[Paper Review] Comparison of MPS based real time evolution algorithms for Anderson Impurity Models

Daniel Bauernfeind, Markus Aichhorn+1 more|arXiv (Cornell University)|Jun 21, 2019

Quantum and electron transport phenomena4 citations

TL;DR

This paper compares Matrix Product State (MPS)-based real-time evolution algorithms—TEBD and TDVP—on Anderson Impurity Models using both star-geometry and Wilson chain-geometry bath representations. Surprisingly, TEBD in star geometry achieves superior efficiency, being up to five times faster than TDVP in chain geometry for the same accuracy, making it the most efficient method for impurity solvers in DMFT calculations.

ABSTRACT

We perform a detailed comparison of two Matrix Product States (MPS) based time evolution algorithms for Anderson Impurity Models. To describe the bath, we use both the star-geometry as well as the commonly employed Wilson chain geometry. For each bath geometry, we use either the Time Dependent Variational Principle (TDVP) or the Time Evolving Block Decimation (TEBD) to perform the time evolution. To apply TEBD for the star-geometry, we use a specially adapted algorithm that can deal with the long-range coupling terms. Analyzing the major sources of errors, one expects them to be proportional to the system size for all algorithms. Surprisingly, we find errors independent of system size except for TEBD in chain geometry. Additionally, we show that the right combination of bath representation and time evolution algorithm is important. While TDVP in chain geometry is a very precise approach, TEBD in star geometry is much faster, such that for a given accuracy it is superior to TDVP in chain geometry. This makes the adapted version of TEBD in star geometry the most efficient method to solve impurity problems.

Motivation & Objective

• To evaluate the accuracy and efficiency of MPS-based real-time evolution algorithms—TEBD and TDVP—for solving Anderson Impurity Models.
• To investigate how bath geometry (star vs. chain) affects algorithmic performance in terms of error scaling and computational cost.
• To determine the optimal combination of time evolution algorithm and bath representation for high-precision, low-cost impurity solver calculations in DMFT.
• To resolve conflicting expectations about error scaling, particularly whether errors grow with system size.

Proposed method

• The study employs both star-geometry and Wilson chain-geometry representations of the bath, with bath parameters derived from the hybridization function.
• For time evolution, TEBD and TDVP are applied to both bath geometries, with a specially adapted TEBD algorithm using swap gates to handle long-range couplings in the star geometry.
• The impurity Green’s function is computed via real-time evolution up to t=15, using DMRG to initialize the MPS ground state for N=59 sites.
• Error analysis is performed by comparing results against reference calculations using TDVP in chain geometry (Δt=0.005, tw=10−13) and TEBD in star geometry (Δt=0.005, tw=10−14).
• Systematic variation of time step (Δt) and truncated weight (tw) allows quantification of error and computation time trade-offs.
• Analytical error estimates for Suzuki-Trotter decomposition are derived for both geometries to understand scaling behavior.

Experimental results

Research questions

• RQ1 Does the bath geometry significantly affect the accuracy and computational cost of MPS-based real-time evolution in impurity models?
• RQ2 Why does TEBD in star geometry show favorable error scaling independent of system size, contrary to expectations?
• RQ3 Is there a combination of time evolution algorithm and bath geometry that achieves both high accuracy and low computational cost for DMFT applications?
• RQ4 How do the errors of TEBD and TDVP scale with system size in star and chain geometries, respectively?
• RQ5 Can the adapted TEBD algorithm for star geometry outperform standard TDVP in chain geometry in terms of efficiency for a given error tolerance?

Key findings

• TEBD in star geometry achieves a computation time up to five times lower than TDVP in chain geometry for the same target error, making it the most efficient method.
• Only TEBD in chain geometry shows error scaling proportional to system size; all other combinations—including TEBD in star geometry—exhibit nearly system-size-independent errors.
• The favorable error scaling in TEBD for star geometry arises because bath states are already close to the single-particle eigenbasis, suppressing dominant error contributions.
• TDVP in chain geometry yields the lowest absolute error for a given set of parameters, but at the cost of significantly higher computation time.
• The adapted TEBD algorithm with swap gates enables efficient time evolution in star geometry without additional computational overhead.
• For DMFT applications requiring large bath sizes to accurately represent hybridization, the system-size-independent error scaling of TEBD in star geometry is a critical advantage.