22.4 Pure Phases. Extremal and Primary KMS States
157
II. (Thermodynamical stability) The representation is defined by a KMS state.
III. (Equilibrium state) The KMS state is the unique translationally invariant state.
The pure phases are defined by extremal KMS states.
For extremal KMS states, condition III can be replaced by the validity of the cluster
property, as in Chap. 16, thanks to Proposition 16.4, which states such an equivalence
for factorial representations. While in the zero temperature case factoriality was
implied by irreducibility, in the non-zero temperature case the equivalence between
extremal and factorial KMS representations is given by the following theorem.
Theorem 22.4 A KMS state Ω is extremal iff its GNS representation π is factorial,
i.e. the centre Z = π(A)
∩ π(A)
consists of multiples of the identity.
Proof.
138 The proof is split into four steps.
1) A KMS state Ω is extremal iff there is no other KMS state ω 1 , which is not a
multiple of Ω, such that
ω 1 ≤ λ Ω, λ > 1.
(22.27)
In fact, if ω 1 exists one has the decomposition
Ω = λ
−1
ω 1 + (Ω − λ
−1
ω 1 ) ≡ ω
1 + ω 2 .
Conversely, if Ω is decomposable in terms of KMS states as Ω = ω 1 + ω 2 , then
clearly there exists ω 1 < Ω.
2) Equation (22.27) implies (the vector Ψ Ω represents Ω)
ω 1 (B
∗ A)
2
≤ ω 1 (B
∗ B) ω 1 (A
∗ A) ≤ λ
2
Ω(B
∗ B) Ω(A
∗ A) =
λ
2
||π(B)Ψ Ω ||
2
||π(A) Ψ Ω ||
2
, ∀A, B ∈ A.
Thus, ω 1 (B
∗ A) defines a bounded (densely defined) sesquilinear form on the
GNS representation space defined by Ω and therefore there exists a unique
bounded operator T
such that
ω 1 (B
∗ A) = (π(B)Ψ Ω , T
π(A)Ψ Ω ).
(22.28)
Furthermore, T
∈ π(A)
since ∀A, B, C, ∈ A
(π(B)Ψ Ω , T
π(C) π(A) Ψ Ω ) = ω 1 (B
∗ C A) = ω 1 ((C
∗ B)
∗ A) =
(π(C
∗ B)Ψ Ω , T
π(A) Ψ Ω ) = (π(B)Ψ Ω , π(C)T
π(A)Ψ Ω ).
3) T
is invariant under time translations
T
t ≡ U (t) T
U (t)
−1
= T
.
(22.29)
138 Here we give a brief sketch; for a detailed proof, see, e.g. O. Bratteli and D.W. Robinson,
Operator Algebras and Quantum Statistical Mechanics, Vol. II, Springer 1981, Proposition 5.3.29.
157
II. (Thermodynamical stability) The representation is defined by a KMS state.
III. (Equilibrium state) The KMS state is the unique translationally invariant state.
The pure phases are defined by extremal KMS states.
For extremal KMS states, condition III can be replaced by the validity of the cluster
property, as in Chap. 16, thanks to Proposition 16.4, which states such an equivalence
for factorial representations. While in the zero temperature case factoriality was
implied by irreducibility, in the non-zero temperature case the equivalence between
extremal and factorial KMS representations is given by the following theorem.
Theorem 22.4 A KMS state Ω is extremal iff its GNS representation π is factorial,
i.e. the centre Z = π(A)
∩ π(A)
consists of multiples of the identity.
Proof.
138 The proof is split into four steps.
1) A KMS state Ω is extremal iff there is no other KMS state ω 1 , which is not a
multiple of Ω, such that
ω 1 ≤ λ Ω, λ > 1.
(22.27)
In fact, if ω 1 exists one has the decomposition
Ω = λ
−1
ω 1 + (Ω − λ
−1
ω 1 ) ≡ ω
1 + ω 2 .
Conversely, if Ω is decomposable in terms of KMS states as Ω = ω 1 + ω 2 , then
clearly there exists ω 1 < Ω.
2) Equation (22.27) implies (the vector Ψ Ω represents Ω)
ω 1 (B
∗ A)
2
≤ ω 1 (B
∗ B) ω 1 (A
∗ A) ≤ λ
2
Ω(B
∗ B) Ω(A
∗ A) =
λ
2
||π(B)Ψ Ω ||
2
||π(A) Ψ Ω ||
2
, ∀A, B ∈ A.
Thus, ω 1 (B
∗ A) defines a bounded (densely defined) sesquilinear form on the
GNS representation space defined by Ω and therefore there exists a unique
bounded operator T
such that
ω 1 (B
∗ A) = (π(B)Ψ Ω , T
π(A)Ψ Ω ).
(22.28)
Furthermore, T
∈ π(A)
since ∀A, B, C, ∈ A
(π(B)Ψ Ω , T
π(C) π(A) Ψ Ω ) = ω 1 (B
∗ C A) = ω 1 ((C
∗ B)
∗ A) =
(π(C
∗ B)Ψ Ω , T
π(A) Ψ Ω ) = (π(B)Ψ Ω , π(C)T
π(A)Ψ Ω ).
3) T
is invariant under time translations
T
t ≡ U (t) T
U (t)
−1
= T
.
(22.29)
138 Here we give a brief sketch; for a detailed proof, see, e.g. O. Bratteli and D.W. Robinson,
Operator Algebras and Quantum Statistical Mechanics, Vol. II, Springer 1981, Proposition 5.3.29.