Discrete Lehmann representation

We explain the implementation of Discrete Lehmann Representation (DLR)[Kaye et al., 2022] in sparse-ir. For both of fermions and bosons, we expand the Green’s function as

\[ G(\tau) = - \sum_{p=1}^L K^\mathrm{L}(\tau, \bar{\omega}_p) c_p \]

for \(0 < \tau < \beta\). The pole positions \(\{\bar{\omega}_1, \cdots, \bar{\omega}_{L}\}\) are chosen as the extrema of \(V'_{L-1}(\omega)\). \(\{K^\mathrm{L}(\tau, \bar{\omega}_p) \}\) forms a non-orthogonal basis set in \(\tau\), which is common for fermions and bosons.

Note

The poles on the real-frequency axis selected for the DLR are based on a rank-revealing decomposition, which offers guaranteed accuracy. Here, we instead select the pole locations based on the zeros of the IR basis functions on the real axis [Shinaoka et al., 2021], which is a heuristic. We do not expect that difference to matter, but please don’t blame the DLR authors if we were wrong :-)

Fermions

For fermions, this is equivalent to modeling the spectral function as

\[ A(\omega) = \rho(\omega) = \sum_{p=1}^L c_p \delta(\omega - \bar{\omega}_p). \]

The choice of the pole positions is heuristic but allows a numerically stable transform between \(\rho_l\) and \(c_p\) through the relation

\[ \rho_l = \sum_{p=1}^L \boldsymbol{V}_{lp} c_p, \]

where the matrix \(\boldsymbol{V}_{lp}~[\equiv V_l(\bar{\omega}_p)]\) is well-conditioned. The Matsubara Green’s function is expanded as

\[ G(\mathrm{i}\nu) = \int \mathrm{d}\omega \frac{A(\omega)}{\mathrm{i}\nu - \omega} = \sum_{p=1}^L \frac{c_p}{\mathrm{i}\nu - \bar{\omega}_p}. \]

Bosons

In the Matsubara-frequency space, The DLR for bosons is defined as follows:

\[ A(\omega) = \sum_{p=1}^L c_p \tanh(\beta\bar{\omega}_p/2) \delta(\omega - \bar{\omega}_p), \]

\[ \rho(\omega) = \frac{A(\omega)}{\tanh(\beta\omega/2)} = \sum_{p=1}^L c_p \delta(\omega - \bar{\omega}_p), \]

\[ G(\mathrm{i}\nu) = \int \mathrm{d}\omega \frac{A(\omega)}{\mathrm{i}\nu - \omega} = \sum_{p=1}^L \frac{c_p \tanh(\beta\bar{\omega}_p/2)}{\mathrm{i}\nu - \bar{\omega}_p}. \]

When transforming data from IR to DLR, one can use the same transformation matrix for fermions and bosons.