Abstract
We present a generalization of the maximum entropy method to the analytic continuation of matrix-valued Green's functions. To treat off-diagonal elements correctly based on Bayesian probability theory, the entropy term has to be extended for spectral functions that are possibly negative in some frequency ranges. In that way, all matrix elements of the Green's function matrix can be analytically continued; we introduce a computationally cheap element-wise method for this purpose. However, this method cannot ensure important constraints on the mathematical properties of the resulting spectral functions, namely positive semidefiniteness and Hermiticity. To improve on this, we present a full matrix formalism, where all matrix elements are treated simultaneously. We show the capabilities of these methods using insulating and metallic dynamical mean-field theory (DMFT) Green's functions as test cases. Finally, we apply the methods to realistic material calculations for LaTiO$_3$, where off-diagonal matrix elements in the Green's function appear due to the distorted crystal structure.
Highlights
In condensed matter physics, response functions are often calculated in imaginary-time formulation, especially when electronic correlations are taken into account
The Wick rotation iτ → t, where τ is the imaginary-time argument and t is the real-time argument (or equivalently iωn → ω, with the nth fermionic Matsubara frequency ωn = (2n + 1)π/β and the real frequency ω), transforms the calculated quantities to real frequencies. This analytic continuation (AC) is not possible straightforwardly, since the kernel of this mapping is ill-conditioned when going from imaginary times to real frequencies
We show how a consistent framework for the analytic continuation of matrix-valued Green’s functions can be constructed on a probabilistic footing
Summary
Response functions are often calculated in imaginary-time formulation, especially when electronic correlations are taken into account This is true for numerical approaches like quantum Monte Carlo [1,2,3], and for perturbative techniques such as the random phase approximation [4,5,6]. The Wick rotation iτ → t, where τ is the imaginary-time argument and t is the real-time argument (or equivalently iωn → ω, with the nth fermionic Matsubara frequency ωn = (2n + 1)π/β and the real frequency ω), transforms the calculated quantities to real frequencies This analytic continuation (AC) is not possible straightforwardly, since the kernel of this mapping is ill-conditioned when going from imaginary times to real frequencies. As a result of the kernel being ill-conditioned, small changes of the input will correspond to largely different outputs, rendering the inversion of this problem highly unstable due to numerical noise, where even an error at the level of machine precision can lead to nonsensical results in practice
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