22.3 KMS States in the Thermodynamical Limit
155
theorem it can be extended to A (the extension of the state will still be denoted by Ω V )
and therefore, as V varies, one gets a sequence {Ω V } of states on A. Since the closed
unit ball of the dual A
∗ of a Banach space A is compact in the weak topology induced
by A, (Alaoglu–Banach theorem), then by the Bolzano–Weierstrass theorem there is
a subsequence Ω V n which is weakly convergent, i.e. ∀A ∈ A one has the existence of
lim
n→∞
Ω V n (A).
In conclusion, by the above compactness argument, one can always find a thermodynamical limit of finite volume states. In general, such a limit will not be unique; different limits correspond to different boundary conditions leading to different phases.
Theorem 22.3
135 Let α
V
t denote the finite volume (algebraic) dynamics defined by
U V (t) = e
i H V t
∈ A, (where H V denotes the finite volume Hamiltonian), and Ω V
denote the KMS (Gibbs) states at inverse temperature β (and with given chemical
potential μ).
If α
V
t converges in norm as V → ∞ on the quasi-local algebra A to a oneparameter group α t of
∗ automorphisms of A and Ω is the weak limit of finite
volume KMS states on A
Ω(A) = lim
n→∞
Ω V n (A), ∀A ∈ A,
(22.23)
then Ω satisfies the KMS condition.
Proof. In order to prove the KMS condition, it is enough to prove that
lim
V →∞
Ω V (B α
V
t (A)) = Ω(B α t (A)), ∀B, A ∈ A.
Now, putting A
V
t ≡ α
V
t (A) we have
|Ω V (B A
V
t ) − Ω(B A t )| ≤ |Ω V (B(A
V
t − A t ))| + |(Ω V − Ω)(B A t )|.
Since Ω V is a continuous functional on A, the first term on the right-hand side is
bounded by ||B|| ||A
V
t − A t ||, which goes to zero as V → ∞ by assumption, and
the second term converges to zero if Ω is the weak limit of Ω V .
One can also show
136 that the general properties of KMS (Gibbs) states derived in
Theorem 22.2 remain valid in the thermodynamical limit, namely the representation
π β defined by a KMS state Ω β has the following properties:
1) there exists an involution operator J , with J
2
= 1, such that
J π(A)
J = π(A)
, J Ψ β = Ψ β ,
(22.24)
135 [HHW].
136 [HHW]; see also Haag’s book (1996) and Hugenholtz’s lectures in London (1972), where one
can also find the connection with the Tomita–Takesaki theory of Von Neumann algebras.
155
theorem it can be extended to A (the extension of the state will still be denoted by Ω V )
and therefore, as V varies, one gets a sequence {Ω V } of states on A. Since the closed
unit ball of the dual A
∗ of a Banach space A is compact in the weak topology induced
by A, (Alaoglu–Banach theorem), then by the Bolzano–Weierstrass theorem there is
a subsequence Ω V n which is weakly convergent, i.e. ∀A ∈ A one has the existence of
lim
n→∞
Ω V n (A).
In conclusion, by the above compactness argument, one can always find a thermodynamical limit of finite volume states. In general, such a limit will not be unique; different limits correspond to different boundary conditions leading to different phases.
Theorem 22.3
135 Let α
V
t denote the finite volume (algebraic) dynamics defined by
U V (t) = e
i H V t
∈ A, (where H V denotes the finite volume Hamiltonian), and Ω V
denote the KMS (Gibbs) states at inverse temperature β (and with given chemical
potential μ).
If α
V
t converges in norm as V → ∞ on the quasi-local algebra A to a oneparameter group α t of
∗ automorphisms of A and Ω is the weak limit of finite
volume KMS states on A
Ω(A) = lim
n→∞
Ω V n (A), ∀A ∈ A,
(22.23)
then Ω satisfies the KMS condition.
Proof. In order to prove the KMS condition, it is enough to prove that
lim
V →∞
Ω V (B α
V
t (A)) = Ω(B α t (A)), ∀B, A ∈ A.
Now, putting A
V
t ≡ α
V
t (A) we have
|Ω V (B A
V
t ) − Ω(B A t )| ≤ |Ω V (B(A
V
t − A t ))| + |(Ω V − Ω)(B A t )|.
Since Ω V is a continuous functional on A, the first term on the right-hand side is
bounded by ||B|| ||A
V
t − A t ||, which goes to zero as V → ∞ by assumption, and
the second term converges to zero if Ω is the weak limit of Ω V .
One can also show
136 that the general properties of KMS (Gibbs) states derived in
Theorem 22.2 remain valid in the thermodynamical limit, namely the representation
π β defined by a KMS state Ω β has the following properties:
1) there exists an involution operator J , with J
2
= 1, such that
J π(A)
J = π(A)
, J Ψ β = Ψ β ,
(22.24)
135 [HHW].
136 [HHW]; see also Haag’s book (1996) and Hugenholtz’s lectures in London (1972), where one
can also find the connection with the Tomita–Takesaki theory of Von Neumann algebras.