[Paper Review] Feynman's solution of the quintessential problem in solid state physics

Kun Chen, Kristjan Haule|arXiv (Cornell University)|Sep 12, 2018

Advanced Chemical Physics Studies34 references4 citations

TL;DR

This paper presents a novel diagrammatic quantum Monte Carlo method that combines Feynman’s variational principle with a sign-blessed diagram grouping technique to solve the uniform electron gas—a cornerstone problem in solid-state physics. By optimizing the screening parameter via the principle of minimal sensitivity and organizing Feynman diagrams into conserving groups, the method achieves unprecedented accuracy in calculating momentum- and frequency-resolved spin and charge response functions, revealing previously undetected fine structures.

ABSTRACT

Two of the most influential ideas developed by Richard Feynman are the Feynman diagram technique and his variational approach. The former provides a powerful tool to construct a systematic expansion for a generic interacting system, while the latter allows optimization of a perturbation theory using a variational principle. Here we show that combining a variational approach with a new diagrammatic quantum Monte Carlo method, both based on the Feynman's original ideas, results in a powerful and accurate solver to the generic solid state problem, in which a macroscopic number of electrons interact by the long range Coulomb repulsion. We apply the solver to the quintessential problem of solid state, the uniform electron gas (UEG), which is at the heart of the density functional theory (DFT) success in describing real materials, yet it has not been adequately solved for over 90 years. While some wave-function properties, like the ground state energy, have been very accurately calculated by the diffusion Monte Carlo method (DMC), the static and dynamic response functions, which are directly accessed by the experiment, and are crucial for development of non-local and non-adiabatic density functionals, remain poorly understood. Our method allows us to calculate the momentum-frequency resolved spin response functions for the first time, and to improve on the precision of the charge response function. The accuracy of both response functions is sufficiently high, so as to uncover previously missed fine structure in these responses. This method can be straightforwardly applied to a large number of moderately interacting electron systems in the thermodynamic limit, including realistic models of metallic and semiconducting solids.

Motivation & Objective

• To address the long-standing challenge of accurately calculating the static and dynamic response functions of the uniform electron gas (UEG), a fundamental problem in solid-state physics.
• To overcome the limitations of traditional perturbation theory and Monte Carlo methods in handling strong correlations and fermionic sign problems.
• To develop a systematic, convergent approach for computing response functions in interacting electron systems using Feynman’s diagrammatic framework.
• To enable high-accuracy, thermodynamically consistent calculations of response functions in realistic metallic and semiconducting materials.

Proposed method

• The method employs a variational starting point with a screened Coulomb interaction and a physical Fermi surface, optimized via the principle of minimal sensitivity (PMS) to accelerate convergence.
• It introduces a novel grouping of Feynman diagrams into 'sign-blessed' conserving groups using Hugenholtz diagrams and free-energy generating functionals, reducing sign cancellation.
• The algorithm systematically generates all topologically distinct polarization diagrams by attaching external vertices to internal propagators in a way that preserves shared propagators and momentum loops.
• The method leverages fermionic crossing symmetry and momentum/frequency loop optimization to minimize the number of independent propagators and reduce statistical variance.
• A binary tree structure is used to efficiently compute the sum of $2^N$ Feynman diagrams belonging to a single Hugenholtz diagram, reducing computational cost from $O(2^N)$ to $O(N)$.
• The approach ensures conservation laws are preserved at the diagrammatic level by constructing polarization diagrams from connected $\ln Z$ diagrams of the previous order.

Experimental results

Research questions

• RQ1 Can a variational diagrammatic Monte Carlo method achieve high-accuracy, convergent results for the response functions of the uniform electron gas?
• RQ2 How can the fermionic sign problem be mitigated in high-order diagrammatic expansions for interacting electron systems?
• RQ3 What is the role of screening and Fermi surface renormalization in accelerating the convergence of response function calculations?
• RQ4 Can the structure of conserving diagram groups be systematically exploited to reduce computational cost and improve statistical efficiency?
• RQ5 What fine structure in the spin and charge response functions of the UEG remains hidden in previous approximations?

Key findings

• The method achieves unprecedented accuracy in calculating the momentum- and frequency-resolved spin response function of the uniform electron gas for the first time.
• The charge response function is computed with higher precision than previous methods, revealing previously undetected fine structure.
• The use of variational screening parameter optimization via the principle of minimal sensitivity leads to dramatically improved convergence of response functions with increasing expansion order.
• The sign-blessed grouping of diagrams reduces statistical variance and enables efficient computation of $2^N$ Feynman diagrams using only $O(N)$ operations via a binary tree structure.
• The method successfully preserves conservation laws at the diagrammatic level, ensuring physical consistency in the computed response functions.
• The approach is generalizable to a wide range of moderately interacting electron systems in the thermodynamic limit, including realistic metallic and semiconducting solids.