[Paper Review] On the pure state $v$-representability of density matrix embedding theory

Fabian M. Faulstich, Raehyun Kim+4 more|arXiv (Cornell University)|Oct 18, 2021

Physics of Superconductivity and Magnetism57 references4 citations

TL;DR

This paper proposes an augmented Lagrangian method (alm-DMET) to relax the non-interacting pure-state v-representability (NI-PS-V) assumption in density matrix embedding theory (DMET), enabling exact matching of high-level and low-level density matrix blocks even when the Aufbau principle fails. By allowing arbitrary orbital occupation profiles while preserving idempotency, the method avoids the spurious gaplessness in the low-level Hamiltonian and achieves improved accuracy in challenging 2D Hubbard and hydrogen models.

ABSTRACT

Density matrix embedding theory (DMET) formally requires the matching of density matrix blocks obtained from high-level and low-level theories, but this is sometimes not achievable in practical calculations. In such a case, the global band gap of the low-level theory vanishes, and this can require additional numerical considerations. We find that both the violation of the exact matching condition and the vanishing low-level gap are related to the assumption that the high-level density matrix blocks are non-interacting pure-state $v$-representable (NI-PS-V), which assumes that the low-level density matrix is constructed following the Aufbau principle. In order to relax the NI-PS-V condition, we develop an augmented Lagrangian method to match the density matrix blocks without referring to the Aufbau principle. Numerical results for 2D Hubbard and hydrogen model systems indicate that in some challenging scenarios, the relaxation of the Aufbau principle directly leads to exact matching of the density matrix blocks, which also yields improved accuracy.

Motivation & Objective

• To address the failure of exact density matrix matching in DMET due to the non-viable NI-PS-V assumption.
• To resolve the issue of vanishing low-level band gaps that arise when the Aufbau principle is violated.
• To develop a robust method that allows exact matching of density matrix blocks without relying on the Aufbau principle.
• To improve numerical stability and accuracy in strongly correlated systems where standard DMET fails.
• To demonstrate the effectiveness of the proposed method on 2D Hubbard and hydrogen chain models under challenging conditions.

Proposed method

• Formulate the correlation potential fitting as a constrained optimization problem using an augmented Lagrangian method to enforce exact density matrix matching.
• Relax the NI-PS-V condition by allowing arbitrary occupation profiles in the low-level 1-RDM, while preserving idempotency (occupation numbers restricted to 0 or 1).
• Use a projected gradient method to solve the constrained optimization, with a penalty parameter and line search for convergence.
• Introduce hyperparameters (e.g., initial penalty, maximum inner iterations) to control convergence and reduce computational cost.
• Bound the number of inner iterations to limit full system diagonalizations and reduce computational overhead.
• Implement an adaptive hyperparameter update strategy every 10 outer iterations to improve convergence behavior.

Experimental results

Research questions

• RQ1 Can exact density matrix matching be achieved in DMET when the high-level 1-RDM blocks are not NI-PS-V?
• RQ2 Does relaxing the Aufbau principle in the low-level 1-RDM construction eliminate the spurious gaplessness in the low-level Hamiltonian?
• RQ3 Can the augmented Lagrangian method improve numerical robustness and convergence in DMET for strongly correlated systems?
• RQ4 How does the performance of the proposed alm-DMET method compare to standard least-squares DMET in terms of accuracy and convergence?
• RQ5 What is the impact of hyperparameter configuration on the convergence and computational cost of alm-DMET?

Key findings

• In challenging cases such as the hole-doped 2D Hubbard model (n = 32/36), the relaxation of the Aufbau principle enabled exact matching of the density matrix blocks.
• The alm-DMET method successfully avoided the vanishing low-level band gap that plagues standard DMET when the NI-PS-V condition fails.
• Numerical results showed improved accuracy in energy and density matrix matching compared to standard least-squares DMET, particularly in strongly correlated regimes.
• The convergence of alm-DMET was independent of the initial RHF guess, as demonstrated on the H36 chain system with different initializations.
• Limiting the number of inner iterations reduced computational cost and stabilized convergence, though optimal hyperparameter tuning remains critical for performance.
• The method achieved identical energy trajectories across different hyperparameter settings, indicating robustness in energy convergence despite varying iteration counts.