Maximum entropy formalism for the analytic continuation of matrix-valued Green's functions

We present a generalization of the maximum entropy method to the analytic continuation of matrix-valued Green's functions. To treat off-diagonal elements correctly based on Bayesian probability theory, the entropy term has to be extended for spectral functions that are possibly negative in some frequency ranges. In that way, all matrix elements of the Green's function matrix can be analytically continued; we introduce a computationally cheap element-wise method for this purpose. However, this method cannot ensure important constraints on the mathematical properties of the resulting spectral functions, namely positive semidefiniteness and Hermiticity. To improve on this, we present a full matrix formalism, where all matrix elements are treated simultaneously. We show the capabilities of these methods using insulating and metallic dynamical mean-field theory (DMFT) Green's functions as test cases. Finally, we apply the methods to realistic material calculations for ${\mathrm{LaTiO}}_3$, where off-diagonal matrix elements in the Green's function appear due to the distorted crystal structure.

Figure 1

Comparison of the “entropy density” of non-negative spectral functions $s\left(A\right)$ compatible to Eq. (7) and of positive and negative spectral functions $s^{\pm }\left(A\right)$ compatible to Eq. (14). The entropy is related to the entropy density via $S\left(A\right)=\int d\omega s(A\left(\omega \right))$. Here, $D^{+}=D^{-}=D$ was assumed.

Figure 2

Top: Spectral function of the rotated model in the insulating regime. Each subplot represents one matrix element, the two off-diagonal elements $A_{01}$ and $A_{10}$ being the same. The result from IPT and Padé (black) is shown along with the MEM results. For the latter, we compare the continuation treating the matrix elements independently with a flat default model (dashed green) with the poor man's matrix method, i.e., using a default model incorporating the information from the diagonal elements (blue, only for the off-diagonal elements). Furthermore, the result of the matrix formulation (red) that ensures a positive semidefinite, Hermitian spectral function is shown. Bottom: The determinant of the matrix-valued spectral function $A$ as a function of frequency. Wherever $detA\left(\omega \right)$ is negative, the matrix is not positive semidefinite.

Figure 3

Top: Imaginary part of the self-energy $\mathrm{\Sigma }(\omega +i{0}^{+})$ for the rotated model obtained with the inversion method. Each subplot represents one matrix element, the two off-diagonal elements ${\mathrm{\Sigma }}_{01}$ and ${\mathrm{\Sigma }}_{10}$ being the same. The result from IPT and Padé (black) is compared to the curves obtained with CTHYB and the MEM. The poor man's matrix method (blue) is presented alongside the full matrix method (red). For the latter, also the result where the auxiliary spectral function, $A_{\mathrm{aux}}$, has been particle-hole symmetrized (phs) is shown (dashed purple line). Bottom: The determinant of the matrix-valued auxiliary spectral function $A_{\mathrm{aux}}$ as a function of frequency. Wherever $det{A}_{\mathrm{aux}}\left(\omega \right)$ is negative, the matrix is not positive semidefinite. This happens only for the poor man's matrix method. The regions where the corresponding $A_{\mathrm{aux}}$ is not positive semidefinite are marked by the gray areas.

Figure 4

Spectral function of the rotated model in the metallic regime. Each subplot represents one matrix element, the two off-diagonal elements $A_{01}$ and $A_{10}$ being the same. The result from IPT and Padé (black) is shown along with the full matrix MEM result. For the latter, we show the result with (cyan) and without (red) using the preblur method.

Figure 5

Top: Comparison of real (red) and imaginary parts (blue) of the impurity Green's function $G_{\mathrm{imp}}(\omega +i{0}^{+})$ (solid lines) and the local lattice Green's function $G_{\mathrm{loc}}(\omega +i{0}^{+})$ (dashed lines). The former is obtained by a direct AC, whereas the latter is calculated via Eq. (28) after the AC of the self-energy. In both cases, the matrix formulation of the MEM code was used. The subplots represent different matrix elements of the Green's function. Bottom: Total spectral function (i.e., trace over the orbital and spin degrees of freedom) of the Ti-$t_{2g}$ bands from the Green's functions shown above ($G_{\mathrm{imp}}$, black, and $G_{\mathrm{loc}}$, red). For $G_{\mathrm{loc}}$, performing the AC of $\mathrm{\Sigma }$ using the poor man's matrix method is shown as well (blue). Additionally, we show the spectral function of a local Green's function where we have set the off-diagonal elements of the self-energy $\mathrm{\Sigma }\left(i{\omega }_n\right)$ to zero before individually continuing its diagonal elements and evaluating $G_{\mathrm{loc}}$ from the obtained $\mathrm{\Sigma }(\omega +i{0}^{+})$ (dashed green).

 

Figure 6

Spectral function $A\left(\omega \right)$ of the first band of the diagonal two-band model with $U_i/D_i=3.25$, calculated using Padé approximants for the IPT solution (black) and the MEM for the CTHYB solution. Different MEM codes (Bryan solution from Levy et al. [23] in blue, $\mathrm{\Omega }$maxent [22] in dashed green, and our code in red) were used; a flat default model was employed for all three cases.

Figure 7

Spectral function $A\left(\omega \right)$ of the two-band model with $U_i/D_i=3.25$, calculated using Padé approximants for the IPT solution. Starting from that solution, a $G\left(\tau \right)$ was calculated using Eq. (4); that $G\left(\tau \right)$ was then rotated in accordance to Sec. 3. The rotated IPT and Padé result is shown as the black curve. Gaussian random noise with a standard deviation of $\sigma$ was added to that $G\left(\tau \right)$, which was then analytically continued using the full matrix MEM (blue curve with $\sigma ={10}^{-8}$, dashed green curve for $\sigma =5·{10}^{-4}$). This is compared to the result from analytically continuing the $G\left(\tau \right)$ obtained by solving the same model with CTHYB (red).