Discrete Lehmann Representation (DLR)¶
Lehmann Representation¶
Any imaginary-time Green's function admits a spectral (Lehmann) representation:
where \(\rho(\omega)\) is the spectral density and \(\xi = -1\) (fermion) or \(+1\) (boson). When \(\rho\) is supported within \([-\omega_\mathrm{max}, \omega_\mathrm{max}]\), we define the dimensionless cutoff \(\Lambda \equiv \beta\omega_\mathrm{max}\).
DLR Approximation¶
The kernel \(K(\tau, \omega)\) is numerically low-rank. The DLR exploits this by approximating \(G(\tau)\) as a sum of \(r\) exponentials:
The DLR frequencies \(\{\omega_l\}\) are selected via interpolative decomposition (pivoted QR) of \(K\) discretized on a composite Chebyshev grid. The same process yields \(r\) imaginary-time nodes \(\{\tau_k\}\) and \(r\) Matsubara frequency nodes \(\{i\nu_{n_k}\}\). The basis is universal: for given \(\Lambda\) and \(\epsilon\), the same \(\{\omega_l\}\) represent any Green's function within that cutoff.
The DLR coefficients \(\widehat{g}_l\) are recovered by solving an \(r \times r\) interpolation problem from samples at the DLR nodes. The expansion transforms analytically to Matsubara frequency:
so that \(\tau \leftrightarrow i\nu_n\) transforms reduce to \(r \times r\) linear algebra instead of \(O(\beta)\)-sized FFTs.
QAssemble Implementation¶
The DLR class constructs separate fermionic and bosonic DLR objects via pydlr. Key attributes and methods:
| Attribute / Method | Description |
|---|---|
tauF, tauB |
Fermionic / bosonic imaginary-time nodes |
omega, nu |
Fermionic / bosonic Matsubara frequency nodes |
FT2F, FF2T |
Fermionic \(\tau \leftrightarrow i\omega_n\) transforms |
BT2F, BF2T |
Bosonic \(\tau \leftrightarrow i\nu_n\) transforms |