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

In the original DLR work [Kaye et al., 2022], the basis is derived by applying an interpolative decomposition (a rank-revealing factorization) to a discretized kernel, without explicitly referring to the IR basis. Here, for both pedagogical and practical reasons, we instead define the DLR starting from the IR singular functions \(U_l(\tau)\) and \(V_l(\omega)\) and then choosing poles \(\{\bar\omega_p\}\) on the real axis, which is also how the sparse-ir implementation constructs the DLR basis. In addition, we choose 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 :-)

Basis functions

Imaginary-time basis functions

The DLR basis functions in the imaginary-time domain are defined using the logistic kernel:

\[ U_p(\tau) = -K^\mathrm{L}(\tau, \bar{\omega}_p) = -\frac{e^{-\tau \bar{\omega}_p}}{1 + e^{-\beta \bar{\omega}_p}}, \]

for \(0 < \tau < \beta\). These basis functions are common for both fermions and bosons, which is a key advantage of using the logistic kernel.

The Green’s function is expanded as:

\[ G(\tau) = \sum_{p=1}^L U_p(\tau) c_p. \]

Matsubara basis functions

The DLR basis functions in the Matsubara-frequency domain differ between fermions and bosons due to the regularization factor:

Fermions:

\[ \hat{U}_p(\mathrm{i}\nu_n) = \frac{1}{\mathrm{i}\nu_n - \bar{\omega}_p}, \]

where \(\nu_n = (2n+1)\pi/\beta\) are fermionic Matsubara frequencies.

Bosons:

\[ \hat{U}_p(\mathrm{i}\nu_n) = \frac{\tanh(\beta \bar{\omega}_p/2)}{\mathrm{i}\nu_n - \bar{\omega}_p}, \]

where \(\nu_n = 2n\pi/\beta\) are bosonic Matsubara frequencies. The factor \(\tanh(\beta \bar{\omega}_p/2)\) is the regularization factor that compensates for the modified spectral function \(\rho(\omega) = A(\omega)/\tanh(\beta\omega/2)\) used with the logistic kernel.

The Matsubara Green’s function is expanded as:

\[ G(\mathrm{i}\nu_n) = \sum_{p=1}^L \hat{U}_p(\mathrm{i}\nu_n) c_p. \]

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.

Bosons

For bosons, the DLR is defined with the modified spectral function:

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

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

Note that \(\rho(\omega)\) has the same form as for fermions, which means the same transformation matrix \(\boldsymbol{V}_{lp}\) can be used for both statistics when transforming between IR and DLR coefficients.