Small divisors, Diophantine numbers and interacting quantum many - PowerPoint PPT Presentation

The interacting Aubry-Andre’ model Basic model; experimentally in cold atoms by Bloch group (Science 2015) If a + x , a − x fermionic operators, x ∈ Z , H = ∑ ( a + x +1 a x + a + x − 1 a − ∑ u cos(2 πω x ) a + x a − ∑ v ( x − y ) a + x a − x a + y a − − ε ( x )+ x + U y x , y x x with v ( x − y ) = δ y − x , 1 + δ x − y , 1 . Equivalent to XXZ chain with a magnetic field cos(2 πω x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The interacting Aubry-Andre’ model Basic model; experimentally in cold atoms by Bloch group (Science 2015) If a + x , a − x fermionic operators, x ∈ Z , H = ∑ ( a + x +1 a x + a + x − 1 a − ∑ u cos(2 πω x ) a + x a − ∑ v ( x − y ) a + x a − x a + y a − − ε ( x )+ x + U y x , y x x with v ( x − y ) = δ y − x , 1 + δ x − y , 1 . Equivalent to XXZ chain with a magnetic field cos(2 πω x ) One imposes a Diophantine condition on the frequency || ω x || ≥ C 0 | x | − τ ∀ x ∈ Z / { 0 } ( ∗ ) || . || is the norm on the one dimensional torus of period 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The interacting Aubry-Andre’ model Basic model; experimentally in cold atoms by Bloch group (Science 2015) If a + x , a − x fermionic operators, x ∈ Z , H = ∑ ( a + x +1 a x + a + x − 1 a − ∑ u cos(2 πω x ) a + x a − ∑ v ( x − y ) a + x a − x a + y a − − ε ( x )+ x + U y x , y x x with v ( x − y ) = δ y − x , 1 + δ x − y , 1 . Equivalent to XXZ chain with a magnetic field cos(2 πω x ) One imposes a Diophantine condition on the frequency || ω x || ≥ C 0 | x | − τ ∀ x ∈ Z / { 0 } ( ∗ ) || . || is the norm on the one dimensional torus of period 1 Diophantine condition says that divisors can be small but x is large. Typically adopted in KAM theory Full measure set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Aubry-Andre’ model In the non interacting case U = 0 the states are obtained by the antisymmetrization (fermions) of the eigenfunctions of almost Mathieu equation − εψ ( x + 1) − εψ ( x − 1) + u cos(2 π ( ω x + θ )) ψ ( x ) = E ψ ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Aubry-Andre’ model In the non interacting case U = 0 the states are obtained by the antisymmetrization (fermions) of the eigenfunctions of almost Mathieu equation − εψ ( x + 1) − εψ ( x − 1) + u cos(2 π ( ω x + θ )) ψ ( x ) = E ψ ( x ) Deeply studied in mathematics Dinaburg-Sinai (1975); Sinai (1987), Froehlich, Spencer, Wittwer (1990); Jitomirskaya (1999); Avila, Jitomirskaya (2006).... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Aubry-Andre’ model In the non interacting case U = 0 the states are obtained by the antisymmetrization (fermions) of the eigenfunctions of almost Mathieu equation − εψ ( x + 1) − εψ ( x − 1) + u cos(2 π ( ω x + θ )) ψ ( x ) = E ψ ( x ) Deeply studied in mathematics Dinaburg-Sinai (1975); Sinai (1987), Froehlich, Spencer, Wittwer (1990); Jitomirskaya (1999); Avila, Jitomirskaya (2006).... For almost every ω, θ the almost Mathieu operator has a)for ε/ u < 1 2 only pps with exponentially decaying eigenfunctions (Anderson localization); b)for ε/ u > 1 2 purely absolutely continuous spectrum (extended quasi-Bloch waves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Interacting model ∑ ( a + x +1 a x + a + x − 1 a − H = − ε ( x ) + x ∈ Λ ∑ u cos(2 π ( ω x )) a + x a − ∑ v ( x − y ) a + x a − x a + y a − x + U y x ∈ Λ x , y with v ( x − y ) = δ y − x , 1 + δ x − y , 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Interacting model ∑ ( a + x +1 a x + a + x − 1 a − H = − ε ( x ) + x ∈ Λ ∑ u cos(2 π ( ω x )) a + x a − ∑ v ( x − y ) a + x a − x a + y a − x + U y x ∈ Λ x , y with v ( x − y ) = δ y − x , 1 + δ x − y , 1 . If a ± x = e ( H − µ N ) x 0 a ± x e − ( H − µ N ) x 0 , x = ( x , x 0 ), N = ∑ x a + x a − x and µ the chemical potential, the Grand-Canonical imaginary time 2-point correlation is y > ≡ S ( x , y ) = Tre − β ( H − µ N ) T { a − x a + y } < T a − x a + Tre − β ( H − µ N ) where T is the time-order product and µ is the chemical potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Interacting model ∑ ( a + x +1 a x + a + x − 1 a − H = − ε ( x ) + x ∈ Λ ∑ u cos(2 π ( ω x )) a + x a − ∑ v ( x − y ) a + x a − x a + y a − x + U y x ∈ Λ x , y with v ( x − y ) = δ y − x , 1 + δ x − y , 1 . If a ± x = e ( H − µ N ) x 0 a ± x e − ( H − µ N ) x 0 , x = ( x , x 0 ), N = ∑ x a + x a − x and µ the chemical potential, the Grand-Canonical imaginary time 2-point correlation is y > ≡ S ( x , y ) = Tre − β ( H − µ N ) T { a − x a + y } < T a − x a + Tre − β ( H − µ N ) where T is the time-order product and µ is the chemical potential. We study the localized regime considering ε, U small and the delocalized regime considering u , U small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Molecular limit x (cos 2 π ( ω x ) − µ ) a + x a − ε = U = 0 molecular limit H = ∑ x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Molecular limit x (cos 2 π ( ω x ) − µ ) a + x a − ε = U = 0 molecular limit H = ∑ x < T a − x a + y > | 0 = δ x , y ¯ g ( x , x 0 − y 0 ) e − ik 0 ( x 0 − y 0 ) g ( x , x 0 − y 0 ) = 1 ∑ ¯ β − ik 0 + cos 2 π ( ω x ) − cos 2 π ( ω ¯ x ) k 0 GS occupation number χ (cos 2 π ( ω x ) ≤ µ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Molecular limit x (cos 2 π ( ω x ) − µ ) a + x a − ε = U = 0 molecular limit H = ∑ x < T a − x a + y > | 0 = δ x , y ¯ g ( x , x 0 − y 0 ) e − ik 0 ( x 0 − y 0 ) g ( x , x 0 − y 0 ) = 1 ∑ ¯ β − ik 0 + cos 2 π ( ω x ) − cos 2 π ( ω ¯ x ) k 0 GS occupation number χ (cos 2 π ( ω x ) ≤ µ ). Let us introduce x ± = ± ¯ x , x ± Fermi coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Molecular limit x (cos 2 π ( ω x ) − µ ) a + x a − ε = U = 0 molecular limit H = ∑ x < T a − x a + y > | 0 = δ x , y ¯ g ( x , x 0 − y 0 ) e − ik 0 ( x 0 − y 0 ) g ( x , x 0 − y 0 ) = 1 ∑ ¯ β − ik 0 + cos 2 π ( ω x ) − cos 2 π ( ω ¯ x ) k 0 GS occupation number χ (cos 2 π ( ω x ) ≤ µ ). Let us introduce x ± = ± ¯ x , x ± Fermi coordinates. If we set x = x ′ + ¯ x ρ , ρ = ± , for small ( ω x ′ ) mod . 1 1 g ( x ′ + ¯ ˆ x ρ , k 0 ) ∼ − ik 0 ± v 0 ( ω x ′ ) mod . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Molecular limit x (cos 2 π ( ω x ) − µ ) a + x a − ε = U = 0 molecular limit H = ∑ x < T a − x a + y > | 0 = δ x , y ¯ g ( x , x 0 − y 0 ) e − ik 0 ( x 0 − y 0 ) g ( x , x 0 − y 0 ) = 1 ∑ ¯ β − ik 0 + cos 2 π ( ω x ) − cos 2 π ( ω ¯ x ) k 0 GS occupation number χ (cos 2 π ( ω x ) ≤ µ ). Let us introduce x ± = ± ¯ x , x ± Fermi coordinates. If we set x = x ′ + ¯ x ρ , ρ = ± , for small ( ω x ′ ) mod . 1 1 g ( x ′ + ¯ ˆ x ρ , k 0 ) ∼ − ik 0 ± v 0 ( ω x ′ ) mod . 1 We assume Diophantine conditions || ω x || ≥ C 0 | x | − τ ( ∗ ) x || ≥ C 0 | x | − τ || ω x ± 2 ω ¯ ∀ x ∈ Z / { 0 } ( ∗∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Small divisors In absence of many body interaction there are only chain graphs, α i = ± ∫ ε n ∑ ∑ dx 0 , 1 ... dx 0 , n ¯ g ( x 1 , x 0 − x 0 , 1 )¯ g ( x 1 + α i , ( x 0 , n − y 0 )) x 1 i ≤ n n ∏ ∑ g ( x 1 + ¯ α k , x 0 , i +1 − x 0 , i ) i =1 k ≤ i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Small divisors In absence of many body interaction there are only chain graphs, α i = ± ∫ ε n ∑ ∑ dx 0 , 1 ... dx 0 , n ¯ g ( x 1 , x 0 − x 0 , 1 )¯ g ( x 1 + α i , ( x 0 , n − y 0 )) x 1 i ≤ n n ∏ ∑ g ( x 1 + ¯ α k , x 0 , i +1 − x 0 , i ) i =1 k ≤ i Propagators g ( k 0 , x ) can be arbitrarily large (small divisors) g ( x ′ ± ¯ x , k 0 ) | ≤ C 0 | x ′ | τ | ˆ Chain graphs are apparently O ( n ! τ ); small divisors accumuluates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Small divisors In absence of many body interaction there are only chain graphs, α i = ± ∫ ε n ∑ ∑ dx 0 , 1 ... dx 0 , n ¯ g ( x 1 , x 0 − x 0 , 1 )¯ g ( x 1 + α i , ( x 0 , n − y 0 )) x 1 i ≤ n n ∏ ∑ g ( x 1 + ¯ α k , x 0 , i +1 − x 0 , i ) i =1 k ≤ i Propagators g ( k 0 , x ) can be arbitrarily large (small divisors) g ( x ′ ± ¯ x , k 0 ) | ≤ C 0 | x ′ | τ | ˆ Chain graphs are apparently O ( n ! τ ); small divisors accumuluates. When U ̸ = 0 there also loops producing additional divergences, absent in KAM or in the non interacting case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Localized regime for IIA Theorem In the spinless interacting Aubry-Andre’ model, assuming (*) and ¯ x verifying (**) if u = 1 , for suitable chemical potential and small ε, U | log | ∆ | | < T a − x a + y > | ≤ Ce − ξ | x − y | 1 + (∆ | x 0 − y 0 ) | ) N with ∆ = (1 + min( | x | , | y | )) − τ , ξ = | log(max( | ε | , | U | )) | . Assuming (**) x half integer the same holds with ∆ replaced by σ = O ( ε 2¯ x ) and ¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Localized regime for IIA Theorem In the spinless interacting Aubry-Andre’ model, assuming (*) and ¯ x verifying (**) if u = 1 , for suitable chemical potential and small ε, U | log | ∆ | | < T a − x a + y > | ≤ Ce − ξ | x − y | 1 + (∆ | x 0 − y 0 ) | ) N with ∆ = (1 + min( | x | , | y | )) − τ , ξ = | log(max( | ε | , | U | )) | . Assuming (**) x half integer the same holds with ∆ replaced by σ = O ( ε 2¯ x ) and ¯ Exponential decay in coordinates signals persistence of localization in presence of interactions in the ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Localized regime for IIA Theorem In the spinless interacting Aubry-Andre’ model, assuming (*) and ¯ x verifying (**) if u = 1 , for suitable chemical potential and small ε, U | log | ∆ | | < T a − x a + y > | ≤ Ce − ξ | x − y | 1 + (∆ | x 0 − y 0 ) | ) N with ∆ = (1 + min( | x | , | y | )) − τ , ξ = | log(max( | ε | , | U | )) | . Assuming (**) x half integer the same holds with ∆ replaced by σ = O ( ε 2¯ x ) and ¯ Exponential decay in coordinates signals persistence of localization in presence of interactions in the ground state. As we said, is crucial that the result is proved by convergent expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Localized regime for IIA Theorem In the spinless interacting Aubry-Andre’ model, assuming (*) and ¯ x verifying (**) if u = 1 , for suitable chemical potential and small ε, U | log | ∆ | | < T a − x a + y > | ≤ Ce − ξ | x − y | 1 + (∆ | x 0 − y 0 ) | ) N with ∆ = (1 + min( | x | , | y | )) − τ , ξ = | log(max( | ε | , | U | )) | . Assuming (**) x half integer the same holds with ∆ replaced by σ = O ( ε 2¯ x ) and ¯ Exponential decay in coordinates signals persistence of localization in presence of interactions in the ground state. As we said, is crucial that the result is proved by convergent expansions. Absence of spin is important in the proof. Localization is believed to persists for any state (MBL). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Localized regime for IIA Theorem In the spinless interacting Aubry-Andre’ model, assuming (*) and ¯ x verifying (**) if u = 1 , for suitable chemical potential and small ε, U | log | ∆ | | < T a − x a + y > | ≤ Ce − ξ | x − y | 1 + (∆ | x 0 − y 0 ) | ) N with ∆ = (1 + min( | x | , | y | )) − τ , ξ = | log(max( | ε | , | U | )) | . Assuming (**) x half integer the same holds with ∆ replaced by σ = O ( ε 2¯ x ) and ¯ Exponential decay in coordinates signals persistence of localization in presence of interactions in the ground state. As we said, is crucial that the result is proved by convergent expansions. Absence of spin is important in the proof. Localization is believed to persists for any state (MBL). Mastropietro PRL (2015); CMP (2016); Comm. Math. Phys. (2017) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Free fermion limit When U = u = 0 , ε = 1 one has the integrable or free fermion limit. k a − k ( − cos k + µ ) a + H = ∑ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Free fermion limit When U = u = 0 , ε = 1 one has the integrable or free fermion limit. k a − k ( − cos k + µ ) a + H = ∑ k . e i k ( x − y ) S 0 ( x , y ) = 1 ∑ β L − ik 0 + cos k − µ k 0 , k µ = cos p F . ± p F Fermi momenta. GS occupation number χ (cos k − µ ≤ 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Free fermion limit When U = u = 0 , ε = 1 one has the integrable or free fermion limit. k a − k ( − cos k + µ ) a + H = ∑ k . e i k ( x − y ) S 0 ( x , y ) = 1 ∑ β L − ik 0 + cos k − µ k 0 , k µ = cos p F . ± p F Fermi momenta. GS occupation number χ (cos k − µ ≤ 0). Close to the singularity cos( k ′ ± p F ) − µ = ± sin p F k ′ + O ( k ′ 2 ) linear dispersion relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Free fermion limit When U = u = 0 , ε = 1 one has the integrable or free fermion limit. k a − k ( − cos k + µ ) a + H = ∑ k . e i k ( x − y ) S 0 ( x , y ) = 1 ∑ β L − ik 0 + cos k − µ k 0 , k µ = cos p F . ± p F Fermi momenta. GS occupation number χ (cos k − µ ≤ 0). Close to the singularity cos( k ′ ± p F ) − µ = ± sin p F k ′ + O ( k ′ 2 ) linear dispersion relation. Momenta measured from the Fermi points conserved up 2 n ω which can be arbitrarily small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA Theorem Assume u , U small, ω Diophantine and p F = n πω with n integer. For C N properly chosen chemical potential | < T a − x a + y > | ≤ 1+(∆ | x − y ) | ) N with ∆ ∼ [ u 2 n ( a n + F )] X n with F = O ( | U | + | u | ) , a n non vanishing and X n = 1 + O ( U ) is a critical exponent. Gaps at p F = n πω mod. 2 π (finite number if ω irrational ). They persists in presence of interactions, but are strongly renormalized via a critical exponent (the ratio of free or interacting goes to zero or infinite as u → 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA Theorem Assume u , U small, ω Diophantine and p F = n πω with n integer. For C N properly chosen chemical potential | < T a − x a + y > | ≤ 1+(∆ | x − y ) | ) N with ∆ ∼ [ u 2 n ( a n + F )] X n with F = O ( | U | + | u | ) , a n non vanishing and X n = 1 + O ( U ) is a critical exponent. Gaps at p F = n πω mod. 2 π (finite number if ω irrational ). They persists in presence of interactions, but are strongly renormalized via a critical exponent (the ratio of free or interacting goes to zero or infinite as u → 0) Results valid uniformly in the ration U / u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA Universal relation 1 X n = 2 − K n K n is the critical exponents in the oscillating part of the density correlation ( K n > 1 if U < 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA Universal relation 1 X n = 2 − K n K n is the critical exponents in the oscillating part of the density correlation ( K n > 1 if U < 0). Validity of bosonization picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA Universal relation 1 X n = 2 − K n K n is the critical exponents in the oscillating part of the density correlation ( K n > 1 if U < 0). Validity of bosonization picture. With other quasi periodic potential (Fibonacci) it was claimed that smallest gaps are closed (Giamarchi et al 1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA Universal relation 1 X n = 2 − K n K n is the critical exponents in the oscillating part of the density correlation ( K n > 1 if U < 0). Validity of bosonization picture. With other quasi periodic potential (Fibonacci) it was claimed that smallest gaps are closed (Giamarchi et al 1999). For chemical exponents in the spectrum with Fermi momentum diophantine power law decay with anomalous exponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA Universal relation 1 X n = 2 − K n K n is the critical exponents in the oscillating part of the density correlation ( K n > 1 if U < 0). Validity of bosonization picture. With other quasi periodic potential (Fibonacci) it was claimed that smallest gaps are closed (Giamarchi et al 1999). For chemical exponents in the spectrum with Fermi momentum diophantine power law decay with anomalous exponent. Mastropietro CMP (1999); PRB (2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstadter model Electrons moving through a magnetic and periodic electrostatic potentials with incommensurate frequencies w ∑ [ t 1 ( a + e 1 e − ieBx 2 / � c a − x + a + e 1 e ieBx 2 / � c a − H = x ) + x + ⃗ ⃗ ⃗ ⃗ x − ⃗ ⃗ ⃗ x e 2 a − e 2 a − x a − e 1 a − t 2 ( a + x + a + ∑ y ) a + x a + v ( ⃗ x − ⃗ x )] + U ⃗ x + ⃗ ⃗ ⃗ x − ⃗ ⃗ ⃗ ⃗ x + ⃗ x + ⃗ ⃗ e 1 x ,⃗ ⃗ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstadter model Electrons moving through a magnetic and periodic electrostatic potentials with incommensurate frequencies w ∑ [ t 1 ( a + e 1 e − ieBx 2 / � c a − x + a + e 1 e ieBx 2 / � c a − H = x ) + ⃗ x + ⃗ ⃗ ⃗ x − ⃗ ⃗ ⃗ x e 2 a − e 2 a − x a − e 1 a − t 2 ( a + x + a + ∑ y ) a + x a + v ( ⃗ x − ⃗ x )] + U ⃗ x + ⃗ ⃗ x − ⃗ ⃗ ⃗ ⃗ ⃗ x + ⃗ ⃗ x + ⃗ e 1 ⃗ x ,⃗ y We consider t 1 = 1 and t 2 = t small. At t 2 = U = 0, uncoupled wires of of fermions labeled by x 2 and energy cos( k 1 − 2 πα x 2 ), eBx 2 / � c = 2 πα ; for a given chemical potential µ = cos p F the ground states is obtained by filling the states k 1 ∈ p x 2 − , p x 2 + with p x 2 ± = ± p F + 2 πα x 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstadter model Electrons moving through a magnetic and periodic electrostatic potentials with incommensurate frequencies w ∑ [ t 1 ( a + e 1 e − ieBx 2 / � c a − x + a + e 1 e ieBx 2 / � c a − H = x ) + x + ⃗ ⃗ ⃗ x − ⃗ ⃗ ⃗ ⃗ x e 2 a − e 2 a − x a − e 1 a − t 2 ( a + x + a + ∑ y ) a + x a + v ( ⃗ x − ⃗ x )] + U ⃗ x + ⃗ ⃗ ⃗ x − ⃗ ⃗ ⃗ x + ⃗ ⃗ x + ⃗ ⃗ e 1 ⃗ x ,⃗ y We consider t 1 = 1 and t 2 = t small. At t 2 = U = 0, uncoupled wires of of fermions labeled by x 2 and energy cos( k 1 − 2 πα x 2 ), eBx 2 / � c = 2 πα ; for a given chemical potential µ = cos p F the ground states is obtained by filling the states k 1 ∈ p x 2 − , p x 2 + with p x 2 ± = ± p F + 2 πα x 2 Assume eBx 2 / � c = 2 πα , p F = n F πα and α diophantine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstardter model Theorem Choosing the chemical potential so that p F = n πα with α Diophantine (*) and t and n F | U | log t small; then the 2-point function decay in x 0 , x 1 faster than any power with rate in x 0 , x 1 ∆ n = [ t n ( a n + R ] with R = O ((max( t , U ))) and a n non vanishing and independent on x 2 No gap is closed for small U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstardter model Theorem Choosing the chemical potential so that p F = n πα with α Diophantine (*) and t and n F | U | log t small; then the 2-point function decay in x 0 , x 1 faster than any power with rate in x 0 , x 1 ∆ n = [ t n ( a n + R ] with R = O ((max( t , U ))) and a n non vanishing and independent on x 2 No gap is closed for small U . Contrary to the previous case there is a condition between U and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sketch of proof in localized regime IAA ∂ 2 The 2-point function is given by y W | 0 ∂ϕ + x ∂ϕ − ∫ e W ( ϕ ) = P ( d ψ ) e − V ( ψ ) −B ( ψ,ϕ ) with P ( d ψ ) a gaussian Grassmann integral with propagator δ x , y ¯ g ( x , x 0 − y 0 ), ¯ g ( x , x 0 ) is the temporal FT of ˆ g ( x , k 0 ) ∫ ∑ ψ + x ψ − x ψ + x + α e 1 ψ − V ( ψ ) = U d x x + α e 1 α = ± ∫ ∫ d x ( ψ + x + e 1 ψ − x + ψ + x − e 1 ψ − d x ψ + x ψ − + ε x ) + ν x ∫ β d x ( ϕ + x ψ − x + ψ + x ϕ − where ∫ d x = ∑ 2 dx 0 , Finally B = ∫ x ) 2 x ∈ Λ − β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof We perform an RG analysis decomposing the propagator as sum of propagators living at γ 2 h − 1 ≤ k 2 x | 2 ≤ γ 2 h +1 , 0 + | ϕ x − ϕ ¯ h = 0 , − 1 , − 2 ... , γ > 1 , ϕ x = cos 2 π ( ω x ) ; this correspond to two regions, around ¯ x + and ¯ x − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof We perform an RG analysis decomposing the propagator as sum of propagators living at γ 2 h − 1 ≤ k 2 x | 2 ≤ γ 2 h +1 , 0 + | ϕ x − ϕ ¯ h = 0 , − 1 , − 2 ... , γ > 1 , ϕ x = cos 2 π ( ω x ) ; this correspond to two regions, around ¯ x + and ¯ x − . This implies that the single scale propagator has the form ρ = ± g ( h ) with | g ( h ) C N ∑ ρ ( x ) | ≤ 1+( γ h ( x 0 − y 0 )) N ; the corresponding ρ Grasmann variable is ψ ( h ) x ,ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof We perform an RG analysis decomposing the propagator as sum of propagators living at γ 2 h − 1 ≤ k 2 x | 2 ≤ γ 2 h +1 , 0 + | ϕ x − ϕ ¯ h = 0 , − 1 , − 2 ... , γ > 1 , ϕ x = cos 2 π ( ω x ) ; this correspond to two regions, around ¯ x + and ¯ x − . This implies that the single scale propagator has the form ρ = ± g ( h ) with | g ( h ) C N ∑ ρ ( x ) | ≤ 1+( γ h ( x 0 − y 0 )) N ; the corresponding ρ Grasmann variable is ψ ( h ) x ,ρ . We integrate the fields with decreasing scale; for instance W (0) (the partition function) can be written as ∫ ∫ ∫ ∫ P ( d ψ ) e V = P ( d ψ 0 ) e V = P ( d ψ ≤− 1 ) e V − 1 ... P ( d ψ ≤− 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof We perform an RG analysis decomposing the propagator as sum of propagators living at γ 2 h − 1 ≤ k 2 x | 2 ≤ γ 2 h +1 , 0 + | ϕ x − ϕ ¯ h = 0 , − 1 , − 2 ... , γ > 1 , ϕ x = cos 2 π ( ω x ) ; this correspond to two regions, around ¯ x + and ¯ x − . This implies that the single scale propagator has the form ρ = ± g ( h ) with | g ( h ) C N ∑ ρ ( x ) | ≤ 1+( γ h ( x 0 − y 0 )) N ; the corresponding ρ Grasmann variable is ψ ( h ) x ,ρ . We integrate the fields with decreasing scale; for instance W (0) (the partition function) can be written as ∫ ∫ ∫ ∫ P ( d ψ ) e V = P ( d ψ 0 ) e V = P ( d ψ ≤− 1 ) e V − 1 ... P ( d ψ ≤− 1 ) The effective potential V h sum of monomials of any order in i ψ ε i dx 0 , 1 ... dx 0 , n W h ∏ ∑ ∫ i , x 0 , i ,ρ i (we have integrated the deltas in x ′ x ′ 1 the propagators). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof According to power counting, the theory is non renormalizable ; all effective interactions have positive dimension, D = 1 and usually this makes a perturbative approach impossible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof According to power counting, the theory is non renormalizable ; all effective interactions have positive dimension, D = 1 and usually this makes a perturbative approach impossible. ε i One has to distinguish among the monomials ∏ i ψ i , x 0 , i ,ρ i in the x ′ effective potential between resonant and non resonant terms. Resonant terms; x ′ i = x ′ . Non Resonant terms x ′ i ̸ = x ′ j for some i , j . (In the non interacting case only two external lines are present). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof According to power counting, the theory is non renormalizable ; all effective interactions have positive dimension, D = 1 and usually this makes a perturbative approach impossible. ε i One has to distinguish among the monomials ∏ i ψ i , x 0 , i ,ρ i in the x ′ effective potential between resonant and non resonant terms. Resonant terms; x ′ i = x ′ . Non Resonant terms x ′ i ̸ = x ′ j for some i , j . (In the non interacting case only two external lines are present). It turns out that the non resonant terms are irrelevant (even if they are relevant according to power counting). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof According to power counting, the theory is non renormalizable ; all effective interactions have positive dimension, D = 1 and usually this makes a perturbative approach impossible. ε i One has to distinguish among the monomials ∏ i ψ i , x 0 , i ,ρ i in the x ′ effective potential between resonant and non resonant terms. Resonant terms; x ′ i = x ′ . Non Resonant terms x ′ i ̸ = x ′ j for some i , j . (In the non interacting case only two external lines are present). It turns out that the non resonant terms are irrelevant (even if they are relevant according to power counting). Roughly speaking, the idea is that if two propagators have similar (not equal) small size ( non resonant subgraphs ) , then the difference of their coordinates is large and this produces a ”gain” as passing from x to x + n one needs n vertices. That is if 2 ) mod1 ∼ Λ − 1 then by the Diophantine condition ( ω x ′ 1 ) mod1 ∼ ( ω x ′ 2Λ − 1 ≥ || ω ( x ′ 1 − x ′ 2 ) || ≥ C 0 | x ′ 1 − x ′ 2 | − τ C Λ τ − 1 2 | ≥ ¯ that is | x ′ 1 − x ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof . w 1 w a w c w b w 2 FIG. 1: A tree ¯ T v with attached wiggly lines representing the external lines P v ; the lines represent propagators with scale ≥ h v connecting w 1 , w a , w b , w c , w 2 , representing the end-points following v in τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof As usual in renormalization theory, one needs to introduce clusters v with scale h v ; the propagators in v have divisors smaller than γ h v (necessary to avoid overlapping divergences). Trees. v ′ is the cluster containing v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof As usual in renormalization theory, one needs to introduce clusters v with scale h v ; the propagators in v have divisors smaller than γ h v (necessary to avoid overlapping divergences). Trees. v ′ is the cluster containing v . v γ − h v ( S v − 1) , v vertex, S v number of Naive bound for each tree ∏ clusters in v . Determinant bounds How we can improve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof As usual in renormalization theory, one needs to introduce clusters v with scale h v ; the propagators in v have divisors smaller than γ h v (necessary to avoid overlapping divergences). Trees. v ′ is the cluster containing v . v γ − h v ( S v − 1) , v vertex, S v number of Naive bound for each tree ∏ clusters in v . Determinant bounds How we can improve? Consider two vertices w 1 , w 2 such that x ′ w 1 and x ′ w 2 are coordinates of the external fields, and let be c w 1 , w 2 the path (vertices and lines) in ¯ T v connecting w 1 with w 2 ; we call | c w 1 , w 2 | the number of vertices in c w 1 , w 2 . The following relation holds, if δ i w = ± 1 it corresponds to an ε end-point and δ i w = (0 , ± 1) is a U end-point ∑ x ′ w 1 − x ′ δ i w w 2 = ¯ x ρ w 2 − ¯ x ρ w 1 + w w ∈ c w 1 , w 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof As usual in renormalization theory, one needs to introduce clusters v with scale h v ; the propagators in v have divisors smaller than γ h v (necessary to avoid overlapping divergences). Trees. v ′ is the cluster containing v . v γ − h v ( S v − 1) , v vertex, S v number of Naive bound for each tree ∏ clusters in v . Determinant bounds How we can improve? Consider two vertices w 1 , w 2 such that x ′ w 1 and x ′ w 2 are coordinates of the external fields, and let be c w 1 , w 2 the path (vertices and lines) in ¯ T v connecting w 1 with w 2 ; we call | c w 1 , w 2 | the number of vertices in c w 1 , w 2 . The following relation holds, if δ i w = ± 1 it corresponds to an ε end-point and δ i w = (0 , ± 1) is a U end-point ∑ x ′ w 1 − x ′ δ i w w 2 = ¯ x ρ w 2 − ¯ x ρ w 1 + w w ∈ c w 1 , w 2 As x i − x j = M ∈ Z and x ′ i = x ′ j then (¯ x ρ i − ¯ x ρ j ) + M = 0, so that ρ i = ρ j as 2¯ x is not integer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof 1 . FIG. 1: A tree τ (only the vertices v ∈ V χ are represented), the corresponding clusters, represented as boxes, and a Feyn- man graph; the propagators have scale h v 1 and h v 2 respec- tively. U U U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof By the Diophantine condition a) ρ w 1 = ρ w 2 the (*); b)if ρ w 1 = − ρ w 2 by (**) v ′ ≥ 2 cv − 1 0 γ h ¯ || ( ω x ′ w 1 ) || 1 + || ( ω x ′ w 2 ) || 1 ≥ || ω ( x ′ w 1 − x ′ w 2 ) || 1 ≥ C 0 ( | c w 2 , w 1 | ) − τ − h ¯ v ′ τ . If two external propagators are small but so that | c w 1 , w 2 | ≥ A γ not exactly equal, you need a lot of hopping or interactions to produce them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof ε n factor we can then extract (we If ¯ ε = max( | ε | , | U | )) from the ¯ ε = ∏ 0 ε 2 h − 1 ) write ¯ h = −∞ ¯ ε N v 2 hv ′ n 4 ≤ ∏ ε ¯ v ∈ L − hv ′ where N v is the number of points in v ; as N v ≥ | c w 1 , w 2 | ≥ A γ τ then − hv ′ 2 hv ′ n 4 ≤ ∏ ε A γ τ ε ¯ ¯ v ∈ L 1 τ / 2 > 1 then where L are the non resonant vertices If γ v ∈ L γ 3 h v S L ≤ C n ∏ v where S L v is the number of non resonant clusters in v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof We localize the resonant terms x = x 0 , i , x with all x ′ i equal L ψ ε 1 x 1 ,ρ ...ψ ε n x n ,ρ = ψ ε 1 x 1 ,ρ ...ψ ε n x 1 ,ρ Note that one has to renormalize monomial of all orders, a potentially very dangerous situation (this is like in KAM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof We localize the resonant terms x = x 0 , i , x with all x ′ i equal L ψ ε 1 x 1 ,ρ ...ψ ε n x n ,ρ = ψ ε 1 x 1 ,ρ ...ψ ε n x 1 ,ρ Note that one has to renormalize monomial of all orders, a potentially very dangerous situation (this is like in KAM). The terms with n ≥ 4 are vanishing by anticommutativity; there are no non-irrelevant quartic terms if the fermions are spinless and r i = r by the diophantine condition (**). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof We localize the resonant terms x = x 0 , i , x with all x ′ i equal L ψ ε 1 x 1 ,ρ ...ψ ε n x n ,ρ = ψ ε 1 x 1 ,ρ ...ψ ε n x 1 ,ρ Note that one has to renormalize monomial of all orders, a potentially very dangerous situation (this is like in KAM). The terms with n ≥ 4 are vanishing by anticommutativity; there are no non-irrelevant quartic terms if the fermions are spinless and r i = r by the diophantine condition (**). We write V h = L V h + R V h . The R V h term is the usual renormalized term in QFT; the bound has an extra γ h v ′ − h v ; then there is an γ h v ′ for each renormalized vertex v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof We localize the resonant terms x = x 0 , i , x with all x ′ i equal L ψ ε 1 x 1 ,ρ ...ψ ε n x n ,ρ = ψ ε 1 x 1 ,ρ ...ψ ε n x 1 ,ρ Note that one has to renormalize monomial of all orders, a potentially very dangerous situation (this is like in KAM). The terms with n ≥ 4 are vanishing by anticommutativity; there are no non-irrelevant quartic terms if the fermions are spinless and r i = r by the diophantine condition (**). We write V h = L V h + R V h . The R V h term is the usual renormalized term in QFT; the bound has an extra γ h v ′ − h v ; then there is an γ h v ′ for each renormalized vertex v . In order to sum over the number of external fieds one uses both the cancellations due to anticommutativity and the diophantine condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof In the invariant tori for KAM the local part is vanishing by remarkable cancellations; here the local part is vanishing if the number of fields is greater than two by anticommutativity and (**). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof In the invariant tori for KAM the local part is vanishing by remarkable cancellations; here the local part is vanishing if the number of fields is greater than two by anticommutativity and (**). There remain the local terms with 2 field which are relevant and produce renormalization of the chemical potential; the flow is controlled by the countertem ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof In the invariant tori for KAM the local part is vanishing by remarkable cancellations; here the local part is vanishing if the number of fields is greater than two by anticommutativity and (**). There remain the local terms with 2 field which are relevant and produce renormalization of the chemical potential; the flow is controlled by the countertem ν . ρ ψ − x is integer there is also a mass term ψ + If 2¯ − ρ producing gaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof In the invariant tori for KAM the local part is vanishing by remarkable cancellations; here the local part is vanishing if the number of fields is greater than two by anticommutativity and (**). There remain the local terms with 2 field which are relevant and produce renormalization of the chemical potential; the flow is controlled by the countertem ν . ρ ψ − x is integer there is also a mass term ψ + If 2¯ − ρ producing gaps. With spin quartic terms are not irrelevant; this suggest tthat new phenomena should appear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case of IAA In the delocalized regime the theory is instead renormalizable. Quartic terms are marginal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case of IAA In the delocalized regime the theory is instead renormalizable. Quartic terms are marginal. There are infinitely many quartic terms; momenta measured from Fermi points verify, ε i = ± 4 4 ∑ ε i k ′ ∑ i = − ε i ρ i p F + 2 n πω + 2 l π i =1 i =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case of IAA In the delocalized regime the theory is instead renormalizable. Quartic terms are marginal. There are infinitely many quartic terms; momenta measured from Fermi points verify, ε i = ± 4 4 ∑ ε i k ′ ∑ i = − ε i ρ i p F + 2 n πω + 2 l π i =1 i =1 In the incommensurate case the r.h.s. can be arbitrarily small.r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case of IAA In the delocalized regime the theory is instead renormalizable. Quartic terms are marginal. There are infinitely many quartic terms; momenta measured from Fermi points verify, ε i = ± 4 4 ∑ ε i k ′ ∑ i = − ε i ρ i p F + 2 n πω + 2 l π i =1 i =1 In the incommensurate case the r.h.s. can be arbitrarily small.r. Only the terns such that the l.h.s vanishes are really marginal; this is due again to the Diophantine condition (there is an high power of couplings u ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case of IAA In the delocalized regime the theory is instead renormalizable. Quartic terms are marginal. There are infinitely many quartic terms; momenta measured from Fermi points verify, ε i = ± 4 4 ∑ ε i k ′ ∑ i = − ε i ρ i p F + 2 n πω + 2 l π i =1 i =1 In the incommensurate case the r.h.s. can be arbitrarily small.r. Only the terns such that the l.h.s vanishes are really marginal; this is due again to the Diophantine condition (there is an high power of couplings u ). Only one effective interaction ψ + + ψ − + ψ + − ψ − − with n = 0 are marginal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case In order to prove the universal relations one has to use Ward identities related to the emerging chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case In order to prove the universal relations one has to use Ward identities related to the emerging chiral symmetry One introduces a reference model and tune its bare parameters so that the fixed points (and the exponents) are the same (this is possible as the model is studied by RG). The model is a Thirring model with non-local interaction and momentum cut-off. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case In order to prove the universal relations one has to use Ward identities related to the emerging chiral symmetry One introduces a reference model and tune its bare parameters so that the fixed points (and the exponents) are the same (this is possible as the model is studied by RG). The model is a Thirring model with non-local interaction and momentum cut-off. Extra symmetries has the effect that closed expression for exponents can be derived by combining WI with SD equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case In order to prove the universal relations one has to use Ward identities related to the emerging chiral symmetry One introduces a reference model and tune its bare parameters so that the fixed points (and the exponents) are the same (this is possible as the model is studied by RG). The model is a Thirring model with non-local interaction and momentum cut-off. Extra symmetries has the effect that closed expression for exponents can be derived by combining WI with SD equations. The momentum cut-off γ N produces correction to the WI which must carefully taken into account. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: delocalized case In order to prove the universal relations one has to use Ward identities related to the emerging chiral symmetry One introduces a reference model and tune its bare parameters so that the fixed points (and the exponents) are the same (this is possible as the model is studied by RG). The model is a Thirring model with non-local interaction and momentum cut-off. Extra symmetries has the effect that closed expression for exponents can be derived by combining WI with SD equations. The momentum cut-off γ N produces correction to the WI which must carefully taken into account. The validity of universal relation is related to a non-perturbative version of the Adler -Bardeen theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Emerging chiral WI At finite N one gets an extra term in the WI (the dot is χ N ( χ N − 1); the contribution to the vertex function can be decomposed (without breaking the determinants) and, for gains due to the long range interaction, the contribution of irreducible terms vanish as N → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: Hofstatter models The Hofstadter model is like a sequence of 1d systems labelled by x 2 , ∑ ε i k ′ ∑ ∑ i = ε i ω i p F + ε i 2 πα x 2 , i mod . 2 π i i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: Hofstatter models The Hofstadter model is like a sequence of 1d systems labelled by x 2 , ∑ ε i k ′ ∑ ∑ i = ε i ω i p F + ε i 2 πα x 2 , i mod . 2 π i i i Again by the Diophantine condition ony the terms not verifying i ε i k ′ ∑ i = 0 are irrelevant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: Hofstatter models The Hofstadter model is like a sequence of 1d systems labelled by x 2 , ∑ ε i k ′ ∑ ∑ i = ε i ω i p F + ε i 2 πα x 2 , i mod . 2 π i i i Again by the Diophantine condition ony the terms not verifying i ε i k ′ ∑ i = 0 are irrelevant. There are several marginal quartic terms (not only 1) an this explain the condition on U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: Hofstatter models The Hofstadter model is like a sequence of 1d systems labelled by x 2 , ∑ ε i k ′ ∑ ∑ i = ε i ω i p F + ε i 2 πα x 2 , i mod . 2 π i i i Again by the Diophantine condition ony the terms not verifying i ε i k ′ ∑ i = 0 are irrelevant. There are several marginal quartic terms (not only 1) an this explain the condition on U . This justify the bosonization approach in which all terms not i ε i k ′ verifying ∑ i = 0 are neglected (Kane et al (2001), to understand FQHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions As in the classical case, the final behavior is determined by behavior to all orders, that is by divergence or convergence of series. Conclusions based on finiteness are dangerous (Poincare’ triviality). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions As in the classical case, the final behavior is determined by behavior to all orders, that is by divergence or convergence of series. Conclusions based on finiteness are dangerous (Poincare’ triviality). We proved persistence of localization in IAA in the ground state. One of the very few cases which this can be established analytically in the thermodynamic limit (KAM cannot deal with infinite particles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .