diff --git a/doc/analytic_continuation/spm.rst b/doc/analytic_continuation/spm.rst index 0a2a4fe0..9cb9cc5b 100644 --- a/doc/analytic_continuation/spm.rst +++ b/doc/analytic_continuation/spm.rst @@ -5,9 +5,10 @@ The sparse-modeling (SpM) method is a method for the analytic continuation of th .. math:: - \Sigma(\tau) = \int_{-\infty}^{\infty} d\omega \frac{e^{-\tau \omega}}{1+e^{-\beta\omega}} \Sigma(\omega) + \Sigma(\tau) = \int_{-\infty}^{\infty} d\omega \frac{e^{-\tau \omega}}{1+e^{-\beta\omega}} \rho(\omega) -where :math:`\Sigma(\tau)` is the self-energy in imaginary time and :math:`\Sigma(\omega)` is the self-energy in real frequency. +where :math:`\Sigma(\tau)` is the self-energy in imaginary time and :math:`\rho(\omega) = -\mathrm{Im}\Sigma(\omega)/\pi` is the spectral function in real frequency. +The real part of the self-energy is obtained by the Kramers-Kronig transformation from the imaginary part. In `the SpM method `_, the kernel matrix of the integral equation, :math:`e^{-\tau\omega}/(1+e^{-\beta\omega})` is decomposed by the singular value decomposition (SVD), and the self-energies :math:`\Sigma(\tau)` and :math:`\Sigma(\omega)` are transformed by the left and right singular vectors, respectively.