Scaling laws for products of random matrices Lois d’´ echelle pour les produits de matrices al´ eatoires Jean-Marc Luck Institut de Physique Th´ eorique (CEA & CNRS) Forum de la Th´ eorie, 3 et 4 avril 2013 1
Outline • A reminder on sums and products of random numbers • Products of random matrices • One-dimensional systems and products of transfer matrices • The ferromagnetic Ising chain in a random magnetic field • The tight-binding model in a random potential • Band-edge scaling law • Ubiquity of scaling laws and their classification • Table of scaling functions 2
A reminder on sums and products of random numbers S N = x 1 + ··· + x N Sum of random numbers: x i random, independent and identically distributed (i.i.d.) var x = x 2 − x 2 = σ 2 x = m and finite Laws of large numbers: S N is asymptotically Gaussian (normal) for large N S N = Nm and var S N = N σ 2 f(S N ) σ N 1/2 • Sum is peaked around its mean value √ • Relative fluctuations fall off as 1 / N S N Nm 3
P N = x 1 ··· x N Product of random numbers: x i i.i.d. positive random numbers P typ = e N ln x • Product peaked around typical value N ln P typ = ln P N = N ln x N • Mean value of product is exponentially deep in the tail P N = x N = e N ln x ≫ P typ ln x < ln x implies N f(P N ) P N P N typ P N 4
Products of random matrices P N = M 1 ··· M N M i i.i.d. random matrices (say real and of size p × p ) ∼ e N γ P typ Again peaked around typical value: N 1 γ = lim N ln | tr P N | is Lyapunov exponent (Furstenberg, 1963) N → ∞ • Describes typical value only (relevant for disordered systems) • Generalizes explicit results ln x (numbers) and ln λ (constant matrices) • Difficult to calculate in general (even in the 2 × 2 case !) 5
One-dimensional systems and products of transfer matrices • One-dimensional system: sequence of independent units • Linear equations for each unit: transfer matrix M 1 M 1 M 2 M 3 M 4 M 5 Transfer-matrix formalism has proved useful in many areas • Optics, electromagnetism, etc. • Statistical Physics Disordered systems: products of random transfer matrices Ferromagnetic Ising chain in a random magnetic field Tight-binding model for an electron in a random potential 6
The ferromagnetic Ising chain in a random magnetic field H = − J ∑ σ n σ n + 1 − ∑ h n σ n n n σ n = ± 1 : classical Ising spins J > 0 : uniform ferromagnetic exchange constant h n : random magnetic fields (i.i.d. random variables) Canonical ensemble : partition function and free energy β = 1 / T : inverse temperature Z N = ∑ F N e − β H F N = − T ln Z N f = lim N → ∞ N { σ n } 7
Transfer-matrix formalism conditioned on the last spin ( σ n = ± ) Partition functions Z ± n � e β ( J + h n ) e β ( − J + h n ) � Z + � � � Z + � n n − 1 = e β ( − J − h n ) e β ( J − h n ) Z − Z − n n − 1 � �� � M ( h n ) Chain of N spins with periodic boundary conditions Z N = tr M ( h N ) ··· M ( h 2 ) M ( h 1 ) 8
( N → ∞ ) Thermodynamic limit • Constant magnetic field ( h n = h ) (Ising, 1924) f = − T ln λ λ is largest eigenvalue of M ( h ) � λ = e β J � sinh 2 β h + e − 4 β J � cosh β h + • Random magnetic field ( h n i.i.d.) f = − T γ γ is Lyapunov exponent of matrix product M ( h N ) ··· M ( h 2 ) M ( h 1 ) 9
Statistics of free energy (ground-state energy) in disordered systems For finite system of N sites Free energy F N = − T ln Z N is a random variable (Brout, 1959) F qu • Quenched average = − T ln Z N N describes properties of typical sample F qu ≈ N f is physical: N F qu ≈ − NT γ given by Lyapunov exponent for 1D system: N F an • Annealed average = − T ln Z N N is unphysical > F qu F an provides (useful) bound N N 10
The tight-binding model in a random potential − e ψ n + 1 + ψ n − 1 + V n ψ n = E ψ n ψ n : amplitude of wavefunction at site n E : energy eigenvalue V n : random site energies (i.i.d. random variables) 11
Transfer-matrix formalism � ψ n � ψ n + 1 � � E − V n � � − 1 = ψ n ψ n − 1 1 0 � �� � M ( V n ) • Constant potential ( V n = V ) ψ n ∼ e i nq Plane waves: tr M ( V ) = E − V = 2cos q Dispersion law: • Static random potential ( V n i.i.d.) � � � ∼ e n γ � ψ typ n γ is Lyapunov exponent of matrix product M ( V N ) ··· M ( V 2 ) M ( V 1 ) γ > 0 on the spectrum 12
Interpretation: localization by disorder (Anderson, 1958) • Eigenfunctions ψ n are all localized in 1D • Their hull falls off exponentially as | ψ n | ∼ exp ( ± n / ξ ) • Localization length ξ = 1 γ 0.4 0.3 0.2 0.1 ψ n 0 −0.1 −0.2 −0.3 −0.4 0 50 100 150 200 n 13
( V = 0 , V 2 ≪ 1 ) Weak-disorder regime Ω = γ + i π H • Complex characteristic exponent: (Cf. Sturm, 1836) γ : Lyapunov exponent Z ∞ E ρ ( E ′ ) d E ′ : H ( E ) = integrated density of states • Perturbative result (Thouless, 1972) E = 2cosh µ (Outside the band) V 2 Ω = µ − 8sinh 2 µ + ··· 14
Behavior of inverse localization length γ = 1 / ξ • Outside band ( | E | > 2 ) γ ≈ µ 1.5 • Inside band ( | E | < 2 , µ → i q ) 1 V 2 γ ≈ 8sin 2 q γ 0.5 β 2 β 2 • Near band edges ( V 2 and µ small) β β 0 γ ∼ β ∼ ( V 2 ) 1 / 3 −4 −2 0 2 4 E E − 2 ∼ β 2 ∼ ( V 2 ) 2 / 3 15
Band-edge scaling law (Halperin, 1965; Derrida & Gardner, 1984) Weak disorder ( V 2 ≪ 1 ) + Band edge ( µ ≪ 1 ) = Scaling behavior Ω = γ + i π H = β G ( x ) x = µ 2 E − 2 � � 1 / 3 , V 2 / 2 β = β 2 = � � 2 / 3 V 2 / 2 G ( x ) = e − 2i π / 3 Ai ′ ( e − 2i π / 3 x ) Ai ( e − 2i π / 3 x ) = Ai ′ ( x )+ iBi ′ ( x ) Ai ( x )+ iBi ( x ) • Non-trivial scaling function G ( x ) mixing real and imaginary parts • Involves logarithmic derivative of special function Here: Ai , Bi : Airy functions 16
Ubiquity of scaling laws and their classification Quite generally: Weak disorder + ‘Interesting point’ (band edge, critical point) = Scaling behavior Our work (CLTT, 2013) • Group SL 2 ( R ) of 2 × 2 real matrices with unit determinant • Continuum limit of matrices close to unity Outcome • Classification of scaling functions describing Ω • In correspondence with special functions • Classification of types of collective behavior of 1D disordered systems 17
Universal model (Comtet, Texier, Tourigny, 2010) Kronig-Penney model with double impurities (point-like scatterers) u n − d 2 ψ d x 2 +( W ( x )+ U ( x )) ψ = ψ α n w n x n : positions of impurities α n = x n − x n − 1 : distances between impurities W ( x ) = ∑ � � δ 2 ( x − x n )+ δ ′ ( x − x n ) w n : supersymmetric potential n U ( x ) = ∑ u n δ ( x − x n ) : scalar potential n 18
Universal transfer matrix M ( α n , w n , u n ) = K ( α n ) A ( w n ) N ( u n ) � cos α − sin α � � e w � � 1 � 0 u K ( α ) = , A ( w ) = , N ( u ) = sin α cos α e − w 0 0 1 This is the Iwasawa decomposition of the group SL 2 ( R ) (Iwasawa, 1949) Subgroup Compact Abelian Nilpotent Disorder Distance Supersymmetric Scalar 19
Continuum limit of matrices close to unity • 3 average values α , w , u • 6 elements of covariance matrix D αα = α 2 − α 2 , D α w = α w − α w , etc. simultaneously small Main outcome • Ω as explicit homogeneous function of these 9 parameters Ω = ··· S ′ ( x ) • involves logarithmic derivative of special function: S ( x ) 20
Table of scaling functions S ( x ) Zeros Disorder Special function Ai ( e − 2i π / 3 x ) 1 quadruple scalar Airy K ν ( x ) 2 double supersymmetric Bessel I i λ ( x ) 2 double distance Bessel W lm ( x ) 1 double + 2 simple general potential Whittaker K ( k ) 4 simple ‘independent’ elliptic 2 F 1 ( α , β ; γ ; x ) 4 simple general hypergeometric • Provides classification of types of collective behavior in 1D disordered systems • Zeros: complex roots of instrumental fourth-degree polynomial • Complex analysis is key novel ingredient 21
To conclude... three references • R. Brout, Statistical-mechanical theory of a random ferromagnetic system Phys. Rev. 115 (1959) 824 • JML, Syst` emes d´ esordonn´ es unidimensionnels Collection Al´ ea Saclay (1992) • A. Comtet, JML, C. Texier, Y. Tourigny, The Lyapunov exponent of products of random 2 × 2 matrices close to the identity J. Stat. Phys. 150 (2013) 13 (arXiv:1208.6430) 22
Appendix: main steps of derivation � X n � � X n − 1 � • Sequence of vectors = M n Y n Y n − 1 � A n � B n M n = M ( α n , w n , u n ) = C n D n Z n = X n • Riccati variables Y n Z n = A n Z n − 1 + B n C n Z n − 1 + D n Ω = lim n → ∞ ln ( C n Z n − 1 + D n + i0 ) 23
• Continuum limit: Z ( n ) → continuous process � d Z = V ( Z ) d n + 2 D ( Z ) d W ( n ) V ( Z ) = − α ( Z 2 + 1 )+ 2 wZ + u + D αα Z ( Z 2 + 1 )+ 2 D ww Z − 4 D α w Z 2 − 2 D α u Z + 2 D wu 2 D ( Z ) = D αα ( Z 2 + 1 ) 2 + 4 D ww Z 2 + D uu − 4 D α w Z ( Z 2 + 1 ) − 2 D α u ( Z 2 + 1 )+ 4 D wu Z 24
• Hilbert transform of invariant distribution Z + ∞ f ( Z ) d Z F ( y ) = y − Z − ∞ • F ( y ) analytic in lower half plane ( Im y < 0 ) • Polynomial differential equation Q ( y ) F ′ ( y )+ R ( y ) F ( y ) = S ( y )+ 2 Ω These two conditions determine both F ( y ) and Ω 25
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