[Paper Review] The plain and simple parquet approximation: single- and multi-boson exchange in the two-dimensional Hubbard model

Friedrich Krien, Anna Kauch|arXiv (Cornell University)|Feb 1, 2022

Physics of Superconductivity and Magnetism51 references22 citations

TL;DR

This paper presents a numerically efficient implementation of the bosonized parquet approximation for the two-dimensional Hubbard model at weak coupling, enabling full momentum and frequency resolution on a 16×16 lattice. By reformulating the parquet equations in terms of bosonic exchange diagrams (single- and multi-boson exchange), the method achieves faster convergence and reduced computational cost, allowing for unbiased evaluation of vertex functions and benchmarking of the truncated unity approximation in the presence of long-ranged antiferromagnetic correlations.

ABSTRACT

The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a $16 imes16$ lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated unity approximation.

Motivation & Objective

• . The paper aims to enable unbiased, large-scale parquet calculations for the Hubbard model by overcoming computational bottlenecks in traditional fermionic formulations.
• It seeks to validate the bosonized parquet formalism as a practical and efficient alternative to standard parquet implementations.
• The study investigates the convergence and accuracy of the truncated unity approximation when applied to bosonized vertex functions.
• It benchmarks the performance of the method in the presence of long-ranged antiferromagnetic correlations, particularly in the spin channel.
• The work aims to provide a reference framework for future studies of vertex corrections, Ward identities, and sum rules in correlated electron systems.

Proposed method

• . The authors employ a reformulation of the parquet equations in terms of U-irreducible vertices (˜Λ = Λ − U), enabling a bosonized description of vertex corrections.
• They separate the vertex functions into single-boson exchange (SBE) and multi-boson exchange (M) contributions, with SBE represented via bolded γ and W quantities.
• The method treats SBE diagrams exactly and uses their asymptotic decay to reduce the number of Matsubara frequencies required in summations.
• The full momentum and frequency dependence of the vertex functions is retained, with parametrization via the truncated unity (TU) approximation tested on M and ∆ functions.
• The algorithm is implemented on a 16×16 square lattice at half-filling, allowing for full resolution of momentum and frequency structures.
• Convergence of the truncated unity is assessed by varying the number of form factors (Nℓ), with comparisons across different parametrization schemes (Φsp, Msp, Msp + ∆spR).

Experimental results

Research questions

• RQ1 . How does the bosonized parquet formalism improve computational efficiency compared to the standard fermionic formulation in large-scale Hubbard model calculations?
• RQ2 To what extent does the truncated unity approximation converge when applied to the bosonized vertex functions, especially in the presence of long-ranged antiferromagnetic correlations?
• RQ3 How do the momentum and frequency dependencies of the vertex functions (Φsp, Msp, ∆sp) behave in the spin channel at weak coupling?
• RQ4 Does the relative importance of Msp compared to ∆sp depend on the correlation length ξ, as measured by the convergence of the truncated unity?
• RQ5 Can the vertex asymptote be effectively extended to low frequencies using the TU approximation in the bosonized framework?

Key findings

• . The bosonized parquet method enables the first full-fermionic parquet calculation on a 16×16 lattice, retaining full momentum and frequency resolution.
• The relative importance of Msp compared to ∆sp remains approximately constant with increasing correlation length ξ, indicating stable convergence of the truncated unity for Msp.
• The truncated unity applied to Msp + ∆spR shows significantly slower convergence at larger U/t (e.g., U=4t), indicating that combining these terms reduces parametrization efficiency.
• The convergence of the truncated unity is fastest when applied only to Msp, consistent with the fact that Msp has weaker momentum dependence than ∆sp.
• The vertex asymptote provides a reliable low-frequency parametrization of the vertex functions, extending its utility beyond high frequencies.
• The method enables unbiased evaluation of vertex functions, facilitating future checks of Ward identities and sum rules in correlated systems.