Projection-Reduction Approach to Optical Conductivities for an Electron-Phonon System and Their Diagram Representation

Utilizing state-dependent projection operators and the Kang-Choi reduction identities, we derive the linear, first, and secondorder nonlinear optical conductivities for an electron system interacting with phonons. The lineshape functions included in the conductivity tensors satisfy “the population criterion” saying that the Fermi distribution functions for electrons and Planck distribution functions for phonons should be combined inmultiplicative forms.The results also contain energy denominator factors enforcing the energy conservation as well as interaction factors describing electron-phonon interaction properly. Therefore, the phonon absorption and emission processes as well as photon absorption and emission processes in all electron transition processes can be presented in an organized manner and the results can be represented in diagrams that can model the quantum dynamics of electrons in a solid.

A proper many-body theory should satisfy the so-called "many-body criteria, " one of the most difficult being "the population criterion, " meaning that the Fermi distribution functions for electrons and the Planck distribution functions for phonons should be combined in multiplicative forms because the electrons and phonons belong to different categories in quantum-statistical physics.The population factors satisfying the population criterion should be combined in a physically acceptable manner with the energy denominator factor to maintain energy conservation and with the interaction factor to interpret the occurrence of local fluctuations due to electron-phonon interactions in such a way that the transition can take place via implicit states.Theories based on the single-electron projection technique starting from the Kubo formalism have failed to meet this criterion.
The many-electron projection techniques are classified into two categories; the first being state-independent or current-dependent [10][11][12][13] and the second being statedependent [14][15][16][17].The conductivity tensors [10,11] produced using the isolation and projection operators, which are stateindependent projection techniques, have limited applications for the following reasons.(a) They do not satisfy the population criterion because the two distribution functions are simply added (or subtracted), such as   + 1/2 ∓ 1/2 ±   , where   and   are the Planck distribution function for a phonon with wave vector  and the Fermi distribution function for an electron with energy   , respectively.(b) They do not satisfy the "projection criterion, " so they are applicable only to cases where the spacing between the energy levels is constant, such as cyclotron resonance phenomena; that is, they are unsuitable for a system with nonuniform energy separation, such as a quantum square well (see the statements below (51)).
Since Franken et al. [18] pioneered the field of nonlinear optical phenomena by the experimental work on optical wave 2 ISRN Condensed Matter Physics mixing, it has attracted considerable research attention [19,20].Most studies focused on the electronic structures and the electromagnetic properties of media with increasing the available laser power [20][21][22][23][24][25][26][27][28][29][30][31][32][33], but there were some problems.Shen [22] did not have explicit forms of the damping terms (relaxation rates) appearing in the conductivity.Suzuki and Ashikawa [28] reported the explicit forms of the damping term but did not satisfy the population criterion.
The present authors introduced an identity relation (the Kang-Choi reduction identity (KCRI) of the first kind) to derive the lineshape formula for a cyclotron transition [12].The population factors appearing in this formula satisfy the population criterion (see [12, equation (27)]).This shows that correct use of the KCRI can satisfy the population criterion.Later, the identity was generalized to cover the higher-order nonlinear optical conductivities [13][14][15][16].The second kind (see [13, equation (4.18)]) was used along with current-dependent projectors, which is called the state-independent projectors, to derive the first-order nonlinear optical conductivity [13].This satisfies the population criterion but has limited applications because it does not satisfy the projection criterion.The corrected extended version of the nonlinear optical conductivity formalism, including the second-order nonlinear optical conductivity, was presented recently by applying the projection-reduction (PR) method by combining the generalized state-dependent projection operator (SDPO) and the KCRI of the 3rd kind (see [14, equation (2.5)]).The axes of the generalized projectors are the creation-annihilation operator pairs (see [14, equation (6.4)]), which makes the formalism applicable to general electron systems; that is, it satisfies the projection criterion.
The aim of this study is to generalize the KCRI to cover higher-order nonlinear optical conductivities and show how physically acceptable forms of the optical conductivities can be obtained by PR method.In addition, a method is introduced to represent the optical conductivity formulae using diagrams, through which a physical intuition to the quantum dynamics of the electrons in a solid can be achieved.

Hamiltonian and Density Operator
To derive the optical conductivities, the Hamiltonian is defined and the density operator is introduced.
2.1.Hamiltonian.We consider the total Hamiltonian,  tot (), for a system of electrons interacting with phonons as follows: tot () =  eq +  int () , where the time-independent part of the system in thermodynamic equilibrium,  eq , is given by Here,   ,   , and  are the electron, phonon, and electronphonon interaction Hamiltonians, respectively,  +  (  ) is the creation (annihilation) operator for an electron in the state |⟩ with energy   ,   ( +  ) is that for a phonon in the state |⟩ with energy ℎ  , |⟩ ≡ |q, ⟩, q is the phonon wave vector,  is the polarization index, and   () is the electron-phonon interaction matrix element defined as   () =   ⟨     exp (q ⋅ r)     ⟩ , where   is the coupling factor that depends on the mode of phonons and r is the electron position vector.
In (1), when a linearly polarized time-dependent electric field with angular frequency , E() = E[exp() + c.c], is applied, the following time-dependent interaction Hamiltonian is considered: where   ,   , and   denote the , , and  components, respectively, of the electron position vector, ()  ≡ ⟨||⟩ for an arbitrary operator , and c.c. means the complex conjugate.

Density Operator.
To determine the general-order nonlinear optical conductivity, we assume that when the timedependent electric field is applied to the system, the total density operator can be split into two parts as follows: where  eq ≡ ( eq ) is the density operator for a system in thermal equilibrium and  int () is the perturbed term by the time-dependent external field.Using the Liouville equation we can obtain Here, we have used the fact that ℎ( eq /) = [ eq ,  eq ] = 0 and  tot () is the Liouville operator corresponding to the total Hamiltonian defined as  tot () ≡  tot () −  tot () for an arbitrary operator, . tot () can be split into two parts for calculation convenience:  tot () =  eq +  int () where  eq and  int (), respectively, correspond to  eq and  int ().

Kang-Choi Reduction Identity
The essential part of this paper is expansion of the optical conductivities (see ( 16), ( 21) and ( 26)) to easily solvable forms.For this purpose, this section introduces various forms of a useful identity, known as the Kang-Choi reduction identity (KCRI).

KCRI of First Kind (KCRI-I).
For arbitrary operators,  and  1 , we consider Considering [ eq ,  eq ] = 0 and the trace property,   () =   (), for the first and fourth terms, the KCRI-I of rank 1 [(KCRI-I) 1 ] can be obtained as follows: which is useful because  and  are separated.Similarly, for arbitrary operators, ,  1 , and  2 , Considering [ eq ,  eq ] = 0 and   () =   () for the first and eighth terms and adding   { eq (− 1  eq  2 +  eq  2  1 which is null, to the right-hand side of (29), the KCRI-I of rank 2 [(KCRI-I) 2 ] can be obtained as follows: Therefore, the KCRI-I of rank  [(KCRI-I)  ] can be inferred as follows: where  = 1, 2, 3, . ... The present form given by ( 32) is the general case.The proof for  = 3 is as follows.The left-hand side (lhs) of ( 32) for  = 3 can be expanded as follows: which is the same as the right-hand side (rhs) of (32) for  = 3.

KCRI of Second Kind (KCRI-II).
The KCRI of the second kind (KCRI-II) is given by the following: which will be used for derivation of the first-order nonlinear optical conductivity formula.Equation ( 34) can be proven by applying the identity on (ℎ −  eq ) −1 ; that is, the lhs of (34) for  = 1, 2, 3, . ... It suffices to prove the case of ( eq ) =  eq , because ( eq ) can be expanded in a Taylor series.The case for  = 3 is dealt with for illustration.Using (KCRI-I) 2 , Applying (KCRI-I) 3 , the rhs of (40) becomes ISRN Condensed Matter Physics Therefore, applying (KCRI-I) 2 in a reverse manner results in rhs of (40) = rhs of (39) for  = 3,  ( eq ) =  eq (42) completing the proof for  = 3.The cases for ( eq ) =  2 eq =  eq ⋅  eq and ( eq ) =  3 eq =  2 eq ⋅  eq can be proven by an iterative operation of (40).Using this lemma, the calculations needed to prove the KCRI for  ≥ 3 are reduced.

KCRI of Third Kind (KCRI-III).
The KCRI of the 3rd kind (KCRI-III) can be expressed as The proof can be obtained as follows.From (37), the lhs of (43) becomes where the KC expansion relation (see (39)) was applied to the last step.Therefore, applying (37) in a reverse manner results in the right-hand side of (43).KCRI-III (see ( 43)) is needed in deriving the second-order nonlinear optical conductivity formula.

State-Dependent Projection Operators
In this section, we calculate the linear, first-, and second-order optical conductivities using the PR method.

Linear Optical Conductivity.
Using the many-electron current density operator   , which is written in terms of the single-electron current operator,   = (ℎ/)/  ( = , , ), as the linear optical conductivity (16) becomes where To calculate  1 (), the first state-dependent projection operators for an arbitrary operator  are introduced as follows: where Applying the identity, 1 =  1 +  1 , to the right-hand side of the Liouville operator in (47) as  eq =  eq ( 1 +  1 ) and using (35) give which can be rearranged as ISRN Condensed Matter Physics 7 Here, we have used  1  +    = 0 and    +    =    +    where   ≡   −   ,  eq =   +  V , and   and  V are the Liouville operators corresponding to   and , respectively.
Note that the projection operators given in (48) are statedependent; that is,  1 projects an arbitrary operator  onto the operator  +    , which depends on the states  and .On the other hand, state-independent projection operators project  into the state-independent current density operator,   .The state-independent projection operator method is applicable only to the case in which     = -number ×   or  1     = 0. Cyclotron phenomenon belongs to this category because for the current operator,  ± =   ±   ,    + = ℎ   + , where   is the cyclotron frequency.This is called the "projection criterion." Equation (48) satisfies this criterion because  1    +    = 0. Taking the ensemble average of (51) gives Here which can be rearranged as where we have used (35) and (KCRI-I) 1 in the second term.To calculate (54) further, we make use of the following relation in the second term of (54): where the first and second terms are canceled out.In this stage, ( eq ) ≈ (  ) is assumed and the terms up to the second order of  V are considered, assuming that the electron-phonon interaction is quite weak.Then, the first term in (54) and the fourth term in (55) are neglected because the ensemble averages of   and  +  are zeros.Therefore, where Equation ( 56) is an easily calculable form because it contains neither  V nor the projection operator in the denominator.

First-Order Nonlinear Optical Conductivity.
By inserting (45) into ( 21), the first-order nonlinear optical conductivity becomes where which can be calculated by defining the second statedependent projection operators for an arbitrary operator, , as follows: where Applying the identity, 1 =  2 +  2 , to the right-hand side of the Liouville operator in (59) as (ℎ 12 −  eq ) −1 = (ℎ 12 −  eq  2 − eq  2 ) −1 and using the identity (35) give the following after a similar procedure as that used in (52): Here where (65) 5.2.1.Calculation of  1 ( 1 ).Applying (35) to (64) with  ≡ ℎ 1 and  ≡  eq results in in the second-order approximation in  V , where we have used the (KCRI-I) where Considering that we obtain, in the second-order approximation in  V , where we have used (37) in the first term of (69), Therefore, inserting (67) and (71) into (63) results in  26), the second-order nonlinear optical conductivity becomes where To calculate (77), the third state-dependent projection operators for an arbitrary operator, , are defined as follows: where Applying the identity, 1 =  3 +  3 , to the right-hand side of the Liouville operator in (77) as (ℎ 123 −  eq ) −1 = (ℎ 123 −  eq  3 −  eq  3 ) −1 and using (35) result in the following after a similar procedure as that used in (52) and (62): Here where ISRN Condensed Matter Physics 9 . Considering (35) with  ≡ ℎ 12 and  ≡  eq , ( 82) can be expanded as follows: where Applying (35) to  11 ( 1 ) again with  ≡ ℎ 1 and  ≡  eq results in in the second-order approximation in  V , where we have used the (KCRI-I) 1 .Equation (87) can be calculated as follows: where we have used in the second-order approximation in  V , where Therefore, inserting (87) and ( 90) into (84) gives (93) Applying KCRI-III to (83) and using in the second-order approximation in  V , where ISRN Condensed Matter Physics Finally, inserting (93) and ( 95) into (81) gives in the second-order approximation in  V , where (100)

Lineshape Functions and Diagram Representations
In this section, the lineshape functions for the electronphonon system are derived.
6.1.Linear Term.In (52) and (56), where   is the Fermi distribution function for an electron in a state, |⟩, with energy   .Inserting (101) into (52) gives Therefore, (46) becomes where the first lineshape function that contains the effect of the electron-phonon interaction is given as which can be calculated as follows: Here, were used, where   is the Planck distribution function for phonons with energy ℎ  .In (105), the "energy denominator factor" is defined as and the "population factors" are defined as The denominator in (103) represents the transition from the initial state  to the final state  with a photon absorption of frequency  and can be represented as Figure 1.If there is no phonon scattering, the lineshape of the conductivity would be like a delta function.On the other hand, the lineshape is broadened as electrons are scattered by phonons, so the lineshape function, Γ (0)  (), is included.Implicit states exist between the initial and final states because they are connected by Γ (0)  (), including the implicit states induced by electronphonon interactions; that is, Γ (0)  () contains the implicit transitions between the initial (or final) and implicit states.An implicit state should be coupled with the initial (or final) state by a proper interaction factor in the case of an implicit transition between the final (or initial) and implicit states.Γ (0)  () must also contain an energy denominator factor enforcing energy conservation in each implicit transition and a population factor justifying the implicit transition.Although Γ (0)  () is called the lineshape function, the lineshape must be calculated from (103), not directly from (105).Γ (0)  () is represented by four diagrams in Figure 2. The two terms correspond to |  ()| 2 and the other two terms correspond to |  ()| 2 .|  ()| 2 is called the "interaction factor, " through which momentum conservation is satisfied.
The physical meaning of the first term in (105) is as follows.|  ()| 2 means that the implicit state  is coupled with the initial state  by a phonon with a wave vector .|  ()| 2 is represented by a vertical dotted line (Figure 2(a)).The states denoted by  are called implicit states because they are included only in lineshape function (see (105)), not in the conductivity (see (103)). (−)   () means that the energy of the implicit state  is determined by the energy of the final state , photon energy ℎ, and phonon energy ℎ  , so energy conservation can be satisfied; that is,   + ℎ =   + ℎ  . (−)  () is represented by a horizontal solid line.The half circle with a rightward (leftward) arrow corresponds to photon absorption (emission) and the upper (lower) half circle corresponds to phonon emission (absorption). + (, ) means that the reverse implicit transition from the final state to the implicit state with photon emission and phonon absorption (  ) should be subtracted from the forward implicit transition from the implicit state to the final state with photon absorption and phonon emission (1 +   ).This means that when an electron-phonon interaction is involved in an electron transition, the electron undergoes an implicit transition from an implicit state, which is coupled with the initial state, to the final state (vice versa) with photon absorption (emission) and phonon emission (absorption).The second term in (105) corresponds to phonon absorption (emission) for a forward (backward) process.The third and fourth terms in (105) correspond to the cases that the implicit state is coupled with the final state.Therefore, the transition from the initial state to the final state occurs via two implicit transitions and the implicit state is coupled with the initial or final states.We can represent the result in the diagram because the lineshape function contains an energy denominator, population, and interaction factors properly.

First-Order
On the other hand, (75) can be calculated as follows: where was used.Therefore, inserting (110) into (62) and using ( 74) and (111), where Here, we have neglected the  13 values because we are interested in the results near the resonance points, ℎ 1 =   1  and ℎ 1 =    1 .Inserting (113) into (58) gives (see appendices (A.1) and (A.2) for details) Considering (A.1) and (A.2) in the appendix, the lineshape functions are calculated as follows:  3(a)).The other one, called a ( 2 |  1 ) process, is  (−2)  ( 1 ,  2 ), which is an exchange process with  1 and  2 being exchanged; that is, it contains a photon absorption of frequency  1 in an intermediate transition from the intermediate state to the final state,  (Figure 3(b)).
The first term,  −1  ( 1 ,  2 ), contains two transition processes.The first is an intermediate transition from the initial state, , to the intermediate state, , with a photon absorption of frequency  1 and lineshape function, Γ (0)  ( 1 ), and the other is a direct transition from the initial state to the final state, , with two photon absorption of frequencies  1 and  2 and lineshape function Γ (11)   ( 1 ,  2 ).Note that ℎ 12 +   = ℎ 1 +   + ℎ 2 +   .The dotted half circle in Figure 3(a) means that the  state is not connected to the  state by any lineshape function.Therefore, no implicit transition exists between the  and  states.The second term,  −2  ( 1 ,  2 ), can be interpreted in a similar manner, where the  state is not connected to the  state by any lineshape function.Figures 4 and 5 show schematic diagrams of the lineshape functions Γ (11)   ( 1 ,  2 ) and Γ (12)   ( 1 ,  2 ), respectively.The linear part Γ (0) () included in   ( 1 ,  2 ) has the same form as (105).
Equation ( 116) can be interpreted as follows (Figure 4).Note that an intermediate state, , exists between the initial state, , and final state, , in the first term of (115) (Figure 3(a)).With reasoning similar to that for the linear conductivity, there are four diagrams; two terms corresponding to |  ()| 2 and the other two terms corresponding to |  ()| 2 .The first term in (116) (Figure 4 116) and the four terms in (117) (Figure 5) can be interpreted in a similar manner.Note that the implicit state in (117) is not connected to the initial state by any lineshape function (see Figure 3(b)).
Equation (125) (Figure 7) can be interpreted by ( 116) and (126) can be interpreted as follows (Figure 8).Like the linear and first-order nonlinear conductivities, there are four diagrams.The first two terms correspond to |  ()| Therefore,  + ( 2 , ) should be subtracted from  + ( 1 , ), not from  + (, ), because the  1 state is also not connected to the initial state, , by any lineshape function (see Figure 6(b)).Equations ( 127) and (128) can be interpreted in a similar manner (Figures 9 and 10).

Concluding Remarks
So far, we presented a method for deriving the nonlinear optical conductivity tensors for a system of electrons interacting with phonons using the PR method.The standard inductive inference will be required for a more rigorous proof of the general form of KCRI.The use of this method in the present problem, however, was beyond the scope of this study because the general-order term contains a large number of elementary factors.Consequently, this study used a simplified version of the induction instead, which states that the proposition KCRI of the th kind is valid for all positive integers, , if it holds for  = 1, 2, and 3.This method is justifiable because the structure of KCRI and the population factor are iterative.
Nonlinear optics is concerned with the phenomena resulting from light-induced changes in the optical properties of a system, the first-order nonlinear part being the simplest example.This study considers the second harmonic generation (SHG) as a proper example, in which the second parts of Γ (11)   ( 1 ,  2 ) and Γ (12)   ( 1 ,  2 ) play an important role with  12 = 2 1 .Note that a small fraction is converted to light at  12 = 2 1 if an incident beam at frequency  1 is focused on a proper sample.The two fractions can be filtered technically using a prism.Although there have been many studies on SHG in systems of electrons interacting with phonons [34][35][36], less attention has been paid to how the absorption and emission of photons and phonons occur in each process.The present work aims to clear up the physical cause and effect of the processes.These results will be improved if radiation-induced inharmonic phonon modes are included in the consideration.This part with a derivation of the highorder nonlinear term will be examined in future.
The projection operators used in this study are generally applicable because they satisfy the projection criterion (see statements below (51)).The lineshape functions in the linear, first-, and second-order nonlinear optical conductivities include the electron distribution functions to satisfy the population criterion as well as energy denominator factors enforcing energy conservation and the interaction factors coupling the implicit state with the initial, final, or intermediate states.Therefore, the phonon emissions and absorptions as well as photon absorptions and emissions in all electron transition processes can be presented in an organized manner and the results can be represented in diagrams, through which a physical intuition to quantum dynamics of electron transition in a solid can be obtained.These diagrams are quite different from the temperature diagrams with imaginary times in the Feynman-like scheme [37][38][39][40][41][42].In the present scheme, the contribution of each   term to the lineshape function (or the damping term) is the sum of all weighted topologically distinct diagrams.Therefore, the present scheme can be categorized as another Feynmanlike scheme.In conclusion, we claim that a theory, which is generally applicable and satisfies the population criterion, can be obtained using the PR method.Formulations of other electron transition phenomena might be obtained using this method.

2 (
2 and the other two terms correspond to |  ()| 2 .The first term (Figure8(a)) represents a transition from the implicit state  to an intermediate state  2 , with two photon absorption of frequencies  1 and  2 . (−)   12 ) (horizontal solid line) represents the transition and |  ()| 2 (vertical dotted line) means that the implicit state is coupled with the initial state, .Note that  + (,  1 ) (dotted circle) should be subtracted from  + (,  2 ) because the intermediate state,  1 , is not connected to the initial state by any lineshape function (see Figure 6(b)) in ( 2 |  1 ) process.The third term (Figure 8(c)) corresponds to a transition from the initial state to the final state, , with three photon absorption of frequencies  1 ,  2 , and  3 .Note that the intermediate state,  2 , is not connected to the final state by any lineshape function (see Figure 6(b)).