Multifractality in simple systems. Eugene Bogomolny

Transcription

1 Multifractality in simple systems Eugene Bogomolny Univ. Paris-Sud, Laboratoire de Physique Théorique et Modèles Statistiques, Orsay, France In collaboration with Yasar Atas

2 Outlook 1 Multifractality Critical models 3 Spin chains models Quantum Ising model XY model Ising model in two fields XXZ model XYZ model 4 Bose-Hubbard model 5 Thermodynamics 6 Summary

3 The simplest model of multifractality Binomial cascade p +r = 1 p r p pr rp r 3 pr rp p p p r p r r r p r 3

4 Typical functions Left σ =. Right σ = 1 After N steps there are N possibilities of σ n =,1, σ = [ ] Ψ σ = p N k r k, k = n σ n x = N N n=1 n (1+σ n ) = Ψ σ = Ψ(x)..15 Ψ(x) x N = 1, p = cos (1).9, r = sin (1).78

5 Multifractal formalism M (= N ) normalized quantities Ψ σ, σ Ψ σ = 1 Rényi entropy Fractal dimensions S R (q,m) = 1 q 1 ln ( D q = lim M σ S R (q,m) lnm Ψ σ q) Localization : D q = Completely delocalized states : Ψ σ 1 M, D q = q 1 Multifractals : D q = non-linear function of q σ Ψ σ q M M τ q τ q = D q (q 1) τ q = increasing and concave function of q

6 Binomial cascade Ψ σ = p N k r k, k = n σ n, M = N Ψ σ q = σ N CN k pq(n k) r qk = (p q +r q ) N k= τ(q) = ln(pq +r q ), D q = ln(pq +r q ) ln (q 1)ln τ q D q q q

7 Singularity spectrum, f(α)-function Ψ σ q e N τ q, N = lnm σ Ψ σ = e E σ, Ψ σ q = σ σ e qe σ e qe ρ(e)de σ e qe σ Density ρ = σ ) δ (E E σ, ρ(e) N en f(e/n ).8 e qe+n f(e/n ) de by saddle point f(α).6.4 q = f (α), τ q = qα f(α), α = E/N α = τ q α

8 3-d Anderson model at metal-insulator transition H = i ε i a i a i a j a i j= adjacent to i ε i =i.i.d. random variables between W/ and W/ Mobility edge : E c (W) E c = W = W c = 16.5 W > W c all states are localized density localized states delocalized states localized states 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 Multifractal wave functions New intermediate type of spectral statistics

9 Critical one-dimensional models Critical power law banded random matrices N N Hermitian matrices H ij = i.i.d. Gaussian variables (real for β = 1 and complex for β = ) with zero mean H ij = and variance H ii = 1/β, H ij = (1+ ) 1 (i j) for i j b Ruijsenaars - Schneider ensemble of random matrices N N unitary matrix related with the Lax matrix of Ruijsenaars - Schneider model M kp = e iφ k 1 e πig N[1 e πi(k p+g)/n ] Φ m = independent random variables (phases) uniformly distributed in [,π] g = parameter

10 Spectral statistics for RS model g = 1/ (top left), 6/5 (top right), 4/3 (bottom left), and 9/4 (bottom right), averaged over 1 realizations of matrices of size N = 71. Solid lines are numerical results, dashed lines indicate the analytical expressions In each panel (from left to right) P(s) = P(1,s) (red), P(,s) (green) and P(3,s) (blue)

11 Fractal dimensions for RS model g =.1 (black),.5 (red),.1 (green),. (blue),.3 (violet),.5 (brown),.7 (maroon),.9 (cyan) Matrix sizes are N = n, 8 n 1 Number of realizations is between 14 for N = 8 and 64 for 1

12 Heisenberg model in external fields with periodic boundary conditions H = N n=1 [ 1+γ σ x nσ x n γ σ y nσ y n+1 + σz nσ z n+1 +λσ z n +ασ x n Basis of z-components of each spin, σ = σ 1,...,σ N, σ j = ±1 ] Any model of N spins- 1 is represented by a M M matrix with M = N Wave function of N spins- 1, HΨ = EΨ Ψ = σ Ψ σ σ Main question is the distribution of coefficients Ψ σ Only ground state wave functions with lowest energy are considered Parameters γ, α, λ are chosen to imply (by the Perron-Frobenius theorem) that Ψ σ Values of parameters are such that GS of the XY models (with = ) are ferromagnetic and for the XYZ models they are anti-ferromagnetic

13 Quantum Ising model Quantum Ising model in transverse field H = N n=1 [ ] σnσ x n+1 x +λσn z Integrable by the Jordan-Wigner transformation 3.6 λ = 1.6 Ψ σ.3 λ = 1 D q λ = 1 1 λ = σ > q

14 Quantum Ising model Exact expressions D + (λ) = 1 1 πln D (λ) = 1 1 πln D 1/ (λ) = 1 D ( 1 λ π π ln ln [ 1+ λ cosu R(λ,u) ]du,... [ 1 λ cosu R(λ,u) ]du,... ), R(λ,u) = 1 λcosu +λ D + D D λ D 1/ D (λ)+d (λ) = {, λ < 1 + ln λ ln, λ > 1 D 1 λ

15 XY model The XY model H = N n=1 [ 1+γ σ x nσ x n γ Integrable by the Jordan-Wigner transformation λ=1.6. σ y nσ y n+1 +λσz n λ=.4, γ=1.4 ] λ=1 Ψ σ.1 λ=.4 D q σ > λ =.8, γ= q γ = 1.4

16 XY model Exact expressions Factorizing field Exact factorized GS wave functions at λ = λ f = 1 γ Ψ = N (cosθ 1 n ±sinθ 1 n ), cos θ = 1 γ 1+γ. n=1 D q = ln(cosq θ +sin q θ) (q 1)ln Exactly as for binomial cascade Limiting values { D..., λ > λ D +..., D = c D Neel..., λ < λ c D ± = 1 1 π [ ln 1± λ cosu ] du πln R (λ,γ,u) D Neel = πln π/ R ± (λ,γ,u) = (λ cosu) +γ sin u ln [ 1 λ +γ (1 +γ )cos u ] du R+ (λ,γ,u)r (λ,γ,u)

17 XY model Asymptotic fractal dimensions for the XY model with anisotropy γ = D D Neel D_.5 D λ c (1.4) λ

18 Ising model in two fields The Ising model in transverse and longitudinal fields H = N n=1 Integrable when α only for λ = 1 [ ] σn x σx n+1 +λσz n +ασx n 1.5 Ψ σ..1 α = 1.4, λ = D q σ > α =.6, λ = 1.5 α = 1.4, λ = α =, λ = q

19 XXZ model The XXZ model in zero fields H = 1 N n=1 Integrable by the coordinate Bethe anzatz [ ] σnσ x n+1 x +σnσ y y n+1 + σz nσn+1 z Ψ σ..1 =.5 D q σ > =.5 =.5 = q 15

20 XXZ model Combinatorial point : = 1 Razumov Stroganov conjecture for odd N = R +1 (proved) : GS energy = 3N/4 The largest coefficient (the one for the Néel configuration) : Ψ 1 max = 3R/ R 5...(3R 1) (R 1) The smallest coefficient corresponds to a half consecutive spins up and other spins down Ψ 1 min = Ψ 1 maxa R σ Ψ σ = 3 R/ Negative moments A R = number of alternating sign matrices = R 1 k= D = 3ln3 ln.377, D 1/ = ln3 ln.79 (3k +1)! (R +k)! lna R R R ln(3 3/4)+O(R) It is a particular case of the emptiness formation probability of a string of n aligned spins with n N Negative moments in anti-ferromagnetic case require a different scaling

21 XYZ model The XYZ model H = N [ 1+γ n=1 σ x n σx n γ Integrable by algebraic Bethe anzatz Special points Factorized field : λ f = (1 ) γ Ψ = N (cosθ 1 n ±sinθ 1 n ), cos θ = 1 γ = γ =.5 n=1 cos θ = 1 γ 1+γ Combinatorial point in zero fields : c = (γ 1)/ 1.5 σ y n σy n+1 + σz n σz n+1 +λσz n. Ψ σ.1 D q σ > = q 15 γ =.6, c =.3 ] =.5

22 Bose-Hubbard model N bosons at L sites H = L j=1 b j = bosonic creation operator at site j [ b j+1 b j +h.c. U ] n j(n j 1), n j = b j b j D q U=. U=.1 U=.5 U=1. U=. U=4. U=6. U=1. U=. HCB q (L = 1,...,, N/L =.5, preliminary calculations of Guillaume Roux)

23 Limiting cases U = = free bosons Ground state wave function on N free bosons at L sites with periodic boundary conditions (n j = the occupation number of site j) Ψ(n 1,n,...,n L ) = N! L N 1 n 1!n!...n L!, n j, n n L = N Standard calculations D q = ln(f q(z c )) νlnz c +qνln(ν/e) (q 1)[(1 +ν)ln(1+ν) νlnν], ν = N L, f q(x) = n= x n (n!) q z c from the saddle point equation f q f q (z c ) ν z c =

24 U = = hard-core bosons Hard-core bosons = fermions. Normalized ground state wave function of N HC bosons (= modulus of fermionic WF) at L sites with periodic boundary conditions (n j determines the momentum of a particle number j, p j = πn j /L) Ψ B (n 1,...,n N ) = 1 L N 1 s<r N e πin r/l e πin s/l ; One has to calculate the sum (Gaudin model) 1 n 1 < n <...n N L S q = 1 N! L L... Ψ B (n 1,...,n N ) q n 1 =1 n N =1 In the continuous limit πn j /L θ j ( L ) N π π S q 1 N! π Ψ B (θ 1,...,θ N ) q dθ 1...dθ N From the Dyson integral S q 1 Γ(1 +qn) N!L (q 1)N Γ N (1 +q)

25 Fractal dimensions of free fermions and HC bosons D q = 1 [ lnγ(1 +q) q lnq ] 1 lnν + c(ν) q 1 { 1 ln(ν), for free fermions and HC bosons c(ν) = ln(1 ν) lnν ln(1 ν), for HC limit of Hubbard model ν For discrete case only special values are known due to Gaudin D 1/ = 1 [ 1 c(ν) πν πν/ ] lntanφdφ, D = 1 c(ν) (1 ln(ν)) For integer k < L N 1, D k in the discrete case coincides with the continuous result If ν = 1/m, D = lnm/c(1/m) D q D q q ν = 1/ q ν = 1/3

26 Thermodynamic analogy Usual thermodynamics of lattice gas Multifractality Z(T) = σ e E σ /T Z R (q) = σ Ψ σ q E σ = classical energy of a Hamiltonian, for example for N sites in 1-dim lattice H = i<j J ij σ i σ j j h j σ j General question : when free energy in thermodynamic limit exists F(T) = lim N 1 N lnz(t) Answer : e.g. pair interaction of not too long range (local interactions) Denote Ψ σ = e E σ then Z R (q) = Z(T = 1/q) Main question : when fractal dimensions exist in the limit N Conjecture τ q = lim N 1 N lnz R(q) For quantum Hamiltonians with local interaction fractal dimensions of (at least) GS wave function do exist

27 Summary Main result Ground state wave functions of practically all standard one-dimensional spin- 1 models are multifractals in the natural spin-z basis For special values of parameters and/or certain dimensions it is possible to derive exact analytical formulas which prove rigorously the existence of fractal dimensions. In other cases one has to rely on numerical calculations The multifractality in spin chains is robust and UNIVERSAL phenomenon. It exists for integrable and non-integrable models, for ferro and anti-ferro magnetic states, as well as for critical and non-critical systems Conjecture Multifractality is a generic property of many-body problems with local interactions