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CRC Press eBooks

Elementary Excitations

Philippe Nozières, David Pines

Published Mar 5, 2018• 2 citations
Publisher

Abstract

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"/>

DOI
10.1201/9780429495717-9

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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"/>
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"/>
A free-electron Fermi gas is a gas of electrons with spin https://www.w3.org/1998/Math/MathML" display="inline"> S = 1 / 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003606857/19698a40-dbd9-4f9a-9cb7-890a1d39adce/content/math8_1.tif "/> . A fermion is a particle with half-integer spin S such as the electron. A boson is a particle with integer spin https://www.w3.org/1998/Math/MathML" display="inline"> S ≥ 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003606857/19698a40-dbd9-4f9a-9cb7-890a1d39adce/content/math8_2.tif "/> . By a gas is meant that there are no interactions between the particles making up the gas. Thus a Fermi gas can exhibit no phase transitions by itself. On the other hand, a free boson gas can exhibit an interesting quantum phase transition into a superfluid state, qualitatively similar to that of the superconducting state of a metal, called Bose-Einstein condensation . In contrast, an electron Fermi liquid would be a system containing electrons that do interact with each other. We will see in the last two chapters of this book that when such interactions occur, magnetic or superconducting transitions associated with the electrons can happen.
The ground state of a free electron gas corresponds to complete filling of the one-electron energy levels of wavevector k and energy ħ 2 k 2 /2m up to a certain energy E F = ћ 2 k F 2 / 2 m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497032/d2b0510a-8f84-4d3a-8776-dff0770e5b14/content/eq458.tif"/> (the Fermi energy). However, in the presence of an attractive interaction, no matter how weak, this state becomes unstable (Cooper, 1957). The instability can be understood by considering two particular electrons of coordinates r 1 and r 2 , the other electrons still being treated as a free electron gas. The only effect of this electron gas is to forbid the two electrons to occupy all states k < k F by the exclusion principle. Let ψ ( r 1 , r 2 ) be the wavefunction of the two electrons. Consider only states where the center of gravity of the pair ( r 1 , r 2 ) is at rest; ψ is then only a function of r 1 – r 2 . Expand ψ in plane waves ( 4-1 ) ψ ( r 1 − r 2 ) = ∑ k g ( k ) e i k ⋅ ( r 1 − r 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497032/d2b0510a-8f84-4d3a-8776-dff0770e5b14/content/eq459.tif"/>