The Anderson Model on the Bethe Lattice: Lifshitz Tails

Transcription

1 The Anderson Model on the Bethe Lattice: Lifshitz Tails Christoph Schumacher TU Chemnitz, Germany 5 September 2016 / A trilateral German-Russian-Ukrainian summer school on Spectral Theory, Differential Equations and Probability joint work with Francisco Hoecker-Escuti Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 1 / 17

2 The Bethe lattice with coordination number 3 The Bethe lattice B k is an infinite simple tree graph of constant degree k Cayley graph of non-abelian group a 0,..., a k a 2 j = 1 introduced 1935 by Hans Bethe B 2 Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 2 / 17

3 The Bethe lattice with coordination number 4 The Bethe lattice B k is an infinite simple tree graph of constant degree k Cayley graph of non-abelian group a 0,..., a k a 2 j = 1 introduced 1935 by Hans Bethe B 3 Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 3 / 17

4 The Bethe lattice with coordination number 5 The Bethe lattice B k is an infinite simple tree graph of constant degree k Cayley graph of non-abelian group a 0,..., a k a 2 j = 1 introduced 1935 by Hans Bethe B 4 Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 4 / 17

5 The Bethe lattice with coordination number 6 The Bethe lattice B k is an infinite simple tree graph of constant degree k Cayley graph of non-abelian group a 0,..., a k a 2 j = 1 introduced 1935 by Hans Bethe B 5 Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 5 / 17

6 The discrete Anderson model Γ: infinite simple undirected graph,e. g. Γ = Z d or Γ = B k The discrete Laplace operator on (the nodes of) Γ: Γ : l 2 (Γ) l 2 (Γ), ( Γ ϕ)(v) := w v ( ϕ(w) ϕ(v) ) Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 6 / 17

7 The discrete Anderson model Γ: infinite simple undirected graph,e. g. Γ = Z d or Γ = B k The discrete Laplace operator on (the nodes of) Γ: Γ : l 2 (Γ) l 2 (Γ), The random potential ( Γ ϕ)(v) := w v ( ϕ(w) ϕ(v) ) V Γ ω : l 2 (Γ) l 2 (Γ), (V Γ ωϕ)(v) := ω v ϕ(v), where ω := (ω v ) v Γ is a vector of non-trivial, bounded, non-negative,i. i. d. random variables with ess inf ω v = 0 Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 6 / 17

8 The discrete Anderson model Γ: infinite simple undirected graph,e. g. Γ = Z d or Γ = B k The discrete Laplace operator on (the nodes of) Γ: Γ : l 2 (Γ) l 2 (Γ), The random potential ( Γ ϕ)(v) := w v ( ϕ(w) ϕ(v) ) V Γ ω : l 2 (Γ) l 2 (Γ), (V Γ ωϕ)(v) := ω v ϕ(v), where ω := (ω v ) v Γ is a vector of non-trivial, bounded, non-negative,i. i. d. random variables with ess inf ω v = 0 The Anderson Hamiltonian on Γ H Γ ω : l 2 (Γ) l 2 (Γ), H Γ ω := Γ + λv Γ ω with coupling constant λ 0 Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 6 / 17

9 The discrete Anderson model Γ: infinite simple undirected graph,e. g. Γ = Z d or Γ = B k The discrete Laplace operator on (the nodes of) Γ: Γ : l 2 (Γ) l 2 (Γ), The random potential ( Γ ϕ)(v) := w v ( ϕ(w) ϕ(v) ) V Γ ω : l 2 (Γ) l 2 (Γ), (V Γ ωϕ)(v) := ω v ϕ(v), where ω := (ω v ) v Γ is a vector of non-trivial, bounded, non-negative,i. i. d. random variables with ess inf ω v = 0 The Anderson Hamiltonian on Γ H Γ ω : l 2 (Γ) l 2 (Γ), H Γ ω := Γ + λv Γ ω with coupling constant λ 0 Ergodicity implies almost sure spectrum Σ = σ(h Γ ω) a. s. Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 6 / 17

10 The Integrated Density of States The Integrated Density of States (IDS) N Γ : R [0, 1], N Γ (E) := E[ δ v, 1 (,E] (H Γ ω)δ v ] evaluates to the expected number of energy levels below the energy threshold E R per unit volume, Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 7 / 17

11 The Integrated Density of States The Integrated Density of States (IDS) N Γ : R [0, 1], N Γ (E) := E[ δ v, 1 (,E] (H Γ ω)δ v ] evaluates to the expected number of energy levels below the energy threshold E R per unit volume, is thermodynamic limit of eigenvalue counting functions(pastur Shubin formula), Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 7 / 17

12 The Integrated Density of States The Integrated Density of States (IDS) N Γ : R [0, 1], N Γ (E) := E[ δ v, 1 (,E] (H Γ ω)δ v ] evaluates to the expected number of energy levels below the energy threshold E R per unit volume, is thermodynamic limit of eigenvalue counting functions(pastur Shubin formula), contains spectral information,e. g. N Γ is a distribution function, and support of the corresponding measure= Σ = σ(h Γ ω) a. s., Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 7 / 17

13 The Integrated Density of States The Integrated Density of States (IDS) N Γ : R [0, 1], N Γ (E) := E[ δ v, 1 (,E] (H Γ ω)δ v ] evaluates to the expected number of energy levels below the energy threshold E R per unit volume, is thermodynamic limit of eigenvalue counting functions(pastur Shubin formula), contains spectral information,e. g. N Γ is a distribution function, and support of the corresponding measure= Σ = σ(h Γ ω) a. s., encodes geometric properties of the underlying space, e. g. E 0 := inf Σ = 0 Γ is amenable. For the Bethe lattice: E 0 := inf Σ = ( k 1) 2 > 0. Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 7 / 17

14 Behavior of the IDS at the bottom of the spectrum Anderson Hamiltonian: H Γ ω := Γ + λv Γ ω N Γ (E 0 + E) Euclidian lattice Γ = Z d Bethe lattice Γ = B k λ = 0 E d/2 E 3/2 λ > 0 exp( E d/2 ) N Γ (E 0 + E) (d = 3) E E Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 8 / 17

15 Behavior of the IDS and the DoS at the bottom of the spectrum Anderson Hamiltonian: H Γ ω := Γ + λv Γ ω N Γ (E 0 + E) Euclidian lattice Γ = Z d Bethe lattice Γ = B k λ = 0 E d/2 E 3/2 λ > 0 exp( E d/2 ) N Γ (E 0 + E) d de N Γ (E 0 + E) (d = 3) E E Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 8 / 17

16 Lifshitz tails For the Bethe lattice: E 0 := inf Σ = ( k 1) 2 > 0. Theorem (Lifshitz tails on the Bethe lattice) Assume ν := lim sup κ 0 κ 1/2 log log P(ω v κ) < 1 (v B k ). Then there exists ε > 0such that, for all E (0, ε), exp ( exp(ε 1 E 1/2 ) ) N B k (E 0 + E) exp ( exp(εe 1/2 ) ) and thus log log log N B k (E 0 + E) lim = 1 E 0 log(e) 2. Note: ν < 1 is an assumption on the distribution of the potential: roughly: P(ω v κ) exp( exp(κ 1/2 )) as κ 0, i. e. small values are not too improbable. Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 9 / 17

17 Behavior of the IDS at the bottom of the spectrum Anderson Hamiltonian: H Γ ω := Γ + λv Γ ω N Γ (E 0 + E) Euclidian lattice Γ = Z d Bethe lattice Γ = B k λ = 0 E d/2 E 3/2 λ > 0 exp( E d/2 ) exp( exp(e 1/2 )) N Γ (E 0 + E) (d = 3) E E Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 10 / 17

18 Behavior of the IDS and the DoS at the bottom of the spectrum Anderson Hamiltonian: H Γ ω := Γ + λv Γ ω N Γ (E 0 + E) Euclidian lattice Γ = Z d Bethe lattice Γ = B k λ = 0 E d/2 E 3/2 λ > 0 exp( E d/2 ) exp( exp(e 1/2 )) N Γ (E 0 + E) d de N Γ (E 0 + E) (d = 3) E E Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 10 / 17

19 Comparison Euclidian lattice and Bethe lattice Tools on Z d : amenability approximation by finite balls: perturbation theory large spectral gap: Fourier transform abelian group: Z d :, B k : X Z d :, B k : X Z d :, B k : X Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 11 / 17

20 Comparison Euclidian lattice and Bethe lattice Tools on Z d : We use amenability approximation by finite balls: perturbation theory large spectral gap: Fourier transform abelian group: Laplace transform of IDS N B k, Tauberian theorem discrete Feynman Kac formula discrete Ismagilov Morgan Sigal formula Z d :, B k : X Z d :, B k : X Z d :, B k : X to reduce Lifshits tails behaviour to properties of ground state energies of Anderson models on finite symmetric rooted trees. Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 11 / 17

21 Finite symmetric rooted trees T L Number of children: k 2 Length of tree: L N root: 0 Level of node v T L : v = dist(0, v) + 1 Advantage w. r. t. B k : explicit formulas for all eigenfunctions and -values 0 T L L Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 12 / 17

22 Estimation of the random ground state energy Consider Anderson model H T L ω := TL + V T L ω on the tree T L the random ground state energy EGS L (ω) := inf ϕ, ϕ =1 HT L ω ϕ Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 13 / 17

23 Estimation of the random ground state energy Consider Anderson model H T L ω := TL + V T L ω on the tree T L the random ground state energy EGS L (ω) := inf ϕ, ϕ =1 HT L ω ϕ Theorem (random ground state energy on trees) Assume ν < 1.Then there are ε > 0 and L > 1 such that for all L > L we have P ( E 0 + (As before: E 0 := ( k 1) 2 ) ε (log L) 2 E L GS E 0 + ε 1 (log L) 2 ) 1 exp( εl). Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 13 / 17

24 Estimation of the random ground state energy Consider Anderson model H T L ω := TL + V T L ω on the tree T L the random ground state energy EGS L (ω) := inf ϕ, ϕ =1 HT L ω ϕ Theorem (random ground state energy on trees) Assume ν < 1.Then there are ε > 0 and L > 1 such that for all L > L we have P ( E 0 + (As before: E 0 := ( k 1) 2 ) Adjacency operator: ε (log L) 2 E L GS E 0 + ε 1 (log L) 2 ) 1 exp( εl). A: l 2 (B k ) l 2 (B k ), (Aϕ)(v) := w v ϕ(w), A = Bk + k + 1 Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 13 / 17

25 The ground state on finite symmetric rooted trees or: radially symmetric eigenfunctions of the adjacency matrix v L v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 14 / 17

26 The ground state on finite symmetric rooted trees or: radially symmetric eigenfunctions of the adjacency matrix radially δ symmetric γ δ function β δ γ δ α δ γ δ β δ γ δ L v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 14 / 17

27 The ground state on finite symmetric rooted trees or: radially symmetric eigenfunctions of the adjacency matrix radially symmetric γ function β γ α γ l 2 (T L ) β Φ γ l 2 ({1,..., L}) 2 δ δ δ δ δ δ δ δ α β 2 2γ δ L v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 14 / 17

28 The ground state on finite symmetric rooted trees or: radially symmetric eigenfunctions of the adjacency matrix α + 2γ = λβ v ΦA = β ka Z Φ α γ γ α + 2γ = λ β 2 2 α β 2 2γ L v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 14 / 17

29 The ground state on finite symmetric rooted trees or: radially symmetric eigenfunctions of the adjacency matrix v sin( π v L+1 ) L v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 14 / 17

30 The ground state on finite symmetric rooted trees or: radially symmetric eigenfunctions of the adjacency matrix v sin( π v L+1 ) 2 ( v 1)/2 λ = 2 k cos( π L+1 ) v sin( π v L+1 ) L v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 14 / 17

31 The ground state on finite symmetric rooted trees or: radially symmetric eigenfunctions of the adjacency matrix 2π v sin( L+1 λ = 2 v ) 2 ( v 1)/2 k cos( 2π L+1 ) v sin( 2π v L+1 ) v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 14 / 17

32 Non-radially symmetric eigenfunctions π( v 1) sin( ) L v + 2 ( v 1)/2 λ = 2 k cos( π L ) π( v 1) sin( ) L v 2 ( v 1)/2 v sin( π( v 1) L ) L v Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 15 / 17

33 Consequences of the spectral properties of trees spectral gap: cos( π L+1 ) cos( π L ) L 3 on B k instead of cos( π L+1 ) cos( 2π L+1 ) L 2 on Z d Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 16 / 17

34 Consequences of the spectral properties of trees spectral gap: cos( π L+1 ) cos( π L ) L 3 on B k instead of cos( π 2π L+1 ) cos( L+1 ) L 2 on Z d low energy eigenfunctions suppress boundary effects Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 16 / 17

35 Consequences of the spectral properties of trees spectral gap: cos( π L+1 ) cos( π L ) L 3 on B k instead of cos( π 2π L+1 ) cos( L+1 ) L 2 on Z d low energy eigenfunctions suppress boundary effects location of the random ground state Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 16 / 17

36 Thank you for your attention! Christoph Schumacher (TU Chemnitz) Lifshitz Tails on the Bethe Lattice Trilateral Summer School 17 / 17