Spectral Functions of Local Operators In The Tomonaga-Luttinger Model

I give a simple general prescription for computing the spectral functions of local operators in the Tomonaga-Luttinger model from the space-time correlation functions. The method is significantly simpler than directly transforming the space-time Greens function and allows a physical interpretation of the singularities encountered in the spectral function. 71.20.-b, 71.27.+a, 79.60.-i Typeset using REVTEX 1 The Tomonaga-Luttinger model [1,2] provides an fascinating example of a soluble model of an interacting Fermion problem with a non-Fermi liquid groundstate. The low energy eigenstates of the system consist of bosonic degrees of freedom representing collective modes of the Fermi surface [3–5]. There are no single particle, Fermionic, low energy eigenexcitations. Correspondingly the spectral functions of local operators in the model display unusual features including multiple singularities with non-trivial power laws. The properties of the single-electron spectral function were studied by Meden and Schönhammer [6] and Voit [7] by Fourier transforming the single-electron Greens function. This procedure is entirely correct; however, it is somewhat unintuitive and the results take on complicated form for the case of models with spin. I give here a prescription for calculating the spectral functions of local operators from the appropriate space-time Greens functions based on the observation that many of the complications encountered in the Fourier transforming of the relevant Greens functions arise from the fact that a general local operator in the Tomonaga-Luttinger model involves the creation of four distinct types of bosons and these complications are circumvented when the independent nature of the different boson types is considered. Using forms suitable for the calculations of spectral functions in the low energy, long wavelength limit, the single electron creation (ψ R or L(x)) and annihilation (ψR or L(x)) operators, the local singlet (ψR,↑(x)ψL,↓(x)) and triplet (ψR,↑(x)ψL,↑(x)) pairing operators and the local charge (ψ R,↑(x)ψL,↑(x)+ψ † R,↓(x)ψL,↓(x)) and spin (ψ † R,↑(x)ψL,↓(x)+ψ † L,↑(x)ψR,↓(x)) density wave operators can all expressed as sums of operators of the form [4,5]: Φ(x;αj) = exp(i ∑ j αjφ † j(x)) exp(i ∑ j αjφj(x)) (1) where φj(x) = (−1) j πx L Nj + i ∑