Single-particle approximation to the canonical density operator of a system of N interacting fermions
type: Item
instance of: scholarly article
title: Single-particle approximation to the canonical density operator of a system of N interacting fermions
zbMATH Open document ID: 0564.46058
DOI: 10.1016/0034-4877(84)90013-2
author: Jan Maćkowiak
published in: Reports on Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: A necessary condition (nc) for the minimum of free energy \(F(\rho)=Tr(\rho H^ N)+\beta^{-1}Tr(\rho \ell n\rho)\) is derived for a system of N interacting fermions (with the Hamiltonian \(H^ N)\) on the set of all density matrices of the form \(\rho =Z^{-1}\exp (-\beta N\Gamma_ 1^ Nh^ 1)\) where \(\Gamma_ 1^ N\) is Kummer's (1,N)- expansion map \((\Gamma^ N_ 1h^ 1=A^ N(h^ 1\otimes I^{\otimes (N-1)})A^ N,\) I being the identity, \(A^ N\) the antisymmetric projection) \(Z=Tr \exp (-\beta N\Gamma_ 1^ Nh^ 1),\) \(\beta >0\), and \(h^ 1\) is a finite-dimensional 1-particle operator. The nc is derived using the well-known variational principle of statistical mechanics which states that the free energy as a functional of the state of the system at fixed temperature \(T=(k\beta)^{-1},\) \(k>0\), and with a fixed number of particles N, reaches its minimum on the canonical density operator. This implies that the necessary condition takes the form \(\delta F(\rho)/\delta h^ 1=0,\) where \(\delta /\delta \rho\) is Fréchet's derivative. It is shown that for values of \(\beta\) in an open circle \(C(0,\beta_ 0)\) in the complex plane there exists a unique bounded analytic in \(\beta\) 1-particle operator \(h^ 1\) satisfying this condition nc. \(h^ 1\) has the form of a power series in \(\beta\) which is uniformly convergent in \(C(0,\beta_ 0)\). The consecutive coefficients of this series are single-particle projections of some definite N- particle operators. The method of calculating these projections, due to S. Pruski, is also presented.
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 3899592
zbMATH Keywords: minimum of free energy
zbMATH Keywords: system of N interacting fermions
zbMATH Keywords: density matrices
zbMATH Keywords: variational principle of statistical mechanics
zbMATH Keywords: canonical density operator
zbMATH Keywords: Fréchet's derivative
zbMATH Keywords: single-particle projections of some definite N-particle operators
MaRDI profile type: Publication
full work available at URL: 0034-4877(84)90013-2
OpenAlex ID: W2010169921
cites work: Theory of Superconductivity
cites work: Some remarks on the ground state of infinite systems in statistical mechanics
cites work: Q5728332
description: scientific article; zbMATH DE number 3899592