Compressing Green's function using intermediate representation between imaginary-time and real-frequency domains

Model-independent compact representations of imaginary-time data are presented in terms of the intermediate representation (IR) of analytical continuation. We demonstrate the efficiency of the IR through continuous-time quantum Monte Carlo calculations of an Anderson impurity model. We find that the IR yields a significantly compact form of various types of correlation functions. This allows the direct quantum Monte Carlo measurement of Green's functions in a compressed form, which considerably reduces the computational cost and memory usage. Furthermore, the present framework will provide general ways to boost the power of cutting-edge diagrammatic/quantum Monte Carlo treatments of many-body systems.

Figure 1

Analytical continuation between real-frequency data $\rho \left(\omega \right)$ and Matsubara-frequency data $G\left(i{\omega }_n\right)$ through the kernel $K$. The intermediate representation is defined in terms of a SVD of $K$.

Figure 2

Upper panel shows the singular values computed for the fermionic and bosonic kernels. The data for $\mathrm{\Lambda }=10000$ are multiplied by a constant for better readability. The lower panel shows the basis functions in the imaginary-time domain $\left[u_l\left(x\right)\right]$ and in the real-frequency domain $\left[v_l\left(y\right)\right]$ computed for the fermionic kernel. The solid gray lines show Legendre polynomials.

Figure 3

Single-particle Green's function computed for the model (13) with $U=4$ and $\beta =100$. Upper panel: Expansion coefficients $G_l$. We show only data for even $l$ since $G_l$ for odd $l$ are zero due to the particle-hole symmetry. Lower panel: $G\left(\tau \right)$ reconstructed from a few small-$l$ coefficients. All the data are averaged along the spin index.

Figure 4

Charge susceptibility computed for the model (13) with $U=4$ and $\beta =100$. The upper panel shows the $\tau$ dependence. The middle and lower panels show the coefficients in terms of the bosonic and fermionic IRs.

Figure 5

Coefficients of generalized spin susceptibilities of the model (13) measured in the mixed representation of bosonic Matsubara frequencies and the fermionic IR. We plot data along the diagonal line $l=l^{\prime }$ (left panel) and those along the line $l^{\prime }=0$ (right panels). The upper and lower panels show data for $\beta =25$ and 100, respectively. We subtract the contributions of the bubble diagram from the data and show data for even $l$ and $l^{\prime }$.

Figure 6

Relative computational time of measurement of two-particle Green's function at bosonic frequency ${\omega }_m=0$ for $\beta =100$. We employed 50 (Legendre), 10 ($\mathrm{\Lambda }=500$), and 20 ($\mathrm{\Lambda }=10000$) basis functions in the measurement.

Figure 7

Basis functions of the IR for the fermionic kernel (left panel) and bosonic kernel (right panel). We plot the differences between the basis functions and the Legendre polynomials.