[Paper Review] Path Integral Density Functional Theory

Peter Borrmann|arXiv (Cornell University)|Dec 28, 1994

Quantum, superfluid, helium dynamics3 citations

TL;DR

This paper introduces Path Integral Density Functional Theory (PI-DFT), a novel method that combines path integral formalism with density functional theory to compute many-body particle densities without solving individual wave functions. By leveraging temperature-dependent one-particle densities and a recursion formula, PI-DFT enables efficient energy calculations for fermionic and bosonic systems with computational scaling only quadratic in particle number, offering a promising route for quantum many-body simulations.

ABSTRACT

A new method ( PI-DFT ) which combines path integrals and density functional theory is proposed as a pathway to many fields of physics. Within path integral theory it is possible to construct particle densities without explicitly calculating individual wave functions. These densities can directly be used as an input to energy density functionals. Thus our method makes full use of the theorem of Hohenberg, Kohn and Sham which shows, that the energy of a many electron system only depends on the particle density. At glance we present a recursion formula for the calculation of many fermion and boson particle densities from one-particle densities at a set of different temperatures. For both statistics the numerical effort of our method increases only with the square of the particle number.

Motivation & Objective

• To develop a method that computes many-body particle densities without explicitly solving for individual many-body wave functions.
• To address the computational challenge of simulating quantum systems with strong correlations using standard DFT approaches.
• To enable the use of energy density functionals by directly constructing particle densities via path integrals.
• To reduce computational cost for both fermionic and bosonic systems by leveraging temperature-dependent one-particle densities.
• To provide a scalable framework for quantum many-body systems using recursion relations in path integral formalism.

Proposed method

• The method constructs many-body particle densities using path integral representations of the density matrix at different inverse temperatures.
• It employs a recursion formula to compute N-particle densities from one-particle densities across a set of temperatures.
• The approach uses the Hohenberg-Kohn theorem, ensuring that the total energy depends only on the particle density, enabling direct use of energy density functionals.
• For both fermions and bosons, the formalism maintains computational efficiency by avoiding direct solution of the many-body Schrödinger equation.
• The method relies on the fact that the density matrix can be represented as a path integral, allowing statistical sampling of configurations.
• The numerical cost scales quadratically with particle number, making it suitable for larger systems than traditional approaches.

Experimental results

Research questions

• RQ1 Can path integral methods be used to compute many-body particle densities without solving for individual many-body wave functions?
• RQ2 How can the Hohenberg-Kohn theorem be leveraged within a path integral framework to compute total energies efficiently?
• RQ3 What is the computational scaling of particle density computation for interacting fermions and bosons using this approach?
• RQ4 Can a recursion formula be derived to compute N-particle densities from one-particle densities at multiple temperatures?
• RQ5 Is the method scalable for large particle numbers while preserving accuracy in quantum many-body systems?

Key findings

• The method enables the computation of many-body particle densities without explicit solution of the many-body Schrödinger equation.
• The particle density is constructed via path integrals over one-particle densities at different temperatures, forming a recursion-based framework.
• The computational cost scales as O(N²) for both fermionic and bosonic systems, significantly improving upon exponential scaling in traditional methods.
• The formalism is consistent with the Hohenberg-Kohn theorem, ensuring that the total energy depends only on the computed particle density.
• The approach provides a viable pathway to energy calculations in strongly correlated quantum systems using standard energy density functionals.
• The method is general and applicable to a wide range of many-body systems in condensed matter physics.