The Sommerfeld-Watson transformation in teaching condensed matter physics: analytical diagonalization of Fermi gas-like models

The SWT is introduced as a method to evaluate sums of the form

where the contour $C$ encloses the real axis poles of $\cot (\pi z)$ but avoids the poles of $f(z)$. This technique is particularly useful when $f(z)$ has a known analytic structure. The authors apply this transformation to diagonalize the Hamiltonians of the Fermi gas and the Anderson model. For the Fermi gas, the Hamiltonian is given by:

where ${ϵ}_{\mathbf{k}}$ is the dispersion relation. The SWT allows the summation over $\mathbf{k}$ to be replaced by an integral over energy, facilitating the diagonalization. For the Anderson model, the Hamiltonian includes a localized impurity level and a conduction band:

The SWT is used to diagonalize this Hamiltonian analytically, yielding the Green's functions and spectral functions. The authors also extend the method to a generalized Anderson model with energy-dependent hybridization $V(ϵ)$ and arbitrary band dispersion ${ϵ}_{\mathbf{k}}$. The analytical results are validated by comparing with numerical diagonalization for specific parameter sets.