The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations

Abstract

The derivation of effective evolution equations is central to the study of non-stationary quantum many-body sytems, and widely used in contexts such as superconductivity, nuclear physics, Bose-Einstein condensation and quantum chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of reduced density matrices, and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov-de-Gennes equations in energy space and discuss conserved quantities.

arXiv comment46 pages, 1 figure; v2: simplified proof of conservation of particle number, additional references; v3: minor clarifications

References in corpus (26)

• Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems
• Mean-Field Dynamics: Singular Potentials and Rate of Convergence
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• Second-order corrections to mean field evolution for weakly interacting Bosons. I
• Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential
• Pair excitations and the mean field approximation of interacting Bosons, I
• Rate of Convergence Towards Hartree Dynamics
• Mean-field Evolution of Fermionic Systems
• Quantitative Derivation of the Gross-Pitaevskii Equation
• Second-order corrections to mean-field evolution of weakly interacting Bosons, II
• Mean field evolution of fermions with Coulomb interaction
• Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction
• Quantum many-body fluctuations around nonlinear Schrödinger dynamics
• Pair excitations and the mean field approximation of interacting Bosons, II
• Minimizers for the Hartree-Fock-Bogoliubov Theory of Neutron Stars and White Dwarfs
• Mean-Field Dynamics of Fermions with Relativistic Dispersion
• Derivation of the nonlinear Schrödinger equation from a many body Coulomb system
• A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics
• On Blowup for time-dependent generalized Hartree-Fock equations
• Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction
• A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on ${\mathbb{T}}^3$ from the dynamics of many-body quantum systems
• On the dynamics of polarons in the strong-coupling limit
• The Grillakis-Machedon-Margetis Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions
• Hartree Corrections in a Mean-field Limit for Fermions with Coulomb Interaction
• The time-dependent Hartree-Fock-Bogoliubov equations for Bosons
• On the Bogolubov-de Gennes Equations

Cited by in corpus (17)

• Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime
• Correlation Energy of a Weakly Interacting Fermi Gas
• Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential
• Mean-field Dynamics for the Nelson Model with Fermions
• Bosonic Collective Excitations in Fermi Gases
• Bosonization of Fermionic Many-Body Dynamics
• Uniform in $N$ Global Well-posedness of the Time-Dependent Hartree-Fock-Bogoliubov Equations in ${\mathbb{R}}^{1+1}$
• Global estimates for the Hartree-Fock-Bogoliubov equations
• Joint quantum-classical Hamilton variation principle in the phase space
• On the Hartree-Fock-Bogoliubov equations
• Dynamics of mean-field Fermi systems with nonzero pairing
• Effective quantum dynamics for magnetic fermions
• Uniform in $N$ estimates for a Bosonic system of Hartree-Fock-Bogoliubov type
• Differential Equations of Quantum Mechanics
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate
• Vortex Lattices and the Bogoliubov-De Gennes Equations
• Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation