The Horizontal Ladder Approximation

The name horizontal ladder approximation was also introduced by Babanov et al.

and denotes the procedure to replace the vertex function Γ1 in the horizontal equa-tion (5.37) by its lowest-order diagram which is simply the interacequa-tion v (or the antisymmetrized interaction ¯V). This procedure leads to a selection of subsets of diagrams contained in the diagrammatic expansion of the vertex function Γ. One of these subsets contains the diagrams known as ladder diagrams which are retained in the “conventional” ladder approximation but in the horizontal ladder approximation in the sense of Babanov et al. a variety of additional diagrams is also retained as can be seen below. If this approximation is imposed, the remaining diagrams can be divided into two subsets:

• Particle-Particle Channel: This set contains only diagrams that describe processes between two particles with the same charge (either one electron with another electron or a hole with another hole).

• Electron-Hole Channel: This set contains only diagrams that describe pro-cesses between an electron and a hole.

The horizontal ladder approximation only retains diagrams from the expansion of Γ, that belong to either one subset. Speaking from a more physical point of view, in the description of a system with the FLEX method there are no processes contained where the interaction of a particle pair with identical charges is combined with the interaction of an electron-hole pair in one single process. For the discussion of the ladder approximation it is convenient to analyze the diagrammatic equations for each channel separately.

Particle-Particle Channel. If we consider only those diagrams with Green-function lines carrying arrows which point into the same direction, replacing the vertex func-tion Γ1 in (5.37) by the interaction line v yields the horizontal ladder approximation for the particle-particle vertex function. The first term of the infinite sum of diagrams can be depicted as

+ +

+ =

PSfrag replacements

Γ ≈ . . . T^pp , (5.38)

where the particle-particle T-matrix T^pp was introduced as the quantity containing the result of the summation over the infinite number of diagrams. All diagrams to

be contained in T^pp are made up of vertical interaction lines connected by a pair of horizontal Green-function lines. Because of this structure the diagrams are called ladder diagrams and the name ladder approximation originates from this subset of diagrams. The infinite sum of ladder diagrams in (5.38) corresponds to a geometric series. Thus, a closed integral equation for the particle-particle T-matrix can be obtained

+

= g1

PSfrag replacements T^pp T^pp . (5.39)

To derive a mathematical formula forT^ppfrom (5.39) the particle-particle propagator Ψ defined in (5.27) can be used to express T^pp in matrix operator notation as

T^pp() = v + vΨ()ˆ T^pp() (5.40)

= v + vΨ()ˆ v + vΨ()ˆ vΨ()ˆ v +vΨ()ˆ vΨ()vˆ Ψ()ˆ v + . . . which can be solved using the geometric series yielding.

T^pp() = v[1 − Ψ()v]⁻¹ . (5.41)

The vertex function in the particle-particle channel is thus approximated by a sum over ladder diagrams yielding the particle-particle T-matrixT^pp. In figure 5.4 the first three lowest-order diagrams form the ladder diagram series are depicted. These ladder diagrams describe the repeated scattering of two particles with the same charge. In diluted electron systems with a short-range repulsive potential the electron-electron ladder diagrams are the leading term in the diagrammatic expansion of the exact self-energy (see i.e. [FW71]). Hence, the electron-electron ladder diagrams from the particle-particle channel yield a good description for the electron system in the limit of small electron densities. The expansion of the self-energy in terms of the electron-electron ladder approximation is conventionally referred to as “the” ladder approximation. The horizontal ladder approximation in the sense of Babanov et al. contains already more types of diagrams in the particle-particle channel namely the hole-hole ladder approximation and in addition to that contains a whole subset of diagrams in the electron-hole channel which are not retained in the conventional ladder approximation. These diagrams belong to the electron-hole channel.

Figure 5.4: First, second and third order ladder diagram

5.4 The FLEX Method 61 Electron-Hole Channel. In this channel the vertex function Γ1 in (5.37) is replaced by the symmetrized interaction defined in (5.23). This also yields an infinite sum over diagrams

+ +

+ =

PSfrag replacements

Γ ≈ . . . T^eh (5.42)

where the electron-hole T-matrixT^eh was introduced. A closed integral equation for this T-matrix is given by

+

= g1

PSfrag replacements T^eh T^eh . (5.43)

I like to further expand the electron-hole T-matrix by inserting the definition of the antisymmetrized interaction (5.37). Furthermore, I like to make use of the underlying cubic symmetry of the crystal. It was already discussed in chapter 4 upon introducing the DFT lattice Green function that the electronic states at one atomic site do not overlap due to the cubic symmetry. Therefore, if an electron and a hole interact the final state of the electron and the hole has to be identical to the initial states for both particles if only those process between particles at the same lattice site are consider. If the electron-hole T-matrix is expanded only in terms of Green functions of particles located at the same lattice site only two subsets of all the diagrams contained in the expansion of T^eh are retained. One subset contains only diagrams with vertical interaction lines further referred to as electron-hole channel 1 (eh1) and the other subset contains only horizontal interaction lines and shall be denoted as electron-hole channel 2 (eh2). Due to the symmetry local processes between an electron and a hole described by diagrams containing horizontal and vertical interaction lines at the same time can not occur.

For each subsets of diagrams a closed integral equation of the same form as equation (5.43) can be derived. Consequently, two electron-hole T-matrices further denoted asT₁^eh and T₂^eh can be introduced by

== + g1

PSfrag replacements T₁^eh T₁^eh (5.44)

for the electron-hole channel 1 and for the electron-hole channel 2 by

+

= g1

PSfrag replacements T₂^eh T₂^eh . (5.45)

I like to point out once more that by replacing the full electron-hole T-matrixT^ehby T₁^ehandT₂^ehan additional approximation is introduced which is identical to neglecting all diagrams in the expansion of T^eh describing inter-atomic processes. However, it

will be demonstrated in the next chapter, that this approximation is justified if the diagrams are finally evaluated within the framework of dynamical mean-field theory.

Figure 5.5 shows some examples of diagrams from the electron-hole channel 1 in the first row and from the electron-hole channel 2 in the second line. The diagrams from the electron-hole channel 1 have the same structure as the diagrams in the particle-particle channel and describe multiple scattering processes between an electron and a hole. The diagrams of the second electron-hole channel are all composed of one interaction lines with a certain number of electron-hole bubble inserted into the line. In the lowest order one electron-hole bubble is inserted as depicted in the first example graph in 5.5 and for higher orders more electron-hole bubbles are inserted.

These diagrams depict the screening of the Coulomb repulsion due to pair interaction fluctuations and have the same topology as the diagrams that yield the self-energy in the GW method. These diagrams are known to give an important contribution to the self-energy of the degenerated high-density electron gas (see i.e. [FW71]). In the limit of high electron density, this subclass of diagrams contribute the leading term to the diagrammatic expansion of the exact self-energy. Since the FLEX method contains both subclasses of diagrams yielding the exact self-energy in the case of low and high electron densities FLEX is also thought to yield an accurate extrapolation for system with intermediate densities.

Figure 5.5: The first row shows example graphs from the eh1-channel, the second row shows two graphs from the eh2-channel

5.4 The FLEX Method 63