Symbolic determinant construction of perturbative expansions

We present a symbolic algorithm for the fully analytic treatment of perturbative expansions of Hamiltonians with general two-body interactions. The method merges well-known analytics with the recently developed symbolic integration tool, algorithmic Matsubara integration, that allows for the evaluation of the imaginary frequency/time integrals. By symbolically constructing Wick contractions at each order of the perturbative expansion we order by order construct the fully analytic solution of the Green's function and self-energy expansions. A key component of this process is the assignment of momentum/frequency conserving labels for each contraction that motivates us to present a fully symbolic Fourier transform procedure which accomplishes this feat. These solutions can be applied to a broad class of quantum chemistry problems and are valid at arbitrary temperatures and on both the real- and Matsubara-frequency axis. To demonstrate the utility of this approach, we present results for simple molecular systems as well as model lattice Hamiltonians. We highlight the case of molecular problems where our results at each order are numerically exact with no stochastic uncertainty.

Figure 1

(a), (b) Real and imaginary parts of the self-energy for ${\mathrm{H}}_2$ in the STO-6g basis with external band indices $a_{\mathrm{ex}}=b_{\mathrm{ex}}=0$. (c), (d) Plots of the real and imaginary parts of ${\mathrm{\Sigma }}^{{\mathrm{H}}_2}$ for $a_{\mathrm{ex}}=b_{\mathrm{ex}}=1$. Here we took $\beta =50.0E_h^{-1}$.

Figure 2

(a), (b) Real and imaginary parts of ${\mathrm{\Sigma }}_{00}$, while (c) and (d) are the components of the self-energy for the second band for $H_2$ (in the STO-6g basis) on the real frequency axes in units of $E_h$. Here we took the regulator $\mathrm{\Gamma }=0.05$.

Figure 3

Spectral function for ${\mathrm{H}}_2$ in the STO-6g basis truncated at fourth order. Inset data is a zoom out of the extra peaks with lower intensity. Here we took the regulator 0.05.

Figure 4

Matsubara self-energy for hydrogen in the cc-pVDZ basis versus the Matsubara frequency with $\beta =50.0$. (a) The real part of the self-energy components (0,0) and (1,1) truncated at second ($n_{\max }=2$) and third ($n_{\max }=3$) orders and (b) are the imaginary counterparts.

Figure 5

(a), (b) Real and imaginary parts of the diagonal elements of the self-energy matrix for $n_{\max }=2,3$ for the Hubbard dimer model. Here we took $t=1.0, U=2.5, \mu =0.70, H=0.30, U_a=0.50, {\mu }_a=0.20, H_a=0.030$, and $\beta =2.0$.

Figure 6

(a) Real part of the (spin up) self-energy of the two dimensional Hubbard model for $t=1.0, U=3.0, \beta =8.33, \vec{k}=(0,\pi )$, and at different values of $\mu$ as indicated, and (b) are the imaginary counterparts.

Figure 7

(a), (b) Real and imaginary parts of the self-energy (truncated at third order) versus the real frequency for the 2D Hubbard model evaluated for the parameter's choice: $U=3t=3.0, \beta =8.33$, and $\vec{k}=(0,\pi )$ with different values of $\mu$ as indicated. We took a Monte Carlo sample of size $1\times {10}^8$ and the regulator $\mathrm{\Gamma }=0.2$.