164
23 Fermi and Bose Gases at Non-zero Temperature
always has a solution for z(β), with z(β) < 1.
On the other hand, for T < T c one has ρ 0 (β) = 0 and, for large V , z(β, V ) ∼
1 − (ρ 0 (β) V )
−1 .
In conclusion, in the thermodynamical limit, (23.2) gives (for simplicity we consider
the case s = 3)
ω(a
∗
( f ) a(g)) = (2π)
−3
d
3 k ˜
g(k) ˜
f (k) z(β) (e
β k
2 − z(β))
−1
, β < β c , (23.13)
= ρ 0 (β) ˜
f (0) ˜
g(0) + (2π)
−3
d
3 k ˜
f (k) ˜
g(k)(e
βk
2 − 1)
−1
, β > β c .
(23.14)
Thus, below the critical temperature one has a condensation of particles in the k = 0
state (Bose–Einstein condensation), i.e. a phase transition between the gas and the
“liquid” phase.
This transition is indeed observed for liquid He
4 , at the critical temperature of
2.18
o K, not so far from the prediction of the free model discussed above, which
gives a critical temperature of 3.14
o K for the density of the liquid Helium (for more
information, see, e.g. K. Huang, Statistical mechanics, Wiley 1987).
23.4 Bose–Einstein Condensation and Symmetry Breaking
Below the critical temperature the equilibrium state ω defined by the infinite volume
limit of (23.2) does not satisfy the cluster property, since ω(a( f )) = 0 and on the
other hand, by putting g a (x) ≡ g(x + a), one has
lim
|a|→∞
ω(a
∗
( f ) a(g a )) = ρ 0 (β) ˜
f (0) ˜
g(0),
(the second term on the r.h.s. of (23.14) vanishes in the limit by the Riemann–
Lebesgue lemma).
Thus, below the critical temperature, ω can be decomposed into primary states,
which can be shown to be labelled by an angle θ and to exhibit the spontaneous
breaking of gauge transformations (defined in Sect. 17.2)
ω θ (a(g)) =
√
ρ 0 e
iθ
˜
g(0).
(23.15)
Such a decomposition can be obtained by appealing to general methods
141 . It can
also be obtained in a rather elementary way by introducing a symmetry breaking coupling with a constant external field j ext = j e
iθ , j > 0, according to the Bogoliubov
strategy
142 discussed in Chap. 20.
141 See J. Cannon, Comm. Math. Phys. 29, 89 (1973) and O. Bratteli and D.W. Robinson, loc. cit.,
Vol. II, pp. 72–73.
142 N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 2, Part 1, Gordon and Breach 1970.
23 Fermi and Bose Gases at Non-zero Temperature
always has a solution for z(β), with z(β) < 1.
On the other hand, for T < T c one has ρ 0 (β) = 0 and, for large V , z(β, V ) ∼
1 − (ρ 0 (β) V )
−1 .
In conclusion, in the thermodynamical limit, (23.2) gives (for simplicity we consider
the case s = 3)
ω(a
∗
( f ) a(g)) = (2π)
−3
d
3 k ˜
g(k) ˜
f (k) z(β) (e
β k
2 − z(β))
−1
, β < β c , (23.13)
= ρ 0 (β) ˜
f (0) ˜
g(0) + (2π)
−3
d
3 k ˜
f (k) ˜
g(k)(e
βk
2 − 1)
−1
, β > β c .
(23.14)
Thus, below the critical temperature one has a condensation of particles in the k = 0
state (Bose–Einstein condensation), i.e. a phase transition between the gas and the
“liquid” phase.
This transition is indeed observed for liquid He
4 , at the critical temperature of
2.18
o K, not so far from the prediction of the free model discussed above, which
gives a critical temperature of 3.14
o K for the density of the liquid Helium (for more
information, see, e.g. K. Huang, Statistical mechanics, Wiley 1987).
23.4 Bose–Einstein Condensation and Symmetry Breaking
Below the critical temperature the equilibrium state ω defined by the infinite volume
limit of (23.2) does not satisfy the cluster property, since ω(a( f )) = 0 and on the
other hand, by putting g a (x) ≡ g(x + a), one has
lim
|a|→∞
ω(a
∗
( f ) a(g a )) = ρ 0 (β) ˜
f (0) ˜
g(0),
(the second term on the r.h.s. of (23.14) vanishes in the limit by the Riemann–
Lebesgue lemma).
Thus, below the critical temperature, ω can be decomposed into primary states,
which can be shown to be labelled by an angle θ and to exhibit the spontaneous
breaking of gauge transformations (defined in Sect. 17.2)
ω θ (a(g)) =
√
ρ 0 e
iθ
˜
g(0).
(23.15)
Such a decomposition can be obtained by appealing to general methods
141 . It can
also be obtained in a rather elementary way by introducing a symmetry breaking coupling with a constant external field j ext = j e
iθ , j > 0, according to the Bogoliubov
strategy
142 discussed in Chap. 20.
141 See J. Cannon, Comm. Math. Phys. 29, 89 (1973) and O. Bratteli and D.W. Robinson, loc. cit.,
Vol. II, pp. 72–73.
142 N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 2, Part 1, Gordon and Breach 1970.