Scalable tensor network algorithm for quantum impurity problems

TL;DR

This work tackles the exponential scaling of GTEMPO when treating many impurity flavors by introducing a multi-flavor extension that integrates out nonessential flavors to form reduced augmented density tensors for two-point observables. The method builds on the impurity path integral and uses reduced ADTs A_pq, computed with on-the-fly zipup contractions and controlled by bond dimensions \\chi and \\chi_2, to obtain accurate Matsubara Green's functions for up to three orbitals (six flavors). Across Toulouse, single-, and multi-orbital Anderson-like models, the approach achieves CTQMC-level accuracy with modest \\chi and \\chi_2, and converges DMFT iterations on the Bethe lattice with errors around 10^-5 to 10^-2, demonstrating scalable applicability to large-scale quantum impurity problems. The results suggest that, due to partial integration of unused flavors, the method preserves accuracy while mitigating the exponential cost, and it holds promise for real-time and non-equilibrium impurity simulations on larger systems.

Abstract

The Grassmann time-evolving matrix product operator method has shown great potential as a general-purpose quantum impurity solver, as its numerical errors can be well-controlled and it is flexible to be applied on both the imaginary- and real-time axis. However, a major limitation of it is that its computational cost grows exponentially with the number of impurity flavors. In this work, we propose a multi-flavor extension of it to overcome this limitation. The key insight is that to calculate multi-time correlation functions on one or a few impurity flavors, one could integrate out the degrees of freedom of the rest flavors before hand, which could greatly simplify the calculation. The idea is particularly effective for quantum impurity problems with diagonal hybridization function, i.e., each impurity flavor is coupled to an independent bath, a setting which is commonly used in the field. We demonstrate the accuracy and scalability of our method for the imaginary time evolution of impurity problems with up to three impurity orbitals, i.e., 6 flavors, and benchmark our results against continuous-time quantum Monte Carlo calculations. Our method paves the way of scaling up tensor network algorithms to solve large-scale quantum impurity problems.

Figures (6)

• Figure 1: (a) Schematical illustration of the zipup algorithm to to calculate the partition function for an impurity problem with $2$ flavors, where the quasi-2D tensor network is contracted (integrating out the pairs of conjugate Grassmann variables) from left to right and the augmented density tensor $\mathcal{A}$ is calculated on the fly as indicated by the dashed $\times$. (b) The scheme used in the multi-flavor GTEMPO method to calculate the partition function based on the reduced ADT $\mathcal{A}_2$ for the second flavor, in which one first calculate $\mathcal{K}\mathcal{I}_{\overline{1}}$ by multiplying $\mathcal{K}$ and $\mathcal{I}_1$ and integrating out the first flavor (b1), and then multiply $\mathcal{K}\mathcal{I}_{\overline{1}}$ and $\mathcal{I}_2$ to obtain $\mathcal{A}_2$ (b2). Again the second multiplication is only performed on the fly similar to (a). The empty circles in (b1) mean that these Grassmann variables do not exist in the corresponding GMPS, while the gray box means that the Grassmann variables inside it will be integrated out after multiplication.
• Figure 2: Mean error of $G(\tau)$ between GTEMPO and the analytical solution, as a function of $\delta \tau$ and $\chi$, for the Toulouse model with $\beta=10$ and $\epsilon_d = 1$.
• Figure 3: $G(\tau)$ for the Toulouse model at (a) $\beta = 10$ and (b) $\beta = 50$. The gray solid lines and the black dashed lines represent the analytical solutions and the GTEMPO results respectively. The left inset in both panels shows the mean error between GTEMPO and the analytical solution as a function of $\chi$, while the right inset shows the absolute error of $G(\tau)$ between GTEMPO and the analytical solution. We have used $\chi=60$ in the main panel (a) and its right inset, and used $\chi=200$ in the main panel (b) and its right inset.
• Figure 4: $G(\tau)$ as a function of $\tau$ for the single-orbital Anderson impurity model with impurity Hamiltonian in Eq.(\ref{['eq:AIM']}), under different parameter settings as shown in the titles. The black dashed lines and the gray solid lines represent the multi-flavor GTEMPO results and the CTQMC results respectively. The left inset in all panels shows the mean error between multi-flavor GTEMPO and CTQMC as a function of $\chi_2$, the right inset shows the absolute error as a function of $\tau$. We have used $\chi=60$ for all the simulations in (a,c), and $\chi=200$ for all the simulations in (b,d). For all the main panels, and their right insets, we have used $\chi_2=60$.
• Figure 5: $G(\tau)$ for the two-orbital (a) and three-orbital (b) impurity problems with impurity Hamiltonian in Eq.(\ref{['eq:Kanamori']}). The gray solid lines in both panels represent the CTQMC results. The red, green and blue dashed lines represent multi-flavor GTEMPO results with $\chi_2=100,200,300$ in (a), and represent multi-flavor GTEMPO results with $\chi_2=300,500,700$ in (b). The insets show the absolute errors between multi-flavor GTEMPO and CTQMC results, where the colored solid line corresponds to multi-flavor GTEMPO result plotted in dashed line with the same color in the main panel.