Chapter 23
Fermi and Bose Gases at Non-zero
Temperature
As an example of symmetry breaking at non-zero temperature we discuss the free
Fermi and Bose gas, starting from finite volume and then discuss the thermodynamical limit.
23.1 Free Fermi and Bose Gases in Finite Volume
We consider a system of free fermions or bosons in a finite volume V with a free
Hamiltonian H 0 defined by periodic boundary conditions (for simplicity, for the
moment we omit the label V which denotes that we are in a finite volume).
139 One
can then use a Fock representation and the non-zero temperature states are the Gibbs
states. In view of the thermodynamical limit to be considered later, it is convenient
to use grand canonical states Ω, (22.2) with the chemical potential μ to be fixed in
such a way that the average density Ω(N )/V takes a given value ρ.
Since H (μ) = H 0 − μ N commutes with the number operator, all the correlation
functions with a different number of creation and annihilation operators vanish. We
adopt the usual statistical mechanics notation by which a( f ), the analog of (3.3), is
antilinear in f , a
∗
( f ) = a( f )
∗ and [a( f ), a
∗
(g)] ∓ = ( f, g), where [, ] ∓ denotes the
commutator/anticommutator and the upper/lower choice refers to the boson/fermion
case.
One easily proves that
e
w H (μ) a
∗
( f ) e
−w H (μ)
= e
−w μ a
∗
(e
w h f ), w = it, −β,
(23.1)
139 Here we give a short and simplified account; for a mathematically more complete treatment, see
O. Bratteli and D.W. Robinson, loc. cit., Vol. II, Sects. 5.2.4, 5.2.5.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_23
159
Fermi and Bose Gases at Non-zero
Temperature
As an example of symmetry breaking at non-zero temperature we discuss the free
Fermi and Bose gas, starting from finite volume and then discuss the thermodynamical limit.
23.1 Free Fermi and Bose Gases in Finite Volume
We consider a system of free fermions or bosons in a finite volume V with a free
Hamiltonian H 0 defined by periodic boundary conditions (for simplicity, for the
moment we omit the label V which denotes that we are in a finite volume).
139 One
can then use a Fock representation and the non-zero temperature states are the Gibbs
states. In view of the thermodynamical limit to be considered later, it is convenient
to use grand canonical states Ω, (22.2) with the chemical potential μ to be fixed in
such a way that the average density Ω(N )/V takes a given value ρ.
Since H (μ) = H 0 − μ N commutes with the number operator, all the correlation
functions with a different number of creation and annihilation operators vanish. We
adopt the usual statistical mechanics notation by which a( f ), the analog of (3.3), is
antilinear in f , a
∗
( f ) = a( f )
∗ and [a( f ), a
∗
(g)] ∓ = ( f, g), where [, ] ∓ denotes the
commutator/anticommutator and the upper/lower choice refers to the boson/fermion
case.
One easily proves that
e
w H (μ) a
∗
( f ) e
−w H (μ)
= e
−w μ a
∗
(e
w h f ), w = it, −β,
(23.1)
139 Here we give a short and simplified account; for a mathematically more complete treatment, see
O. Bratteli and D.W. Robinson, loc. cit., Vol. II, Sects. 5.2.4, 5.2.5.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_23
159