Chapter 3

Equilibrium: KMS states

In a quantum system with Hamiltonian H such that Tr exp( H) is finite for some >0, the Gibbsian rule is as follows: The system in thermal equilibrium is in a state given by a density matrix ⇢ on , H 1 H H ⇢ = Z() e ,Z():=Tre .

Among its many properties, we concentrate on an a priori rather coincidental properties. Let ! denote the state associated to the density matrix ⇢ , and ⌧ (A)=exp(itH)A exp( itH). Con- t sider the function F (A, B; t)=Z() 1Tr(exp( i(t i)H)A exp(itH)B). Using the cyclicity of the trace, F(A, B; t)=!(A⌧t(B)). On the other hand, F(A, B; t) can be analytically contin- ued into the complex plane to t +i to give F (A, B; t +i)=Z() 1Tr(exp( itH)A exp(i(t + i)H)B)=!(⌧t(B)A). Hence, there is an analytic function F(A, B; z) defined on the strip z C :0 Imz with boundary values { 2   }

F(A, B; t)=!(A⌧t(B)),F(A, B; t +i)=!(⌧t(B)A). (3.1)

This turns our to be the property that extends naturally to the algebraic setting.

3.1 Definition

It will be useful to first introduce some terminology.

Definition 11. Apair( ,⌧ ) is a C*-dynamical system if is a C*-algebra with an identity A t A and R t ⌧t is a strongly continuous group of *-automorphisms of , namely 3 7! A ⌧ (A) ⌧ (A) 0(✏ 0) k t+✏ t k! ! for all A . 2A t It follows from the strong continuity that ⌧t is generated by a *-derivation, ⌧t(A)=e A: Proposition 22. Let : ,A (A)=t 1(⌧ (A) A), let t A!A 7! t t

D():= A :limt(A) exists , { 2A t 0+ } ! and define

: D() !A 1 A (A)= lim t (⌧t(A) A). 7! t 0+ !

15 Then, is a closed, densely defined map such that

1 D() and (1) = 0, 2 (AB)=(A)B + A(B),

(A⇤)=(A)⇤.

In fact, just as there is a one-to-one correspondence between self-adjoint generators and strongly continuous unitary groups on a Hilbert space, there a correspondence between *- derivations and strongly continuous groups of *-automorphisms on a C*-algebra. This is Hille- Yosida’s theorem. Recall that A is an analytic element for a derivation if A D(n) for all tn n 2 n N and 1 A < , for 0 toperator on A A generates a strongly continuous groups of *-automorphisms if and only if

is a *-derivation has a dense set of analytic elements A + (A) A , R,A D(). k kk k 8 2 2

In the case of quantum mechanics with a finite number of degrees of freedom, ⌧t(A)= exp(itH)A exp( itH) is strongly continuous if and only if H is bounded, in which case it is also norm continuous (see exercises). The associated derivation := i[H, ] is bounded and · everywhere defined. In fact, as a consequence of the closed graph theorem, an everywhere defined derivation necessarily generates a norm-continuous *-automorphism.

Definition 12. Let ( ,⌧ ) be a C*-dynamical system. A state ! on is a (⌧,)-KMS state A t A for >0 if, for any A, B , there exists a function F(A, B, z),analyticinS := z C : 2A { 2 0 < Imz< ,continuousonS ,andsatisfyingtheKMSboundarycondition(3.1). } We shall say that A is an analytic element for ⌧ if the map A ⌧ (A) extends to an analytic t 7! t function on C. Theorem 24. Let ( ,⌧ ) be a C*-dynamical system. A state ! on is a (⌧,)-KMS state if A t A and only if there exists a dense, ⌧-invariant *-subalgebra of analytic elements for ⌧ such that D t

!(BA)=!(A⌧i(B)). (3.2)

The following proposition shows that the condition of analyticity is never a true restriction.

Proposition 25. Let ( ,⌧ ) be a C*-dynamical system, and let be its generator. For any A t A and m N, let 2A 2 m mt2 A := ⌧ (A)e dt. m ⇡ t r ZR Then

1. Am is analytic for ⌧t

2. Am is analytic for

3. The *-subalgebra ⌧ := Am : A ,m N is dense A { 2A 2 }

16 2 2 1 Proof. First of all, ⌧t(A) exp( mt )= A exp( mt ) L (R), so that Am is well-defined, k k k k 2 A and A A . Moreover, m 2A k mkk k m m(t s)2 ⌧ (A )= ⌧ (A)e dt. s m ⇡ t r ZR The right-hand-side extends to an analytic function, with r.h.s. A exp( m(Imz)2), which k kk k can be used to extend ⌧s(Am)to⌧z(Am) for all z C. Moreover, by dominated convergence, 2 n n 1+n d m d m(t s)2 m mt2 ⌧ (A ) = ⌧ (A) e dt = ⌧ (A)H (t)e dt dsn s m ⇡ t dsn ⇡ t n s=0 r ZR s=0 r ZR n were Hn are the Hermite polynomials, so that Am D( ) for all n N, and Am is analytic 2 2 for . Finally,

1 t2 An A = ⌧t/pn(A) A e dt 0(n ) p⇡ R ! !1 Z ⇣ ⌘ by the strong continuity of ⌧t and dominated convergence. Proof of Theorem 24. Necessity. Let A ,B and ! be a (⌧,)-KMS state. Then z 2A 2A⌧ 7! G(z)=!(A⌧z(B)) is analytic and G(t)=F(A, B; t) for t R. Hence z G(z) F(A, B; z) 2 7! is analytic on S, continuous on S R and vanishes on R. By the Schwarz reflection principle, [ it extends to an analytic function on the double strip S S that vanishes on R. Hence it [ equals zero everywhere, and by continuity also on S, that is F(A, B; z)=!(A⌧z(B)) for all z S . In particular, setting z =i yields ! (BA)=! (A⌧ (B)). 2 i Suciency. First, for A, B , z F (A, B; z):=!(A⌧z(B)) is analytic on C.Since 2D 7! ⌧ (B) , t 2D F (A, B; t)=!(A⌧t(B)),F(A, B; t +i)=!(A⌧i(⌧t(B))) = !(⌧t(B)A), by (3.2). Now, !(A⌧ (B)) A ⌧ iImz(B) so that F (A, B; z) is bounded on S and Hadamard’s | z |k kk k three lines theorem yields sup F (A, B; z) A B . For arbitrary A, B ,letAn z S k kk k 2A ! A, B B,withA ,B .Since2 n ! n n 2D F (A ,B ; z) F (A ,B ; z)=F (A A ,B ; z)+F (A ,B B ; z), n n m m n m n m n m so that F (An,Bn; z) is uniformly Cauchy in S. Its limit is therefore analytic on S and continuous on its closure, and it still satisfies the KMS boundary condition.

Clearly, the Gibbs state on a finite dimensional Hilbert space satisfies the KMS condition. As we shall see later, it is also the unique KMS state in this case1. The following theorem shows that a KMS state passes the simplest test for an equilibrium state: it is invariant under time evolution. Mathematically, this is also useful as it implies the unitary implementability of the dynamics in the GNS representation. Proposition 26. Let ( ,⌧ ) be a C*-dynamical system and let ! be a (⌧,)-KMS state. Then A t ! ⌧t = ! for all t R. 2 Proof. Let A . The function z g(z)=!(⌧ (A)) is analytic. By Theorem 24, 2A⌧ 7! z g(z +i)=!(1⌧i(⌧z(A))) = !(⌧z(A)1) = g(z). Hence, g is a periodic function along the imaginary axis, and moreover, g(t+i↵) ⌧ (A) = | |k t+i↵ k ⌧i↵(A) sup0 ⌧i(A) , which is finite. Hence, g is analytic and bounded on C, so that k k   k k it is constant by Liouville’s theorem. This extends to all observables by continuity.

1This shows once again that there cannot be a phase transition for quantum spin systems in finite volume.

17 3.2 The energy-entropy balance inequality

Just as the Gibbs state is characterised by the variational principle as being a minimiser of the free energy, general KMS states are equivalently defined by satisfying the energy-entropy balance inequality (EEB). In this section, ( ,⌧ ) is a C*-dynamical system, and is the generator of ⌧ . A t t We start with a simple observation.

Lemma 27. If a state ! over is such that i!(A⇤(A)) R for all A D(), then ! ⌧t = !. A 2 2 Proof. Since !(B⇤(B)) is purely imaginary, and (B⇤)=(B)⇤, !((B⇤B)) = !(B⇤(B)) + !(B⇤(B)) = 0. Hence, with the continuity of !, t !(⌧ (A⇤A)) !(A⇤A)= !((⌧ (A⇤A)))ds =0. t s Z0 Hence, the statement holds for all positive elements of , and further extends to all of by A A noting that any observable is a linear combination of four positive elements. ˇ Let f be the Fourier transform of f Cc1(R). By Paley-Wiener’s theorem, f is analytic in n 2 C and f(z) Cn(1 + z )exp(R Im(z) ) for all n N. Let | | | | | | 2 ⌧ (A):= f(t)⌧ (A)dt D() f t 2 ZR since it is analytic for . Let ! be a ⌧ -invariant state and H = dP () is the GNS Hamil- t R tonian satisfying H⌦= 0. We have R itH !(A⇤⌧ (A)) = f(t) ⇡(A)⌦, e ⇡(A)⌦ dt = fˇ()dµ () (3.3) f h i A ZR ZR where dµ ()= ⇡(A)⌦,dP()⇡(A)⌦ is the spectral measure associated with ⇡(A)⌦. Simi- A h i lary, !(⌧ (A)A )= fˇ()d⌫ ()whered⌫ ()= ⇡(A )⌦,dP( )⇡(A )⌦ . Moreover, the f ⇤ R A A h ⇤ ⇤ i analyticity of z f(z)!(A ⌧ (A)) and the KMS condition yield 7! R ⇤ z

!(A⇤⌧f (A)) = f(t +i)!(⌧t(A)A⇤)dt. ZR The right hand side is also equal to fˇ()exp()d⌫ (). Since this and (3.3) hold for any R A test function fˇ, we obtain R dµ A ()=e. (3.4) d⌫A Theorem 28. Astate! over is a (⌧,)-KMS state if and only if A !(A⇤A) i!(A⇤(A)) !(A⇤A)ln !(AA⇤) for all A D(). 2 Proof. We only prove . First observe that !(A (A)) = i ⇡(A)⌦,H⇡(A)⌦ and ) ⇤ h i ⇡(A)⌦,H⇡(A)⌦ dµA() h i = R . ⇡(A)⌦,⇡(A)⌦ dµ () h i R R A By Jensen’s inequality, R

dµA() exp( )dµA() d⌫A() !(AA ) exp R R = R = ⇤ dµ ()  dµ () dµ () !(A A) ✓ R R A ◆ R R A RR A ⇤ if ! is a (⌧,)-KMS stateR by (3.4). Hence,R exp(i!(A (A))/R!(A A)) !(AA )/!(A A). ⇤ ⇤  ⇤ ⇤

18 n Corollary 29. Let be a C*-algebra with a unit and ⌧ n N be a sequence of strongly con- A { } 2 tinuous one-parameter groups of automorphisms of such that A ⌧ n(A) ⌧ (A)(n ) t ! t !1 for all A ,t R, where ⌧t is a strongly continuous one-parameter group of automorphisms 2A 2 n of .If !n n N is a sequence of (⌧ ,)-KMS states, then any weak-* limit point of !n is a A { } 2 { } (⌧,)-KMS state.

Proof. See exercises.

n ⇤n Simple example. ⌧ = ⌧ the dynamics of a quantum spin system in a finite volume ⇤n,such that ⇤ as n 2, generated by a Hamiltonian H . The unique KMS state is the n ! !1 ⇤n Gibbs state with density matrix Z () 1 exp( H ). If the infinite volume dynamics exists, n ⇤n ⌧ ⇤n (A) ⌧ (A), then the limiting thermodynamic states are (⌧ ,)-KMS states. ! 3.3 Passivity and stability

Definition 13. Let ( ,⌧ ) be a C*-dynamical system. A state ! on is a passive state if A t A i!(U (U)) 0 for any U ( ) D(). Here, ( ) is the connected component of the ⇤ 2U0 A \ U0 A identity in the set of all unitary elements of . A Proposition 30. If ! is a (⌧,)-KMS state, then ! is passive.

Proof. Choose A = U ( ) D() in the EEB inequality. 2U0 A \ In order to have equivalence in the proposition above, one needs to require complete passivity, N N N namely that ! is passive as a state on the tensored system ( , ⌧t) for all N N. ⌦i=1 ⌦i=1A ⌦i=1 2 Interpretation in the case dim( ) < ,where!(A)=Tr(⇢ A)where⇢ = Z() 1 exp( H) H 1 with Z()=Trexp( H), the Gibbs state. Consider a time dependent Hamiltonian H(t)= H(t) ,t [0,T] such that H(0) = H(T )=H, an let U be the associated unitary evolution on ⇤ 2 [0,T]. The change in energy between t = 0 and t = T is given by

W := Tr(U⇢ U ⇤H) Tr(⇢ H)=Tr(⇢ U ⇤[H, U]) = i! (U ⇤(U)) 0 since the KMS state is passive. Passivity expresses a basic thermodynamic fact: the total work done by the system on the environment in an arbitrary cyclic process, W , is non-positive on average.

Definition 14. Let ( ,⌧ ) be a C*-dynamical system with generator 0.Alocal perturbation A t V of 0 is given by V = 0 +i[V, ],D(V )=D(0), · for a V = V . ⇤ 2A Using Thm 23, one can show that V generates a strongly continuous group of automor- V d 0 V 0 V phisms ⌧ .Sinceds ⌧ s(⌧s (A)) = ⌧ s(i[V,⌧s (A)]), we have Duhamel’s formula t V 0 0 V ⌧t (A)=⌧t (A)+ ⌧t s(i[V,⌧s (A)])ds, Z0 2 ⇤n ⇤m if n m,and x , n0 N such that x ⇤n n n0 ⇢  8 2 9 2 2 8

19 which can be solved iteratively yielding Dyson’s expansion

V 0 1 0 0 0 0 ⌧t (A)=⌧t (A)+ i[⌧t1 (V ), i[⌧t2 (V ), i[⌧tk (V ),⌧t (A)] ]]dt1 dtk. (3.5) 0 t1 tk t ··· ··· ··· Xk=1 Z  ···  In fact, writing V for C, the series is norm convergent for all C,t R,A and 2 2 2 2A defines an analytic function ⌧ V (A). 7! t The unitary element solving

i@ V =V ⌧ 0(V ), V =1, t t t t 0 V V V 0 has the following intertwining property ⌧t (A)t =t ⌧t (A). Solving the di↵erential equation iteratively again yields the expansion

V 1 k 0 0 t =1+ i ⌧t1 (V ) ⌧tk (V )dt1 dtk. 0 t1 tk t ··· ··· Xk=1 Z  ···  In fact, all above results continue to hold for a time dependent perturbation Vt.Acyclic perturbation of a C*-dynamical system is a norm-di↵erentiable family [0,T] t V = V 3 7! t t⇤ 2A such that V = V = 0, V D() and (dV /dt)=dV /dt. 0 T t 2 t t Definition 15. The work performed on the system along a cyclic V , t [0,T] is t 2 T dV W := ! ⌧ V t dt. t dt Z0 ✓ ◆ where ! is the initial state of the system.

By the boundary condition, 0 = T @ ! ⌧ V (V ) dt, so that 0 t t t R T W = ! ⌧ V 0 (V ) dt (3.6) t t Z0 V 0 since (Vt)= (Vt). Also note that by the first law of thermodynamics, this also equals the total heat given by the system to the environment.

0 Lemma 31. Let ( ,⌧t) be a C*-dynamical system with generator and R t Vt = V ⇤ A 3 7! t 2A be a norm-di↵erentiable local perturbation such that V =0if t ( , 0] [T, ), V D(0) t 2 1 [ 1 t 2 and 0(dV /dt)=d0V /dt. Then W = i!(V 0(V )). t t T T ⇤ V 0 0 V 0 V Proof. Under the given assumption, t D( ), and (t ) is di↵erentiable with d (t )/dt = 0 V 2 (dt /dt) (without proof). But then

T V 0 V V 0 V V 0 V i!( ( ⇤)) = ! i@ ( ) ( ⇤)+ i@ ( ⇤) T T t t t t t t Z0 T T V 0 0 V V 0 0 V V 0 0 V = ! ⌧ (V ) ( ⇤) (⌧ (V ) ⇤) = ! ⌧ ( (V )) ⇤ . t t t t t t t t t Z0 Z0 Conclude by (3.6).

Theorem 32. Under the assumptions of the previous lemma, if ! is a (⌧,)-KMS state for some , then W 0.

20 Proof. By Lemma 31, W = i!(V 0(V )). Since V is unitary and ! is a (⌧,)-KMS state, T T ⇤ T W 0 by passivity, Proposition 30. We now consider a cyclic machine working between two reservoirs at inverse temperature . The C*-dynamical system is given by = and ⌧ 0 = ⌧ ⌧ , with generator 1  2 A A1 ⌦A2 1 ⌦ 2 0 = 1+1 . The initial state is ! = ! ! where ! is a (⌧ , )-KMS state, and it is 1 ⌦ ⌦ 2 1 ⌦ 2 i i i a(, 1)-KMS state for the dynamics := ⌧ ⌧ with generator = 1+1 . t 1,1t ⌦ 2,2t 1 1 ⌦ ⌦ 2 2 The machine is represented by a cyclic perturbation V temporarily coupling the reservoirs. t 2A The total work on the system decomposes in W = Q1 + Q2 where V V V V Q = i!( ( 1)( ⇤)),Q= i!( (1 )( ⇤)) 1 T 1 ⌦ T 2 T ⌦ 2 T are the amounts of heat given to both reservoirs. Now, V V V V Q + Q = i!( ( 1+1 )( ⇤)) = i!( ( ⇤)) 0, 1 1 2 2 T 1 1 ⌦ ⌦ 2 2 T T T or Q (T T ) WT . Assuming now that Q < 0 (heat pumped out of the hot reservoir) 1 2 1 1 1 W T T 1 2 Q  T 1 1 which is Carnot’s statement of the second law of thermodynamics, namely a bound on the eciency of a cyclic machine initially at equilibrium (ratio of the work performed by the system to the heat pumped out of the hot reservoir). Stability of the thermal equilibrium refers to a number of results revolving around the fact that the dynamics applied to a state ‘close to thermal’ drives the system back to equilibrium. In fact, under additional assumption, it can be shown that this property is equivalent to the KMS condition. The first result is about structural stability, and can be proved by perturbation theory in the line of (3.5). Proposition 33. Let ( ,⌧ ) be a C*-dynamical system, and ! a (⌧,)-KMS state on . Then, A t A for every local perturbation V , there is a (⌧ V ,)-KMS state !V and 1. !V is !-normal 2. there is C>0 such that ! !V C V k k k k 3. the map ! !V is a bijection from the set of (⌧,)-KMS states onto the set of (⌧ V ,)- 7! KMS states See exercises for a proof in the finite dimensional case. Note in particular that local perturba- tions cannot induce a phase transition. Dynamical stability needs more assumptions to hold, usually in the form of asymptotic abelianness of the dynamical system, namely [A,⌧ (B)] 0 in some sense. t ! V Theorem 34. Let V = V ⇤ and let ! be a (⌧ ,)-KMS state, and let !˜ be a weak-* 0 2A 0 accumulation point of ! ⌧t as t .Iflimt [V,⌧t (A)] =0for all A , then !˜ is a !1 !1 k k 2A (⌧ 0,)-KMS state. Proof. By lower semicontinuity of (u, v) u ln(u/v), we have 7! 0 !˜(A⇤A) 0 ! ⌧t (A⇤A) 0 V 0 !˜(A⇤A)ln lim inf ! ⌧t (A⇤A)ln lim inf i!(⌧t (A)⇤ (⌧t (A))) !˜(AA ) t 0 t ⇤  !1 ! ⌧t (AA⇤)  !1 0 0 0 0 = i!˜(A⇤ (A)) + lim inf !(⌧t (A)⇤[V,⌧t (A)]) = i!˜(A⇤ (A)) t !1 by the EEB inequality, the decomposition V = 0 +i[V, ] and 0 ⌧ 0 = ⌧ 0 0. ·

21 Note that the theorem does not state whether the limit of ! ⌧ 0 exists. However, it does so t in two simple cases. Firstly, if there is a unique (⌧ 0,)-KMS state, since then all accumulation points of ! ⌧ 0 must be equal. Secondly, if [V,⌧0(A)] decays fast enough: t t V 0 Proposition 35. Let V = V ⇤ and let ! be a ⌧ -invariant state. Then ! := limt ! ⌧t 2A ± !±1 exists (in the weak*-topology) if and only if R t !([V,⌧0(A)]) is integrable at for all 3 7! t ±1 A . 2A d V 0 V 0 Proof. Integrating ds ⌧ s(⌧s (A)) = ⌧ s(i[V,⌧s (A)]) and using the invariance of ! yields

t2 !(⌧ 0 (A)) !(⌧ 0 (A)) = i !([V,⌧0(A)])ds t2 t1 s Zt1 for all A . 2A In particular, a sucient condition for the existence of the limit is the integrability of the map R t [V,⌧0(A)] . We finally state a sharp result. Let ! be an arbitrary reference state. 3 7! k t k (A) For any self-adjoint element V of a norm-dense *-subalgebra ,thereisa > 0 A0 ⇢A V such that [V,⌧V (A)] ds < , ,A . k s k 1 | | V 2A0 ZR (S) For any self-adjoint element V of a norm-dense *-subalgebra ,thereisa > 0 A0 ⇢A V such that if ,thereexistsa⌧ V -invariant, !-normal state such that | | V T V 1 V V !+ := lim ! ⌧t dt exists, and lim ! ! =0 t T 0 k k !1 Z0 ! Theorem 36. Assume that ! is a factor state and that (A) holds. Then (S) holds if and only if ! is a (⌧ 0,)-KMS state for some .

V V In that case, by a variant of Theorem (34), !+ is a (⌧ ,)-KMS state.

3.4 On the set of KMS states

Let ( ,⌧ ) be a C*-dynamical system with an identity. For any >0, let ( ) be the set of A t S A all (⌧,)-KMS states. The physical intuition is as follows: for small , there is a unique thermal state, corresponding to the high temperature phase. As grows, the set of ( ) becomes S A non trivial, and any state can be decomposed into pure thermodynamic phases. This picture is made mathematically precise in the following theorem:

Theorem 37. Let ( ,⌧ ) be a C*-dynamical system with an identity, and let ( ) be the set A t S A of all (⌧,)-KMS states, for >0. Then,

1. ( ) is convex and weakly-* compact S A 2. The normal extension of ! to ⇡( ) is a KMS state A 00 3. ! ( ) is an extremal point if and only if ! is a factor state, 2S A and if !0 is an !-normal, extremal KMS state, then ! = !0 4. ⇡(A) ⇡(A) consists of time-invariant elements 0 \ 00

22 5. If ! ( ) is such that the GNS Hilbert space is separable, there is a unique prob- 2S A ability measure µ on ( ), which is concentrated on the extremal points, such that S A ! = ( ) ⌫dµ(⌫) S A Proof. (Sketch,R incomplete) (1). If ! ,! ( ), A, B , with associated analytic functions 1 2 2S A 2A F (A, B, ), F (A, B, ), then the analytic function F (A, B, )+(1 )F (A, B, ) has ,1 · ,1 · ,1 · ,2 · boundary values associated to ! +(1 )! so that ( ) is convex. Moreover, the EEB 1 2 S A inequality implies that ( ) is a weakly-* closed subset of the weakly-* compact set ( ), S A E A hence ( ) is weakly-* compact. S A (2) Follows by density of ⇡( )in⇡( ) in the weak topology, Corollary 7. A A 00 (3) If ! is not a factor state, then there exists a projection 1 = P ⇡(A) ⇡(A) .We 6 2 0 \ 00 first claim that !(P ) = 0. Otherwise 0 = !(P )= P ⌦ 2 so that P ⌦= 0. But then, for 6 k k any A, B , !(A PB)= ⇡(A)⌦,P⇡(B)⌦ =0sinceP ⇡(A) , and hence P =0by 2A ⇤ h i 2 0 cyclicity of ⌦. Now, ! = !(P )! + !(1 P )! ,where! (A)=!(PA)/!(P ) and ! (A)= 1 2 1 2 !((1 P )A)/!(1 P ), is a non-trivial decomposition of !. Moreover, !(P )! (BA)=!(PBA)= 1 !(BPA)=!(PA⌧ (B)) = !(P )! (A⌧ (B)) so that ! is not extremal in ( ). i 1 i S A (4). Let C ⇡(A) ⇡(A) and consider the normal extension of ! to ⇡(A) . Repeating 2 0 \ 00 00 the proof of Proposition 26, t !(⌧ (A B)C) is constant. Hence, t !(⌧ (A )C⌧ (B)) = 7! t ⇤ 7! t ⇤ t ⇡(A)⌦,U ⇡(C)U ⇡(B)⌦ = !(A ⌧ (C)B) is constant for all A, B . h t⇤ t i ⇤ t 2A (5). That any KMS state can be decomposed into extremal KMS states follows from con- vexity and Krein-Milman’s theorem. Uniqueness is more involved.

The algebra ⇡(A) ⇡(A) contains both microscopic and macroscopic observables. Ele- 00 ments in the centre ⇡(A) ⇡(A) induce ‘superselection rules’: If S = S ⇡(A) ⇡(A) with 0 \ 00 ⇤ 2 0 \ 00 S = 1, the Hilbert space decomposes into components on which S is a constant multiple 6 · of the identity, while these components with di↵erent ‘quantum numbers’ are not connected by any observable. In the case of KMS states, (3) above states that such observables associ- ated with quantum numbers are constant in time. Furthermore, in a factor, any such S is a constant multiple of the identity. Hence, by (2) above, extremal KMS states associate fixed, non-fluctuating values to all quantum numbers: they are ‘macroscopically pure’ states. Theorem 38. Let ( ,⌧ ) be a C*-dynamical system, and let ! be a faithful (⌧,)-KMS state, A t for >0. Let ↵ be a *-automorphism of . Then, A 1. ! ↵ is a (↵ 1 ⌧ ↵,)-KMS state 2. If ! ↵ = !, then ↵ ⌧t = ⌧t ↵ for all t R 2 3. If ↵ ⌧t = ⌧t ↵ for all t R, then ! ↵ is a (⌧,)-KMS state 2 Proof. Let F be the analytic function associated to !.ThenF↵(A, B; z):=F (↵(A),↵(B); z) is an analytic function in S, continuous on S and such that, for t R, 2 1 F (A, B; t)=!(↵(A)⌧ (↵(B))) = (! ↵)(A(↵ ⌧ ↵)(B)) ↵ t t 1 F (A, B; t +i)=!(⌧ (↵(B))↵(A)) = (! ↵)((↵ ⌧ ↵)(B)A) ↵ t t which shows that ! ↵ is a (↵ 1 ⌧ ↵,)-KMS state. In order to prove (2), we use the fact that the ⌧-group with respect to which a ! is a KMS state is unique3.But! is simultaneously a(⌧,)-KMS state and by (1) a (↵ 1 ⌧ ↵,)-KMS state, hence ⌧ = ↵ 1 ⌧ ↵. Finally, (3) t t follows immediately from (1).

3 1 In the case dim( ) < , a faithful state is given by a ⇢>0, which determines uniquely H := ln ⇢ and H 1 hence the dynamics.

23 3.5 Symmetries

Definition 16. Let ( ,⌧ ) be a C*-dynamical system. A *-automorphism ↵ of is a symmetry A t A if ↵ ⌧t = ⌧t ↵ for all t R. 2 In this case, and by 1 above, if ! is a (⌧,)-KMS state, then so is ! ↵. Hence, in the presence of a symmetry the set ( ) is invariant under ⌧ for any fixed >0. In particular, if there is a S A unique (⌧,)-KMS state, then it is itself invariant and one says that the symmetry is unbroken at . If, on the other hand, there is a (⌧,)-KMS state which is not invariant, then the symmetry is said to be broken and there is more than one equilibrium state at , indicating a phase transition. Examples are the breaking of rotational SU(2)-symmetry in magnetic transitions, translational Rd symmetry in liquid-solid transitions. Here is a general criterion for the absence of symmetry breaking: (A) There is a sequence U of unitary elements of the algebra such that U D() and n 2A n 2 lim ↵(A) U ⇤AUn =0,A. n n !1 k k 2A (Bi) There is M such that (U ) M k n k (Bii) All (⌧,)-KMS states are ↵2-invariant and there is M such that U (U )+U (U ) M k n⇤ n n n⇤ k If (A) holds, one says that ↵ is almost inner. Theorem 39. Let ↵ be a symmetry of ( ,⌧ ). If (A) and either (Bi) or (Bii) are satisfied, A t then all (⌧,)-KMS states are ↵-invariant for all >0. Note that the symmetry can still be broken in the ground state, = . 1 Proof. Let ! be a (⌧,)-KMS state, H the associated Hamiltonian such that H⌦= 0 and let ˇ H = dP (). For any bounded interval I R,let hn n N be a sequence of real-valued Cc1 ⇢ { } 2 functions supported on intervals [a ,b ], with b a 1 and such that hˇ ()2 = 1 for R n n | n n| n n all I. Let An := ⌧ (A), which is analytic for ⌧t. First of all, 2 hn P bn iHs iHt 2 !(A⇤ A )= h (t)h ( s) e ⇡(A)⌦, e ⇡(A)⌦ dtds = hˇ () dµ () n n n n h i n A Z Zan as well as !(A A )= bn hˇ ()2d⌫ (). Hence, by the measure-theoretic KMS property, n n⇤ an n A !(A A ) exp(a )!(A A ) and further n⇤ n n Rn n⇤ !(An⇤ An) !(An⇤ An)ln an!(An⇤ An). !(AnAn⇤ )

bn 2 Similarly, i!(A⇤ (An)) = hˇn() dµA(), and hence, n an R i!(A⇤ (A )) b !(A⇤ A ) n n  n n n We further write the EEB inequality for the observable U ⇤ An, n, m N, namely m 2 !(An⇤ An) !(An⇤ An)ln i!(An⇤ Um(Um⇤ )An) i!(An⇤ (An)) !(Um⇤ AnAn⇤ Um)  and use the two inequalities above to obtain (note the position of * in the numerator!)

!(AnAn⇤ ) !(An⇤ An)ln i!(An⇤ Um(Um⇤ )An)+(bn an)!(An⇤ An) (3.7) !(Um⇤ AnAn⇤ Um) 

24 and b a 1. n n  Assumption (Bi). Since !(A U (U )A ) (U ) !(A A ) M!(A A ), (3.7) yields | n⇤ m m⇤ n |k m k n⇤ n  n⇤ n (M+1) !(A A⇤ ) e !(U ⇤ A A⇤ U ) n n  m n n m and letting m , !(A A ) e(M+1)(! ↵)(A A ). Summing over n, we have proved that !1 n n⇤  n n⇤ there exists a constant C = C(,M) such that

!(AA⇤) C(! ↵)(AA⇤),  which extends to all A . By the remark after Lemma 9, there is a T ⇡! ↵( )0 such that 2A 2 A !(A)= T ⌦! ↵,⇡! ↵(A)T ⌦! ↵ , which shows that ! is (! ↵)-normal. If ! is an extremal h i KMS state, then ! ↵ is also extremal so that they must be equal by Theorem 37(3). Since this holds for all extremal KMS state, the general result holds by decomposition, Theorem 37(5). Assumption (Bii). We repeat the procedure above with the state ! ↵, sum (3.7) and the similar bound with U U , proceed as above and obtain m $ m⇤ 2 (M+2) 2 ((! ↵)(A A⇤ )) e !(A A⇤ )(! ↵ )(A A⇤ ). (3.8) n n  n n n n Hence, (! ↵)(A) C˜!(A). Hence (! ↵)is!-normal and the conclusion holds as above. 

25