4 December 2017 joint with N. Brownlowe, J. Ramagge and N. Stammeier

Transcription

1 Equilibrium University of Oslo 4 December 2017 joint with N. Brownlowe, J. Ramagge and N. Stammeier

2 The KMS condition for finite systems Finite quantum systems: a time evolution on M n (C) is given by a one-parameter group of automorphisms σ t (a) = e ith ae ith, where t R, a M n (C) and H is a self-adjoint matrix.

3 The KMS condition for finite systems Finite quantum systems: a time evolution on M n (C) is given by a one-parameter group of automorphisms σ t (a) = e ith ae ith, where t R, a M n (C) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕ G (a) = Tr(ae βh ). It satisfies Tr(e βh ) ϕ G (ab) = ϕ G (bσ iβ (a)), (1) for a, b M n (C) analytic, i.e. t σ t (a) extends to an entire function on C.

4 The KMS condition for finite systems Finite quantum systems: a time evolution on M n (C) is given by a one-parameter group of automorphisms σ t (a) = e ith ae ith, where t R, a M n (C) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕ G (a) = Tr(ae βh ). It satisfies Tr(e βh ) ϕ G (ab) = ϕ G (bσ iβ (a)), (1) for a, b M n (C) analytic, i.e. t σ t (a) extends to an entire function on C. Partition function of (M n (C), σ) is β Tr(e βh ).

5 The KMS condition for finite systems Finite quantum systems: a time evolution on M n (C) is given by a one-parameter group of automorphisms σ t (a) = e ith ae ith, where t R, a M n (C) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕ G (a) = Tr(ae βh ). It satisfies Tr(e βh ) ϕ G (ab) = ϕ G (bσ iβ (a)), (1) for a, b M n (C) analytic, i.e. t σ t (a) extends to an entire function on C. Partition function of (M n (C), σ) is β Tr(e βh ). (1) - the KMS condition, cf. Haag-Hugenholtz-Winnick (1967): equilibrium for a state on a C -algebra with time evolution.

6 KMS states By analogy with finite systems and the Gibbs state, extend the notions of KMS β state, partition function, inverse temperature.

7 KMS states By analogy with finite systems and the Gibbs state, extend the notions of KMS β state, partition function, inverse temperature. A C -algebra, σ : R Aut(A) time evolution, ϕ a state on A. 1 ϕ is KMS β (at inverse temperature β [0, )) if ϕ(ab) = ϕ(bσ iβ (a)) for all a, b A a, the dense -subalgebra of analytic elements.

8 KMS states By analogy with finite systems and the Gibbs state, extend the notions of KMS β state, partition function, inverse temperature. A C -algebra, σ : R Aut(A) time evolution, ϕ a state on A. 1 ϕ is KMS β (at inverse temperature β [0, )) if ϕ(ab) = ϕ(bσ iβ (a)) for all a, b A a, the dense -subalgebra of analytic elements. 2 A state ϕ is a ground state if for all a, b with b analytic, the function z ϕ(aσ z (b)) is bounded in the upper-half plane.

9 KMS states By analogy with finite systems and the Gibbs state, extend the notions of KMS β state, partition function, inverse temperature. A C -algebra, σ : R Aut(A) time evolution, ϕ a state on A. 1 ϕ is KMS β (at inverse temperature β [0, )) if ϕ(ab) = ϕ(bσ iβ (a)) for all a, b A a, the dense -subalgebra of analytic elements. 2 A state ϕ is a ground state if for all a, b with b analytic, the function z ϕ(aσ z (b)) is bounded in the upper-half plane. 3 KMS if ϕ = w lim ϕ n as β n and ϕ n is KMS βn.

10 KMS states By analogy with finite systems and the Gibbs state, extend the notions of KMS β state, partition function, inverse temperature. A C -algebra, σ : R Aut(A) time evolution, ϕ a state on A. 1 ϕ is KMS β (at inverse temperature β [0, )) if ϕ(ab) = ϕ(bσ iβ (a)) for all a, b A a, the dense -subalgebra of analytic elements. 2 A state ϕ is a ground state if for all a, b with b analytic, the function z ϕ(aσ z (b)) is bounded in the upper-half plane. 3 KMS if ϕ = w lim ϕ n as β n and ϕ n is KMS βn. References: Bratteli-Robinson, Pedersen, Connes-Marcolli.

11 KMS strictly subset of ground states Theorem (Laca-Raeburn (2010)) C (N N ) is the universal C -algebra generated by isometries s and {v p p prime}, subject to the relations 1 v p s = s p v p ; 2 v p v q = v q v p, 3 v p v q = v q v p when p q, 4 s v p = s p 1 v p s, and 5 v p s k v p = 0 for 1 k < p. Dynamics: σ t (s) = s and σ t (v p ) = p it v p.

12 KMS strictly subset of ground states Theorem (Laca-Raeburn (2010)) C (N N ) is the universal C -algebra generated by isometries s and {v p p prime}, subject to the relations 1 v p s = s p v p ; 2 v p v q = v q v p, 3 v p v q = v q v p when p q, 4 s v p = s p 1 v p s, and 5 v p s k v p = 0 for 1 k < p. Dynamics: σ t (s) = s and σ t (v p ) = p it v p.then, for β < 1, there are no KMS states, if β [1, 2], there is a unique KMS β state;

13 KMS strictly subset of ground states Theorem (Laca-Raeburn (2010)) C (N N ) is the universal C -algebra generated by isometries s and {v p p prime}, subject to the relations 1 v p s = s p v p ; 2 v p v q = v q v p, 3 v p v q = v q v p when p q, 4 s v p = s p 1 v p s, and 5 v p s k v p = 0 for 1 k < p. Dynamics: σ t (s) = s and σ t (v p ) = p it v p.then, for β < 1, there are no KMS states, if β [1, 2], there is a unique KMS β state; if β (2, ], the KMS β states are parametrised by probability measures on T while the ground states are parametrised by states on the Toeplitz C -algebra generated by a single isometry.

14 KMS strictly subset of ground states Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Let S be a right monoid and N : S N homomorphism such that S is admissible. Consider the time evolution σ t (v s ) = Ns it v s. If β c R is such that the function ζ N (β) := n (β 1), n Irr(N(S)) converges for β β c, then for (C (S), R, σ) we have

15 KMS strictly subset of ground states Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Let S be a right monoid and N : S N homomorphism such that S is admissible. Consider the time evolution σ t (v s ) = Ns it v s. If β c R is such that the function ζ N (β) := n (β 1), n Irr(N(S)) converges for β β c, then for (C (S), R, σ) we have 1 β [0, 1): no KMS β state;

16 KMS strictly subset of ground states Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Let S be a right monoid and N : S N homomorphism such that S is admissible. Consider the time evolution σ t (v s ) = Ns it v s. If β c R is such that the function ζ N (β) := n (β 1), n Irr(N(S)) converges for β β c, then for (C (S), R, σ) we have 1 β [0, 1): no KMS β state; 2 β [1, β c ]: unique KMS β if action S c S/S c essentially free, where S c S sub of elements having with any t in S;

17 KMS strictly subset of ground states Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Let S be a right monoid and N : S N homomorphism such that S is admissible. Consider the time evolution σ t (v s ) = Ns it v s. If β c R is such that the function ζ N (β) := n (β 1), n Irr(N(S)) converges for β β c, then for (C (S), R, σ) we have 1 β [0, 1): no KMS β state; 2 β [1, β c ]: unique KMS β if action S c S/S c essentially free, where S c S sub of elements having with any t in S; 3 β (β c, ]: KMS β states parametrised by normalised traces on C (S c ) ;

18 KMS strictly subset of ground states Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Let S be a right monoid and N : S N homomorphism such that S is admissible. Consider the time evolution σ t (v s ) = Ns it v s. If β c R is such that the function ζ N (β) := n (β 1), n Irr(N(S)) converges for β β c, then for (C (S), R, σ) we have 1 β [0, 1): no KMS β state; 2 β [1, β c ]: unique KMS β if action S c S/S c essentially free, where S c S sub of elements having with any t in S; 3 β (β c, ]: KMS β states parametrised by normalised traces on C (S c ) ; 4 Ground states: parametrised by states on C (S c ).