Discrete Lehmann representation of imaginary time Green's functions

We present an efficient basis for imaginary time Green's functions based on a low-rank decomposition of the spectral Lehmann representation. The basis functions are simply a set of well-chosen exponentials, so the corresponding expansion may be thought of as a discrete form of the Lehmann representation using an effective spectral density which is a sum of functions. The basis is determined only by an upper bound on the product beta omega(max), with the inverse temperature andmax an energy cutoff, and a user-defined error tolerance . The number r of basis functions scales as O(log beta omega(max)) log(1/epsilon)). The discrete Lehmann representation of a particular imaginary time Green's function can be recovered by interpolation at a set of r imaginary time nodes. Both the basis functions and the interpolation nodes can be obtained rapidly using standard numerical linear algebra routines. Due to the simple form of the basis, the discrete Lehmann representation of a Green's function can be explicitly transformed to the Matsubara frequency domain, or obtained directly by interpolation on a Matsubara frequency grid. We benchmark the efficiency of the representation on simple cases, and with a high-precision solution of the Sachdev-Ye-Kitaev equation at low temperature. We compare our approach with the related intermediate representation method, and introduce an improved algorithm to build the intermediate representation basis and a corresponding sampling grid.

Automated summary

We present an efficient basis for imaginary time Green's functions based on a low-rank decomposition of the spectral Lehmann representation. The basis functions are simply a set of well-chosen exponentials, and the corresponding expansion can be considered as a discrete form of the Lehmann representation. The basis is determined by an upper bound on the product beta omega(max), the inverse temperature and energy cutoff, and a user-defined error tolerance. The number of basis functions scales logarithmically with the product beta omega(max) and the reciprocal of the error tolerance. The basis functions and interpolation nodes can be obtained rapidly using standard numerical linear algebra routines. The discrete Lehmann representation of the Green's function can be transformed to the Matsubara frequency domain or obtained directly by interpolation on a Matsubara frequency grid.