Luttinger Hamiltonian

In semiconductors with diamond stracture (e.g. Si, Ge) or zinc blende stmcture (e.g. GaAs), the electronic stracture near the top of valence hands having S = 3/2 (S is the total angular momentum of the atomic orbital) can be described by the Luttinger Hamiltonian, ... 

The remainder of this section is devoted to a more detailed derivation of (5) from (3). We first transform (3) into a Luttinger Hamiltonian, following Luther and Peschel,6 to first order in the coefficient Jz of PjPj (This is, in fact, the limit of accuracy of the Luther-Peschel replacement of the HI Hamiltonian by a Luttinger model. The equivalence breaks down completely as Jz approaches unity where the HI model is singular.) We drop the constant magnetic field term, because when the Hamiltonian is cast in the Luttinger form it is evident that this term merely alters the fermi level and cannot affect the exponents. 

Our next step is to transform the Luttinger Hamiltonian (10) into the F.fetov-Larkin long-wavelength Hamiltonian (5). This particular equivalence is actually mentioned by Efetov and Larkin in a later section of their paper. The transformation is accomplished by putting... 

One-channel quantum wires can be described by the Tomonaga-Luttinger model with the Hamiltonian... 

In the analyses of conventional ZB semiconductors, we frequently assume a symmetric parabolic band for the conduction band state, and the Luttinger-Kohn Hamiltonian is used to describe the valence band states. In general, the effective Hamiltonian is derived from a k.p perturbation theory or from the theory of invariants developed by Pikus and Bir. In the latter theory, the operator form of the effective Hamiltonian can easily be constructed from symmetry consideration alone. Within this framework, the lowest two conduction bands and the upper six valence bands are described to the second order of k. The invariant forms of the Hamiltonians are written as follows [26,27] ... 

For WZ compounds, we must consider hexagonal symmetry in the effective Hamiltonian. The Luttinger-Kohn Hamiltonian is constructed under the condition of cubic symmetry and the form reflects cubic crystal symmetry. Thus, in the analysis of WZ nitrides, we must use a k-dependent parabolic band for the conduction band state and Bir-Pikus Hamiltonians for the valence band states. The Hamiltonians for the upper six valence bands and the lowest two conduction bands are given by [28]... 

In the analyses of zincblende (ZB) compounds, the Luttinger-Kohn Hamiltonian is used to describe the valence band states. The Hamiltonian is written as [1,2]... 

Here ji are the Luttinger parameters, and is a spin-orbit splitting energy. L and a denote orbital and spin angular momentum operators, respectively. [Lj,Lj] is defined as [Lj,Lj] = (LjLj + LjLj)/2. The summation of i,j runs through x,y,z. In the analysis of wurtzite (WZ) compounds, the Bir-Pikus Hamiltonian for the valence band states is used. The Hamiltonian is given by [3]... 

In the analyses of conventional zincblende (ZB) semiconductors, we frequently assume a parabolic band for the conduction bands, and the 6 x 6 Luttinger-Kohn Hamiltonians are used to describe the upper valence bands [1,2], In treating the valence bands together with the conduction bands on an equal footing, as when estimating the momentum matrix elements, we often make use of the 8 x 8 Kane Hamiltonian [3], However, the form of the Hamiltonians reflects the crystal symmetry, and Kane Hamiltonians are constructed under the condition of cubic symmetry. For wurtzite (WZ) materials, therefore, we must consider hexagonal symmetry in the effective Hamiltonian. Let us consider the 8 x 8 k.p Hamiltonian for WZ structure [4,5],... 

What immediately comes out of these equations at the one-loop level is the fact that g (f) is decoupled from the set of couplings (2g2 gi)( ) and Reverting to the expression (21) for the Hamiltonian, this means that long-wavelength charge and spin correlations are decoupled, a key feature of a Luttinger liquid known as spin-charge separation. 

The harmonic part of the phase Hamiltonian corresponds to the Tomanaga-Luttinger model with no backscattering and Umklapp terms. It is exactly solvable, the spectrum shows only collective excitations and all the properties of... 

Given the values of the Luttinger liquid parameters, we can now look at the properties of the system using the harmonic part of the Hamiltonian. Thus the... 

This appendix is based on the one in the paper by Baldereschi and Lipari [1] and it outlines the fundamental tensor properties of Py and Jy introduced in Sect. 5.3, with reference to Luttinger s Hamiltonian for holes in the J = 3/2 VB. In an orthogonal reference frame, a tensor of rank k can be reduced... 

Here we outline the general formulas for recombination of an electron-hole pair belonging to Luttinger liquid rings in conduction and valence bands, (see Fig, 3). We start with the two-component Luttinger liquid model on a ring [108] with Hamiltonian H1 + H2 + Hint, where Hj describe noninter-... 

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