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

TY - JOUR

T1 - The Dirac–Frenkel Principle for Reduced Density Matrices, and the Bogoliubov–de Gennes Equations

AU - Benedikter, Niels

AU - Sok, Jérémy

AU - Solovej, Jan Philip

PY - 2018/4/1

Y1 - 2018/4/1

N2 - The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, 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.

AB - The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, 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.

U2 - 10.1007/s00023-018-0644-z

DO - 10.1007/s00023-018-0644-z

M3 - Journal article

SN - 1424-0637

VL - 19

SP - 1167

EP - 1214

JO - Annales Henri Poincare

JF - Annales Henri Poincare

IS - 4

ER -