Resonant Deloc. on the Complete Graph Michael Aizenman Princeton University Cargese, 4 Sept. 2014 Based on: M.A. - S. Warzel: “Extended states ...” / “Resonant delocalization for random Schrödinger operators on tree graphs", (2011,2013) M.A. - M. Shamis - S. Warzel: “Partial delocalization on the complete graph” (2014) 1 / 17
Random Schrödinger operators - the question of spectral characteristics Single quantum particle on regular graph G (e.g. Z d ) H ( ω ) := − ∆ + λ V ( x ; ω ) on ℓ 2 ( G ) (Anderson ’58, Mott - Twose ’61,...) ◮ discrete Laplacian: (∆ ψ )( x ) := � dist ( x , y )= 1 ψ ( y ) − n ( x ) ψ ( x ) ◮ Disorder parameter: λ > 0 ◮ V ( x ; · ) , x ∈ G , i.i.d. rand. var., e.g. abs. cont distr. P ( V ( 0 ) ∈ dv ) Of particular interest: Localization and delocalization under disorder 2 / 17
Random Schrödinger operators - the question of spectral characteristics Single quantum particle on regular graph G (e.g. Z d ) H ( ω ) := − ∆ + λ V ( x ; ω ) on ℓ 2 ( G ) (Anderson ’58, Mott - Twose ’61,...) ◮ discrete Laplacian: (∆ ψ )( x ) := � dist ( x , y )= 1 ψ ( y ) − n ( x ) ψ ( x ) ◮ Disorder parameter: λ > 0 ◮ V ( x ; · ) , x ∈ G , i.i.d. rand. var., e.g. abs. cont distr. P ( V ( 0 ) ∈ dv ) Of particular interest: Localization and delocalization under disorder (“steelpan”, Trinidad and Tobago) 2 / 17
Random Schrödinger operators - the question of spectral characteristics Single quantum particle on regular graph G (e.g. Z d ) H ( ω ) := − ∆ + λ V ( x ; ω ) on ℓ 2 ( G ) (Anderson ’58, Mott - Twose ’61,...) ◮ discrete Laplacian: (∆ ψ )( x ) := � dist ( x , y )= 1 ψ ( y ) − n ( x ) ψ ( x ) ◮ Disorder parameter: λ > 0 ◮ V ( x ; · ) , x ∈ G , i.i.d. rand. var., e.g. abs. cont distr. P ( V ( 0 ) ∈ dv ) Of particular interest: Localization and delocalization under disorder Currently, delocalization remains less understood. Possible mechanisms: ◮ continuity (?) (trees: [K’96, ASiW’06]) ◮ quantum diffusion (?) [EY’00] ◮ resonant delocalization [AW’11, AShW ’14] (“steelpan”, Trinidad and Tobago) 2 / 17
Eigenfunction hybridization (tunneling amplitude vs. energy gaps) � E 1 � τ Two-level system H = Reminder from QM 101: τ ∗ E 2 τ . Energy gap: ∆ E := E 1 − E 2 Tunneling amplitude: ◮ Case | ∆ E | ≫ | τ | : Localization ψ 1 ≈ ( 1 , 0 ) , ψ 2 ≈ ( 0 , 1 ) . ◮ Case | ∆ E | ≪ | τ | : Hybridized eigenfunctions 1 1 ( 1 , 1 ) , ( 1 , − 1 ) . ψ 1 ≈ √ ψ 2 ≈ √ 2 2 Heuristic explanation of the abs. cont. spectrum on tree graphs: (A-W ‘11) e − L λ ( E ) R (typ.) Tunnelling amp. for states with energy E at distances R : Since the volume grows exponentially fast as K R , extended states will form in spectral regimes with L λ ( E ) < log K . M.A., S. Warzel, JEMS 15 : 1167-1222 (2013), PRL 106 : 136804 (2011) EPL 96 : 37004 (2011) [The implications include a surprising correction of the standard picture of the phase diagram: absence of a mobility edge for the Anderson Hamiltonian on tree graphs at weak disorder (Aiz-Warzel, EPL 2011).] 3 / 17
Quasimodes & their tunnelling amplitude Definition: 1. A quasi-mode (qm) with discrepancy d for a self-adjoint operator H is a pair ( E , ψ ) s.t. � ( H − E ) ψ � ≤ d � ψ � . 2. The pairwise tunnelling amplitude , among orthogonal qm’s of energy close to E may be defined as τ jk ( E ) in � e j + σ jj ( E ) � − 1 τ jk ( E ) P jk ( H − E ) − 1 P jk = . τ kj ( E ) e k + σ kk ( E ) (the “Schur complement” representation). Seems reasonable to expect: If the typical gap size for quasi-modes is ∆( E ) , the condition for resonant delocalization at energies E + Θ(∆ E ) is: ∆( E ) ≤ | τ jk ( E ) | . Question: how does that work in case of many co-resonating modes? 4 / 17
Example: Schrödinger operator on the complete graph (of M sites) H M = −| ϕ 0 �� ϕ 0 | + κ M V with: √ ◮ � ϕ 0 | = ( 1 , 1 , . . . , 1 ) / M , ◮ V 1 , V 2 , . . . V M iid standard Gaussian rv’s, i.e. 1 e − v 2 / 2 , ̺ ( v ) = √ 2 π 2 log M . ◮ κ M := λ �� Remarks: ◮ Choice of ( κ M ) motivated by: inProb max { V 1 , ..., V M } � 2 log M + o ( 1 ) . = ◮ The spectrum of H for M → ∞ : → [ − λ, λ ] ∪ {− 1 , 0 } (on the ‘macroscopic scale’) . σ ( H M ) − ◮ Eigenvalues interlace with the values of K M V ◮ Studied earlier by Bogachev and Molchanov (‘89), and Ossipov (‘13) - both works focused on localization. 5 / 17
Two phase transitions for H M = −| ϕ 0 �� ϕ 0 | + λ √ 2 log M V Quasi-modes: | ϕ 0 � (extended), and | δ j � j = 1 , ..., M (localized). 1. A transition at the spectral edge (1 st -order), at λ = 1 : - λ -1 0 λ < 1 : E 0 = − 1 + o ( 1 ) , (the ground state is extended) Ψ 0 ≈ ϕ 0 - λ -1 0 λ > 1 : E 0 = − λ + o ( 1 ) , Ψ 0 ≈ δ argmin ( V ) (the ground state is localized except for ‘avoided crossings’) (Similar first order trans. in QREM and ... were studied [num. & rep.] by Jörg, Krzakala, Kurchan, Maggs ’08, Jörg, Krzakala, Semerjian, Zamponi ’10, ... More on the subject in the talks of Leticia Cugliandolo and Simone Warzel) 2. Emergence of a band of semi-delocalized states: of main interest here √ at energies near E = − 1 , for λ > 2 . A similar band near E = 0 is found for all λ > 0 . 6 / 17
Helpful tools: I. the characteristic equation Proposition The eigenvalues of H M intertwine with the values of κ V . The spectrum of H M consists of the collection of energies E for which M 1 1 � κ M V ( x ) − E = 1 , (1) F M ( E ) := M x = 1 and the corresponding eigenfunctions are given by: Const . (2) ψ E ( x ) = κ M V ( x ) − E . Proof: “rank one” perturbation theory = ⇒ for any z ∈ C \ R : 1 1 1 1 κ M V − z + [ 1 − F M ( z )] − 1 (3) H M − z = κ M V − z | ϕ 0 �� ϕ 0 | κ M V − z , In particular, � ϕ 0 , ( H M − z ) − 1 ϕ 0 � = ( F M ( z ) − 1 − 1 ) − 1 . The spectrum and eigenfunctions are given by the poles and residues of this “resolvent”. 7 / 17
The scaling limit Zooming onto scaling windows centered at a sequence of energies E M with: lim and |E M − E| ≤ C / ln M , M →∞ E M = E ∈ [ − λ, λ ] , u n , M := E n , M − E M ω n , M := κ M V j − E M denote , . ∆ M ( E M ) ∆ M ( E M ) rescaled eigenvalues rescaled potential values Questions of interest: 1. the nature of the limiting point process of the rescaled eigenvalues (including: extent of level repulsion (?), and relation to rescaled potential values) 2. the nature of the corresponding eigenfunctions (extended versus localized, and possible meaning of these terms). 8 / 17
Results (informal summary) Theorem 1 [Bands of partial delocalization (A., Shamis, Warzel)] If either ◮ E = 0, λ > 0; or √ ◮ E = − 1, and λ > 2, ( ց ̺ ’s Hilbert transform) � 1 − κ − 1 � and additionally the lim exists: lim M →∞ M ∆ M ( E ) M ̺ ( E M /κ M ) =: α then: I. the eigenvalues within the scaling window are delocalized in ℓ 1 sense , localized in ℓ 2 sense. II. the rescaled eigenvalue point process converges in distribution to the Šeba point process at level α [defined below]. √ Theorem 2 [A non-resonant delocalized state for λ < 2] √ For λ < 2, there is a sequence of energies satisfying lim M →∞ E M = − 1 such that within the scaling windows centered at E M : 1. There exists one eigenvalue for which the corresponding eigenfunction ψ E is ℓ 2 -delocalized [. . . ] 2. All other eigenfunctions in the scaling window are ℓ 2 -localized [. . . ] Elsewhere localization (Theorem 3 – not displayed here). 9 / 17
Key elements of the proof ◮ Rank-one perturbation arguments yield the characteristic equation: 1 1 � κ M V n − E = 1 Eigenvalues : ( ∗ ) M n 1 Eigenvectors : ψ j , E = up to normalization κ M V j − E ◮ To study the scaling limit we distinguish between the head contribution in (*), S M ,ω ( u ) , and the tail sum, transforming (*) into: S M ,ω ( u ) = M ∆ M ( E ) − T M ,ω ( u ) := − R M ,ω ( u ) with 1 [ | ω n | ≥ ln M ] � T M ,ω ( u ) = ω M , n − u n ◮ Prove & apply some general results concerning limits of random Pick functions (aka Herglotz - Nevanlinna functions). In particular: the scaling limit of a function such as R M ,ω ( u ) is either: i. constant ⇒ Šeba process & semi-delocalization, ii. singular (+ ∞ ) or ( −∞ ) ⇒ localization, or iii. singular with transition ⇒ localization + single deloc. state √ ( E = − 1 , λ < 2) 10 / 17
Putting it all together (with details in appended slides) 1. Proofs of Theorems 1 - 3 (the spectral characteristics of H M ,ω ) 1 1 � κ M V n − E = 1 Eigenvalues : ( ∗ ) Recall: M n 1 Eigenvectors : ψ j , E = up to normalization κ M V j − E distinguishing head S M ,ω ( u ) versus tail contributions, rewrite (*) as: S M ,ω ( u ) = M ∆ M ( E ) − T M ,ω ( u ) 1 [ | ω n |≤ ln M ] 1 [ | ω n |≥ ln M ] with and , S M ,ω ( u ) = � T M ,ω ( u ) = � n ω M , n − u n ω M , n − u apply the general results on such functions. 2. The heuristic criterion for resonant delocalization “checks out” yields the correct answer. 3. The localization criteria require some discussion ( ℓ 2 versus ℓ 1 ). 4. Comment on operators with many mixing modes (crossover to random matrix asymptotics) 11 / 17
Thank you for your attention Alternatively - some further details are given below 12 / 17
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