Large Deviation Principles for Weakly Interacting Fermions N. J. B. - PowerPoint PPT Presentation

Large Deviation Principles for Weakly Interacting Fermions N. J. B. Aza Departamento de F ´ ısica Matem´ atica, Universidade de S˜ ao Paulo Joint work with J.-B. Bru, W. de Siqueira Pedra and L. C. P. A. M. M¨ ussnich October 08, 2016

Large Deviation Theory and Quantum Lattice Systems Lebowitz–Lenci–Spohn ’00, Gallavotti–Lebowitz–Mastropietro ’02, Netoˇ cny–Redig ’04, Lenci–Rey-Bellet ’05, Hiai–Mosonyi–Ogawa ’07, Ogata ’10, Ogata–Rey-Bellet ’11, de Roeck–Maes–Netoˇ cny–Sch¨ utz ’15

Large Deviation Theory and Quantum Lattice Systems Lebowitz–Lenci–Spohn ’00, Gallavotti–Lebowitz–Mastropietro ’02, Netoˇ cny–Redig ’04, Lenci–Rey-Bellet ’05, Hiai–Mosonyi–Ogawa ’07, Ogata ’10, Ogata–Rey-Bellet ’11, de Roeck–Maes–Netoˇ cny–Sch¨ utz ’15 Observe that for ρ a state on the C ∗ –algebra A and A ∈ A a selfadjoint element, there is a unique probability measure µ ρ, A on R such that µ ρ, A ( spec ( A )) = 1 and, for all continuous functions f : R → C , � ρ ( f ( A )) = f ( x ) µ ρ, A ( d x ) . R µ A . = µ ρ, A is the measure associated to ρ and A . For a sequence of selfadjoints { A l } l ∈ R + of A , and a state ρ , we say that these satisfy a Large Deviation Principle (LDP), with scale | Λ l | , if, for all Borel measurable Γ ⊂ R , 1 1 − inf Γ I ( x ) ≤ lim inf | Λ l | log µ A l (Γ) ≤ lim sup | Λ l | log µ A l (Γ) ≤ − inf x ∈ Γ I ( x ) x ∈ ˚ l →∞ l →∞

Large Deviation Theory and Quantum Lattice Systems To find an LDP we desire to use the G¨ artner–Ellis Theorem (GET) to µ A l , through the scaled cumulant generating function 1 | Λ l | log ρ ( e s | Λ l | A l ) , f ( s ) = lim s ∈ R . l →∞ If f exists and is differentiable, then the good rate function I is the Legendre–Fenchel transform of f . In the case of lattice fermions we represent f as a Berezin–integral and analyse it using “tree expansions”. The scale | Λ l | will be then the volume of the boxes Λ l : Λ l . = { ( x 1 , . . . , x d ) ∈ Z d : | x 1 | , . . . , | x d | ≤ l } ∈ P f ( Z d ) . For lattice fermions, A is the CAR C ∗ –algebra generated by the identity ✶ and { a s , x } s , x ∈ L . L . = S × Z d where S is the set of Spins of single fermions. However, our proofs do not depend on the particular choice of S .

Large Deviation Theory and Quantum Lattice Systems CAR : { a x , a ∗ { a x , a x ′ } = 0 , x ′ } = δ x , x ′ ✶ . A Λ ⊂ A is the C ∗ –subalgebra generated ✶ and { a x } x ∈ Λ . Λ ∈ A + ∩ A Λ and An interaction Φ is a map P f ( Z d ) → A s.t. Φ Λ = Φ ∗ Φ ∅ = 0. Φ is of finite range if for Λ ∈ P f ( Z d ) and some R > 0 , diam Λ > R → Φ Λ = 0. For any interaction Φ , we define the space average K Φ ∈ A Λ l by l . 1 K Φ � = Φ Λ . l | Λ l | Λ ∈ P f ( Z d ) , Λ ∈ Λ l

Main Result Note that finite range interactions define equilibrium (KMS) states of A . Theorem (A., Bru, M¨ ussnich, Pedra) Let β > 0 and consider any finite range translation invariant interaction Ψ = Ψ 0 + Ψ 1 . If the interparticle component Ψ 1 ( Ψ 0 is the free part) is small enough (depending on β ), then any invariant equilibrium state ρ of Ψ and the sequence of averages K Φ of ANY translation invariant l interaction Φ , have an LDP and s �→ f ( s ) is analytic at small s.

Main Result Remarks 1 Note that, in contrast to previous results, we do not impose β to be small or Φ (defining K Φ l ) to be an one–site interaction. 2 Uniqueness of KMS states is not used. 3 Use C ∗ –algebras formalism and Grassmann algebras. 4 Determinant bounds or study of Large Determinants. 5 Direct representation of f by Berezin–integrals. In particular we do not use the correlation functions. 6 Beyond the LDP, the analyticity of f ( · ) together with the Bryc Theorem implies the Central Limit Theorem for the system.

Main Result Sketch of the proof. 1 | Λ l | log tr ( e − β H l ′ e sK l ) 1 f ( s ) = lim l →∞ lim . tr ( e − β H l ′ ) l ′ →∞

Main Result Sketch of the proof. 1 | Λ l | log tr ( e − β H l ′ e sK l ) 1 f ( s ) = lim l →∞ lim . tr ( e − β H l ′ ) l ′ →∞ 2 From a Feynmann–Kac–like formula for traces, we write the KMS state as a Berezin–integral tr ∧ ∗ H ( e − β H l ′ e sK l ) � W ( n ) l , l ′ . l ′ ( H ( n ) ) e = lim d µ C ( n ) tr ∧ ∗ H ( e − β H ( 0 ) l ′ ) n →∞

Main Result Sketch of the proof. 1 | Λ l | log tr ( e − β H l ′ e sK l ) 1 f ( s ) = lim l →∞ lim . tr ( e − β H l ′ ) l ′ →∞ 2 From a Feynmann–Kac–like formula for traces, we write the KMS state as a Berezin–integral tr ∧ ∗ H ( e − β H l ′ e sK l ) � W ( n ) l , l ′ . l ′ ( H ( n ) ) e = lim d µ C ( n ) tr ∧ ∗ H ( e − β H ( 0 ) l ′ ) n →∞ 3 The covariance C ( n ) satisfies: l ′ � m � � m � � � ��� m � a ) ( k a ) � � ( ϕ ∗ C ( n ) ϕ ( k b ) � � ϕ ∗ � � � � det � ≤ a � H ∗ � ϕ b � H . � l ′ � b a , b = 1 a = 1 b = 1

Main Result Sketch of the proof. 1 | Λ l | log tr ( e − β H l ′ e sK l ) 1 f ( s ) = lim l →∞ lim . tr ( e − β H l ′ ) l ′ →∞ 2 From a Feynmann–Kac–like formula for traces, we write the KMS state as a Berezin–integral tr ∧ ∗ H ( e − β H l ′ e sK l ) � W ( n ) l , l ′ . l ′ ( H ( n ) ) e = lim d µ C ( n ) tr ∧ ∗ H ( e − β H ( 0 ) l ′ ) n →∞ 3 The covariance C ( n ) satisfies: l ′ � m � � m � � � ��� m � a ) ( k a ) � � ( ϕ ∗ C ( n ) ϕ ( k b ) � � ϕ ∗ � � � � det � ≤ a � H ∗ � ϕ b � H . � l ′ � b a , b = 1 a = 1 b = 1 Use Brydges–Kennedy Tree expansions (BKTE) to verify GET. BKTE are solution of an infinite hierarchy of coupled ODEs. . . End

Perspectives and Questions Perspectives: 1 Quantum Hypothesys Testing? Open problems, e.g., study thermodynamic limit of the relative entropy between equilibrium state ω β Λ ∈ A Λ and translation invariant state ω Λ . 2 Related problems to our approach. 3 . . .

Perspectives and Questions Perspectives: 1 Quantum Hypothesys Testing? Open problems, e.g., study thermodynamic limit of the relative entropy between equilibrium state ω β Λ ∈ A Λ and translation invariant state ω Λ . 2 Related problems to our approach. 3 . . . Open Questions: 1 LDP for time correlation (transport coefficients)? 2 Systems in presence of disorder? 3 What about LDP for commutators of averages i [ K Φ 1 , K Φ 2 ] in place of simple averages K Φ ? (Also related to transport) 4 . . .

Thank you!

Supporting facts 1 For any invertible operator C ∈ B ( H ) and ξ ∈ ∧ ∗ ( H ⊕ ¯ H ) , the Gaussian H ⊕ ¯ � d µ C ( H ) : ∧ ∗ � � Grassmann integral: → C 1 with covariance C , is H defined by � � d µ C ( H ) ξ . d ( H ) e � H , C − 1 H � ∧ ξ. = det ( C ) ϕ m ∈ ¯ � d µ C ( H ) 1 = 1 and for any m , n ∈ N and all ¯ ϕ 1 , . . . , ¯ H , 2 ϕ 1 , . . . , ϕ n ∈ H , � ϕ k ( C ϕ l )] m d µ C ( H ) ¯ ϕ 1 · · · ¯ ϕ m ϕ 1 · · · ϕ m = det [ ¯ k , l = 1 δ m , n 1 3 For all N ∈ N and A 0 , . . . , A N − 1 ∈ B ( ∧ ∗ H ) , � N − 1 H ( k ) �� � N − 1 � � � � E ( N ) � κ ( k ) ( A k ) Tr ∧ ∗ H ( A 0 · · · A N − 1 ) 1 = , d H k = 0 k = 0 N − 1 � H ( 0 ) , H ( 0 ) � + � H ( 0 ) , H ( N − 1 ) � + ( � H ( k ) , H ( k ) �−� H ( k ) , H ( k − 1 ) � ) � . where E ( N ) = e , k = 1 H κ ( k ) . ( 0 , 0 ) ◦ κ : B ( ∧ ∗ H ) → ∧ ∗ ( H ( k ) ⊕ ¯ = κ ( k , k ) H ( k ) ) and for ( i , j ) : ∧ ∗ ( H ( i ) ⊕ ¯ H ( j ) ) → ∧ ∗ ( H ( k ) ⊕ ¯ i , j , k , l ∈ { 0 , . . . , N } , κ ( k , l ) H ( l ) ) .