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Spectral reconstruction techniques, their shortcomings and relevance to the electric conductivity coefficient

C. Andratschke, B. B. Brandt, E. Garnacho-Velasco, L. Pannullo, S. Singh, A. Dean M. Valois

Abstract

Spectral reconstruction is a well studied numerically ill-posed problem which arises due to the relation of the Euclidean correlator to the spectral function via an inhomogeneous Fredholm equation of the first kind. Several different methods are on the market to resolve this issue, each taking different approaches and assumptions. In this proceedings we focus on implementing and testing a machine learning framework for spectral reconstruction, as well as implementing a novel method of estimating the behavior of the spectral function in the vicinity of vanishing frequency, which we denote as multipoint method, and compare these methods to well established spectral reconstruction techniques from the literature using mock data. As a physics application, we apply the reconstruction techniques to quenched lattice data for the correlation function in the vector channel at non-zero external magnetic field to extract the spectral function and the electric conductivity through its behaviour at vanishing frequency via a Kubo formula.

Spectral reconstruction techniques, their shortcomings and relevance to the electric conductivity coefficient

Abstract

Spectral reconstruction is a well studied numerically ill-posed problem which arises due to the relation of the Euclidean correlator to the spectral function via an inhomogeneous Fredholm equation of the first kind. Several different methods are on the market to resolve this issue, each taking different approaches and assumptions. In this proceedings we focus on implementing and testing a machine learning framework for spectral reconstruction, as well as implementing a novel method of estimating the behavior of the spectral function in the vicinity of vanishing frequency, which we denote as multipoint method, and compare these methods to well established spectral reconstruction techniques from the literature using mock data. As a physics application, we apply the reconstruction techniques to quenched lattice data for the correlation function in the vector channel at non-zero external magnetic field to extract the spectral function and the electric conductivity through its behaviour at vanishing frequency via a Kubo formula.
Paper Structure (9 sections, 16 equations, 4 figures)

This paper contains 9 sections, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spectral reconstruction
  3. Unsupervised machine learning
  4. Multipoint method
  5. Mock data
  6. Results
  7. Mock data
  8. Lattice calculation
  9. Conclusions

Figures (4)

  • Figure 1: Left panel: Sketch of the unsupervised neural network structure. The shaded area signifies a softplus activation function. Right panel: Reconstruction of $\rho(\omega)$ as in Wang:2021cqw versus the direct extraction of $\rho(\omega)/\omega$ for a mock Breit-Wigner spectral function as described in Sec. \ref{['Mock data']}.
  • Figure 2: Comparison of the smearing kernels with $N_t=36$ obtained from the unregularized BG method and the multipoint (MP) with different numbers of used correlator points.
  • Figure 3: Comparison of the performance of the machine learning method ("unsupervised"), the Gaussian method, BG, and MEM on the mock data described in Sec. \ref{['Mock data']} with $N_\tau = 36$ and a noise level of $A=3$ (top) and $A=4$ (bottom).
  • Figure 4: Left: $\rho_{33}(\omega)/\omega$, at $N_b=6$, obtained using various spectral reconstruction methods, including the Gaussian method via the fredipy package fredipyHorak:2021syv. Right: Longitudinal electric conductivity as a function of the magnetic field strength. Here, the error bars only include statistical errors. We observe qualitative agreement with results from full QCD staggered simulations Astrakhantsev_2020.