsequent theoretical work, the coherent andspectroscopic nature of the photon-assisted current noiseof noninteracting electrons has been studied based on thescattering approach and the Green’s function (GF) ap-proach. A detection scheme for finite-frequency currentnoises under an AC field using a resonant circuit has beenproposed theoretically. Photon-assisted current noiseshave also been measured in various systems such as dif-fusive metals, diffusive normal metal-superconductorjunctions, quantum point contacts, and tunnel junc-tions. Recently, the time-resolved current noises havebeen measured to evaluate the quantum purity of elec-trons emitted from on-demand electron sources.
Although the scattering theory has clarified theproperties of photon-assisted current noises of noninter-acting electrons, it is of limited use to describe the effectsof the Coulomb interaction. A quantum dot (QD) is atypical system in which the Coulomb interaction signif-icantly affects the transport properties. This raises the question of whether qualitative features of the photon-assisted current noises obtained in noninteracting elec-tron systems should or should not be altered in thepresence of the Coulomb interaction. One sophisticatedway to tackle this problem is a perturbative expansionwith respect to the Coulomb interaction. For a reliabledescription of nonequilibrium transport in the presenceof the Coulomb interaction, we need to carefully con-sider the charge conservation law, which is equivalent togauge invariance. Hershfield showed that the vertex cor-rections are essential for satisfying the gauge invarianceof current noises, and derived their explicit expressionsfor zero-frequency noises under stationary bias voltageswithin the self-consistent Hartree-Fock (SCHF) approxi-mation. Recently, Ding and Dong discussed the charge-conserving approximation for time-dependent trans-port quantities, and studied the finite-frequency currentnoises of the same system. We note that a similarcalculation using the charge-conserving approximationwas performed for a QD system with superconductingleads. However, the vertex corrections of the currentnoises through a QD system under time-dependent ex-ternal fields have not yet been studied.The purpose of this paper is to study effects ofthe Coulomb interaction on the photon-assisted cur-rent noises using the gauge-invariant approximationscheme. We consider a single-level QD system under aperiodically-oscillating external field, and derive the ex-plicit expressions of the vertex corrections of the photon-assisted current noises within the SCHF approximation.Using these expressions, we examine the features of thevertex corrections under the AC field.This paper is organized as follows: In Sec. II, we pro-vide our model and describe a gauge-invariant formula-tion of the time-dependent transport phenomena. Wepresent the expressions of the vertex functions withinthe SCHF approximation, and introduce Floquet-GFsto describe transport in the periodically driven system.In Sec. III, we introduce the generalized distribution,which captures characteristic features of the AC field.In Sec. IV, an effective temperature is introduced to un-derstand the properties of zero-frequency current noisesunder the AC field. In Sec. V, the frequency dependenceof photon-assisted current noises is discussed in detail.We show that the vertex corrections are sensitive to thedynamics induced by the AC field. The conclusions aregiven in Sec. VI.
II. FORMULATION
In this section, we describe our model and the gen-eral formulation for time-dependent transport of inter-acting electrons. According to this scheme, the gaugeinvariance of transport quantities is guaranteed by self-consistently determined vertex functions.
We showthe explicit expressions of the vertex functions within theSCHF approximation, and introduce Floquet-GFs. A. Model
We consider the single-impurity Anderson model witha time-dependent external field in order to describe thegeneral properties of photon-assisted transport througha QD system. The model Hamiltonian is given as H = X σ ( ǫ d + eA σ ( t )) ˆ d † σ ˆ d σ , + X α = L,R X k ,σ ( ǫ k + ev ασ ( t ))ˆ c † α k σ ˆ c α k σ , + X α = L,R X k ,σ (cid:16) t α e ieA ασ ( t ) ˆ d † σ ˆ c α k σ + h . c . (cid:17) , + U ˆ n ↑ ˆ n ↓ . (1)Here, ˆ d σ and ˆ c α k σ are the electron annihilation operatorsof a single-level QD and reservoirs α (= L, R ), respec-tively. The electron spin is denoted by σ (= ↑ , ↓ ), and themomentum of electrons in the reservoirs is denoted by k .The first term of the Hamiltonian describes an isolatedQD with an energy level, ǫ d , under a gate voltage, A σ ( t ),whereas the second term represents noninteracting elec-tron reservoirs with scalar potentials, v ασ ( t ). The thirdterm describes electron tunneling between the QD andthe leads, where hopping amplitudes and vector poten-tials are denoted by t α and A ασ ( t ), respectively. All theexternal fields, A σ ( t ), A ασ ( t ), and v ασ ( t ), are assumedto be classical variables. The Coulomb interaction in theQD is included in the last term. In this paper, we set thespeed of light, c , and the Dirac constant, ~ , to unity, anduse the electron charge, e (= − | e | < ǫ dσ = − U/
2) by an AC field; A σ ( t ) = ǫ sin(Ω t ) and A Lσ ( t ) = A Rσ ( t ) = 0 [see Fig. 1(a)]. Thebias voltage is assumed to be stationary and applied sym-metrically; v Lσ ( t ) = − v Rσ ( t ) = V /
2. The chemical po- (a) (b)
FIG. 1. (a) Schematic picture of the system. The energy levelof the QD is sinusoidally modulated by the AC field. (b) TheKeldysh contour, C . The upper (lower) branch is denoted by C − (+) . tentials of the leads are taken to be zero. We are notconcerned with spin-dependent transport in this paper.We study nonequilibrium electron transport by imple-menting the Keldysh path integral formalism. Allthe quantities are defined on the Keldysh contour, C ,which is composed of the forward-time path, C − , andthe backward-time path, C + [see Fig. 1(b)]. The argu-ment on the Keldysh contour is denoted by the Greekalphabet τ . We use the external fields defined on theKeldysh contour, A σ ( τ ), A ασ ( τ ), and v ασ ( τ ), which areallowed to have different values on the different branches C − and C + . The partition function, Z [ A ], is written as Z [ A ] = Z D [ ¯ dd ] e i ( S + S U ) , (2) S ≡ Z dτ dτ ′ X σ ¯ d σ ( τ ) G − dσ ( τ, τ ′ ) d σ ( τ ′ ) , (3) S U ≡ − Z dτ U n d ↑ ( τ ) n d ↓ ( τ ) , (4)where d σ and ¯ d σ are the Grassmann fields of electronsin the QD. The partition function, Z [ A ], is a functionalof the external fields, A σ ( τ ), A ασ ( τ ), and v ασ ( τ ), whichare symbolically expressed as the argument, A . G dσ isthe unperturbed GF of the dot electron, which includesthe effect of hybridization between the QD and the leads: G − dσ ( τ, τ ′ ) ≡ g − dσ ( τ, τ ′ ) − Σ σ ( τ, τ ′ ) , (5)Σ σ ( τ, τ ′ ) = X α = L,R Σ ασ ( τ, τ ′ ) , (6)Σ ασ ( τ, τ ′ ) = | t α | e ie ( A ασ ( τ ) − A ασ ( τ ′ )) X k g α k σ ( τ, τ ′ ) . (7)Here, g dσ ( τ, τ ′ ) and g α k σ ( τ, τ ′ ) are the GFs of elec-trons for an isolated dot without a Coulomb interaction( U = 0) and those of isolated leads, respectively. Non-perturbative effects of the external fields are included inthese unperturbed GFs.The doubled degrees of freedom of the external gaugefields, A ± α ) σ ( t ), can describe both the time-evolutionand the statistical correlation. To preserve the normal-ization and the causality structure of the partition func-tion, the gauge fields, A − α ) σ ( t ) and A +0( α ) σ ( t ), must beequated to one another in the last step of calculations; A − α ) σ ( t ) = A +0( α ) σ ( t ) = A α ) σ ( t ). Note that the physi-cal external fields, A α ) σ ( t ), are left as finite quantities sothat nonperturbative effects of the time-dependent fieldcan be discussed. B. Formulation of time-dependent transport
A systematic formulation of transport under time-dependent fields can be developed using two-particle-irreducible (2PI) formalism.
The 2PI effective actionfor the present system is written asΓ[ G ; A ] = − i Sp (cid:2) ln (cid:0) G dσ | A =0 G − dσ (cid:1)(cid:3) − i Sp (cid:2) G dσ G − dσ − (cid:3) + Γ [ G ; A ] , (8)where G is an abbreviation of the nonequilibrium GF G dσ ( τ, τ ′ ). The various components of the GF on the realtime axis are summarized in Appendix A. The symbolSp denotes the summation of all the internal indices: theintegration on the Keldysh contour and the summationof the spin. The functional Γ is the sum of the 2PIvacuum diagrams with the internal lines set to G , andcorresponds to the Luttinger-Ward functional. The Dyson equation can be derived by differentiatingEq. (8) with respect to the propagator and setting thesource fields to zero: G − dσ ( τ, τ ′ ) = G − dσ ( τ, τ ′ ) − Σ Uσ ( τ, τ ′ ) , (9)where the self-energy is defined asΣ Uσ ( τ, τ ′ ) ≡ − i δ Γ [ G ; A ] δG dσ ( τ ′ , τ ) . (10)Current noises can be concisely written in terms of thecurrent vertex functionsΓ σα ′ σ ′ ( τ , τ ; τ ′ ) ≡ δ σσ ′ Γ α ′ σ ′ ( τ , τ ; τ ′ ) + Γ Uσα ′ σ ′ ( τ , τ ; τ ′ ) , (11) where the bare and dressed parts of the current vertexfunction are defined as e Γ α ′ σ ′ ( τ , τ ; τ ′ ) ≡ − δG − dσ ′ ( τ , τ ) δA α ′ σ ′ ( τ ′ ) , (12) e Γ Uσα ′ σ ′ ( τ , τ ; τ ′ ) ≡ δ Σ Uσ ( τ , τ ) δA α ′ σ ′ ( τ ′ ) , (13)respectively. The bare vertex function is obtained asΓ α ′ σ ′ ( τ , τ ; τ ′ ) = − i [ δ ( τ ′ , τ ) − δ ( τ , τ ′ )] × Σ α ′ σ ′ ( τ , τ ) , (14)whereas the dressed vertex function is related to the self-energy by the relation δG dσ ( τ , τ ) δA α ′ σ ′ ( τ ′ ) = e Z dτ dτ G dσ ( τ , τ )Γ σα ′ σ ′ ( τ , τ ; τ ′ ) × G dσ ( τ , τ ) . (15)The gauge invariance of the transport quantities is guar-anteed by the vertex functions self-consistently deter-mined using these relations. The current-current correlation function on theKeldysh contour is defined by the functional derivativeof the 2PI effective action as D ασα ′ σ ′ ( τ, τ ′ ) ≡ − i δ Γ[ G ; A ] δA ασ ( τ ) δA α ′ σ ′ ( τ ′ ) (cid:12)(cid:12)(cid:12)(cid:12) A s = K =0 = e [ h j ασ ( τ ) j α ′ σ ′ ( τ ′ ) i−h j ασ ( τ ) ih j α ′ σ ′ ( τ ′ ) i ] . (16)The current-current correlation function is written as thesum of the two terms: D ασα ′ σ ′ ( τ, τ ′ ) = D αα ′ σ ( τ, τ ′ ) δ σσ ′ + D Uασα ′ σ ′ ( τ, τ ′ ) , (17)where D αα ′ σ ( τ, τ ′ ) ≡ − ie δ αα ′ Z dτ [ G dσ ( τ ′ , τ )Γ ασ ( τ , τ ′ ; τ ) − Γ ασ ( τ ′ , τ ; τ ) G dσ ( τ , τ ′ )]+ e Z dτ dτ dτ dτ G dσ ( τ , τ )Γ α ′ σ ( τ , τ ; τ ′ ) G dσ ( τ , τ )Γ ασ ( τ , τ ; τ ) , (18) D Uασα ′ σ ′ ( τ, τ ′ ) ≡ e Z dτ dτ dτ dτ G dσ ( τ , τ )Γ Uσα ′ σ ′ ( τ , τ ; τ ′ ) G dσ ( τ , τ )Γ ασ ( τ , τ ; τ ) . (19)The former is called the bare part of the current-currentcorrelation and the latter is its vertex correction. These are the formal expressions of the current noises, whichhave been obtained by Ding and Dong using the fullcounting statistics.The symmetrized current noises are defined as S ασα ′ σ ′ ( t, t ′ ) ≡ S αα ′ σ ( t, t ′ ) δ σσ ′ + S Uασα ′ σ ′ ( t, t ′ ) , (20)where S αα ′ σ ( t, t ′ ) ≡ D − +0 αα ′ σ ( t, t ′ ) + D + − αα ′ σ ( t, t ′ ) , (21) S Uασα ′ σ ′ ( t, t ′ ) ≡ D − + Uασα ′ σ ′ ( t, t ′ ) + D + − Uασα ′ σ ′ ( t, t ′ ) . (22)The vertex corrections, which cannot be expressed solelyin terms of the one-body quantity, are essential for satis-fying the gauge invariance of the current noises. C. Vertex functions within the SCHFapproximation
In this paper, we use the SCHF approximation.
The self-energy is given by the Hartree term asΣ Uσ ( τ , τ ) = − iU G d ¯ σ ( τ , τ ) δ ( τ , τ ) , (23)where δ ( τ, τ ′ ) is the Dirac delta function on the Keldyshcontour. The dressed vertex functions can be written asΓ Uσα ′ σ ′ ( τ , τ ; τ ′ ) ≡ ˜Γ Uσα ′ σ ′ ( τ , τ ′ ) δ ( τ , τ ) , (24)because the incoming and outgoing fermion lines meet atthe same vertex in the SCHF approximation. We definethe bare parts of the density-density, density-current, andcurrent-density correlation functions as χ nnσ ( τ, τ ′ ) ≡ − iG dσ ( τ, τ ′ ) G dσ ( τ ′ , τ ) , (25) χ nασ ( τ, τ ′ ) ≡ i Z dτ dτ G dσ ( τ, τ ) × Γ ασ ( τ , τ ; τ ′ ) G dσ ( τ , τ ) , (26) χ αnσ ( τ, τ ′ ) ≡ χ nασ ( τ ′ , τ ) . (27)Furthermore, we define the polarization function, M σ ( τ, τ ′ ), which satisfies the integral equation as (cid:0)(cid:2) − U χ nn ¯ σ χ nnσ (cid:3) M σ (cid:1) ( τ, τ ′ ) = δ ( τ, τ ′ ) , (28)where ( AB ) ( τ, τ ′ ) ≡ R dτ A ( τ, τ ) B ( τ , τ ′ ). Using thesefunctions, the dressed vertex functions are written as˜Γ U ¯ σασ ( τ, τ ′ ) = − U ( M σ χ nασ ) ( τ, τ ′ ) , (29)˜Γ Uσασ ( τ, τ ′ ) = − U ( M σ χ nn ¯ σ χ nασ ) ( τ, τ ′ ) . (30) D. Floquet-GFs
We define the Fourier transformation of the GFs withrespect to the relative time, t r ≡ t − t ′ , as G νν ′ dσ ( ω, T ) ≡ Z ∞−∞ dt r e iωt r G νν ′ dσ ( T + t r / , T − t r / , (31) (a) (b) FIG. 2. Diagrammatic representation of (a) Floquet-GF and(b) Hartree term. The Floquet indices are allocated at eachendpoint of the GFs. where ν and ν ′ are the Keldysh indices and T ≡ ( t + t ′ ) / t → t + T p , where T p ≡ π/ Ω is theperiod of the AC field. Then, the Wigner representationof the GF is defined as (cid:16) G νν ′ dσ (cid:17) n ( ω ) ≡ T p Z T p / − T p / dT e in Ω T G νν ′ dσ ( ω, T ) . (32)The Wigner representation is suitable for gaining phys-ical insights because of its clear interpretation. In par-ticular, the zeroth mode of the Wigner representation( G νν ′ dσ ) ( ω ) corresponds to the GF averaged over one pe-riod of the AC field.The Fourier indices can be efficiently handled if the GFis transformed into the Floquet representation as (cid:16) G νν ′ dσ (cid:17) mn ( ω ) ≡ (cid:16) G νν ′ dσ (cid:17) m − n (cid:18) ω + m + n (cid:19) , (33)where the frequency ω is folded into the first time-Brillouin zone, i.e., − Ω / ≤ ω < Ω /
2. In this paper,we use bold letters for matrices in the Floquet represen-tation. The Wigner and Floquet representations of otherquantities (self-energy, current noises, etc.) can be de-fined in the same way.Feynman rules in a driven system are analogous tothose in a time-translationally invariant system, ex-cept for additional Floquet indices. The Floquet-GF( G νν ′ dσ ) mn ( ω ) is expressed by a propagator with Floquetindices n and m at the initial and terminal points, re-spectively [Fig. 2(a)]. The internal Floquet indices aresummed up at all the vertices of the diagram. For in-stance, the Hartree term [Fig. 2(b)] includes the summa-tion of the additional index n .Various equations can be expressed in a simple matrixform using the Floquet representation. In particular, theexact equations of the retarded and lesser Floquet-GF ofthe QD can be derived from Eq. (9) as G rdσ ( ω ) = G r dσ ( ω ) + G r dσ ( ω ) Σ rUσ ( ω ) G rdσ ( ω ) , (34) G − + dσ ( ω ) = G rdσ ( ω ) (cid:0) Σ − +0 σ ( ω ) + Σ − + Uσ ( ω ) (cid:1) G adσ ( ω ) . (35)Eqs. (34) and (35) are called the Dyson and Keldyshequation, respectively.The relations of the Floquet-GFs are inherited fromthe GFs defined on the real-time axis. The advancedFloquet-GF is determined by the relation G adσ ( ω ) =( G rdσ ) † ( ω ). The lesser and greater Floquet-GFs havethe relations G − + dσ ( ω ) = − (cid:0) G − + dσ (cid:1) † ( ω ) and G + − dσ ( ω ) = − (cid:0) G + − dσ (cid:1) † ( ω ), respectively. III. GENERALIZED DISTRIBUTIONFUNCTION AND SPECTRAL FUNCTION
In this section, we define the generalized distributionfunction to understand the characteristic properties ofdriven systems. In addition, we calculate the spectralfunction within the SCHF approximation. Throughoutthis paper, we assume ∆ L = ∆ R = 1, ǫ d = − U/ β = 20, and Ω = 5in all numerical calculations. A. Generalized distribution function
The tunneling self-energy (7) describes nonequilibriumelectron tunneling between the leads and the QD. Thetunneling self-energy is calculated as Σ r ασ ( ω ) = − i ∆ α , (36) Σ − +0 ασ ( ω ) = i ∆ α J † ˜ f α ( ω ) J , (37)in the wide band limit. The line width is defined as∆ α ≡ π | t α | ρ , where ρ is the DOS of a conduc-tion electron at the Fermi energy. The unitary ma-trix, J , is defined using the Bessel function, J m ( ǫ / Ω),as ( J ) mn ≡ i m − n J m − n ( ǫ / Ω), and is a unit matrix.The diagonal matrix, ˜ f α , is defined as ( ˜ f α ) mn ( ω ) ≡ f ( ω + m Ω − ev ασ + ǫ d ) δ mn , where f ( ω ) = ( e βω + 1) − .The lesser component of the tunneling self-energy canbe rewritten as Σ − +0 ασ ( ω ) = i ∆ α f α ( ω ) , (38) f α ( ω ) ≡ J † ˜ f α ( ω ) J , (39)by introducing the generalized nonequilibrium distribu-tion function, f α ( ω ). We note that f α ( ω ) coincides with˜ f α ( ω ) in the absence of the AC field ( ǫ = 0) because J becomes a unit matrix.In this paper, the effect of the AC field is fully in-cluded in the generalized distribution function, f α ( ω ),while the unperturbed dot GFs are not modified. Fea-tures of f α ( ω ) can be clearly seen in the Wigner represen-tation. The zeroth mode of the generalized distributionfunction, ( f α ) ( ω ), is written as the weighted sum of theFermi distribution function:( f α ) ( ω ) = X J m (cid:16) ǫ Ω (cid:17) f ( ω + m Ω − ev α + ǫ d ) . (40)Figure 3(a) displays ( f L ) ( ω ) at ǫ d = − U/ − V = 0 for ǫ = 0, 4, and 8. The width of each step is equal to the external driving frequency, Ω, and itsheight at ω = n Ω ( n ∈ Z ) is the square of the Besselfunction, (cid:12)(cid:12) J n ( ǫ Ω ) (cid:12)(cid:12) . We note that the multi-step struc-ture of the generalized distribution function originatesfrom the coherent nature of electrons under the AC field.The first mode of the generalized distribution function, i ( f L ) ( ω )(= − i ( f L ) − ( ω )), is shown in Fig. 3(b) for ǫ = 4 and 8, whose contribution is not small in compar-ison with the zeroth mode.The Floquet-GFs for U = 0, which are hereafter calledthe unperturbed Floquet-GFs, are calculated as G r dσ ( ω ) = 1 ω + m Ω + i ∆ / , G − +0 dσ ( ω ) = i ∆ G r dσ ( ω ) f ( ω ) G a dσ ( ω ) , (41)with ∆ ≡ ∆ L + ∆ R and f ( ω ) ≡ ∆ L f L ( ω ) + ∆ R f R ( ω )∆ L + ∆ R . (42)By utilizing the Floquet-GFs, the lesser component iswritten in a pseudo-equilibrium form even with an ACfield. B. Spectral function
The self-energy within the SCHF approximation isgiven by the Hartree diagrams [Fig. 2(b)];( Σ rUσ ) mn ( ω ) = − iU X n Z Ω2 − Ω2 dω π ( G − + d ¯ σ ) n + m − n,n ( ω ) , (43)( Σ − + Uσ ) mn ( ω ) = 0 . (44)We note that the integrand of the retarded self-energy isthe full Floquet-GF. Using these expressions, the Dysonequation (34) and the Keldysh equation (35) are simpli-fied to G rdσ ( ω ) = [ − G r dσ ( ω ) Σ Uσ ( ω )] − G r dσ ( ω ) , (45) G − + dσ ( ω ) = i ∆ G rdσ ( ω ) f ( ω ) G adσ ( ω ) , (46)respectively. The Floquet-GFs are determined by solvingEqs. (43)-(46) self-consistently.The Wigner representation of the spectral function isdefined as ( A σ ) n ( ω ) ≡ − π Im ( G rdσ ) n ( ω ) , (47)where ( G rdσ ) n is the Wigner representation of the re-tarded GF. Figure 4 shows ( A σ ) ( ω ), ( A σ ) ( ω ), and( A σ ) − ( ω ) for ǫ d = − U/ − . V = 0, and ǫ = 8.The zeroth mode of the spectral function, ( A σ ) ( ω ), hasa Lorentzian spectral function of the QD with a levelbroadening ∆. The different oscillating modes of thespectral function ( A σ ) ± ( ω ) have a peak (dip) around (a) (b) FIG. 3. (Color online) (a) The zeroth mode of the generalized distribution function, ( f L ) ( ω ), for ǫ = 0, 4, and 8. (b) Thefirst mode of the generalized distribution function, i ( f L ) ( ω ). The amplitude of the AC field is taken to be ǫ = 4 and 8.Parameters are as follows: ∆ L = ∆ R = 1, ǫ d = − U/ − β = 20, V = 0, and Ω = 5. -0.100.10.20.30.40.50.60.70.80.91-10 -8 -6 -4 -2 0 2 4 6 8 10 FIG. 4. The three different modes of the spectral function:( A σ ) ( ω ), ( A σ ) ( ω ), and ( A σ ) − ( ω ). Parameters are as fol-lows: ∆ L = ∆ R = 1, ǫ d = − U/ − . β = 20, V = 0, ǫ = 8, and Ω = 5. ω = ± Ω / not necessarilyhold true in nonequilibrium systems, while a spectral mo- mentum sum rule, Z dω ( A σ ) n ( ω ) = δ n , (48)can be confirmed by our result. IV. PHOTON-ASSISTED CURRENT NOISE ATZERO FREQUENCY
In this section, we study the zero-frequency currentnoise under the AC field. We introduce an effective tem-perature to characterize the effects of photon absorptionand emission processes on the current noises.The photon-assisted current noises in the Floquet rep-resentation are given by the sum of the bare part and thevertex correction as S ασα ′ σ ′ ( ω ) = S αα ′ σ ( ω ) δ σσ ′ + S Uασα ′ σ ′ ( ω ) . (49)The explicit expression of the bare part can be obtainedby straightforward calculation from Eq. (18) as S αα ′ σ ( ω ) /e = δ αα ′ (cid:2)(cid:0) G − + dσ ◦ Σ + − ασ (cid:1) ( ω ) + (cid:0) Σ − +0 ασ ◦ G + − dσ (cid:1) ( ω ) (cid:0) G + − dσ ◦ Σ − +0 ασ (cid:1) ( ω ) + (cid:0) Σ + − ασ ◦ G − + dσ (cid:1) ( ω ) (cid:3) − (cid:2)(cid:0) G rdσ Σ − +0 α ′ σ ◦ Σ + − α ′ σ G adσ (cid:1) ( ω ) + (cid:0) G rdσ Σ + − α ′ σ ◦ Σ − +0 α ′ σ G adσ (cid:1) ( ω )+ (cid:0) Σ + − ασ G adσ ◦ G rdσ Σ − +0 ασ (cid:1) ( ω ) + (cid:0) Σ − +0 ασ G adσ ◦ G rdσ Σ + − ασ (cid:1) ( ω ) − α ∆ α ′ (cid:0) G rdσ f α ′ − f α G adσ + G − + dσ ◦ G rdσ f α − f α ′ G adσ + G − + dσ (cid:1) ( ω ) (cid:3) , (50)where the circle denotes the convolution in the Floquet representation,( A ◦ B ) mn ( ω ) ≡ X m ,n Z Ω2 − Ω2 dω π ( A ) m ,n ( ω )( B ) − n + n − N , − m + m − N ( − ω + ω + N Ω) . (51)The integer N is chosen so that the argument of the function B is reduced into the first time-Brillouin zone, FIG. 5. (Color online) Zero-frequency photon-assisted currentnoise for ǫ = 0, 2, 4, 6, 8, and 12. Parameters are as follows:∆ L = ∆ R = 1, ǫ d = − U/ − β = 20, and Ω = 5. i.e., − Ω2 ≤ − ω + ω + N Ω < Ω2 .In the SCHF approximation, the vertex corrections tothe photon-assisted current noises for the parallel andanti-parallel spins are calculated as S Uασα ′ σ ( ω ) /e = iU (cid:2) χ r αnσ ( ω ) M rσ ( ω ) χ r nn ¯ σ ( ω ) χ K nα ′ σ ( ω )+ χ r αnσ ( ω ) M rσ ( ω ) χ K nn ¯ σ ( ω ) χ a nα ′ σ ( ω )+ χ r αnσ ( ω ) M Kσ ( ω ) χ a nn ¯ σ ( ω ) χ a nα ′ σ ( ω )+ χ K αnσ ( ω ) M aσ ( ω ) χ a nn ¯ σ ( ω ) χ a nα ′ σ ( ω ) (cid:3) , (52)and S Uα ¯ σα ′ σ ( ω ) /e = iU (cid:2) χ r αn ¯ σ ( ω ) M rσ ( ω ) χ K nα ′ σ ( ω )+ χ r αn ¯ σ ( ω ) M Kσ ( ω ) χ a nα ′ σ ( ω )+ χ K αn ¯ σ ( ω ) M aσ ( ω ) χ a nα ′ σ ( ω ) (cid:3) , (53)respectively. The expressions of the vertex correctionterms in the absence of the AC field have already beenobtained for the zero-frequency noise and for the finite-frequency noise. The present results are straightforwardextensions of these previous works in consideration of thefinite AC field.Figure 5 shows the spin-averaged zero-frequency cur-rent noise, ¯ S LL ≡ (1 / P σσ ′ ( S LσLσ ′ ) (0), as a functionof the bias voltage, V , for ǫ d = − U/ −
1. We can seetwo significant features: (1) the singularities at the biasvoltage corresponding to the driving frequency Ω and(2) the remarkable enhancement of the zero-bias currentnoise by the AC field. The singularities of ¯ S LL at Ω re-flects the structures of the generalized distribution func-tion [see Fig. 3(a)]. This feature is consistent with theprevious work, which used the energy-independent scat-tering matrix. The enhancement of the zero-bias noiseby the AC field can be understood in terms of an effectivetemperature.Roughly speaking, the multi-step structure of the gen-eralized distribution function can be approximated by the Fermi distribution function with a modified temperature.In this paper, we define an effective temperature by anextrapolated fluctuation dissipation relation: T eff ≡ ¯ S LL k B ¯ G , (54)where k B is the Boltzmann constant. The linear con-ductance is defined as ¯ G ≡ lim V → ( ¯ I/V ) by the time-averaged current¯ I = e ∆ L ∆ R ∆ L + ∆ R X m Z Ω2 − Ω2 dω π ( A σ ( f L − f R )) mm ( ω ) . (55)We observe that the complete definition of the effectivetemperature has been open to discussion. In Fig. 6(a), the generalized distribution function,( f L ) , is compared with the Fermi distribution function, f eq L , with T = T eff for ǫ d = − U/ − V = 0. Theoverall broadening of the multi-step structures is well-approximated by the Fermi distribution with the corre-sponding effective temperatures for ǫ = 2 and 6, while f eq L no longer coincides with ( f L ) for ǫ = 8.In Fig. 6(b), we show S LL and the equilibrium ther-mal noise, S eq , with T = T eff as a function of ǫ for ǫ d = − U/ − V = 0. These quantities agreewell with each other for ǫ < Ω, indicating that the effec-tive temperature works well for the weak AC field. Forthe strong AC field, S LL oscillates as a function of ǫ due to the coherent nature of electrons under the strongAC field. This oscillation has already been obtained innoninteracting electron systems. The present resultindicates that the quantum oscillation in photon-assistedcurrent noises is robust against the Coulomb interaction.The oscillation of S eq can be explained by the dependenceof the effective temperature, T eff , on ǫ [see the inset ofFig. 6(b)]. T eff is significantly enhanced at ǫ / Ω = 2 . .
52, 8 .
65, and 11 .
7, which correspond to the zeros of thezero-order Bessel function, J ( ǫ / Ω). The difference inthe positions of the peaks and dips between S LL and S eq implies the limit of the present definition of the effectivetemperature. V. PHOTON-ASSISTED CURRENT NOISES ATFINITE FREQUENCIES
In this section, we study the frequency dependence ofvarious correlation functions, polarization functions, andphoton-assisted current noises employing the SCHF ap-proximation. The effects of the AC field on these quanti-ties are discussed from the point of view of the effectivetemperature and photon absorption and emission.
A. Correlation functions
The vertex corrections to the photon-assisted currentnoises are written in terms of the bare correlation func- (a) (b)
FIG. 6. (Color online) (a) Staircase-structured generalized distribution function, ( f L ) , compared with the smooth equilibriumFermi distribution function, f eq L , at T = T eff . The external field amplitudes are for ǫ = 2 , 6, and 8. (b) Comparison betweenthe photon-assisted thermal noise and the equilibrium thermal noise evaluated at the corresponding T eff . The dependence ofthe effective temperature, T eff , on the amplitude of the external AC field, ǫ , is plotted in the inset. Parameters are as follows:∆ L = ∆ R = 1, ǫ d = − U/ − β = 20, V = 0, and Ω = 5. -0.4-0.3-0.2-0.10-10 -8 -6 -4 -2 0 2 4 6 8 10 (a) (b) (c) (d) -0.3-0.2-0.100.10.2-10 -8 -6 -4 -2 0 2 4 6 8 10-0.3-0.2-0.100.10.2-10 -8 -6 -4 -2 0 2 4 6 8 10 -0.4-0.200.20.40.60.811.21.41.6-10 -8 -6 -4 -2 0 2 4 6 8 10-0.4-0.200.20.40.60.811.21.41.6-10 -8 -6 -4 -2 0 2 4 6 8 10 -0.100.10.20.30.40.50.60.70.80.9-10 -8 -6 -4 -2 0 2 4 6 8 10 FIG. 7. (Color online) The retarded and the Keldysh components of the charge-charge correlation function [(a) and (b)] andthe polarization function [(c) and (d)] for ǫ = 0, 4, and 8. Parameters are as follows: ∆ L = ∆ R = 1, ǫ d = − U/ − β = 20, V = 0, and Ω = 5. tions and the polarization function. We discuss theirproperties under the AC field in this subsection.In the Floquet representation, the retarded and lessercomponents of the charge-charge correlation function Eq. (25) are written as χ r nnσ ( ω ) = − i (cid:2) ( G rdσ ◦ G − + dσ )( ω ) + ( G − + dσ ◦ G adσ )( ω ) (cid:3) , (56) χ − +0 nnσ ( ω ) = − i ( G − + dσ ◦ G + − dσ )( ω ) , (57)respectively. We focus on the retarded and Keldysh com-ponents (see Appendix A) in the following discussions.In the parameter regime under consideration, dominantcontributions come from the zeroth mode of the cor-relation functions, which are denoted by ¯ χ r nnσ ( ω ) ≡ ( χ r nnσ ) ( ω ) and ¯ χ K nnσ ( ω ) ≡ (cid:0) χ K nnσ (cid:1) ( ω ).In Fig. 7(a), we show the retarded component of thecharge-charge correlation function, ¯ χ r nnσ ( ω ), for ǫ d = − U/ − V = 0. The real (imaginary) part of¯ χ r nnσ ( ω ) is an even (odd) function with respect to ω (see Appendix B for symmetry relations of various corre-lation functions), and has a dip (peak-and-dip) structurearound ω = 0. These structures are reduced by the ACfield due to the rise of the effective temperature discussedin Sec. IV.In Fig. 7(b), we show the Keldysh component of thecharge-charge correlation function, ¯ χ K nnσ ( ω ), for thesame parameters. The Keldysh component, ¯ χ K nnσ ( ω ),is purely imaginary and even with respect to ω . In theabsence of the AC field, the Keldysh component has twodip structures around ω = ± ∆ / χ K nn (0), reflecting the rise of the effectivetemperature. The overall Lorentzian dip structure is in-sensitive to the AC field.In the SCHF approximation, the dressed vertex func-tions include a RPA-type polarization function, which describes the dynamical screening effect. The retardedand lesser components of the polarization function areobtained from Eq. (28) as M rσ ( ω ) = (cid:2) − U χ r nn ¯ σ ( ω ) χ r nnσ ( ω ) (cid:3) − , (58) M − + σ ( ω ) = U M rσ ( ω ) (cid:2) χ r nn ¯ σ ( ω ) χ − +0 nnσ ( ω )+ χ − +0 nn ¯ σ ( ω ) χ a nnσ ( ω ) (cid:3) M aσ ( ω ) , (59)respectively. We denote the diagonal elements of the re-tarded and Keldysh polarization functions by ¯ M rσ ( ω ) ≡ ( M rσ ) ( ω ) and ¯ M Kσ ( ω ) ≡ (cid:0) M Kσ (cid:1) ( ω ), respectively.Figure 7(c) and figure 7(d) show ¯ M rσ ( ω ) and ¯ M Kσ ( ω ),respectively, for the same parameters. The polariza-tion functions have peak or dip structures around zero-frequency ( ω = 0), which come from those in the corre-sponding charge-charge correlation functions. The peakof the real part of the retarded polarization function at ω = 0 is suppressed by the AC field, because the dynami-cal screening effect is weakened by the rise of the effectivetemperature. The reduction of the Keldysh componentof the polarization function occurs due to the same rea-son. As the amplitude of the AC field increases, ¯ M rσ and¯ M Kσ approach 1 and 0, respectively.We need to calculate the bare parts of the charge-current and current-charge correlation functions [seeEqs. (26) and (27)] to evaluate vertex corrections to thecurrent noises. Their retarded and lesser components arewritten in the Floquet representation as χ r αnσ ( ω ) = i ∆ α [( G rdσ ◦ G rdσ f α )( ω ) − ( f α G adσ ◦ G adσ )( ω ) + i χ r nnσ ( ω )] , (60) χ − +0 αnσ ( ω ) = − i ∆ α [ G − + dσ ◦ G rdσ ( − f α ) ( ω ) + (cid:0) f α G adσ ◦ G + − dσ (cid:1) ( ω )] − ∆ α χ − +0 nnσ ( ω ) , (61) χ r nασ ( ω ) = i ∆ α [( G rdσ f α ◦ G adσ )( ω ) − ( G rdσ ◦ f α G adσ )( ω )] , (62) χ − +0 nασ ( ω ) = − (cid:0) χ − +0 αnσ (cid:1) † ( ω ) . (63)The retarded and Keldysh components of the zerothmode of the current-charge correlation function are de-noted by ¯ χ r αnσ ( ω ) ≡ ( χ r αnσ ) ( ω ) and ¯ χ K αnσ ( ω ) ≡ (cid:0) χ K αnσ (cid:1) ( ω ), respectively.In Fig. 8(a), we show the retarded component of thecurrent-charge correlation function, ¯ χ r Lnσ ( ω ), for ǫ d = − U/ −
1, and V = 0. The real (imaginary) partof ¯ χ r Lnσ ( ω ) is an even (odd) function with respect to ω . The absolute value of ¯ χ r Lnσ ( ω ) tends to be reducedby the AC field because of the rise of the effective tem-perature, whereas the peaks develop at ω = ± Ω due tophoton absorption and emission. In Fig. 8(b), we showthe Keldysh component, ¯ χ K Lnσ ( ω ), for the same param-eters. The real part of ¯ χ K Lnσ ( ω ) is an odd function withrespect to ω . The imaginary part vanishes at V = 0.The overall structure is insensitive to the AC field, whilea small structure develops at ± Ω. B. Vertex corrections
In this paper, we focus on the zeroth mode of thephoton-assisted current noises, ¯ S αα ′ σ ( ω ) ≡ ( S αα ′ σ ) ( ω )and ¯ S Uασα ′ σ ′ ( ω ) ≡ ( S Uασα ′ σ ′ ) ( ω ). First, we con-sider the unbiased system ( V = 0), in which the rela-tions ¯ S LLσ ( ω ) = ¯ S RRσ ( ω ) and ¯ S LRσ ( ω ) = ¯ S RLσ ( ω )hold. The parallel-spin current noise is written as thesum of the bare term and the vertex correction term,¯ S ασα ′ σ ( ω ) = ¯ S αα ′ σ ( ω ) + ¯ S Uασα ′ σ ( ω ). In contrast, theantiparallel-spin current noise is written only in terms ofthe vertex correction term, ¯ S α ¯ σα ′ σ ( ω ) = ¯ S Uα ¯ σα ′ σ ( ω ), be-cause electrons with different spins cannot correlate witheach other without the Coulomb interaction. This indi-cates that we can directly evaluate the vertex correction, S Uα ¯ σα ′ σ ( ω ), if the spin-dependent current noises can bemeasured using, for example, a spin filter.0 (a) -0.2-0.100.1-10 -8 -6 -4 -2 0 2 4 6 8 10-0.2-0.100.1-10 -8 -6 -4 -2 0 2 4 6 8 10 (b) -0.4-0.3-0.2-0.100.10.20.30.4-10 -8 -6 -4 -2 0 2 4 6 8 10 FIG. 8. (Color online) (a) The retarded and (b) Keldysh components of the current-charge correlation function for ǫ = 0, 4,and 8. Parameters are as follows: ∆ L = ∆ R = 1, ǫ d = − U/ − β = 20, V = 0, and Ω = 5. (a) (b)(c) (d) -0.6-0.5-0.4-0.3-0.2-0.100.10.2 0 2 4 6 8 10 12 14 1600.20.40.60.811.2 0 2 4 6 8 10 12 14 16-0.03-0.02-0.0100.010.02 0 2 4 6 8 10 12 14 16 -0.06-0.04-0.0200.020.040.06 0 2 4 6 8 10 12 14 16 FIG. 9. (Color online) The noise spectra for different amplitudes of the external field, ǫ = 0, 4, and 8. The auto-correlationfunction, ( S LL ) , and the cross-correlation function, ( S LR ) , are shown in (a) and (b), respectively. The vertex correctionsto the current noise are displayed in the case of (c) parallel spins ( S LσLσ ) and (d) anti-parallel spins ( S LσL ¯ σ ) . Parametersare as follows: ∆ L = ∆ R = 1, ǫ d = − U/ − β = 20, V = 0, and Ω = 5. The auto-correlation function of photon-assisted cur-rents, ¯ S LLσ ( ω ), is shown in Fig. 9(a) for ǫ d = − U/ − V = 0. In the absence of the AC field, the auto-correlation function increases monotonically and reachesa finite value as the frequency increases. This behav- ior is consistent with a previous study. The AC fieldinduces a step-like structure in the noise spectra, andeach step starts to rise at integral multiples of the driv-ing frequency, Ω = 5. These features indicate that thepresent model behaves the same way as the multi-level1 (a) (b) (c)
FIG. 10. (Color online) The noise spectra of the averaged current in the presence of a bias voltage ( V = 2) for differentamplitudes of the external field, ǫ = 0, 4, and 8. The bare part is shown in (a), while the vertex corrections for theparallel spins and the anti-parallel spins are shown in (b) and (c), respectively. Parameters are as follows: ∆ L = ∆ R = 1, ǫ d = − U/ − β = 20, V = 2, and Ω = 5. QD system without the AC field because of the ap-pearance of the Floquet sidebands. In contrast to thebehavior of the zero-frequency noise, ¯ S LLσ (0), the auto-correlation function at high frequencies is suppressed bythe AC field. In Fig. 9(b), we show the cross-correlationfunction, ¯ S LRσ ( ω ), for the same parameters. At zerofrequency, the cross-correlation function is related to theauto-correlation function as ¯ S LRσ (0) = − ¯ S LLσ (0), dueto charge conservation. In the absence of the AC field,¯ S LRσ ( ω ) is negative at low frequencies and positive athigh frequencies. The application of the AC field shifts¯ S LRσ ( ω ) in the negative direction and induces the dipstructures at the integral multiples of the driving fre-quency.In Fig. 9(c) and 9(d), we show the vertex corrections, S ULσLσ ′ ( ω ), for the parallel spins ( σ ′ = σ ) and the anti-parallel spins ( σ ′ = ¯ σ ), respectively. For ǫ = 4 and 8,the peaks appear at integral multiples of the frequency ofthe AC field, and the spectra of the vertex corrections arestrongly frequency-dependent due to the rich structureof the retarded current-charge correlation function [seeFig. 8(a)]. On the other hand, the vertex correctionsare suppressed on the whole by the application of theAC field, because the rise of the effective temperatureweakens the dynamical screening effect. These effects ofthe AC field on the vertex corrections are expected to begeneral in interacting electron systems.To see effects of the bias voltage on the current noise,we consider the noise spectra associated with the aver-aged current, ( I L ( t ) − I R ( t )) /
2. The bare and vertexcorrection parts of the averaged current noise are givenby S σ ( ω ) ≡ ( S LLσ ( ω ) + S RRσ ( ω ) − S LRσ ( ω ) − S RLσ ( ω )) / S U tot σσ ′ ( ω ) = ( S ULσLσ ′ ( ω ) + S URσRσ ′ ( ω ) − S ULσRσ ′ ( ω ) − S URσLσ ′ ( ω )) /
4, respectively.In Fig. 10(a), we show the zeroth mode of the barepart of the noise associated with the averaged current,¯ S σ ( ω ) ≡ ( S σ ) ( ω ), for ǫ d = − U/ − V = 2. The three curves correspond to different am-plitudes of the AC field; ǫ = 0, 4, and 8. The increase ofthe zero-frequency noise is due to the rise of the effectivetemperature. In the high-frequency regime, the depen- dence of the noise on ǫ is weak, due to the cancellationof the auto- and cross-correlation functions. Figure 10(b)and 10(c) show the vertex correction parts, ¯ S U tot σσ ( ω ) ≡ ( S U tot σσ ) ( ω ) and ¯ S U tot σ ¯ σ ( ω ) ≡ ( S U tot σ ¯ σ ) ( ω ), respec-tively, for the same parameters. Due to the assumedparticle-hole symmetry, the vertex corrections associatedwith the correlation of the averaged current completelyvanishes in the absence of the external AC field, whichis consistent a the previous study. However, the vertexcorrections become finite when both the AC field and thebias voltage are simultaneously applied. As seen in thefigures, ¯ S U tot σσ ( ω ) ( ¯ S U tot σ ¯ σ ( ω )) has a clear peak (dip)at the driving frequency, Ω = 5, which is caused by thecurrents resonantly driven by the external AC field. Thisresonant character cannot be explained by the concept ofthe effective temperature. VI. CONCLUSION
In this paper, we investigated current noises througha QD system under an AC field to study effects thatthe Coulomb interaction had on photon-assisted currentnoises. We employed the gauge-invariant formalism andthe Floquet-GF method, and have derived explicit ex-pressions for the vertex corrections of the photon-assistedcurrent noises within the SCHF approximation.We first focused on the zero-frequency photon-assistedcurrent noise in Sec. IV. The bias-voltage dependence ofthe zero-frequency photon-assisted current noises has twofeatures: singularities at the integral multiples of the fre-quency of the AC field and a remarkable enhancementof the zero-bias noise. These two features are the directconsequences of the multi-step structure of the general-ized distribution functions defined in Sec. III. An effectivetemperature was introduced as a good indicator of the en-hancement of the zero-bias noise under the AC field. Thequantum coherence of electrons induced by the AC fieldmanifests itself in the oscillatory behavior of the zero-frequency current noise as a function of the amplitudeof the AC field. These features are robust against the2Coulomb interaction, and are in line with previous the-oretical and experimental results on noninteracting elec-trons.We next studied the frequency dependence of thephoton-assisted current noises in Sec. V. The AC fieldaffects the vertex correction in two ways: (1) The vertexcorrection is reduced by the AC field because the screen-ing effect in the QD is suppressed by the rise of the effec-tive temperature, and (2) The frequency dependence ofthe vertex corrections shows resonant structures at theinteger multiples of the driving frequency due to photonabsorption and emission processes. We expect that theseare general features of interacting electron systems undera strong AC field.Our study will offer a useful viewpoint for understand-ing photon-assisted transport of other phenomena, suchas the Coulomb blockade and the Kondo effect.
We finally point out that the present results serve as astarting point of the full counting statistics of inter-acting electrons under time-dependent external fields.
ACKNOWLEDGMENTS
We are grateful for helpful discussions with R. Sakanoand T. Jonckheere. This work was supported by JSPSKAKENHI Grant Numbers 24540316 and 26220711.T.J.S. acknowledges financial support provided by theAdvanced Leading Graduate Course for Photon Science(ALPS).
Appendix A: Definitions of GFs on the real-time axis
The GF of the QD electron is projected onto the realtime axis by specifying the branches, and is denoted by G νν ′ dσ ( t, t ′ ) = G dσ ( τ, τ ′ ) for τ ∈ C ν and τ ′ ∈ C ν ′ . TheGFs G − + dσ ( t, t ′ ) and G + − dσ ( t, t ′ ) are called the lesser andgreater GFs, respectively. The retarded, advanced, andKeldysh component of the GF are defined as G rdσ ( t, t ′ ) ≡ G −− dσ ( t, t ′ ) − G − + dσ ( t, t ′ ) , (A1) G adσ ( t, t ′ ) ≡ ( G rdσ ( t ′ , t )) ∗ , (A2) G Kdσ ( t, t ′ ) ≡ G − + dσ ( t, t ′ ) + G + − dσ ( t, t ′ ) , (A3)respectively. All the GFs are determined by the retardedGF, G rdσ ( t, t ′ ), and the lesser GF, G − + dσ ( t, t ′ ). The samerelations hold for the other correlation functions. Appendix B: Symmetry Relations of the bare partsof correlation functions
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