FRACTAL DIMENSIONS FOR CERTAIN CRITICAL RANDOM MATRIX ENSEMBLES. Eugène Bogomolny

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1 FRACTAL DIMENSIONS FOR CERTAIN CRITICAL RANDOM MATRIX ENSEMBLES Eugène Bogomolny Univ. Paris-Sud, CNRS, LPTMS, Orsay, France In collaboration with Olivier Giraud VI Brunel Workshop on PMT Brunel, 7-8 December 200

2 Outline Critical random matrix ensembles Perturbation series for fractal dimensions Strong multifractality Weak multifractality Conjecture: χ = D /d Summary

3 Well accepted conjectures Berry, Tabor (977): Integrable systems = Poisson statistics 0.8 p(s)=exp( s) ( +Ε)Ψ=0 p(s) s Bohigas, Giannoni, Schmit (984): Chaotic systems = Random Matrix Statistics 0.8 p(s)=π/2 s exp( π s 2 /4) ( +Ε)Ψ=0 p(s) s

4 3-d Anderson model at metal-insulator transition 3-d Anderson model H = ε i a i a i i a j a i j= adjacent to i ε i =i.i.d. random variables between W/2 and W/2 Mobility edge: E c (W) E c = 0 W = W c = 6.5 delocalized states density localized states localized states W > W c all states are localized 0 E c E c E > E c. States are localized. Poisson statistics of eigenvalues E < E c. States are delocalized. Random matrix statistics E = E c. States are neither localized or delocalized. Fractal wave functions. New intermediate type of spectral statistics. Shklovskii et al (993)

5 Spectral characteristics of 3-d Anderson model at metal-insulator transition Nearest neighbor distribution Critical, W=6.5 Insulator, W=00 Metal, W= s 5 Number variance Insulator, W=00 Critical, W=6.5 Metal, W= L

6 Characteristic features of intermediate statistics Level repulsion at small distances as for the RMT p(s) 0 when s 0 Exponential decrease of p(s) at large distances as for the Poisson p(s) e as when s Linear asymptotics of the number variance Σ 2 (L) (n(l) n(l)) 2 χl when L χ = spectral compressibility χ = for Poisson, χ = 0 for the RMT Multi-fractal character of eigenfunctions Ψ 2q L (q )D q when L D q = 0 for the Poisson, D q = for the RMT

7 Random matrix models of intermediate statistics Typically: M ij = ε j δ ij + V (i j) V (i j) g i j α ε j = i.i.d. random variables between W/2 and W/2 States i and j are in resonances provided ε j ε i V (i j) Number of resonances connected with a cite i N resonances (i) j V (i j) Levitov (990) α > = localization α < = delocalization α = = intermediate statistics

8 Critical power law banded random matrices (Mirlin et al (996)) N N Hermitian matrices whose elements, H ij, are i.i.d. Gaussian variables (real for β = and complex for β = 2) with zero mean H ij = 0 and the variance H ii 2 = /β and H ij 2 = ( + (i j)2 b 2 ) for i j Perturbation series: (Mirlin, Evers (2000)) b : D q = q/(2πβb), χ = /(2πβb). D q = qχ b : (c β = for β =, c β = π/ 8 for β = 2) D q = 4bc β Γ(q /2) πγ(q), χ = 4bc β D q = Γ(q /2) πγ(q) ( χ), D = χ Absence of universality for spectral statistics

9 Ruijsennars-Schneider ensemble (E.B., Schmit, Giraud (2009)) Ruijsenaars - Schneider classical integrable model N H(p, q) = cos(p j ) ( sin 2 πa sin 2 µ j= k j 2 (q j q k ) Ruijsenaars - Schneider ensemble of random matrices ) /2 N N unitary matrix related with the Lax matrix of this model M kp = e iφ k e 2πia N[ e 2πi(k p+a)/n ] Φ m = independent random variables (phases) uniformly distributed in [0, 2π] a = parameter All spectral correlation functions can be calculated analytically using the transfer operator approach Everything strongly depends on integer part of a

10 0 < a < Poisson distribution shifted by a p(n, s) = p(s) p(, s) = a e (s a)/( a), s > a, p(s) = 0, s < a ( a) n (n )! e (s na)/( a), s > na, p(n, s) = 0, s < na p(s) 4 p(n,s) s s a = 0., 0.2,..., 0.9 a = 0.5

11 < a < 2 p(s) p(, s) = 0 for s > a p(2, s) = 0 for s < a and s > 2a p(3, s) = 0 for s < a and s > 3a a = 4/3 p(s) = 8 ( 64 s2, 0 < s < a, p(2, s) = s 8 ) 52 s3 e 3s/4, 4/3 < s < 8/3 ( 3 4 p(3, s) = 8 32 s s3) e 3s/ s2, 4/3 < s < 8/3 ( s 8 52 s3) e 3s/4 + 9e 3s/2 4, 8/3 < s <

12 2 < a < 3 Tedious calculations and complicated expressions p(s) p(, s) = 0 for s > a p(2, s) = 0 for s > a p(3, s) = 0 for s < a and s > 2a p(n,s) 2 p(n,s) s s a = 9/4 a = 8/3

13 Fractal dimensions Fractal dimensions are not yet accessible for analytical calculations Perturbation series = the only analytical way to them The Ruijsenaars-Schneider ensemble: M mn = e iφ m e 2πia ) N ( e 2πi(m n+a)/n Perturbation series are possible around all integer points a = k. a = k + ǫ ( ) M mn = M mn (0) πi(n ) + ǫ + ǫm mn () + O(ǫ 2 ) N where M (0) mn = e iφ m δ n, m+k M mn () = e iφ m πe πi(m n+k)/n ( δ n, m+k ) N sin(π(m n + k)/n) (δ n, m+k = when n m + k mod N and 0 otherwise)

14 Perturbation series for strong multifractality: D q a M (0) mn is diagonal degenerate perturbation series Unperturbed eigenfunctions Ψ (0) j (α) = δ jα Unperturbed eigenvalues λ α = e iφ α The first order = the contributions of 2 2 sub-matrices M mm M mn eiφ m e iφ m h πe πi(m n)/n, h = a e iφ n h e iφ n N sin(π(m n)/n) M nm M nn Eigenvalues,λ,2 = e iθ,2 and eigenfunctions of this matrix are θ,2 = Φ m,n ± (η), u = h h (η), v = (η) h (η) where η = 2 (Φ m Φ n ), (η) = sin 2 η + h 2 sinη

15 Mean moments of eigenfunctions I q = Nρ(E) N j,α= Ψj (α) 2q δ(e E α ). ρ(e) = the total mean eigenvalue density. For RSE: ρ(e) = /2π Fractal dimensions I q N (q )D q I q = Nρ(E) m n =m [ ] ρ(φ m )ρ(φ n )dφ m dφ n u 2q + v2q δ(e θ ) The dominant contribution is from region η h I q = 2ρ(E) [ h 2q + (η) 2q ] dη N ( h (η)) q m n =m Substitute η = h t and extend the integration over t from to I q = J(q) πn m n =m h

16 where J(q) = One gets [ ] ( ( + e 2t ) + πγ q q ( + e 2t ) 2 cosh(t)dt = q Γ(q ) ) N j= π N sin(πj/n) = 2 lnn + 2(γ + ln2 lnπ) Finally when N ( ) Γ q 2 I q 2a lnn π Γ(q ) By definition I q N (q )D q, therefore in the first order in a ( ) Γ q 2 D q = 2a π Γ(q)

17 Perturbation series for week multifractality: D q When a = k + ǫ and k 0 the unperturbed matrix M (0) mn = e iφ m δ n, m+k is the shift matrix and its eigenfunctions are extended The case k = Eigenvalues λ(α) and eigenfunctions Ψ (0) n (α) of M (0) mn are λ(α) = e i Φ+2πiα/N, Ψ (0) n (α) = N e is n(α), S n (α) = 2π n N α(n ) (Φ j Φ), Φ = j= N Φ j /N j= The expansion of the exact eigenfunctions into a series of unperturbed eigenfunctions Ψ n (α) = Ψ (0) n (α) + β C αβ Ψ (0) n (β).

18 The first order in ǫ = a C αβ = ǫ At the leading order in ǫ mn Ψ(0) m (β)m mnψ () (0) n (α) λ(α) λ(β) N Ψ n (α) 2q = N q[ + n= W(α) = N N [ n= β q(q ) 2 ] W(α), ] 2 e is n(β) is n (α) C αβ + c.c.. Direct (but tedious) calculations show strong cancellations and W(α) = ǫ 2 π2 N 3 N β= N n= sin 2 (πβn/n) sin 2 (πn/n) sin 2 (πβ/n). When N the diverging terms correspond to two regions: (i) β N with n/n = O(), (ii) n N with β/n = O() W(α) 2ǫ2 N N n= g(n/n) sin 2 (πn/n), g(y) = β= sin 2 (πβy) β 2 = π2 2 y( y)

19 Fractal dimensions for RSE The remaining sum over n can be transformed into an integral over y and when N W(α) 2ǫ 2 lnn + O(). Combining all terms together one finds D q = q( a) 2. For k 2 calculations are more tedious but one can prove that D q = q For comparison when a (a k)2 k 2 when a k D q = 2a ( ) Γ q 2 πγ(q)

20 Spectral compressibility for RSE 0 < a < χ = χ = ( a) 2. χ 2a, a < a < 2 ( a 2 4 4a( a)z2 + a 2 sinh 2 ) z sinh sinh 2 2 z z (2z sinh2z) 2 z 2 where z is the solution of a = z is real when < a < 4/3 z is imaginary when 4/3 < a < For a = 4/3, χ = /9 2z 2 z sinh2z z 2 + sinh 2 z z sinh2z

21 where and χ = 2 < a < 3 [ a(sin 2 (a 3) 2 (a 2)z 2 6(a 2)z 2 sin 2 z z + z 2 z sin2z) 2 (a 3)(a )(2a 5)z 3 sin2z + 2(a 2)(cos2z + 2)(a )(a 2)z 2 sin 2 z ] 2a(a 2)(2a 3)z cosz sin 3 z + a(a ) 2 sin 4 z x = a sin 2 z + (a 2)z 2 + ( a)z sin2z (a ) sin 2 z + (a 3)z 2 + (2 a)z sin2z From exact expressions it follows χ e x x = sinz z ez/tan z 2a a (a k) 2 k 2 a k and k

22 Fractal dimensions for CrBRME and RSE CrBRME RSE Weak multifractality /b a k D q = q 2π β b χ = 2π β b D q = q (a k)2 k 2 χ = (a k)2 k 2 Strong multifractality b a ( ) ) Γ q D q = 4bc 2 (q β π Γ(q) D q = 2a Γ 2 χ = 4bc β π Γ(q) χ = 2a (c β = for β =, c β = π/ 8 for β = 2) Universal formulas ( ) D q = Γ q 2 π Γ(q) ( χ), D q = qχ, D = χ

23 Conjecture: χ = D /d, (E.B. and Giraud (200)) Wave function entropy (information dimension): n Ψ n(α) 2 ln Ψ n (α) 2 D lnn Chalker, Kravtsov, Lerner (996): χ = /2 D 2 /2d χ, -D χ, -D a χ (solid line) and D (red circles) for RSE

24 Critical banded random matrices χ, -D χ, -D b b χ (black circles) and D (red error bars) for PLBRM with β = 2 Perturbation series: χ = /(4πb) for b, χ = π 2b 4(2/ 3 )π 2 b 2 for b

25 Critical banded random matrices χ, -D χ, -D b b χ (black circles) and D (red error bars) for PLBRM with β = Perturbation series: χ = /(2πb) for b, χ = 2 2b for b

26 Critical ultrametric matrices (Fyodorov, Ossipov, Rogriguez (2009)) 2 K 2 K Hermitian matrices with independent Gaussian variables with zero mean and H nn 2 = W 2. H mn 2 = 2 2 d mn J 2, d mn = the ultrametric distance between m and n along the binary tree /2 /2 /2 /2 /2 /2 /2 /2 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /4 /2 /2 /4 /4 /4 /4 /4 /4 /4 /4 /4 /2 /2 /2 /2 /2 /2

27 Perturbation series for ultrametric ensemble π J Γ(q /2) π J D q =, χ =, 2ln2W π Γ(q) 2 ln2w D q = Γ(q /2) π Γ(q) ( χ), χ = D χ, -D χ, -D J/W J/W χ and D for ultrametric matrices

28 Higher dimensional conjecture: χ = D /d Standard two-dimensional critical model: MIT in the quantum Hall effect via the Chalker-Coddington network model (Evers et al. (2008)) D =.7405 ± Conjecture: χ c = D /2 χ c = ± (Klesse, Metzler (997)) χ = 0.24 ± Standard three dimensional critical model: MIT in 3-d Anderson model (Rodriguez et al. (200)) D =.93 ± 0.0 (Rodriguez et al. (2009)) D 0 = ± 0.06 Symmetry: D = 2d D 0 D =.973 ± 0.06 Conjecture: χ c = D /3 χ c (Ndawana et al. (2002)) χ = 0.28 ± 0.06 (Ndawana et al. (2002)) χ = 0.32 ± 0.03

29 Summary Large varieties of intermediate statistics Lax matrices of integrable classical systems give new soluble ensembles of random matrices with intermediate statistics Absence of universality for spectral statistics Perturbation series for fractal dimensions Conjecture: χ = D /d New perspectives for intermediate statistics

30 Compressibility for ultrametric ensemble By definition: χ = + lim L lim N F L,N, F L,N = ρ L/ ρ L/ ρ [R 2 (E + s/2, E s/2) ρ 2 ]ds. Here R 2 (E, E 2 ) is the two-point correlation function R 2 (E, E 2 ) = N m n δ(e λ m )δ(e 2 λ n ), ρ = Nρ(E) is the mean density. In the first order of perturbation series it is sufficient to consider 2 2 sub-matrix H mm H nm H mn H nn and then to perform the summation of all pairs m n. Eigenvalues: λ,2 = ξ ± η 2 + h 2, h = H mn, ξ = (H mm + H nn )/2, η = (H mm H nn )/2.

31 R 2 (E + s/2, E s/2) = = n m δ(e+s/2 ξ η 2 +h 2 )δ(e s/2+ξ η 2 +h 2 )ρ(ξ+η)ρ(ξ η)2dξdη 2ρ 2 (E) n =m δ(s 2 η 2 + h 2 )dη = ρ 2 (E) n =m s θ(s 2h) s2 4h2 Integrating over s gives (assuming η H mn ) F L,N = 2ρ ( L ) 2 4 Hmn N ρn 2 n =m 2L. The first term exists only when H mn L/(2ρN). H mn = z J 2 d mn/2 z is the random complex variable Re z) = Im z = 0 and (Rez) 2 = (Imz) 2 = / 2. Ultrametric distance d mn = 2k and each value of k is degenerate 2 k times (criticallity)

32 F L,N 2ρ K i=i 0 2 i ( L ρn with i 0 such that L/(2ρN) J z /2 i 0. ) 2 4 (J z 2 i ) 2 2L Therefore K 2 i = 2 K 2 i0 = N 2J z N Lρ i=i 0 K [ F L,N 2ρ 2 i ( L ρn i=i 0 ) 2 4 (J z 2 i ) 2 L ρn ] 2J z Change i to 2J z /2 i = L/(xρN). Then F(L, N) 4ρ J [ xm z ( ] /x ln2 2 )dx where x m = L/(4ρJ z ). In the limit L, x m. ( /x 2 )dx = π/2, z = π/2, and ρ(0) = / 2πW: χ = π J 2 ln2w