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Josef Kaufmann
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We present the Python package ana_cont for the analytic continuation of fermionic and bosonic many-body Greens functions by means of either the Pade approximants or the maximum entropy method. The determination of hyperparameters and the implementation are described in detail. The code is publicly available on GitHub, where also documentation and learning resources are provided.
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A method for analytic continuation of imaginary-time correlation functions (here obtained in quantum Monte Carlo simulations) to real-frequency spectral functions is proposed. Stochastically sampling a spectrum parametrized by a large number of delta-functions, treated as a statistical-mechanics problem, it avoids distortions caused by (as demonstrated here) configurational entropy in previous sampling methods. The key development is the suppression of entropy by constraining the spectral weight to within identifiable optimal bounds and imposing a set number of peaks. As a test case, the dynamic structure factor of the S=1/2 Heisenberg chain is computed. Very good agreement is found with Bethe Ansatz results in the ground state (including a sharp edge) and with exact diagonalization of small systems at elevated temperatures.
We describe Rabacus, a Python package for calculating the transfer of hydrogen ionizing radiation in simplified geometries relevant to astronomy and cosmology. We present example solutions for three specific cases: 1) a semi-infinite slab gas distribution in a homogeneous isotropic background, 2) a spherically symmetric gas distribution with a point source at the center, and 3) a spherically symmetric gas distribution in a homogeneous isotropic background. All problems can accommodate arbitrary spectra and density profiles as input. The solutions include a treatment of both hydrogen and helium, a self-consistent calculation of equilibrium temperatures, and the transfer of recombination radiation. The core routines are written in Fortran 90 and then wrapped in Python leading to execution speeds thousands of times faster than equivalent routines written in pure Python. In addition, all variables have associated units for ease of analysis. The software is part of the Python Package Index and the source code is available on Bitbucket at https://bitbucket.org/galtay/rabacus . In addition, installation instructions and a detailed users guide are available at http://pythonhosted.org//rabacus .
Bayesian parametric analytic continuation (BPAC) is proposed for the analytic continuation of noisy imaginary-time Greens function data as, e.g., obtained by continuous-time quantum Monte Carlo simulations (CTQMC). Within BPAC, the spectral function is inferred from a suitable set of parametrized basis functions. Bayesian model comparison then allows to assess the reliability of different parametrizations. The required evidence integrals of such a model comparison are determined by nested sampling. Compared to the maximum entropy method (MEM), routinely used for the analytic continuation of CTQMC data, the presented approach allows to infer whether the data support specific structures of the spectral function. We demonstrate the capability of BPAC in terms of CTQMC data for an Anderson impurity model (AIM) that shows a generalized Kondo scenario and compare the BPAC reconstruction to the MEM as well as to the spectral function obtained from the real-time fork tensor product state impurity solver where no analytic continuation is required. Furthermore, we present a combination of MEM and BPAC and its application to an AIM arising from the ab initio treatment of SrVO$_3$.
The Quantum Monte Carlo (QMC) method can yield the imaginary-time dependence of a correlation function $C(tau)$ of an operator $hat O$. The analytic continuation to real-time proceeds by means of a numerical inversion of these data to find the response function or spectral density $A(omega)$ corresponding to $hat O$. Such a technique is very sensitive to the statistical errors in $C(tau)$ especially for large values of $tau$, when we are interested in the low-energy excitations. In this paper, we find that if we use the flat histogram technique in the QMC method, in such a way to make the {it histogram of} $C(tau)$ flat, the results of the analytic continuation for low-energy excitations improve using the same amount of computational time. To demonstrate the idea we select an exactly soluble version of the single-hole motion in the $t-J$ model and the diagrammatic Monte Carlo technique.
We present SpM, a sparse modeling tool for the analytic continuation of imaginary-time Greens function, licensed under GNU General Public License version 3. In quantum Monte Carlo simulation, dynamic physical quantities such as single-particle and magnetic excitation spectra can be obtained by applying analytic continuation to imaginary-time data. However, analytic continuation is an ill-conditioned inverse problem and thus sensitive to noise and statistical errors. SpM provides stable analytic continuation against noise by means of a modern regularization technique, which automatically selects bases that contain relevant information unaffected by noise. This paper details the use of this program and shows some applications.
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