Improved estimator for the self-energy function

Recently, Hafermann et al. proposed efficient measurement procedures for the self-energy and vertex functions within the CT-HYB algorithm^1,2. In their method, some higher-order correlation functions (related to the quantities being sought through the equation of motion) are measured. For the case of interactions of density-density type, the segment algorithm is available³. Thus, the additional correlators can be obtained essentially at no additional computational cost. When the calculations are completed, the required self-energy function and vertex function can be evaluated analytically.

The improved estimator for the self-energy function can be expressed in the following form: $$\begin{equation} \Sigma_{ab}(i\omega_n) = \frac{1}{2} \sum_{ij} G^{-1}_{ai}(i\omega_n) (U_{jb} + U_{bj}) F^{j}_{ib}(i\omega_n), \end{equation}$$ where $U_{ab}$ is the Coulomb interaction matrix element. The expression for the new two-particle correlator $F^{j}_{ab}(\tau - \tau')$ reads $$\begin{equation} F^{j}_{ab}(\tau-\tau') = -\langle \mathcal{T} d_{a}(\tau) d^{\dagger}_{b}(\tau') n_{j}(\tau') \rangle, \end{equation}$$ and $F^{j}_{ab}(i\omega_n)$ is its Fourier transform. The actual measurement formula is $$\begin{equation} F^{j}_{ab}(\tau - \tau') = -\frac{1}{\beta} \left\langle \sum_{\alpha\beta = 1}^{k} \mathcal{M}_{\beta\alpha}\delta^{-}(\tau-\tau', \tau^{e}_{\alpha} - \tau^{s}_{\beta}) n_{j}(\tau^s_\beta)\delta_{a,\alpha}\delta_{b,\beta} \right\rangle. \end{equation}$$ As one can see, this equation looks quite similar to the one for imaginary-time Green's function. Thus we use the same method to measure $F^{j}_{ab}(\tau - \tau')$ and finally get the self-energy function via the first equation. Here, the matrix element $n_{j}(\tau^s_\beta)$ (one or zero) denotes whether or not flavor $j$ is occupied (whether or not a segment is present) at time $\tau^s_\beta$.

This method can be combined with the orthogonal polynomial representation⁴ as introduced in the previous subsection to suppress fluctuations and filter out the Monte Carlo noise. Using this technique, we can obtain the self-energy and vertex functions with unprecedented accuracy, which leads to an enhanced stability in the analytical continuations of those quantities². In the iQIST software package, we only implemented the improved estimator for the self-energy function. Note that when the interaction matrix is frequency-dependent, the first equation should be modified slightly¹.

Reference

¹ Hartmut Hafermann, Phys. Rev. B 89, 235128 (2014)

² Hartmut Hafermann, Kelly R. Patton, and Philipp Werner, Phys. Rev. B 85, 205106 (2012)

³ Philipp Werner, Armin Comanac, Luca de’ Medici, Matthias Troyer, and Andrew J. Millis, Phys. Rev. Lett. 97, 076405 (2006)

⁴ Lewin Boehnke, Hartmut Hafermann, Michel Ferrero, Frank Lechermann, and Olivier Parcollet, Phys. Rev. B 84, 075145 (2011)