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KMS States 15 3.1 Definition TMP Master Core Module: Mathematical Statistical Physics I Sven Bachmann & Robert Helling July 7, 2015 Contents 1 Introduction 2 1.1 The Ising model: Peierls argument . 2 2 C*-algebras, states and representations 3 2.1 C*-algebras . 3 2.2 Representations and states . 5 2.3 Examples: Quantum spin systems, the CCR and CAR algebras . 8 2.3.1 Quantum spin systems . 8 2.3.2 Fermions: the CAR algebra . 10 2.3.3 Bosons: the CCR algebra . 11 2.3.4 Fock spaces and the Fock representation . 12 3 Equilibrium: KMS states 15 3.1 Definition . 15 3.2 The energy-entropy balance inequality . 18 3.3 Passivity and stability . 19 3.4 On the set of KMS states . 22 3.5 Symmetries . 24 4 Ideal quantum gases 26 4.1 The ideal Fermi gas . 26 4.2 The ideal Bose gas & Bose-Einstein condensation . 26 5 Renormalisation 27 5.1 The renormalisation idea . 27 5.2 Block spin transformation in the Ising model . 27 6 Phase transitions in quantum spin systems 28 6.1 The theorem of Mermin & Wagner . 29 6.2 Existence of a phase transition in the Heisenberg model . 31 7 Computer simulations 36 1 Chapter 1 Introduction 1.1 The Ising model: Peierls argument 2 Chapter 2 C*-algebras, states and representations 2.1 C*-algebras Let A be an associative algebra over C. A is a normed algebra if there is a norm A 3 x 7! kxk 2 R+ such that kxyk ≤ kxkkyk. A complete normed algebra is a Banach algebra. A mapping x 7! x∗ of A into itself is an involution if (x∗)∗ = x; (x + y)∗ = x∗ + y∗; (xy)∗ = y∗x∗; (λx)∗ = λx∗: An algebra with an involution is a *-algebra. Definition 1. A Banach *-algebra A is called a C*-algebra if kx∗xk = kxk2; x 2 A: Proposition 1. Let A be a C*-algebra. 1. kx∗k = kxk; 2. If A does not have an identity, let Ae be the algebra obtained from A by adjoining an identity 1. Then Ae is a C*-algebra with norm k · k defined by kλy + xyk kλ1 + xk = sup ; λ 2 C: y6=0 kyk Proof. Exercise. In the following, A will always denote a C*-algebras with an identity if not specified other- wise. Definition 2. The spectrum Sp(x) of x 2 A is the set Sp(x) := fλ 2 C : x − λ1 is not invertible in Ag : 3 If jλj > kxk, then the series λ−1 P (x/λ)n is norm convergent and sums to (λ1 − x)−1. n2N Hence, Sp(x) ⊂ Bkxk(0). Assume now that x 2 A is a self-adjoint element and that a + ib 2 2 2 2 Sp(x), a; b 2 R. Then a+i(b+t) 2 Sp(x+it1). Since kx+it1k = kx+it1kkx−it1k = kx +t 1k ≤ kxk2 + t2, and by the remark above, ja + i(b + t)j2 ≤ kxk2 + t2, and further 2bt ≤ kxk − a2 − b2 Qn for all t 2 R, so that b = 0. For any polynomial P over C, P (µ) − λ = A i=1(µ − zi), and Qn P (x) − λ1 = A i=1(x − zi) 2 A for any x 2 A. Hence, λ 2 Sp(P (x)) iff zj 2 Sp(x) for a 1 ≤ j ≤ n. Since P (zj) = λ, we have that λ 2 Sp(P (x)) iff λ 2 P (Sp(x)). We have proved Proposition 2. Let x 2 A. 1. Sp(x) ⊂ Bkxk(0); 2. if x = x∗, then Sp(x) ⊂ [−kxk; kxk]; 3. if xx∗ = x∗x, i.e. x is normal, then kxk = supfjλj : λ 2 Sp(x)g; 4. for any polynomial P , Sp(P (x)) = P (Sp(x)); The proof of 3. is left as an exercise. Note that the condition holds in particular for x = x∗ An element x 2 A is positive if it is self-adjoint and Sp(x) ⊂ R+. Proposition 3. Let x 2 A, x 6= 0. The following are equivalent: 1. x is positive; 2. there is a self-adjoint z 2 A such that x = z2; 3. there is y 2 A such that x = y∗y; Proof. (3) ) (2) by choosing y = z. (3) ) (1) since z2 is self-adjoint and since, by Proposition 2, Sp(z2) ⊂ [0; kzk2]. To show (1) ) (3), we note that1 for any µ > 0, 1 Z 1 p 1 1 2 µ = λ − dλ (2.1) π 0 λ λ + µ p R 1 −1 −1 Since x is positive, (x+λ1) is invertible for all λ > 0 so that z := π 0 λ λ − (x + λ1) dλ is well defined as a norm convergent integral, and x = z2. Using again (2.1) with µ = 1, we have that p kxk1=2 Z 1 λ kxk1=21 − z = (^x + λ1)−1(^x − 1)dλ, x^ = xkxk−1 π 0 λ + 1 Butx ^ positive implies Sp(^x) ⊂ [0; 1], hence Sp(1 − x^) ⊂ [0; 1] and k1 − x^k ≤ 1. Moreover, since Sp((^x + λ1)−1) = (Sp(^x + λ1))−1 ⊂ [(1 + λ)−1; λ−1], we have k(^x + λ1)−1k ≤ λ−1 for λ > 0. Hence, k1 − zkxk−1=2k ≤ 1 so that Sp(z) ⊂ [0; 2kxk1=2] and finally z is positive. It remains to prove (2) ) (1). Since x = y∗y is self-adjoint, x2 is positive and we denote by jxj its positive square root defined by the integral above. Then x± := (jxj ± x)=2 is positive and ∗ x−x+ = x+x− = 0. Decomposing yx− = s + it, with self-adjoint s; t, we have (yx−) (yx−) + ∗ 2 2 ∗ 3 (yx−)(yx−) = 2(s +t ) ≥ 0. But −(yx−) (yx−) = −x−(−x−+x+)x− = x− is positive, so that ∗ ∗ ∗ (yx−)(yx−) is positive. On the other hand, Sp((yx−)(yx−) )[f0g = Sp((yx−) (yx−))[f0g ⊂ ∗ R−, hence (yx−) (yx−) = 0 so that x− = 0, and finally x = x+ is positive. 1Convergence follows from the asymptotics O(s−1=2) as s ! 0 and O(s−3=2) as s ! 1, while the change of p variables λ = µξ yields immediately that the integral is µ, up to a constant. 4 Definition 3. A *-morphism between two *-algebras A and B is a linear map π : A!B ∗ ∗ such that π(A1A2) = π(A1)π(A2) and π(A ) = π(A) , for all A; A1;A2 2 A. It is called a *-isomorphism if it is bijective. A *-isomorphism A!A is an automorphism. Proposition 4. Let A; B be two C*-algebras and π : A!B a *-morphism. Then kπ(x)kB ≤ kxkA, and the range fπ(A): A 2 Ag is a *-subalgebra of B. Proof. If x is self-adjoint, so is π(x) and kπ(x)k = supfjλj : λ 2 Sp(π(x))g. If x−λ1 is invertible, then 1 = π((x−λ1)−1(x−λ1)) = π((x−λ1)−1)π(x−λ1) so that π(x)−λ1 is invertible, whence Sp(π(x)) ⊂ Sp(x), we have that kπ(x)k ≤ supfjλj : λ 2 Sp(x)g = kxk. The general case follows from kπ(x)k2 = kπ(x∗x)k ≤ kx∗xk = kxk2. Let Γ be a locally compact Hausdorff space, and let C0(Γ) be the algebra, under pointwise multiplication, of all complex valued continuous functions that vanish at infinity. Theorem 5. If A is a commutative C*-algebra, then there is a locally compact Hausdorff space Γ such that A is *-isomorphic to C0(Γ). Proof. See Robert's lectures. In classical mechanics, the space Γ is usually referred to as the phase space. If A is a commutative C*-algebra with an identity, then A is isomorphic to C(K), the algebra of continuous functions on a compact Hausdorff space K. Let U be a *-subalgebra of L(H). The commutant U 0 is the subset of L(H) of operators that commute with every element of U, and so forth with U 00 := (U 0)0. In particular, U ⊂ U 00, and further U 0 = U 000. Definition 4. A von Neumann algebra or W*-algebra on H is a *-subalgebra U of L(H) such 00 0 that U = U. Its center is Z(U) := U\U , and U is a factor if Z(U) = C · 1. Theorem 6. Let U be a *-subalgebra of L(H) such that UH = H. Then U is a von Neumann algebra iff U is weakly closed. Note that UH = H is automatically satisfied if 1 2 U. Furthermore, for any *-subalgebra U 00 of L(H), let U be its weak closure for which U = U by the theorem. Since U ⊂ U, we have 0 0 0 U ⊂ U . Furthermore, if x 2 U , and y 2 U, with U 3 yn * y, then x commutes with yn for all 0 0 00 n and hence with y, so that U 0 ⊂ U . It follows that U = U 0, whence U = U 00 and so: Corollary 7. U is weakly dense in U 00, namely U = U 00. 2.2 Representations and states Definition 5. Let A be a C*-algebra and H a Hilbert space. A representation of A in H is a *-morphism π : A!L(H). Moreover, 1. Two representations π; π0in H; H0 are equivalent if there is a unitary map U : H!H0 such that Uπ(x) = π0(x)U; 2. A representation π is topologically irreducible if the only closed subspaces that are invari- ant under π(A) are f0g and H; 3.
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