[Paper Review] Charge conserving approximation for excitation properties of crystalline materials

B. Rosenstein, Dingping Li|arXiv (Cornell University)|Jun 11, 2018

Advanced Chemical Physics Studies44 references3 citations

TL;DR

This paper proposes a charge-conserving, non-perturbative approximation scheme called the Covariant Cubic Approximation (CCA) for calculating electron excitation properties in crystalline solids. By truncating the Dyson-Schwinger equations of a downfolded low-energy electronic model and enforcing covariance to preserve Ward identities, CCA achieves high accuracy—particularly at the third-order approximant—while remaining computationally feasible for realistic materials, as demonstrated on benchmark models including hBN.

ABSTRACT

A charge conserving approximation scheme determining the excitations of crystalline solids is proposed. Like other such approximations, it relies on "downfolding" of the original microscopic model to a simpler electronic model on the lattice with pairwise interactions. A systematic truncation of the set of Dyson - Schwinger equations for correlators of the low energy (downfolded) model of a material, supplemented by a "covariant" calculation of correlators lead to a converging series of approximates. The covariance preserves all the Ward identities among correlators describing various condensed matter probes. It is shown that the third order approximant of this kind beyond classical and gaussian (Hartree - Fock) is precise enough and due to several fortunate features the complexity of calculation is surprisingly low so that a realistic material computation is feasible. Focus here is on the electron field correlator describing the electron (hole) excitations measured in photoemission and other probes. The scheme is tested on several solvable benchmark models

Motivation & Objective

• To develop a systematically improvable, manifestly charge-conserving approximation for electron excitation spectra in crystalline solids.
• To overcome limitations of GW and other approaches that violate Ward identities and fail in strongly correlated systems.
• To enable accurate, first-principles computation of single-particle excitations such as those probed by photoemission, using a non-perturbative scheme.
• To demonstrate feasibility of the method on realistic materials by balancing accuracy and computational cost.
• To provide a viable, universal 'beyond-GW' framework that preserves fundamental symmetries and conservation laws.

Proposed method

• The method begins with a DFT-based downfolding to a low-energy effective model on the lattice, retaining only relevant bands and pairwise interactions.
• It truncates the set of Dyson-Schwinger equations for correlators in the downfolded model to generate a converging series of approximations.
• Covariance is enforced via a covariant calculation of correlators to preserve all Ward identities, ensuring charge conservation.
• The third-order approximant, termed the Covariant Cubic Approximation (CCA), is derived and shown to be both accurate and computationally tractable.
• The scheme requires solving large but sparse systems of linear chain equations in frequency-momentum space, with matrix density ~2.4×10⁻⁴ in the hBN example.
• Analytic continuation to real frequencies is performed by solving the equations across all Matsubara frequencies and k-points.

Experimental results

Research questions

• RQ1 Can a non-perturbative, charge-conserving approximation be systematically constructed for electron excitation spectra in crystalline solids?
• RQ2 Does enforcing covariance in the approximation preserve Ward identities and thus ensure charge conservation in correlated electron systems?
• RQ3 Is the third-order approximant in the truncated Dyson-Schwinger series sufficiently accurate for realistic materials while remaining computationally feasible?
• RQ4 Can this method outperform standard GW and other 'beyond-GW' approaches in strongly correlated regimes where Ward identity violations cause inaccuracies?
• RQ5 What is the computational scaling and memory requirement of solving the resulting chain equations for realistic materials like hBN?

Key findings

• The third-order approximant of the CCA scheme is sufficiently precise for accurate description of excitation properties, as validated on solvable benchmark models.
• The method preserves all Ward identities through covariance, ensuring fundamental conservation laws are respected, unlike many existing 'beyond-GW' approaches.
• For hexagonal boron nitride (hBN), the number of chain equations is ~2.6×10⁸, but the matrix density is low (~2.4×10⁻⁴), enabling sparse matrix solvers.
• The computational cost is surprisingly low due to favorable structural features, making first-principles calculations on realistic materials feasible with current hardware.
• The scheme requires solving the chain equations n times (n ≈ 32,768 for hBN) to cover all frequencies and k-points, but exact solvers like LAPACK are viable.
• The method is applicable to photoemission and other probes by directly computing the electron Green's function, which describes single-particle (hole) excitations.