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Continuity of a Function01:23

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A function is continuous at a point a if three conditions are met: the function is defined at a, the limit of the function as x approaches a exists, and this limit equals the function’s value. Mathematically, this is written asThis definition ensures the graph of the function does not exhibit any breaks, holes, or jumps at that point. Discontinuities occur when any of these conditions fail. A removable discontinuity exists when the two-sided limit exists but the function is either undefined...
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Nevanlinna Analytical Continuation.

Jiani Fei1, Chia-Nan Yeh1, Emanuel Gull1

  • 1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA.

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Summary
This summary is machine-generated.

This study introduces a new method for quantum simulations, improving the accuracy of spectral functions. It overcomes limitations in existing techniques, enabling higher resolution in real-frequency calculations.

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Area of Science:

  • Quantum Mechanics
  • Computational Physics
  • Materials Science

Background:

  • Finite temperature quantum system simulations yield imaginary frequency Green's functions.
  • These functions correspond to real-frequency spectral functions, crucial for experiments.
  • Current methods struggle with the imaginary-to-real frequency transformation, causing feature loss or unphysical results.

Purpose of the Study:

  • To develop a more accurate method for spectral function calculation from imaginary frequency Green's functions.
  • To overcome the limitations of existing methods in resolving high-frequency features and ensuring physical results.
  • To enhance the resolution of real-frequency calculations in finite-temperature quantum simulations.

Main Methods:

  • Utilizing the analytic "Nevanlinna" structure of the Green's function.
  • Implementing a continued fraction expansion consistent with the analytic structure.
  • Applying the method to synthetic data and the band structure of silicon.

Main Results:

  • The new method produces intrinsically positive and normalized spectral functions.
  • Accurate resolution of sharp, smooth, and multipeak synthetic data.
  • Precise resolution of high energy features in silicon's band structure.
  • Uncovering previously unresolved features in correlated systems.

Conclusions:

  • Explicitly respecting the analytic structure of Green's functions is key to accurate spectral function calculations.
  • The developed continued fraction expansion method significantly improves spectral feature resolution.
  • This work overcomes a major limitation in finite-temperature quantum simulations, enhancing predictive power.