Nonequilibrium Perturbative Formalism and Spectral Function for the Anderson Model

The present work is based on the nonequilibrium perturbative formalism. There the self-energies are derived up to the forth-order. In consequence, it proves that the nonequilibrium (real-time) perturbative expansion can be connected with the Matsubara imaginary-time perturbative expansion for equilibrium. As the numerical results, the Kondo resonance still disappears for bias voltage exceeding the Kondo temperatures, as observed in experiments of two terminal systems.


Nonequilibrium Perturbative Formalism
The basic idea on the nonequilibrium perturbation theory grounded on the time-contour which starts and ends at t = −∞ via t = ∞ has been proposed by Schwinger. [ 1] After that, the frame of the nonequilibrium perturbation theory has been built up using the nonequilibrium Green's functions given after the time-contour by Keldysh. [ 2] The perturbative equation is expressed in matrix form: where The nonequilibrium Green's functions are given in the Heisenberg representation by Here, the time ordering operator T arranges in chronological order andT is the anti time ordering operator which arranges in the reverse of chronological order. The angular brackets denote thermal average in nonequilibrium. The present work is undertaken on the basis of the nonequilibrium perturbative formalism, Eq. ( 1 ). The retarded and advanced self-energies up to the fourth-order are formulated. Then it is confirmed that the nonequilibrium perturbative expansion can be connected with the Matsubara imaginary-time perturbative expansion for equilibrium.

The Kondo effect
The Kondo effect [ 3] was discovered forty years ago and after that, the Kondo physics has been clarified from Landau's Fermi liquid theory [ 4], the renormalization group [ 5], scaling [ 6], etc.. Besides, generalized Kondo problem, that of more than one channel or one impurity has been investigated. [ 7,8] Then, the Kondo effect in electron transport through a quantum dot has been predicted theoretically at the end of 1980s [ 9] and after a decade, this phenomenon has been observed. [ 10] The Kondo effect has been studied theoretically using the Anderson model and the predictions have been confirmed experimentally. In the Kondo regime, the conductance has been observed to reach the unitarity limit and the Kondo temperatures estimated from observation [ 11] are in excellent agreement with the expression derived using the Anderson model. [ 12] Furthermore, the Kondo effect in a quantum dot has been studied for nonequilibrium system where the bias voltage is applied. [ 13] The Yamada-Yosida theory, [ 14] perturbation theory for equilibrium based on the Fermi liquid theory [ 4] has been extended to nonequilibrium system and it has been shown that for bias voltage higher than the Kondo temperatures, the Kondo resonance disappears in the spectral function with the second-order self-energy of the Anderson model. [ 15] This is in good agreement with the experiments of two terminal systems that it has been observed that the Kondo effect is suppressed when source-drain bias voltage is comparable to or exceeds the Kondo temperatures. [ 16,17] In the present work, using self-energies derived up to the fourth-order, the behavior of Kondo resonance is investigated for nonequilibrium state caused by bias voltage. The numerical results are still that the Kondo resonance is broken, as supported by two-terminal experiments.

Formalism
A thermal average can be obtained on the basis of the nonequilibrium perturbation theory. [ 1,2,18,19,20,21,22] It is assumed that we can know only the state at t = −∞, that is, initially at t = −∞, the system is equilibrium and/or noninteracting state. The perturbation is turned on at t = −∞ and introduced adiabatically and then, brought wholly into the system at t = 0; around t = 0, the system is regarded as stationary nonequilibrium and/or interacting state. After that, the perturbation is taken away adiabatically and vanishes at t = ∞. When the time evolution of the state is irreversible, then, the state at t = ∞ cannot be well-defined. The time evolution is therefore, performed along the real-time contour which starts and ends at t = −∞, as illustrated in Fig. 1.

S matrix is defined by
HereH I is perturbation term in interaction representation. For thermal equilibrium, the statistical operator ( density matrix ) is written in Gibbs form for the grand canonical ensemble by Equation ( 8 ) is not valid exactly for nonequilibrium. We have no specific limitations upon the statistical operator. The statistical operator can generally be expressed in Schrödinger representation by [ 20,22] Here, P m is probability that the system is in state m and |m S (t) > is the state in Schrödinger representation. ̺ S satisfies the Liouville equation by The statistical operator in the interaction representation is given by̺(t) = e iH0t/ ̺ S (t)e −iH0t/ and satisfies As a matter of course, ̺ S (0) = ̺(0) =̺(0). Here ̺(t) is in the Heisenberg representation. The time evolution is expressed using S matrix bỹ The thermal average in the Heisenberg representation at t = 0 can be obtained, for example by [ 19,22] where . . . av = Tr[̺(−∞). . .]. Then, the thermal average is derived by following the ordinary procedure via the Wick's theorem.

Relation of Self-Energy
After the perturbative expansion is executed, the retarded and advanced self-energies are formulated. According to the definition, the retarded and advanced Green's functions are given by Here, the curly brackets denote anticommutator. The Dyson's equations for the retarded and advanced Green's functions are given by As the necessity to Eqs. ( 15 ) and ( 16 ), the self-energies Σ r and Σ a are also required to be retarded and advanced functions in time, respectively. In accordance with the ordinary procedure of nonequilibrium perturbative formalism, [ 2,19] there and using this, then, The relationship for self-energies ought to be obtained here by comparison of Eq. ( 1 ) with Eqs. ( 15 ) and ( 16 ): 3. Expressions of Self-Energy for Anderson model

Anderson Model
We consider equilibrium and nonequilibrium stationary states. Nonequilibrium state is caused by finite bias voltage, that is, the difference of chemical potentials; after bias voltage was turned on, long time has passed enough to reach stationary states. Since the states are stationary, Hamiltonian has no time dependence. The system is described by the Anderson model connected to leads. The impurity with on-site energy E 0 and the Coulomb interaction U is connected to the left and right leads by the mixing matrix elements, v L and v R . The Anderson Hamiltonian is given by d † (d) is creation (annihilation) operator for electron on the impurity, andĉ † L andĉ † R (ĉ L andĉ R ) are creation (annihilation) operators in the left and right leads, respectively. σ is index for spin. The chemical potentials in the isolated left and right leads are µ L and µ R , respectively. The applied voltage is, therefore defined by eV ≡ µ L − µ R .
We consider that the band-width of left and right leads is large infinitely, so that the coupling functions, Γ L and Γ R can be taken to be independent of energy, E. On-site energy E 0 is set canceling with the Hartree term, i.e. the first-order contribution to self-energy for electron correlation: Σ Accordingly, the Fourier components of the noninteracting ( unperturbed ) Green's functions reduce to where Γ = (Γ L + Γ R )/2. Hence, the inverse Fourier components can be written by In addition, from Eq. ( 1 ), we have f L and f R are the Fermi distribution functions in the isolated left and right leads, respectively. By Eqs. ( 22 ) and ( 23 ), the nonequilibrium state is introduced as the superposition of the left and right leads. Then, the effective Fermi distribution function can be expressed by [ 15] f

Self-Energy
The retarded and advanced self-energies are derived up to the forth-order. Equations ( 17 ) and ( 18 ) are divided into retarded and advanced terms in time: Then it is found that for the self-energies drawn using the Wick's theorem, It leads to As a consequence, the retarded and advanced self-energies are obtained as retarded and advanced functions of time, respectively: In addition, it proves Hence the following relations still stand: Σ r − Σ a = Σ < − Σ > and furthermore As the results, the second-order self-energy is written by Here g ± (t) = g r (t) + g a (t), that is, g + (t) = g r (t) = −iθ(t)e −Γt for t≥0 and g − (t) = g a (t) = iθ(−t)e Γt for t < 0. Additionally, g < (t) and g > (t) are the inverse Fourier components of Eqs. ( 22 ) and ( 23 ). Figure 2 shows the diagram for the second-order self-energy. As shown numerically later, the second-order contribution coincide with those derived by Hershfield et al. [ 15]. In the symmetric equilibrium case, the asymptotic behavior at low energy is expressed by the exact results based on the Bethe ansatz method. [ 23,24]  The third-order terms corresponding to the diagram in Fig. 3(a) are expressed by Figure 3(b) illustrates the diagram for the following terms: Equations ( 28 )

Self-Energy
The third-order terms, Eqs. ( 28 )-( 31 ) cancel under electron-hole symmetry not only at equilibrium but also at nonequilibrium: Σ (3) pp (E). As a consequence, the third-order contribution to self-energy vanishes in the symmetric case. It agrees with the results of Refs. [14,24,26] based on the Yamada-Yosida theory that all odd-order contributions except the Hartree term vanish for equilibrium in the symmetric single-impurity Anderson model; probably, it is just the same with nonequilibrium state. On the other hand, the third-order terms contribute to the asymmetric system where electron-hole symmetry breaks and furthermore, the third-order terms for spin-up and for spin-down contribute respectively when the spin degeneracy is lifted for example, by magnetic field. For the fourth-order contribution, three terms which constitute each of four groups contribute equivalently under electron-hole symmetry. Moreover, to the asymmetric system, the terms brought by the diagrams of Figs. 4(a) and 4(b) contribute equivalently and the terms by the diagrams of Figs. 4(j) and 4(k) make equivalent contribution, and the rest, the eight terms contribute respectively. Further, the twenty-four terms for spin-up and spin-down take effect severally in the presence of magnetic field.
The second-order and the fourth-order contributions to self-energy for zero temperature symmetric Anderson model are shown in Figs. 5(a) and 5(b) and in Figs. 6(a) and 6(b), respectively. Equation ( 27 ) [ 15]. In comparison of Figs. 6(a) and 6(b) with Figs. 5(a) and 5(b), it is found that the fourth-order contribution for equilibrium has the same but narrow curves at low energy with those of the second-order contribution. In addition, the broad curves are attached at high energy for the fourth-order self-energy. ( The higher-order contribution is, the more the curves must oscillate as a function of energy. ) When the voltage, eV /Γ exceeds ∼2.0, the behavior of curves of self-energy changes distinctly and comes to present striking contrasts to that for the second-order contribution. Especially, the curve for the imaginary part of the fourth-order contribution rises up with maximum at E = 0. On the other hand, for the second-order contribution, a valley appears with minimum at energy of zero−it is quite the contrary. Moreover, from these results, it is expected that the sixth-order contribution to imaginary part of self-energy has minimum at E = 0. Because of these, the perturbative expansion is hard to converge for eV /Γ > ∼2.0, as mentioned later.
Besides, the current conservation is mentioned. In Ref. [15], it is shown that the continuity of current entering and leaving the impurity stands exactly at any strength of U within the approximation up to the second-order for the symmetric single-impurity Anderson model. In comparison of Figs. 6(a) and 6(b) with Figs. 5(a) and 5(b), it is found that curves of fourth-order self-energy have the symmetry similar to those of the second-order. From this, it is anticipated that the current conservation are satisfied perfectly with approximation up to the fourth-order in the single-impurity system where electron-hole symmetry holds. The continuity of current can be maintained perfectly in single-impurity system as far as electron-hole symmetry stands. On the other hand, current comes to fail to be conserved with increasing U in asymmetric single-impurity case and in two-impurity case.

Spectral Function
The spectral function with the second-order self-energy is generally known. It is plotted for U/Γ = 10.0 and zero temperature in Fig. 7. For equilibrium, the Kondo peak at energy of zero is very sharp and the two-side broad peaks appear at E≃ ±U/2. The curve for eV = 0 is identical with that shown in Ref. [24]. As eV becomes higher than the Kondo temperatures, k B T K [ 27], the Kondo peak becomes lower and finally vanishes, while the two-side broad peaks rise at E≃ ±U/2.    Figure 8 shows the spectral function with the self-energy up to the fourth-order for equilibrium and zero temperature. With strengthening U , two-side narrow peaks come to occur in the vicinity of E = ±U/2 in addition to the Kondo peak. At U large enough, the Kondo peak becomes very acute and two-side narrow peaks rise higher and sharpen; the energy levels for the atomic limit are produced distinctly. The fourth-order self-energy has the same but narrow curves as functions of energy with those of the second-order and those curves make the peaks at E = ±U/2.
For the present approximation up to the fourth-order, the Kondo peak at E = 0 reaches the unitarity limit and the charge, n corresponds to 1/2, that is, the Friedel sum rule is correctly satisfied: [ 28] ρ(E f ) = sin 2 (π n )/πΓ, where ρ(E f ) is the local density of states at the Fermi energy. Here, the discussions should be made on the ranges of U in which the present approximation up to the fourth-order stands. From the results, it is found that the approximation within the fourth-order holds up to U/Γ ∼5.0 and is beyond the validity for U/Γ> ∼6.0. In addition, the curve for imaginary part of the fourth-order contribution is positive partly, as shown in Fig. 6(b) and as a consequence, the curve of the spectral function becomes negative partly for too large U . In such a case, the present approximation is out of validity and the higher-order terms are required. Next, the results for nonequilibrium and zero temperature are shown. The expression for the Friedel sum rule, Eq. ( 33 ) does not stand for nonequilibrium, since the charge cannot be expressed with respect to the local density of states. All the same, the Kondo peak reaches the unitarity limit and n = 1/2 in the symmetric and noninteracting case. The spectral functions with the selfenergy up to the fourth-order are plotted for eV /Γ = 0.5 and eV /Γ = 1.0 in Figs. 9, respectively. When U is strengthened and eV exceeds k B T K ( approximately, k B T K /Γ ∼0.5 for U/Γ = 3.5 and k B T K /Γ ∼0.3 for U/Γ= 5.0 ), the Kondo peak for eV /Γ = 0.5 falls in and instead, the twoside narrow peaks remain to sharpen in the vicinity of E = ±U/2. For eV /Γ = 1.0, the Kondo peak becomes broad and disappears for U large enough. The two-side peaks is generated small in the vicinity of E = ±U/2. The Kondo resonance is quite broken for bias voltage exceeding the Kondo temperatures; this accords with the experimental results of two terminal systems that the Kondo effect is suppressed when source-drain bias voltage is comparable to or exceeds the Kondo temperatures, eV ≥k B T K . [ 16,17] For eV /Γ > ∼2.0, the Kondo peak does not lower even when eV is much larger than k B T K . The perturbative expansion is hard to converge on account of the imaginary part of the self-energy for eV /Γ > ∼2.0, as described before; thereby, the higher-order contribution to self-energy is probably needed for high voltage.
In the present work, nonequilibrium state is represented as the superposition of the two leads and the effective Fermi distribution function, Eq. ( 24 ) is qualitatively similar to that for finite temperatures. From the analogy in the Fermi distribution function, it is inferred that there are nonequilibrium fluctuations similar to thermal fluctuations. [ 29] Because of the effective Fermi distribution function, not only for the second-order but also for the fourth-order, the Kondo resonance is destroyed, qualitatively the same as for finite temperatures. In contrast, if the finite voltage state is expressed as two localized states, the numerical results of the Kondo peak splitting can be obtained. All the same, for two terminal systems, the Kondo resonance splitting may not take place for finite bias voltage.
Summary: The present work is based on the nonequilibrium perturbative formalism. Here the self-energies are derived and then it is indicated that the nonequilibrium ( real-time ) perturbative expansion can be related to the Matsubara imaginary-time perturbative expansion for equilibrium. As the numerical results, the Kondo peak disappears as bias voltage exceeding the Kondo temperatures. Because of the analogy of the effective Fermi distribution function for nonequilibrium with that for finite temperatures, the present result is qualitatively similar to that for finite temperatures. This characteristic appears in the experiments of two terminal systems.
Additionally, Figure 4(c) shows the diagram for the following terms: Next, the terms brought from diagram in Fig. 4(d) are expressed by The terms for diagram in Fig. 4(e) are written by In addition, Figure 4(f) denotes the diagram for the following terms: Next, the terms formulated from diagram illustrated in Fig. 4(g) are expressed by Figure 4(h) illustrates the diagram for the following terms: Σ a(4) Besides, the terms formulated from the diagram in Fig. 4(i) are written by Σ r(4) i Σ a(4) i × g ± (t 2 ) sgn(t 3 ) g > (t 2 + t 3 )g > (t 3 )g < (−t 3 ) −g < (t 2 + t 3 )g < (t 3 )g > (−t 3 ) .