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CRC Press eBooks

Neutral Fermi Liquids

Philippe Nozières, David Pines

Published Mar 5, 2018• 0 citations
Publisher

Abstract

Let us consider a noninteracting Fermi gas in equilibrium at a temperature T . The probability that a single particle has energy ϵ is given by the well-known expression f ( ϵ ) = 1 1 + exp [ ( ϵ − μ ) / k T ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429495717/2cc6e3e4-f702-48c8-b407-59783ef3d6a6/content/eq1.tif"/>

DOI
10.1201/9780429495717-2

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Let us consider a noninteracting Fermi gas in equilibrium at a temperature T . The probability that a single particle has energy ϵ is given by the well-known expression f ( ϵ ) = 1 1 + exp [ ( ϵ − μ ) / k T ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429495717/2cc6e3e4-f702-48c8-b407-59783ef3d6a6/content/eq1.tif"/>
The nonequilibrium theory described in the previous chapters reduces to a particularly simple form for a system of fermions very close to zero temperature. To see this, let us define a “local occupation number” f ( p , ω ; R , T ) $ f(p, \omega ;R, T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math12_1.tif"/> by writing 12.1 g < ( p , ω ; R , T ) = a ( p , ω ; R , T ) f ( p , ω ; R , T ) g<(p, ω;R, T)=a(p, ω;R, T)f(p, ω;R, T) g^{ < } (p,~\omega ;R,~T) = a(p,~\omega ;R,~T)f(p,~\omega ;R,~T) g<(p, ω;R, T)=a(p, ω;R, T)f(p, ω;R, T) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math12_1.tif"/>
The eigenstates of a gas of non-interacting bosons are symmetrized combinations of N different single particle states. As for the case of the corresponding fermion system, the eigenstates may be characterized by a distribution function, n p , equal to 1 if the state p is occupied, zero otherwise. The system energy is ( 2.1 ) E = ∑ n n p p 2 2 m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429495717/2cc6e3e4-f702-48c8-b407-59783ef3d6a6/content/eq14_v2.tif"/>
To obtain an expression for the Fermi energy εF, deˆned in Section 12.3, consider a system of N fermions of spin 12 and mass m in a container of volume V at T = 0 K. In zero-applied magnetic ˆeld, the mS = ± 12 spin states are degenerate, and it is convenient to introduce a degeneracy factor in expressions that involve the density of single particle states. e fermions ˆll the lowest energy states up to the energy εF = µ. By equating the number of particles to the number of states in the range 0 to εF, an expression for εF is obtained. In general, for fermions, N n er r r= ∑ = ∑ +−1 1/( ).( )b e m Conversion of the sum to an integral gives N f= ∞∫2 0 r e e e( ) ( ) ,d (13.1) where ρ(ε) is the density of states and the factor 2 is the spin degeneracy factor. f ee b e m( ) = +−1 1/( )( ) is the Fermi function, which at T = 0 K has the form f(ε) = 1 for e eʺ F and f(ε) = 0 for e e> F. is may be seen in Figure 12.1. At T = 0 K, Equation 13.1 may therefore be written as N = ∫2 0 r e ee ( ) .dF (13.2) From Equation 4.14, the density of single particle states for particles in a box is r e p e( ) ( / )( / ) / /= V m4 22 2 3 2 1 2 , and substitution in Equation 13.2 gives N V m V m = ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ e pF = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 m N V .