Density of States for Random Band Matrices in d = 2

Transcription

1 Density of States for Random Band Matrices in d = 2 via the supersymmetric approach Mareike Lager Institute for applied mathematics University of Bonn Joint work with Margherita Disertori ZiF Summer School Randomness in Physics and Mathematics, August 2016

2 1 Model 2 3

3 The model: Random Band matrices model for conducting properties of disordered materials (similar to Random Schrödinger operator) d 1, Λ Z d finite, H : Λ Λ C hermitian random matrix matrix elements: Gaussian random variables H ii N R (0, J ii ), H ij N C (0, J ij ), for i < j, with < order relation on Z d covariance J ij := ( W 2 + 1) 1 ij e i j /W for i j > W, R Λ Λ discrete Laplacian on Λ with periodic b.c. W 1 band width

4 Special cases Model Gaussian unitary ensemble (GUE): W Λ, J ij = 1 N Λ Z d : extended eigenvectors, a.c. spectrum Diagonal case: W = 0, J ij = δ ij Λ Z d : localized eigenvectors, pure point spectrum

5 Special cases Model Gaussian unitary ensemble (GUE): W Λ, J ij = 1 N Λ Z d : extended eigenvectors, a.c. spectrum Diagonal case: W = 0, J ij = δ ij Λ Z d : localized eigenvectors, pure point spectrum RBM interpolates between GUE and diagonal case

6 Density of States Model quantity of interest: averaged density of states probability density of eigenvalues ρ Λ (E) := 1 Λ E δ λj (E) = 1 π Λ lim E[Im Tr G + ] ε 0 j with E R, λ j (random) eigenvalues of H and Green s function G + := ((E + iε) 1 H) 1

7 Result in d = 2 Model Theorem (Disertori, L. 2016) Let d = 2 and I = {E : η < E 1.8}. For each fixed α (0, 1), there exist W 0 (α) such that for all W W 0 (α) and E I ρ Λ (E) ρ SC (E) W 2 e K(ln W )α, n E ρ Λ(E) C n n n 0 (W ), where C n and K are independent of Λ and W and lim W n 0 (W ) =. ρ SC is Wigner s semicircle law { 1 ρ SC (E) = π 1 E 2 4 if E 2, 0 if E > 2. Both estimates are uniform in Λ and hold also for Λ Z 2.

8 Other known results Model d = 3 [Disertori, Pinson, Spencer 2002] ρ Λ (E) ρ SC (E) W 2 d = 1 [M. Shcherbina, T. Shcherbina 2016] ρ Λ (E) ρ SC (E) W 1

9 Rough idea of the proof in d = 2 dual representation via the supersymmetric approach as integral over 2 Λ real variables

10 Rough idea of the proof in d = 2 dual representation via the supersymmetric approach as integral over 2 Λ real variables saddle point analysis and complex translation to the saddle points (direct outcome: semicircle law)

11 Rough idea of the proof in d = 2 dual representation via the supersymmetric approach as integral over 2 Λ real variables saddle point analysis and complex translation to the saddle points (direct outcome: semicircle law) estimates in a finite cube of volume W 2 region correction dominant saddle O(W 2 )e K(ln W )α Brascamp-Lieb inequality second saddle O(e c ln W ) decay from Fermionic need exp. decay in W integral large field region O(e cw ) small probability argument

12 Rough idea of the proof in d = 2 dual representation via the supersymmetric approach as integral over 2 Λ real variables saddle point analysis and complex translation to the saddle points (direct outcome: semicircle law) estimates in a finite cube of volume W 2 (ln W ) α region correction dominant saddle O(W 2 e K(ln W )α ) Brascamp-Lieb inequality second saddle O(e c(ln W )1+α ) decay from Fermionic integral large field region O(e cw ) small probability argument

13 Rough idea of the proof in d = 2 dual representation via the supersymmetric approach as integral over 2 Λ real variables saddle point analysis and complex translation to the saddle points (direct outcome: semicircle law) estimates in a finite cube of volume W 2 (ln W ) α region correction dominant saddle O(W 2 e K(ln W )α ) Brascamp-Lieb inequality second saddle O(e c(ln W )1+α ) decay from Fermionic integral large field region O(e cw ) small probability argument infinite volume cluster expansion with complex covariance

14 Dual representation via supersymmetry 1 Λ E[Tr G + ] = = dm e 1 2 i,j Λ J 1 ij Str[M i M j ] j Λ Sdet[E ε M j ] 1 a 0 dadb e 1 2 ((a,j 1 a)+(b,j 1 b)) j Λ E ε ib j E ε a j det [ ] J 1 F 2π a 0, ( ) aj ρ M j = j supermatrix, ρ j ib j E ε = E + iε { a j, b j R ρ j, ρ j Grassman variables Str M j = a j ib j, Sdet M j = det[a j ρ j b 1 j ρ j ] det b 1 j 1 F (a, b) ij = δ ij (E ε a j )(E ε ib j )

15 saddle point analysis Model (a, J 1 a) = j a2 j + W 2 i j (a i a j ) 2 a j a constant ( e Λ 2 (a2 +b 2 ) E ib E a ) Λ = e Λ ( a 2 2 +ln(e a)+ b2 2 ln(e ib) ), critical points a s ± = E r ± ie i and b s ± = ie r ± E i, where E = E r ie i = E 2 i 1 E 2 4 Im a Ē E ε Re a Im b Re b E ie iē

16 obtaining the semicircle law Lemma By a complex deformation a j a j + a s + and b j b j + b s + 1 lim ε 0 Λ E[Tr G + ] =a s + + dµ B (a, b)r(a, b)a 0, where dµ B (a, b) is Gaussian with B := ( W 2 + (1 E 2 )) 1 and R(a, b) := det[1 + DB] e V(a,b), where D ij = D ij (a, b) = δ ij D j (a, b) is diagonal and V(a, b) = j Λ V (a j ) V (ib j ), V (x) = 1 0 x 3 (1 t) 2 (Ē tx) 3 dt.

17 obtaining the semicircle law Lemma By a complex deformation a j a j + a s + and b j b j + b s + 1 lim ε 0 Λ E[Tr G + ] =a s + + dµ B (a, b)r(a, b)a 0, Remember: where dµ B (a, b) is Gaussian with B := ( W 2 + (1 E 2 )) 1 and ρ Λ (E) = 1 π Λ lim ε 0 E[Im Tr G + ] R(a, b) := det[1 + DB] e V(a,b), a s + = E 2 i 1 E 2 4 where D ij = D ij (a, b) = δ ij D j (a, b) is diagonal and ρ SC (E) = 1 V(a, b) = π 1 E 2 4 if E 2, 1 estimate remainder x V (a j ) dµ V B (a, (ib b)r(a, j ), V (x) b)a = 0 (cluster 3 (1 t) expansion) 2 dt. j Λ 0 (Ē tx) 3

18 Thank you for your attention! Any questions?