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

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Small divisors, Diophantine numbers and interacting quantum many body systems

Vieri Mastropietro

Universit´ a di Milano

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Incommensurate structures

System of electrons subject to incommensurate disorder or potentials have rather unusual properties due the generically fractal structure

• f their spectrum.

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Incommensurate structures

System of electrons subject to incommensurate disorder or potentials have rather unusual properties due the generically fractal structure

• f their spectrum.

Physical examples are quasi-random optical lattices realizing an interacting Aubry-Andre’ model (e.g. Bloch at al) or Graphene realization of Hofstadter model (e.g. Kim et al.).

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Incommensurate structures

System of electrons subject to incommensurate disorder or potentials have rather unusual properties due the generically fractal structure

• f their spectrum.

Physical examples are quasi-random optical lattices realizing an interacting Aubry-Andre’ model (e.g. Bloch at al) or Graphene realization of Hofstadter model (e.g. Kim et al.). In absence of interaction: Harper or almost Mathieu Schroedinger

• equation. Delocalized or localized states depending on the size of

the potential. The spectrum have infinitely many gaps (Cantor set).

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Incommensurate structures

System of electrons subject to incommensurate disorder or potentials have rather unusual properties due the generically fractal structure

• f their spectrum.

Physical examples are quasi-random optical lattices realizing an interacting Aubry-Andre’ model (e.g. Bloch at al) or Graphene realization of Hofstadter model (e.g. Kim et al.). In absence of interaction: Harper or almost Mathieu Schroedinger

• equation. Delocalized or localized states depending on the size of

the potential. The spectrum have infinitely many gaps (Cantor set). Mathematical difficulty: Small divisors problem, very similar to the

• ne appearing in classical mechanics.

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Small divisors

Small divisors are expressions like (⃗ ω · ⃗ k)−1 ⃗ k ∈ Zd/{0} appear in perturbative expansion.

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Small divisors

Small divisors are expressions like (⃗ ω · ⃗ k)−1 ⃗ k ∈ Zd/{0} appear in perturbative expansion. In these kind of problems a new physical behavior appears not as a divergence of a graph but as the failure of convergence of the series (see e.g. the series for prime integrals in perturbed Hamiltonian systems and Poincare’ triviality; or Birkoff series)

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Small divisors

Small divisors are expressions like (⃗ ω · ⃗ k)−1 ⃗ k ∈ Zd/{0} appear in perturbative expansion. In these kind of problems a new physical behavior appears not as a divergence of a graph but as the failure of convergence of the series (see e.g. the series for prime integrals in perturbed Hamiltonian systems and Poincare’ triviality; or Birkoff series) In certain cases, as in KAM tori, the small divisors can be controlled and series are convergent.

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Small divisors

Small divisors are expressions like (⃗ ω · ⃗ k)−1 ⃗ k ∈ Zd/{0} appear in perturbative expansion. In these kind of problems a new physical behavior appears not as a divergence of a graph but as the failure of convergence of the series (see e.g. the series for prime integrals in perturbed Hamiltonian systems and Poincare’ triviality; or Birkoff series) In certain cases, as in KAM tori, the small divisors can be controlled and series are convergent. KAM methods and number theoretical properties have been used for the construction of the single particle spectrum of Harper equation (construction of eigenstates of Almost Mathieu is very close to construction of KAM tori).

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Interactions

In presence of interactions between particles, unavoidable in real experiments, the single particle description is not valid and one cannot use classical KAM methods.

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Interactions

In presence of interactions between particles, unavoidable in real experiments, the single particle description is not valid and one cannot use classical KAM methods. Several basic questions:

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Interactions

In presence of interactions between particles, unavoidable in real experiments, the single particle description is not valid and one cannot use classical KAM methods. Several basic questions: Do localization persists in presence of interaction (MBL, Iyer, Oganesyan, Refael, Huse 2012)? (cold atoms exp., Bloch et al)

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Interactions

In presence of interactions between particles, unavoidable in real experiments, the single particle description is not valid and one cannot use classical KAM methods. Several basic questions: Do localization persists in presence of interaction (MBL, Iyer, Oganesyan, Refael, Huse 2012)? (cold atoms exp., Bloch et al) Do some gaps of the spectrum close (Giamarchi 1999)?

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Interactions

In presence of interactions between particles, unavoidable in real experiments, the single particle description is not valid and one cannot use classical KAM methods. Several basic questions: Do localization persists in presence of interaction (MBL, Iyer, Oganesyan, Refael, Huse 2012)? (cold atoms exp., Bloch et al) Do some gaps of the spectrum close (Giamarchi 1999)? New gaps open corresponding to FQHE(Teo Kane et al 2011 ) ? (graphene experiments)

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Interactions

Small divisors can produce effects which can be seen only at a non perturbative level. KAM cannot deal with infinitely many particles.

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Interactions

Small divisors can produce effects which can be seen only at a non perturbative level. KAM cannot deal with infinitely many particles. I review some results using non-perturbative RG and some number theoretical properties in the thermodynamic limit and T = 0.

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Interactions

Small divisors can produce effects which can be seen only at a non perturbative level. KAM cannot deal with infinitely many particles. I review some results using non-perturbative RG and some number theoretical properties in the thermodynamic limit and T = 0. Direct RG methods have been used for proving convergence of KAM Lindsdet series: tree expansion

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Interactions

Small divisors can produce effects which can be seen only at a non perturbative level. KAM cannot deal with infinitely many particles. I review some results using non-perturbative RG and some number theoretical properties in the thermodynamic limit and T = 0. Direct RG methods have been used for proving convergence of KAM Lindsdet series: tree expansion In presence of many body interaction, insted of trees one have also loop grahs.

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Interactions

Small divisors can produce effects which can be seen only at a non perturbative level. KAM cannot deal with infinitely many particles. I review some results using non-perturbative RG and some number theoretical properties in the thermodynamic limit and T = 0. Direct RG methods have been used for proving convergence of KAM Lindsdet series: tree expansion In presence of many body interaction, insted of trees one have also loop grahs. Sall divisors+determinant bounds+WI

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The interacting Aubry-Andre’ model

Basic model; experimentally in cold atoms by Bloch group (Science 2015)

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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 =

−ε( ∑

x

(a+

x+1ax+a+ x−1a− x )+

∑

x

u cos(2πωx)a+

x a− x +U

∑

x,y

v(x−y)a+

x a− x a+ y a− y

with v(x − y) = δy−x,1 + δx−y,1. Equivalent to XXZ chain with a magnetic field cos(2πωx)

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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 =

−ε( ∑

x

(a+

x+1ax+a+ x−1a− x )+

∑

x

u cos(2πωx)a+

x a− x +U

∑

x,y

v(x−y)a+

x a− x a+ y a− y

with v(x − y) = δy−x,1 + δx−y,1. Equivalent to XXZ chain with a magnetic field cos(2πωx) One imposes a Diophantine condition on the frequency ||ωx|| ≥ C0|x|−τ ∀x ∈ Z/{0} (∗) ||.|| is the norm on the one dimensional torus of period 1

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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 =

−ε( ∑

x

(a+

x+1ax+a+ x−1a− x )+

∑

x

u cos(2πωx)a+

x a− x +U

∑

x,y

v(x−y)a+

x a− x a+ y a− y

with v(x − y) = δy−x,1 + δx−y,1. Equivalent to XXZ chain with a magnetic field cos(2πωx) One imposes a Diophantine condition on the frequency ||ωx|| ≥ C0|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.

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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(2π(ωx + θ))ψ(x) = Eψ(x)

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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(2π(ωx + θ))ψ(x) = Eψ(x) Deeply studied in mathematics Dinaburg-Sinai (1975); Sinai (1987), Froehlich, Spencer, Wittwer (1990); Jitomirskaya (1999); Avila, Jitomirskaya (2006)....

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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(2π(ωx + θ))ψ(x) = Eψ(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)

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Interacting model

H = −ε( ∑

x∈Λ

(a+

x+1ax + a+ x−1a− x ) +

∑

x∈Λ

u cos(2π(ωx))a+

x a− x + U

∑

x,y

v(x − y)a+

x a− x a+ y a− y

with v(x − y) = δy−x,1 + δx−y,1.

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Interacting model

H = −ε( ∑

x∈Λ

(a+

x+1ax + a+ x−1a− x ) +

∑

x∈Λ

u cos(2π(ωx))a+

x a− x + U

∑

x,y

v(x − y)a+

x a− x a+ y a− y

with v(x − y) = δy−x,1 + δx−y,1. If a±

x = e(H−µN)x0a± x e−(H−µN)x0, x = (x, x0), N = ∑ x a+ x a− x and µ

the chemical potential, the Grand-Canonical imaginary time 2-point correlation is < Ta−

x a+ y >≡ S(x, y) = Tre−β(H−µN)T{a− x a+ y }

Tre−β(H−µN) where T is the time-order product and µ is the chemical potential.

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Interacting model

H = −ε( ∑

x∈Λ

(a+

x+1ax + a+ x−1a− x ) +

∑

x∈Λ

u cos(2π(ωx))a+

x a− x + U

∑

x,y

v(x − y)a+

x a− x a+ y a− y

with v(x − y) = δy−x,1 + δx−y,1. If a±

x = e(H−µN)x0a± x e−(H−µN)x0, x = (x, x0), N = ∑ x a+ x a− x and µ

the chemical potential, the Grand-Canonical imaginary time 2-point correlation is < Ta−

x a+ y >≡ S(x, y) = Tre−β(H−µN)T{a− x a+ y }

Tre−β(H−µN) 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.

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Molecular limit

ε = U = 0 molecular limit H = ∑

x(cos 2π(ωx) − µ)a+ x a− x

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Molecular limit

ε = U = 0 molecular limit H = ∑

x(cos 2π(ωx) − µ)a+ x a− x

< Ta−

x a+ y > |0 = δx,y ¯

g(x, x0 − y0) ¯ g(x, x0 − y0) = 1 β ∑

k0

e−ik0(x0−y0) −ik0 + cos 2π(ωx) − cos 2π(ω¯ x) GS occupation number χ(cos 2π(ωx) ≤ µ).

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Molecular limit

ε = U = 0 molecular limit H = ∑

x(cos 2π(ωx) − µ)a+ x a− x

< Ta−

x a+ y > |0 = δx,y ¯

g(x, x0 − y0) ¯ g(x, x0 − y0) = 1 β ∑

k0

e−ik0(x0−y0) −ik0 + cos 2π(ωx) − cos 2π(ω¯ x) GS occupation number χ(cos 2π(ωx) ≤ µ). Let us introduce x± = ±¯ x, x± Fermi coordinates.

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Molecular limit

ε = U = 0 molecular limit H = ∑

x(cos 2π(ωx) − µ)a+ x a− x

< Ta−

x a+ y > |0 = δx,y ¯

g(x, x0 − y0) ¯ g(x, x0 − y0) = 1 β ∑

k0

e−ik0(x0−y0) −ik0 + cos 2π(ωx) − cos 2π(ω¯ x) GS occupation number χ(cos 2π(ωx) ≤ µ). Let us introduce x± = ±¯ x, x± Fermi coordinates. If we set x = x′ + ¯ xρ, ρ = ±, for small (ωx′)mod.1 ˆ g(x′ + ¯ xρ, k0) ∼ 1 −ik0 ± v0(ωx′)mod.1

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Molecular limit

ε = U = 0 molecular limit H = ∑

x(cos 2π(ωx) − µ)a+ x a− x

< Ta−

x a+ y > |0 = δx,y ¯

g(x, x0 − y0) ¯ g(x, x0 − y0) = 1 β ∑

k0

e−ik0(x0−y0) −ik0 + cos 2π(ωx) − cos 2π(ω¯ x) GS occupation number χ(cos 2π(ωx) ≤ µ). Let us introduce x± = ±¯ x, x± Fermi coordinates. If we set x = x′ + ¯ xρ, ρ = ±, for small (ωx′)mod.1 ˆ g(x′ + ¯ xρ, k0) ∼ 1 −ik0 ± v0(ωx′)mod.1 We assume Diophantine conditions ||ωx|| ≥ C0|x|−τ (∗) ||ωx ± 2ω¯ x|| ≥ C0|x|−τ ∀x ∈ Z/{0} (∗∗)

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Small divisors

In absence of many body interaction there are only chain graphs, αi = ± εn ∑

x1

∫ dx0,1...dx0,n¯ g(x1, x0 − x0,1)¯ g(x1 + ∑

i≤n

αi, (x0,n − y0))

n

∏

i=1

¯ g(x1 + ∑

k≤i

αk, x0,i+1 − x0,i)

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Small divisors

In absence of many body interaction there are only chain graphs, αi = ± εn ∑

x1

∫ dx0,1...dx0,n¯ g(x1, x0 − x0,1)¯ g(x1 + ∑

i≤n

αi, (x0,n − y0))

n

∏

i=1

¯ g(x1 + ∑

k≤i

αk, x0,i+1 − x0,i) Propagators g(k0, x) can be arbitrarily large (small divisors) |ˆ g(x′ ± ¯ x, k0)| ≤ C0|x′|τ Chain graphs are apparently O(n!τ); small divisors accumuluates.

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Small divisors

In absence of many body interaction there are only chain graphs, αi = ± εn ∑

x1

∫ dx0,1...dx0,n¯ g(x1, x0 − x0,1)¯ g(x1 + ∑

i≤n

αi, (x0,n − y0))

n

∏

i=1

¯ g(x1 + ∑

k≤i

αk, x0,i+1 − x0,i) Propagators g(k0, x) can be arbitrarily large (small divisors) |ˆ g(x′ ± ¯ x, k0)| ≤ C0|x′|τ 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.

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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 | < Ta−

x a+ y > | ≤ Ce−ξ|x−y|

| log |∆| 1 + (∆|x0 − y0)|)N with ∆ = (1 + min(|x|, |y|))−τ, ξ = | log(max(|ε|, |U|))|. Assuming (**) and ¯ x half integer the same holds with ∆ replaced by σ = O(ε2¯

x)

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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 | < Ta−

x a+ y > | ≤ Ce−ξ|x−y|

| log |∆| 1 + (∆|x0 − y0)|)N with ∆ = (1 + min(|x|, |y|))−τ, ξ = | log(max(|ε|, |U|))|. Assuming (**) and ¯ x half integer the same holds with ∆ replaced by σ = O(ε2¯

x)

Exponential decay in coordinates signals persistence of localization in presence of interactions in the ground state.

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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 | < Ta−

x a+ y > | ≤ Ce−ξ|x−y|

| log |∆| 1 + (∆|x0 − y0)|)N with ∆ = (1 + min(|x|, |y|))−τ, ξ = | log(max(|ε|, |U|))|. Assuming (**) and ¯ x half integer the same holds with ∆ replaced by σ = O(ε2¯

x)

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.

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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 | < Ta−

x a+ y > | ≤ Ce−ξ|x−y|

| log |∆| 1 + (∆|x0 − y0)|)N with ∆ = (1 + min(|x|, |y|))−τ, ξ = | log(max(|ε|, |U|))|. Assuming (**) and ¯ x half integer the same holds with ∆ replaced by σ = O(ε2¯

x)

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).

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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 | < Ta−

x a+ y > | ≤ Ce−ξ|x−y|

| log |∆| 1 + (∆|x0 − y0)|)N with ∆ = (1 + min(|x|, |y|))−τ, ξ = | log(max(|ε|, |U|))|. Assuming (**) and ¯ x half integer the same holds with ∆ replaced by σ = O(ε2¯

x)

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)

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Free fermion limit

When U = u = 0, ε = 1 one has the integrable or free fermion limit. H = ∑

k(− cos k + µ)a+ k a− k .

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Free fermion limit

When U = u = 0, ε = 1 one has the integrable or free fermion limit. H = ∑

k(− cos k + µ)a+ k a− k .

S0(x, y) = 1 βL ∑

k0,k

eik(x−y) −ik0 + cos k − µ µ = cos pF. ±pF Fermi momenta. GS occupation number χ(cos k − µ ≤ 0).

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Free fermion limit

When U = u = 0, ε = 1 one has the integrable or free fermion limit. H = ∑

k(− cos k + µ)a+ k a− k .

S0(x, y) = 1 βL ∑

k0,k

eik(x−y) −ik0 + cos k − µ µ = cos pF. ±pF Fermi momenta. GS occupation number χ(cos k − µ ≤ 0). Close to the singularity cos(k′ ± pF) − µ = ± sin pFk′ + O(k′2) linear dispersion relation.

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Free fermion limit

When U = u = 0, ε = 1 one has the integrable or free fermion limit. H = ∑

k(− cos k + µ)a+ k a− k .

S0(x, y) = 1 βL ∑

k0,k

eik(x−y) −ik0 + cos k − µ µ = cos pF. ±pF Fermi momenta. GS occupation number χ(cos k − µ ≤ 0). Close to the singularity cos(k′ ± pF) − µ = ± sin pFk′ + O(k′2) linear dispersion relation. Momenta measured from the Fermi points conserved up 2nω which can be arbitrarily small.

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Delocalized regime for IAA

Theorem Assume u, U small, ω Diophantine and pF = nπω with n integer. For properly chosen chemical potential | < Ta−

x a+ y > | ≤ CN 1+(∆|x−y)|)N with

∆ ∼ [u2n(an + F)]Xn with F = O(|U| + |u|), an non vanishing and Xn = 1 + O(U) is a critical exponent. Gaps at pF = nπω mod. 2π (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 pF = nπω with n integer. For properly chosen chemical potential | < Ta−

x a+ y > | ≤ CN 1+(∆|x−y)|)N with

∆ ∼ [u2n(an + F)]Xn with F = O(|U| + |u|), an non vanishing and Xn = 1 + O(U) is a critical exponent. Gaps at pF = nπω mod. 2π (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 Xn = 1 2 − Kn Kn is the critical exponents in the oscillating part of the density correlation (Kn > 1 if U < 0).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA

Universal relation Xn = 1 2 − Kn Kn is the critical exponents in the oscillating part of the density correlation (Kn > 1 if U < 0). Validity of bosonization picture.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Delocalized regime for IAA

Universal relation Xn = 1 2 − Kn Kn is the critical exponents in the oscillating part of the density correlation (Kn > 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 Xn = 1 2 − Kn Kn is the critical exponents in the oscillating part of the density correlation (Kn > 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 Xn = 1 2 − Kn Kn is the critical exponents in the oscillating part of the density correlation (Kn > 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 H = ∑

⃗ x

[t1(a+

⃗ x+⃗ e1e−ieBx2/ca− ⃗ x + a+ ⃗ x−⃗ e1eieBx2/ca− ⃗ x ) +

t2(a+

⃗ x+⃗ e2a− ⃗ x + a+ ⃗ x−⃗ e2a− ⃗ x )] + U

∑

⃗ x,⃗ y

v(⃗ x − ⃗ y)a+

x a− ⃗ x a+ ⃗ x+⃗ e1a− ⃗ x+⃗ e1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstadter model

Electrons moving through a magnetic and periodic electrostatic potentials with incommensurate frequencies w H = ∑

⃗ x

[t1(a+

⃗ x+⃗ e1e−ieBx2/ca− ⃗ x + a+ ⃗ x−⃗ e1eieBx2/ca− ⃗ x ) +

t2(a+

⃗ x+⃗ e2a− ⃗ x + a+ ⃗ x−⃗ e2a− ⃗ x )] + U

∑

⃗ x,⃗ y

v(⃗ x − ⃗ y)a+

x a− ⃗ x a+ ⃗ x+⃗ e1a− ⃗ x+⃗ e1

We consider t1 = 1 and t2 = t small. At t2 = U = 0, uncoupled wires of of fermions labeled by x2 and energy cos(k1 − 2παx2), eBx2/c = 2πα ; for a given chemical potential µ = cos pF the ground states is obtained by filling the states k1 ∈ px2

−, px2 + with

px2

± = ±pF + 2παx2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstadter model

Electrons moving through a magnetic and periodic electrostatic potentials with incommensurate frequencies w H = ∑

⃗ x

[t1(a+

⃗ x+⃗ e1e−ieBx2/ca− ⃗ x + a+ ⃗ x−⃗ e1eieBx2/ca− ⃗ x ) +

t2(a+

⃗ x+⃗ e2a− ⃗ x + a+ ⃗ x−⃗ e2a− ⃗ x )] + U

∑

⃗ x,⃗ y

v(⃗ x − ⃗ y)a+

x a− ⃗ x a+ ⃗ x+⃗ e1a− ⃗ x+⃗ e1

We consider t1 = 1 and t2 = t small. At t2 = U = 0, uncoupled wires of of fermions labeled by x2 and energy cos(k1 − 2παx2), eBx2/c = 2πα ; for a given chemical potential µ = cos pF the ground states is obtained by filling the states k1 ∈ px2

−, px2 + with

px2

± = ±pF + 2παx2

Assume eBx2/c = 2πα, pF = nFπα and α diophantine.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstardter model

Theorem Choosing the chemical potential so that pF = nπα with α Diophantine (*) and t and nF|U| log t small; then the 2-point function decay in x0, x1 faster than any power with rate in x0, x1 ∆n = [tn(an + R] with R = O((max(t, U))) and an non vanishing and independent on x2 No gap is closed for small U.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hofstardter model

Theorem Choosing the chemical potential so that pF = nπα with α Diophantine (*) and t and nF|U| log t small; then the 2-point function decay in x0, x1 faster than any power with rate in x0, x1 ∆n = [tn(an + R] with R = O((max(t, U))) and an non vanishing and independent on x2 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

The 2-point function is given by

∂2 ∂ϕ+

x ∂ϕ− y W |0

eW (ϕ) = ∫ P(dψ)e−V (ψ)−B(ψ,ϕ) with P(dψ) a gaussian Grassmann integral with propagator δx,y ¯ g(x, x0 − y0), ¯ g(x, x0) is the temporal FT of ˆ g(x, k0) V (ψ) = U ∫ dx ∑

α=±

ψ+

x ψ− x ψ+ x+αe1ψ− x+αe1

+ε ∫ dx(ψ+

x+e1ψ− x + ψ+ x−e1ψ− x ) + ν

∫ dxψ+

x ψ− x

where ∫ dx = ∑

x∈Λ

∫ β

2

− β

2 dx0, Finally B =

∫ dx(ϕ+

x ψ− x + ψ+ x ϕ− x )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof

We perform an RG analysis decomposing the propagator as sum of propagators living at γ2h−1 ≤ k2

0 + |ϕx − ϕ¯ x|2 ≤ γ2h+1,

h = 0, −1, −2..., γ > 1, ϕx = cos 2π(ωx) ; 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 γ2h−1 ≤ k2

0 + |ϕx − ϕ¯ x|2 ≤ γ2h+1,

h = 0, −1, −2..., γ > 1, ϕx = cos 2π(ωx) ; this correspond to two regions, around ¯ x+ and ¯ x−. This implies that the single scale propagator has the form ∑

ρ=± g (h) ρ

with |g (h)

ρ (x)| ≤ CN 1+(γh(x0−y0))N ; the corresponding

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 γ2h−1 ≤ k2

0 + |ϕx − ϕ¯ x|2 ≤ γ2h+1,

h = 0, −1, −2..., γ > 1, ϕx = cos 2π(ωx) ; this correspond to two regions, around ¯ x+ and ¯ x−. This implies that the single scale propagator has the form ∑

ρ=± g (h) ρ

with |g (h)

ρ (x)| ≤ CN 1+(γh(x0−y0))N ; the corresponding

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(dψ)eV = ∫ P(dψ≤−1) ∫ P(dψ0)eV = ∫ P(dψ≤−1)eV −1...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof

We perform an RG analysis decomposing the propagator as sum of propagators living at γ2h−1 ≤ k2

0 + |ϕx − ϕ¯ x|2 ≤ γ2h+1,

h = 0, −1, −2..., γ > 1, ϕx = cos 2π(ωx) ; this correspond to two regions, around ¯ x+ and ¯ x−. This implies that the single scale propagator has the form ∑

ρ=± g (h) ρ

with |g (h)

ρ (x)| ≤ CN 1+(γh(x0−y0))N ; the corresponding

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(dψ)eV = ∫ P(dψ≤−1) ∫ P(dψ0)eV = ∫ P(dψ≤−1)eV −1... The effective potential V h sum of monomials of any order in ∑

x′

1

∫ dx0,1...dx0,nW h ∏

i ψεi x′

i ,x0,i,ρi (we have integrated the deltas in

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. One has to distinguish among the monomials ∏

i ψ

εi

x′

i ,x0,i,ρi in the

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. One has to distinguish among the monomials ∏

i ψ

εi

x′

i ,x0,i,ρi in the

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. One has to distinguish among the monomials ∏

i ψ

εi

x′

i ,x0,i,ρi in the

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

• f their coordinates is large and this produces a ”gain” as passing

from x to x + n one needs n vertices. That is if (ωx′

1)mod1 ∼ (ωx′ 2)mod1 ∼ Λ−1 then by the Diophantine condition

2Λ−1 ≥ ||ω(x′

1 − x′ 2)|| ≥ C0|x′ 1 − x′ 2|−τ

that is |x′

1 − x′ 2| ≥ ¯

CΛτ −1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof

.

w1 wa wb wc w2

• FIG. 1: A tree ¯

Tv with attached wiggly lines representing the external lines Pv; the lines represent propagators with scale ≥ hv connecting w1, wa, wb, wc, w2, 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 hv; the propagators in v have divisors smaller than γhv (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 hv; the propagators in v have divisors smaller than γhv (necessary to avoid overlapping divergences). Trees. v ′ is the cluster containing v. Naive bound for each tree ∏

v γ−hv(Sv−1), v vertex, Sv number of

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 hv; the propagators in v have divisors smaller than γhv (necessary to avoid overlapping divergences). Trees. v ′ is the cluster containing v. Naive bound for each tree ∏

v γ−hv(Sv−1), v vertex, Sv number of

clusters in v. Determinant bounds How we can improve? Consider two vertices w1, w2 such that x′

w1 and x′ w2 are coordinates

• f the external fields, and let be cw1,w2 the path (vertices and lines)

in ¯ Tv connecting w1 with w2; we call |cw1,w2| the number of vertices in cw1,w2. The following relation holds, if δi

w = ±1 it corresponds to

an ε end-point and δi

w = (0, ±1) is a U end-point

x′

w1 − x′ w2 = ¯

xρw2 − ¯ xρw1 + ∑

w∈cw1,w2

δiw

w

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof

As usual in renormalization theory, one needs to introduce clusters v with scale hv; the propagators in v have divisors smaller than γhv (necessary to avoid overlapping divergences). Trees. v ′ is the cluster containing v. Naive bound for each tree ∏

v γ−hv(Sv−1), v vertex, Sv number of

clusters in v. Determinant bounds How we can improve? Consider two vertices w1, w2 such that x′

w1 and x′ w2 are coordinates

• f the external fields, and let be cw1,w2 the path (vertices and lines)

in ¯ Tv connecting w1 with w2; we call |cw1,w2| the number of vertices in cw1,w2. The following relation holds, if δi

w = ±1 it corresponds to

an ε end-point and δi

w = (0, ±1) is a U end-point

x′

w1 − x′ w2 = ¯

xρw2 − ¯ xρw1 + ∑

w∈cw1,w2

δiw

w

As xi − xj = M ∈ Z and x′

i = x′ j then (¯

xρi − ¯ xρj) + M = 0, so that ρi = ρj as 2¯ x is not integer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof

1 . U U U

• FIG. 1: A tree τ (only the vertices v ∈ Vχ are represented),

the corresponding clusters, represented as boxes, and a Feyn- man graph; the propagators have scale hv1 and hv2 respec- tively.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some idea of the proof

By the Diophantine condition a) ρw1 = ρw2 the (*); b)if ρw1 = −ρw2 by (**) 2cv −1

0 γh¯

v′ ≥

||(ωx′

w1)||1 + ||(ωx′ w2)||1 ≥ ||ω(x′ w1 − x′ w2)||1 ≥ C0(|cw2,w1|)−τ

so that |cw1,w2| ≥ Aγ

−h¯ v′ τ . If two external propagators are small but

not exactly equal, you need a lot of hopping or interactions to produce them

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof

If ¯ ε = max(|ε|, |U|)) from the ¯ εn factor we can then extract (we write ¯ ε = ∏0

h=−∞ ¯

ε2h−1) ¯ ε

n 4 ≤

∏

v∈L

εNv2hv′ where Nv is the number of points in v ; as Nv ≥ |cw1,w2| ≥ Aγ

−hv′ τ

then ¯ ε

n 4 ≤

∏

v∈L

¯ εAγ

−hv′ τ

2hv′

where L are the non resonant vertices If γ

1 τ /2 > 1 then

≤ C n ∏

v∈L γ3hvSL

v where SL

v is the number of non resonant clusters

in v.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof

We localize the resonant terms x = x0,i, x with all x′

i equal

Lψε1

x1,ρ...ψεn xn,ρ = ψε1 x1,ρ...ψεn x1,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 = x0,i, x with all x′

i equal

Lψε1

x1,ρ...ψεn xn,ρ = ψε1 x1,ρ...ψεn x1,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 ri = r by the diophantine condition (**).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof

We localize the resonant terms x = x0,i, x with all x′

i equal

Lψε1

x1,ρ...ψεn xn,ρ = ψε1 x1,ρ...ψεn x1,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 ri = r by the diophantine condition (**). We write V h = LV h + RV h. The RV h term is the usual renormalized term in QFT; the bound has an extra γhv′−hv ; then there is an γhv′ for each renormalized vertex v.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof

We localize the resonant terms x = x0,i, x with all x′

i equal

Lψε1

x1,ρ...ψεn xn,ρ = ψε1 x1,ρ...ψεn x1,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 ri = r by the diophantine condition (**). We write V h = LV h + RV h. The RV h term is the usual renormalized term in QFT; the bound has an extra γhv′−hv ; then there is an γhv′ 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. If 2¯ x is integer there is also a mass term ψ+

ρ ψ− −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. If 2¯ x is integer there is also a mass term ψ+

ρ ψ− −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, εi = ±

4

∑

i=1

εik′

i = − 4

∑

i=1

εiρipF + 2nπω + 2lπ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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, εi = ±

4

∑

i=1

εik′

i = − 4

∑

i=1

εiρipF + 2nπω + 2lπ 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, εi = ±

4

∑

i=1

εik′

i = − 4

∑

i=1

εiρipF + 2nπω + 2lπ 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, εi = ±

4

∑

i=1

εik′

i = − 4

∑

i=1

εiρipF + 2nπω + 2lπ 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 γN 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 γN 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 χN(χN − 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 x2, ∑

i

εik′

i =

∑

i

εiωipF + ∑

i

εi2παx2,i mod.2π

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: Hofstatter models

The Hofstadter model is like a sequence of 1d systems labelled by x2, ∑

i

εik′

i =

∑

i

εiωipF + ∑

i

εi2παx2,i mod.2π Again by the Diophantine condition ony the terms not verifying ∑

i εik′ i = 0 are irrelevant.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ideas of proof: Hofstatter models

The Hofstadter model is like a sequence of 1d systems labelled by x2, ∑

i

εik′

i =

∑

i

εiωipF + ∑

i

εi2παx2,i mod.2π Again by the Diophantine condition ony the terms not verifying ∑

i εik′ 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 x2, ∑

i

εik′

i =

∑

i

εiωipF + ∑

i

εi2παx2,i mod.2π Again by the Diophantine condition ony the terms not verifying ∑

i εik′ 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 verifying ∑

i εik′ i = 0 are neglected (Kane et al (2001), to

understand FQHE

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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).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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). The spectrum has still infinitely many gaps in extended regime, renormalized by exponents. Uniform in the ratio between hopping and interaction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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). The spectrum has still infinitely many gaps in extended regime, renormalized by exponents. Uniform in the ratio between hopping and interaction. In the Hofstadter model gaps also persist but our results are not uniform in the ratio.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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). The spectrum has still infinitely many gaps in extended regime, renormalized by exponents. Uniform in the ratio between hopping and interaction. In the Hofstadter model gaps also persist but our results are not uniform in the ratio. Confirmation of bosonization in the extended regime in IAA or Hosftadter.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions

Open problems. In IAA is believed that all eigenstes are localized, not only the ground state. One has to solve a small divisor problem for prime integrals (the corresponding series in Hamiltonian system would be diverging). Imbrie (2016) partial result.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions

Open problems. In IAA is believed that all eigenstes are localized, not only the ground state. One has to solve a small divisor problem for prime integrals (the corresponding series in Hamiltonian system would be diverging). Imbrie (2016) partial result. In Hofstadter there are several instable directions, possibly producing extra gaps. Confirmation of validity of bosonization approach for weak couplings. FQHE.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions

Open problems. In IAA is believed that all eigenstes are localized, not only the ground state. One has to solve a small divisor problem for prime integrals (the corresponding series in Hamiltonian system would be diverging). Imbrie (2016) partial result. In Hofstadter there are several instable directions, possibly producing extra gaps. Confirmation of validity of bosonization approach for weak couplings. FQHE. Random disorder (Giamarchi Schultz 1988; Imbrie (2016) partial result for MBL)

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Conclusions

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