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23 Fermi and Bose Gases at Non-zero Temperature
Now we discuss the limit j → 0 by distinguishing two cases:
1) T > T c . In this case, for any given ρ, eq. (23.18), with j = 0, always has a solution for z = z(β), with 0 < z(β) < 1. Thus, by choosing z so that z(β, ∞, j) →
z(β), when j → 0, one gets in this limit
ρ(β, j) → ρ, ρ 0 (β, j) → 0.
Thus, ω j (a( f )) → 0, independently of the way je
iθ
→ 0, and therefore there
is a unique phase.
2) T < T c . In this case, ρ − ρ
(β) > 0 and therefore one must have
ρ 0 (β, j) ≡ β
2 j
2
/(ln z)
2 j→0
−→ ρ − ρ
(β) ≡ ρ 0 (β) > 0.
This requires to choose z in such a way that z(β, ∞, j) → 1, as j → 0; it suffices
to take z(β, ∞, j) = exp[−β j (ρ 0 (β))
−1/2
]. Then
lim
j→0
ω j (a( f )) = (ρ 0 (β))
1/2 e
iθ ˜
f (0) ≡ ω θ (a( f )).
(23.20)
Thus, the limit depends on the phase θ of the external field and one has a oneparameter family of equilibrium states ω θ , θ ∈ [0, 2π). As it is easy to see, all
such states satisfy the cluster property; therefore, they are primary states on
the Weyl algebra and define pure phases. Clearly, each state ω θ is not invariant
under gauge transformations, which are therefore spontaneously broken in each
representation defined by ω θ .