Formalism of Nonequilibrium Perturbation Theory and Kondo Effect             

  
  
Formalism of Nonequilibrium Perturbation Theory
and Kondo Effect   
  
  
  
  
  

Mami Hamasaki

Abstract

The formalism of nonequilibrium perturbation theory was constructed by Schwinger and Keldysh and then was developed with the diagrammatical technique by Lifshitz and Pitaevskii. Until now there has been widespread application to various researches in physics, condensed matter, plasmas, atoms and molecules, nuclear matter etc.. In spite of this, the formalism has not been established as perturbation theory. For example, there is no perturbative method to derive arbitrary self-energy properly. In addition, the connection with other formalism, e.g., the Matsubara imaginary-time perturbative formalism is uncertain. Although there must be the relationship between self-energies in the perturbative formalism, such basic problems remain to be solved. The solution is given by the present work. The real-time perturbative expansion is performed on the basis of the adiabatic theorem. As the results, the requirements of self-energies as functions in time are demonstrated and the formulated self-energies meet the known relations. Besides, it gives exact agreement with functions derived by perturbative expansion in imaginary-time and analytical continuity. As a consequence, it implies that the present formalism can be generalized.

Next, using the formulated self-energies, the behavior of the Kondo resonance is investigated for nonequilibrium states caused by bias voltage. As numerical results, the Kondo peak disappears when voltage exceeds the Kondo temperatures; it is supported by experiments for two terminal systems. Over ten years, it has been being waited in expectation that the Kondo peak splits owing to bias voltage as a candidate for two channel Kondo effect. Nevertheless, it has not been observed in two terminal systems by experiments. Here, it is discussed why the Kondo peak splitting may not arise in normal two terminal systems.

Chapter 1 Introduction

1.1 Nonequilibrium Perturbative Formalism
( Schwinger-Keldysh Formalism )

The basic idea on the nonequilibrium perturbation theory was proposed by Schwinger in 1961.[1] That included the essentials for the nonequilibrium perturbation theory. The idea of time reversal is the basis of the theory. The real time-contour has the positive and reverse time directions, so that it starts and ends at by way of . In addition, the matrix form in the nonequilibrium Green’s functions is written after the real time-contour. After that, there are two main developments in the nonequilibrium perturbation theory. One is the expansion of the formalism by means of the equation of motion in the Green’s function by Kadanoff and Baym[2]; there their time-contour includes imaginary-time path. Another is that the formalism was extended as the frame of the nonequilibrium perturbation theory by Keldysh in 1965.[3] The formalism is constructed using density matrix and S-matrix after the time-contour and the Dyson’s equation in the matrix form of the nonequilibrium Green’s functions on the basis of the idea of Schwinger. The Dyson’s equation for the Keldysh formalism is given by

(1.1)

where

Then the Keldysh formalism was studied in further detail and generalized as the formalism of the nonequilibrium Green’s functions and the perturbative method with the help of the diagrammatic technique by Lifshitz and Pitaevskii.[4]

The main progress in the formalism of the nonequilibrium Green’s functions and the nonequilibrium perturbation theory is summarized with the aid of chart as follows:

Basic idea on the nonequilibrium perturbation theory ( the real time-contour and the matrix form ) was proposed by Schwinger in 1961    

                                            

Formalism was constructed after the time-contour and the matrix form by Keldysh in 1965    

Formalism was expanded using the equation of motion by Kadanoff and Baym (1961-62)  

 

                                            

\@getcirc\@circlefnt

\@ovvert32\@ovvert01\@ovhorz\@ovhorz to be continued!

Formalism was developed with the diagrammatical technique by Lifshitz and Pitaevskii    

the present work    

\@getcirc\@circlefnt

\@ovvert32\@ovvert01\@ovhorz\@ovhorz to be continued!

The connections between these ways have also been investigated and those have been confirmed in accordance together.

Recently, the formalism of the nonequilibrium Green’s functions and the nonequilibrium perturbative method have been applied widely to the various fields: condensed matter, plasmas, atoms and molecules, nuclear matter etc..[5] Especially, the application to the problems of mesoscopic systems: the transports in quantum dots and quantum wire, has worked with great success[6-8]; for instance, electrical current and current noise are expressed in terms of the nonequilibrium Green’s functions. For current,

(1.2)

where denotes the hopping matrix element. This reduces to the Landauer formula:[9]

(1.3)

where and are the Fermi distribution functions in the isolated left and right leads, respectively. For the current noise at zero-frequency from the autocorrelation function of the current[10]:

(1.4)

It reduces to the Khlus-Lesovik formula for one channel at zero-frequency for shot noise:[11]

(1.5)

Here, the transmission probability through a noninteracting system in Eqs. ( 1.3 ) and ( 1.5 ) is written by

(1.6)

and mean the coupling functions with the left and the right leads, respectively, and and are retarded and advanced Green’s functions, severally, as described in detail later. The expression for current noise also reduces to the Johnson-Nyquist noise for thermal noise: ( here, signifies conductance, and is the Boltzmann’s constant and denotes temperature ).[12] These are excellently compatible with the values observed by experiments.

Nonetheless, the formalism of the nonequilibrium perturbation theory has not been completed yet; the perturbative methods with the diagrammatic technique remain to be clarified well. In particular, despite of that the definition of the nonequilibrium Green’s functions is given in time, the real-time perturbative expansion on the adiabatic theorem[13]the basics of method of the perturbative expansionis almost unknown. The general formalism to formulate arbitrary self-energy has still not been established and the connection with other formalism, for instance, the Matsubara imaginary-time perturbative formalism[14] is obscure. For this reason, the present work succeeds to the Keldysh formalism and the diagrammatical method of Lifshitz and Pitaevskii to make progress in the formalism of the real-time perturbative expansion based on the adiabatic theorem. If this matrix form equation Eq. ( 1.1 ) works exactly as the Dyson’s equation, this equation must be convertible into the Dyson’s equations for retarded and advanced Green’s functions. Accordingly, there must be the relations between self-energies. However, since the definition of the self-energies for this Dyson’s equation in matrix form is not given, the self-energies drawn from perturbative expansion have not been made clear.

In the present work, thus, the solution on the relations between self-energies is given. Using the solution, the retarded and advanced self-energies are derived from the self-energies in matrix form. Then, the derived retarded and advanced self-energies meet the conditions required as functions in time and the generally known relations on nonequilibrium Green’s functions are fulfilled. Additionally, the formulated self-energies are in agreement with those derived by perturbative expansion in imaginary-time and analytical continuity. Thereby, the solution is in accordance with the generally known formation. As a result, it infers that the formalism of the nonequilibrium perturbation theory can be generalized.

1.2 Kondo Effect

The Kondo effect was discovered forty years ago[15]; the phenomenon of the minimum of the electrical resistivity in metals was explained in view of the interaction between conduction electrons and impurity by Kondo. After that, the Kondo physics was clarified from Landau’s Fermi liquid theory[16], the renormalization group[17] and scaling[18]. Besides, the generalized Kondo problem with more than one channel or one impurity was proposed.[19] It has then been investigated in further detail.[20-23] Especially, the resistivity has been expressed for the multichannel Kondo effect by the conformal field theoretical work,[20-22] in agreement with experiments, as mentioned in Chapter 4.

Moreover, the Kondo effect in electron transport through a quantum dot was predicted theoretically at the end of 1980s[24-26]. After a decade, finally, this phenomenon was observed.[27] The Kondo effect was studied theoretically by use of the Anderson model. From scaling theory[18], the Kondo temperatures

(1.7)

Here means the band-width and the Coulomb interaction , and is the coupling function with leads, corresponding to the density of states for conduction electron. The Kondo temperatures correspondent to the strength of the Kondo coupling decrease with increasing , as found from Eq. ( 1.7 ). The predictions from theoretical work using Anderson model were confirmed experimentally. In the Kondo regime, the conductance was observed to reach the unitarity limit and the Kondo temperatures estimated from observation[28] are in excellent agreement with the expression derived by the use of scaling theory for the asymmetric Anderson model[29]:

(1.8)

where on-site energy is . The perturbative approach, the Yamada-Yosida theory[30]the perturbation theory for equilibrium based on the Fermi liquid theory[16] with the Matsubara imaginary-time perturbative method[14] is quite successful.

Furthermore, the Kondo effect in a quantum dot was studied for nonequilibrium system where bias voltage is applied.[31] We have to know not only the Kondo effect but also the nonequilibrium state caused by bias voltage. The Yamada-Yosida theory was extended to nonequilibrium systems with the help of the Keldysh formalism and the Kondo effect in nonequilibrium system was studied. As the results, it was 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.[8] This results have been discussed little. After that, on experiments, it has been observed that the Kondo effect is suppressed when source-drain bias voltage is comparable to or exceeds the Kondo temperatures.[32,33] The numerical results of the present work are also consistent with those. For the Kondo effect in nonequilibrium systems, it has been expected that the Kondo peak splits by bias voltage and that the two separated energy levels made in a quantum dot by the Kondo coupling act as two channels for two channel Kondo effect. In order to search that, every efforts have been done for many years. Such the phenomenon has however, never been observed for a quantum dot connected with two normal leads. In the present paper, it is discussed the reason why the Kondo peak is just broken and the Kondo peak splitting may not take place in simple two terminal systems with leads of the continuous energy states.

Chapter 2 Nonequilibrium Perturbation Theory

A thermal average can be gained on the basis of the nonequilibrium perturbation theory.[1,3,4,13,34-37]

The perturbation theory is based on the adiabatic theorem ( called the Gell-Mann and Low’s theorem[13] ). The Hamiltonian is given by

(2.1)

and are unperturbed and perturbed terms, respectively. Here, assuming that we can know only the state of at , thereby, initially at , so that the system is equilibrium and/or noninteracting state. The perturbation is turned on at and introduced adiabatically. Then the perturbation is brought wholly into the system at ; around , the system is regarded as stationary nonequilibrium and/or interacting state. After that, the perturbation is taken away adiabatically and disappears at . When the time evolution of the state is reversible, the state at can be expressed using the state at by adding the phase factor.

Now, let us consider the nonequilibrium state. If the time evolution of the state is irreversible for the nonequilibrium state, then, the state at is not well-defined; when the perturbation is removed entirely at , the state does not come back to the same state as at . In this case, the ordinary perturbative method should be improved; the time evolution should return to the well-defined state at . Accordingly, the time evolution is performed along the real-time contour which starts and ends at as illustrated in Fig. 2.1. It is the extension of the Gell-Mann and Low’s theorem.[13]

Figure 2.1: time contour which starts and ends at through

2.1 S-matrix (S-operator)

The Hamiltonian is given by Eq. ( 2.1 ). The time evolution in the interaction representation is expressed in terms of S-matrix by

(2.2)

S-matrix is defined by

(2.3)

and has the following properties:

(2.4)
(2.5)
(2.6)
(2.7)

Equation ( 2.7 ) can be solved formally by

(2.8)
(2.9)

Here, the time ordering operator arranges in chronological order and is the anti time ordering operator which arranges in the reverse of chronological order. For the time evolution along the time contour as in Fig. 2.1, S-matrices, Eqs. (2.8) and (2.9) are required for paths on the branch from to and on the branch from to , respectively.

In the same way, S-matrix for imaginary-time perturbative formalism is defined by

(2.10)

and is also written by

(2.11)

These are the requisites for the Matsubara imaginary-time perturbative formalism.[14]

2.2 Matsubara Imaginary-Time Perturbative Formalism

For thermal equilibrium, the statistical operator ( density matrix ) is written in Gibbs form for the grand canonical ensemble by

(2.12)

By rearranging Eq. ( 2.10 ), we have

(2.13)

By substitution of Eq. ( 2.13 ) into Eq. ( 2.12 ), the thermal average for equilibrium is obtained by

(2.14)

By insertion of Eq. ( 2.11 ) in Eq. ( 2.14 ), the perturbative expansion is executed for the Matsubara imaginary-time perturbative formalism[14] using the Bloch and De Dominicis’s theorem[38-41]. The Matsubara Green’s function is defined by

(2.15)

The functions in terms of the Matsubara Green’s function in imaginary-time are converted into those in the Matsubara frequency by the Fourier transformation for the Matsubara Green’s function:

(2.16)

The Matsubara frequency, for fermion and for boson, ; thereby, the Matsubara Green’s function is periodic.

After that, the analytical continuity is performed. For fermion, when the Taylor expansion for the function around poles on imaginary axis is done, then, , hence, , and as approximation, . The residue theorem yields the conversion of sum of the functions in the Matsubara frequency into the contour integral by

(2.17)

It should be noted that the contour in integral surrounds the poles on imaginary axis. After that, for example, when ), then, the contour integral with contour enclosing , a pole on real axis is executed by

(2.18)

where is the Fermi distribution function.

In the same way, in boson case, for poles on imaginary axis, , i.e. , and approximately, , we have

(2.19)

Then, by changing the contour integral around poles on imaginary axis into the contour integral parallel with real axis ( e.g. ), we have the functions written in terms of retarded and advanced Green’s functions in energy. For high-order perturbation theory, the analytical continuation is so complicated. As the method of analytical continuation, Éliashberg’s method[42] is known.[4,41]

2.3 Nonequilibrium Perturbative Formalism

2.3.1 Nonequilibrium Real-Time Perturbative Formalism

For nonequilibrium, Equation ( 2.12 ) is not exact. We should note that there are no specific limitations upon the statistical operator. The von Neumann’s statistical operator is expressed independently of whether the states are at thermal equilibrium or nonequilibrium by[35,37]

(2.20)

in the Schrödinger representation. Here, is probability that the system is in state and is the state in the Schrödinger representation. It satisfies the Liouville equation by

(2.21)

The statistical operator in the interaction representation is given by

(2.22)

and also obeys the Liouville equation by

(2.23)

As a matter of course,

(2.24)

where is statistical operator in the Heisenberg representation. The time evolution is written by means of S-matrix by

(2.25)

These are the properties of the von Neumann’s statistical operator. Although the von Neumann’s statistical operator Eq. ( 2.20 ) is not an explicit expression, it is still considered to be the same type as Eq. ( 2.14 ) so as to execute the perturbative expansion.

The thermal average for nonequilibrium is drawn in view of the analogy with the imaginary-time perturbative method for the Matsubara Green’s function using S-matrix and the time evolution of the statistical operator, Eq.( 2.25 ). Thus the thermal average of the operators in the Heisenberg representation at can be brought, for example by[3,4,34,37]

where means . denotes an arbitrary operator in the interaction representation. Here, and . In this case, the operators and in Eq.( 2.26 ) both are on the branch from to of the time contour in Fig. 2.1.

Using the last expression in Eq.( 2.26 ), the real-time perturbative expansion is executed diagrammatically by the help of the Wick’s theorem. On the diagrammatical perturbative expansion, the diagrammatical methods written by Lifshitz and Pitaevskii.[4] are extended.

Figure 2.2: The extension of the diagrammatic technique of Lifshitz and Pitaevskii. brought as the sum over terms going from time to time by way of times and .
Figure 2.3: brought as the sum of terms passing by way of times, and .

There the summation over terms in all times is taken. For example, the third-order self-energy is brought as the sum over terms going from time to time by way of times and as illustrated in Fig. 2.2. See Fig.2.1 again. It should be noted that the times are on the branch from to of the time contour in Fig. 2.1 and the times are on the branch from to of the time contour. In addition, for the fourth-order self-energy, the sum of terms passing through times, and is taken, as shown in Fig. 2.3.

2.3.2 Schwinger-Keldysh Formalism

The Dyson’s equation for the Keldysh formalism is given by

(2.27)

where

The nonequilibrium Green’s functions are defined by

(2.28)
(2.29)
(2.30)
(2.31)

As shown in Eq. ( 2.26 ), the operators in Eq. ( 2.28 ) both are on the branch from to of the time contour in Fig. 2.1. In the same way, those in Eq. ( 2.29 ) both are on the branch from to of the time contour, and those in Eq. ( 2.30 ) are on the branch and on the branch, respectively. In Eq. ( 2.31 ), they are on the branch and on the branch, respectively.

Additionally, retarded and advanced Green’s functions are defined by

(2.32)
(2.33)

Here, the curly brackets signifies anticommutator. The Dyson’s equations for retarded and advanced Green’s functions are given by

(2.34)
(2.35)

As the necessity to Eqs. ( 2.34 ) and ( 2.35 ), the self-energies and must be retarded and advanced functions in time, respectively.

In accordance with the ordinary procedure of nonequilibrium perturbative formalism,[3,4, 34,35,37] for the Dyson’s equation for the Keldysh formalism, Eq. ( 2.27 ), the transformation is carried out: by use of

and

From the definition of Green’s functions,

in other words,

is called the Keldysh Green’s function. For the self-energies part, the following relationship is required:

(2.36)
(2.37)
(2.38)

The above is known in general. Here, it is uncertain whether or not the requirements of self-energies as functions in time are fulfilled. It is because the definition of the self-energies in matrix form is not given, as mentioned earlier.

The solution is given from the present work.[43] That is explained as follows: in the right-hand sides of Eqs. ( 2.36 ), ( 2.37 ) and ( 2.38 ), the functions, and are deduced from perturbative expansion with the Wick’s theorem. At this time, the directions in time are not defined in the functions and . This is due to the following functions in dependence upon time:

(2.39)
(2.40)

For this reason, for the right-hand sides of Eqs. ( 2.36 ), ( 2.37 ) and ( 2.38 ), the directions in time must necessarily be taken into consideration.

As the main point, the terms in the right-hand sides are taken for sum of retarded and advanced terms:

(2.41)
(2.42)
(2.43)

For self-energy functions derived by the present perturbative expansion via the Wick’s theorem, the following relations are found:

(2.44)
(2.45)
(2.46)
(2.47)

these relations have never been known in general. When these are substituted into Eqs. ( 2.41 ), ( 2.42 ) and ( 2.43 ), then, the advanced term of Eq. ( 2.41 ) and the retarded term of Eq. ( 2.42 ) are canceled:

(2.48)
(2.49)

so that Equations ( 2.41 ) and ( 2.42 ) reduce to

(2.50)
(2.51)

The retarded and advanced self-energies are acquired as retarded and advanced functions in time, respectively; they are the requirements. In addition, Equation ( 2.43 ) is certainly reproduced.

Then, it leads to

(2.52)
(2.53)

they consist with the expressions in the review by Rammer and Smith.[44] The formalism in the review of Rammer and Smith is different from the nonequilibrium perturbative expansion, so that the present method is confirmed to connect with the other formalism. By performing the Fourier transformation for Eqs. ( 2.52 ) and ( 2.53 ), we have

(2.54)

it proves that the relation stands. Since Equation ( 2.54 ) is widely known, Equations ( 2.52 ) and ( 2.53 ) should generally hold as functions in time.

Besides,

(2.55)
(2.56)

and

(2.57)

still work.

As mentioned above, the present solution is in accordance with the generally known relations. It indicates that the present solution has validity.

Chapter 3 Expressions of Self-Energy for Anderson model

3.1 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, the Hamiltonian has no time dependence. The system is described by the Anderson model linking to leads. The impurity ( the quantum dot ) with on-site energy and the Coulomb interaction is connected to the left and right leads by the mixing matrix elements, and . The system is illustrated below.

U=0
Left Lead

          U0 \@getcirc\@circlefnt\@ovvert32\@ovvert01\@ovhorz\@ovhorz Quantum Dot           U=0 Right Lead

The Anderson Hamiltonian is given by

(3.1)

() is creation (annihilation) operator for electron on the impurity, and and ( and ) are creation (annihilation) operators in the left and right leads, respectively. is index for spin. The chemical potentials in the isolated left and right leads are and respectively. The applied voltage is, therefore defined by .

We consider that the band-width of left and right leads is large infinitely, so that the coupling functions, and can be taken to be independent of energy, . On-site energy is set being canceled with the Hartree term, i.e. the first-order contribution to self-energy for electron correlation, as mentioned later.

Accordingly, the Fourier components of the noninteracting ( unperturbed ) Green’s functions reduce to

(3.2)
(3.3)

where . Hence, the inverse Fourier components can be written by

(3.4)
(3.5)

In addition, by solving the Dyson’s equation Eq. ( 2.27 ), we have

(3.6)
(3.7)

and are the Fermi distribution functions in the isolated left and right leads, respectively. By Eqs. ( 3.6 ) and ( 3.7 ), the nonequilibrium state is introduced effectively as the superposition of the left and right leads. In this case, the effective Fermi distribution function can be expressed by[8]

(3.8)

This effective Fermi distribution function is reasonable because it is considered that leads have the continuous energy states and the two chemical potentials are not two localized states.

3.2 Self-Energy

The expressions for self-energies of the Coulomb interaction of the Anderson model are formulated by the method of the nonequilibrium perturbation theory based on the adiabatic theorem, explained in Chapter 2. Practically, the perturbative expansion is done with respect to the Coulomb interaction term of Eq. ( 3.1 ) using the last expression of Eq. ( 2.26 ) in view of the Dyson’s equation Eq. ( 2.27 ) and each diagram. In and obtained by the present solution, Equations (2.44)-(2.47) are satisfied. Every formulated and are in the relation of the complex conjugate each other.

3.2.1 First-Order Contribution

For the Anderson model, the first-order contribution, the Hartree term can be written by

(3.9)

where is charge density.

3.2.2 Second-Order Contribution

The second-order self-energies are expressed by

(3.10)