The general setting A toy model for quantum diffusion A supersymmetric non linear sigma model for quantum diffusion Margherita DISERTORI joint work with T. Spencer and M. Zirnbauer Laboratoire de Math´ ematiques Rapha¨ el Salem CNRS - University of Rouen (France)
The general setting A toy model for quantum diffusion ◮ the general setting : Anderson localization, random matrices and sigma models ◮ a toy model for quantum diffusion
The general setting A toy model for quantum diffusion Disordered conductors Anderson localization : disorder induced localization for conducting electrons The framework quantum mechanics: lattice field model, finite volume Λ ⊂ Z d Hamiltonian describing the system: matrix H ∈M Λ ( C ), H ∗ = H ψ eigenvector with � ψ � 2 = 1: | ψ j | 2 = P (electron at j ) then | ψ j | ∼ const. ∀ j → quantum diffusion (conductor) | ψ j | ∼ δ jj 0 → localization (insulator) disorder → H random with some probability law P ( H ) dH transition local/ext
The general setting A toy model for quantum diffusion Some models Random Schr¨ odinger Band Matrix H ∗ = H , H ij i.r.v. not i.d. H = − ∆ + λ ˆ V � � ˆ V ij = δ ij V j V j i . i . d . r . v . N C (0 ,J ij ) 1 /W | i − j |≤ W H ij ∼ J ij ∼ λ ∈ R ∆ discete Lapl . N R (0 ,J ii ) 0 | i − j | >W λ =0 H = − ∆: ext . W ≥| Λ | H ∼ GUE: ext . H ≃ λ ˆ λ ≫ 1 V : local . W ∼ 0 H ∼ diagonal: local . 0 <λ ≪ 1 , Λ → Z d W ≫ 1 , Λ → Z d d =1 , 2 local . d =1 , 2 local . d =3 ext . d =3 ext .
The general setting A toy model for quantum diffusion signatures of quantum diffusion Green’s Function : G ε, Λ ( E,x,y )=( E + iε − H Λ ) − 1 ( x,y ) , x,y ∈ Λ , E ∈ R , ε> 0 we have to study �| G ǫ, Λ ( E ; x,y ) | 2 � H Localization (a) | x − y |≫ 1 : � | G ε, Λ ( E,x,y ) | 2 � ≤ K ε e −| x − y | /ξE uniformly in ε, Λ (b) x = y : � | G ε, Λ ( E,x,x ) | 2 � ≥ K uniformly in ε, Λ ε Diffusion (a) | x − y |≫ 1 : lim Λ → Z d � | G ε, Λ ( E,x,y ) | 2 � ≃ ( − ∆+ ε ) − 1 x,y (b) x = y : � | G ε, Λ ( E,x,x ) | 2 � ≤ K, ∀ ε | Λ | =1
The general setting A toy model for quantum diffusion Some interesting quantities in physics • Conductivy (Kubo formula) � = 0 insulator (loc.)) x i x j ε 2 � | G ε, Λ ( E, 0 ,x ) | 2 � → σ ij ( E ) = lim 1 lim � π > 0 conductor (diff.) ε → 0 Λ →∞ x ∈ Λ • Inverse participation ratio ψ E localized 1 1 � ψ E � 4 | ψ E ( x ) | 4 ∼ 4 = � | supp( ψ E ) | ∼ 1 ψ E extended x ∈ Λ | Λ | � insulator P Λ ( E )= � ρ Λ ( E ) � ψ E � 4 4 � x ∈ Λ ε � | G ε, Λ ( E,x,x ) | 2 � K> 0 � − − − − → = lim � ρ Λ ( E ) � π | Λ |� ρ Λ ( E ) � conductor ε → 0 Λ → Z d 0
The general setting A toy model for quantum diffusion Techniques multiscale analysis, cluster expansion, renormalization : good in the localization regime transfert matrix : applies to 1 dimension delocalization regime : no general technique
The general setting A toy model for quantum diffusion Supersymmetric approach F. Wegner, K. Efetov 1. change of representation: algebraic operations involving fermionic and bosonic variables � SUSY | G ǫ ( E ; x, y ) | 2 � � = dµ ( { Q j } ) O ( Q x , Q y ) H ◮ Q j j ∈ Λ (small) matrix containing both fermionic and bosonic elements ◮ dµ ( { Q j } ) strongly correlated → saddle analysis 2. restriction to the saddle manifold → non linear sigma model 3. control the fluctuations around the saddle manifold
The general setting A toy model for quantum diffusion Saddle analysis: analytic tools new integration variables ◮ slow modes along the saddle manifold → non linear sigma model ◮ fast modes away from the saddle manifold fast modes slow modes saddle manifold NLSM is believed to contaqin the low energy physics
The general setting A toy model for quantum diffusion non linear sigma model � e − F ( ∇ Q ) e − εM ( Q ) dµ ( Q ) → dµ saddle ( Q ) = dQ j δ ( Q 2 j − Id) jλ Λ features ◮ saddle is non compact | Λ | → 0 as Λ → Z d 1 ◮ no mass: ε = ◮ internal symmetries (from SUSY structure) main problem: obtain the correct ε behavior hard to exploit the symmetries → try something “easier”
The general setting A toy model for quantum diffusion A nice SUSY model for quantum diffusion vector model (no matrices), Zirnbauer (1991) → expected to have same features of exact SUSY model for random band matrix main advantages ◮ after integrating out Grassman variables measure is positive ◮ symmetries are simpler to exploit ⇒ good candidate to develop techniques to treat quantum diffusion
The general setting A toy model for quantum diffusion The model after integrating out the Grassman variables j dt j e − tj ] e −B ( t ) det 1 / 2 [ D ( t )] dµ ( t )= [ � t j ∈ R , j ∈ Λ ◮ B ( t ) = β � (cosh( t j − t j ′ ) − 1) + ε � (cosh t j − 1) <j,j ′ > j ∈ Λ =( t, ( − β ∆+ ε ) t )+ higher order terms ◮ D ( t ) > 0 positive quadratic form : tj + tj ′ + ε � e tj ( f j − f j ′ ) 2 j f 2 � ( f, D ( t ) f ) = β e j <j,j ′ > Observable : current-current correlation O xy = e t x D ( t ) − 1 xy e t y
The general setting A toy model for quantum diffusion Qualitative behavior of the observable ◮ β ≫ 1 → the field is constant t j ≃ t ∀ j O ( t ) xy = e tx + ty D ( t ) − 1 xy ≃ [ − β ∆+ ǫe − t ] − 1 ( x,y ) ◮ saddle analysis to determine t : ◮ in d = 1, εe − t ∼ 1 /β ◮ in d = 2, εe − t ∼ e − β ◮ in d = 3, t ∼ 0. so in 1d and 2d we obtain a mass : −| x − y | √ mβ 1 O ( t ) xy ≃ − β ∆+ mβ ( x,y ) ≤ e
The general setting A toy model for quantum diffusion Results phase transition in d = 3. Localization ∀ β at d = 1 and for β ≪ 1 at d = 2 Thm 1 t x ∼ constant ∀ x ◮ Diffusion ( d = 3, β ≫ 1) Thm 2 t x ∼ 0 ∀ x Thm 3 O ( t ) xy dµ ( t ) ∼ ( − ∆+ ε ) − 1 � xy No analogous result for random Schr¨ odinger. ◮ Localization � d = 1 ∀ β O ( t ) xy dµ ( t ) ≤ 1 ε e −| x − y | /ξβ for Thm 4 � d = 2 , 3 β ≪ 1 Same result as in random Schr¨ odinger.
The general setting A toy model for quantum diffusion Fluctuations of t : t x ∼ const ∀ x Thm 1 M. Disertori, T. Spencer, M. Zirnbauer 2010 For β ≫ 1 � [cosh( t x − t y )] m dµ Λ ,β ( t ) ≤ 2 uniformly in Λ and ε . This bound holds ◮ ∀ x, y ∈ Λ , and ∀ m ≤ β 1 / 8 in d = 3 (published) ◮ ∀� x − y � < e β 1 / 3 and ∀ m ≤ β 1 / 3 in d = 2 (unpublished) β β ◮ ∀� x − y � < ln β and ∀ m ≤ | x − y | in d = 1 (unpublished)
The general setting A toy model for quantum diffusion t x ∼ 0 Thm 2 M. Disertori, T. Spencer, M. Zirnbauer 2010 For β ≫ 1 and d = 3 the field t x remains near zero ∀ x . More precisely � [cosh( t x )] p dµ Λ ,β,ε ( t ) ≤ 2 ∀ p ≤ 4 ∀ x ∈ Λ 1 uniformly in Λ and ε ≥ | Λ | 1 − α , with α = 1 / ln β . (optimal value would be α = 0).
The general setting A toy model for quantum diffusion Diffusion � � e t x + t y D ( t ) − 1 Set: �O xy � = O ( t ) xy dµ ( t ) = xy dµ ( t ). Thm 3 M. Disertori, T. Spencer, M. Zirnbauer 2010 For d = 3 and β ≫ 1 we have �O� ∼ ( − β ∆ Λ + ε ) − 1 . More precisely the exists constant C > 0 such that 1 C [ f ; ( − β ∆ Λ + ε ) − 1 f ] ≤ [ f ; �O� f ] ≤ C [ f ; ( − β ∆ Λ + ε ) − 1 f ] 1 ∀ f : Λ → R + ., uniformly in Λ and ε ≥ | Λ | 1 − α .
The general setting A toy model for quantum diffusion Localization Thm 4 M. Disertori, T. Spencer 2010 The correlation between t x and t y decays exponentially ◮ ∀ β > 0 at d = 1 , ◮ pour β ≪ 1 at d = 2 , 3 . xy dµ ( t ) ≤ 1 − | x − y | � e t x + t y D ( t ) − 1 lβ �O xy � = ε e ∀ x, y ∈ Λ . uniformly in Λ and ε > 0 .
The general setting A toy model for quantum diffusion Bound on the t fluctuations : sketch of the proof ◮ bound on short scale fluctuations: Ward identites ◮ conditional bound on large scale fluctuations: Ward identites ◮ unconditional bound on large scale fluctuations: previous bounds plus induction on scales
The general setting A toy model for quantum diffusion Ward identities � � 1 − m � � cosh m ( t x − t y ) SUSY ⇒ 1 = β C xy 1 ◮ 0 < C xy := ( δ x − δ y ) M ( t ) ( δ x − δ y ) ( f j − f j ′ ) 2 A xy ( jj ′ ) ◮ ( f, M ( t ) f ) = � <j,j ′ > ◮ local conductance: A xy ( jj ′ ) = e t j + t j ′ − t x − t y cosh( t x − t y ) > 0 Problem: A xy ( jj ′ ) can be very small!
The general setting A toy model for quantum diffusion Short scale fluctuations: | x − y |≤ 10 C xy ≤ 1 for all t configurations: � � � � � � cosh m ( t x − t y ) 1 − m ≥� cosh m ( t x − t y ) � 1 − m 1= β C xy β 1 � cosh m ( t x − t y ) � ≤ ⇒ ≤ 2 1 − m β as long as m ≤ β/ 2
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