Supersymmetry and Random Matrices: Avoiding the SaddlePoint - PowerPoint PPT Presentation

F AKULT ¨ AT F ¨ UR P HYSIK Supersymmetry and Random Matrices: Avoiding the Saddle–Point Approximation Thomas Guhr SuSy and Random Matrices — in Honor of Tom Spencer Institut Henri Poincaré, Paris, April 2012 Paris, April 2012

Outline • Gaussian Unitary Ensemble and Supersymmetric Itzykson–Zuber Integral • Supersymmetric Harish-Chandra Integral (beyond the Unitary Case) • Crossover Transitions and Dyson’s Brownian Motion in Superspace • Some Applications • Chiral Random Matrix Theory and Yet Another Group Integral • A Certain Class of Matrix Bessel Functions • “Supersymmetry without Supersymmetry” • Supersymmetry for Arbitrary Invariant Probability Densities Paris, April 2012

Acknowledgments thank you for collaboration: thank you for discussion: Heiner Kohler Martin Zirnbauer Mario Kieburg Jacobus Verbaarschot Thomas Wilke Yan Fyodorov Johan Grönqvist Grigori Olshanski Christian Recher Vera Serganova Tilo Wettig Isaiah Kantor Hans Weidenmüller Poul Damgaard Hans–Jürgen Sommers Kim Splittorff Hans–Jürgen Stöckmann Burkhard Seif Thomas Papenbrock Paris, April 2012

Introduction Efetov (early 80’s): supersymmetric non–linear σ model for disordered systems, connection to RMT established very broad range of applications Efetov, Schwiete, Takahashi (2004): new foundation by superbosonization Verbaarschot, Zirnbauer (1985): supersymmetric non–linear σ model for two–point function starting from RMT (zero dimensions) − → weak disorder, large matrix dimension saddle–point approximation, Goldstone modes, coset manifold some cases: direct and exact solution possible, finite matrix dimension, structural information, applications Paris, April 2012

Gaussian Unitary Ensemble and Supersymmetric Itzykson–Zuber Integral Paris, April 2012

Generating and Correlation Functions Gaussian ensemble ( β = 1 , 2 , 4 ) of N × N random matrices H � � ∂ k � R ( β ) Z ( β ) k –level correlations k ( x 1 , . . . , x k ) = k ( x + J ) � � k � p =1 ∂J p J =0 generating function obeys the identity � k � det( H − x p − J p ) Z ( β ) d [ H ] exp( − tr H 2 ) k ( x + J ) = det( H − x − p + J p ) p =1 � d [ σ ] exp( − str σ 2 )sdet − N ( σ − x − − J ) = where σ is a 2 k × 2 k or 4 k × 4 k supermatrix − → drastic reduction of dimensions Paris, April 2012

GUE Generator — Eigenvalues and Angles diagonalization σ = u − 1 su with u ∈ U( k/k ) and s = diag ( s 11 , . . . , s k 1 , is 12 , . . . , is k 2 ) � � 1 d [ σ ] = B 2 k/k ( s ) d [ s ]d µ ( u ) , B k/k ( s ) = det s p 1 − is q 2 p,q =1 ,...,k generating function, r = x + J � � � − str ( u − 1 su + r ) 2 � Z (2) d [ s ] B 2 sdet − N s − k ( r ) = k/k ( s ) d µ ( u ) exp � � − str ( s + r ) 2 � 1 sdet − N s − = 1 + d [ s ] B k/k ( s ) exp B k/k ( r ) everything compact, hyperbolic symmetry not needed TG (1991) Paris, April 2012

Correlation Functions J p derivatives trivial R k ( x 1 , . . . , x k ) = det [ C N ( x p , x q )] p,q =1 ,...,k well–known kernel is found to be a double integral − ( s p 1 + x p ) 2 + ( is q 2 + x q ) 2 � � is q 2 � N � � + ∞ + ∞ � ds p 1 ds q 2 C N ( x p , x q ) = exp s − s p 1 − is q 2 p 1 −∞ −∞ � N − 1 � � x 2 q − x 2 = exp ϕ n ( x p ) ϕ n ( x q ) p n =0 determinant structure is a built–in feature of supersymmetry Paris, April 2012

SuSy Harish-Chandra–Itzykson–Zuber Integral unitary supergroup U( k 1 /k 2 ) � dµ ( u ) exp( i str u − 1 sur ) = det[exp( is p 1 r q 1 )] det[exp( is p 2 r q 2 )] B k 1 /k 2 ( s ) B k 1 /k 2 ( r ) = exp( i str sr ) + permutations B k 1 /k 2 ( s ) B k 1 /k 2 ( r ) � B k 1 /k 2 ( s ) = ∆ k 1 ( s 1 )∆ k 2 ( is 2 ) � with p,q ( s p 1 − is q 2 ) , ∆ k 1 ( s 1 ) = ( s p 1 − s q 1 ) p<q for k 1 /k 2 = N/ 0 , 0 /N result in ordinary space recovered TG, J. Math. Phys. 32 (1991) 336 Alfaro, Medina, Urrutia, J. Math. Phys. 36 (1995) 3085, received 28 Nov. 1994, hep-th/9412012 TG, Commun. Math. Phys. 176 (1996) 555, received 22 Nov. 1994 Paris, April 2012

Radial Laplacian and Separability supersymmetric Itzykson–Zuber integral is eigenfunction of ∆ s ψ ( s, r ) = − str r 2 ψ ( s, r ) , with ψ ( s, r ) = ψ ( r, s ) and with radial Laplacian k 1 k 2 � � 1 ∂ k 1 /k 2 ( s ) ∂ 1 ∂ k 1 /k 2 ( s ) ∂ B 2 B 2 ∆ s = + B 2 B 2 k 1 /k 2 ( s ) ∂s p 1 ∂s p 1 k 1 /k 2 ( s ) ∂s p 2 ∂s p 2 p =1 p =1 separability, generalizes ordinary case k i 2 � � ∂ 2 f ( s ) 1 ∆ s B k 1 /k 2 ( s ) = f ( s ) ∂s 2 B k 1 /k 2 ( s ) pi i =1 p =1 eigenfunctions of flat Laplacian trivial Paris, April 2012

Supersymmetric Harish-Chandra Integral (beyond the Unitary Case) Paris, April 2012

Ordinary Harish–Chandra Integral G compact semi–simple Lie group, a , b fixed elements in Cartan subalgebra H 0 of Lie algebra of G � � � � 1 exp (tr w ( a ) b ) tr U − 1 aUb exp dµ ( U ) = |W| Π( a )Π( w ( b )) G w ∈W Π( a ) product of all positive roots of H 0 , W Weyl reflection group everything stays in the space of the Lie group and its algebra ! Harish–Chandra (1957) Paris, April 2012

Supersymmetric Harish–Chandra Integral supersymmetric Itzykson–Zuber integral: case of U( k 1 /k 2 ) − → most interesting remaining case is UOSp( k 1 / 2 k 2 ) conjecture: Serganova (1992) and Zirnbauer (1996) proof: TG, Kohler (2002) Laplacian ∆ A over Lie superalgebra uosp( k 1 / 2 k 2 ) construct radial part ∆ a over Cartan subalgebra identify Harish–Chandra integrals as eigenfunctions of ∆ a ∆ a is separable ! − → solution of eigenequation trivial proof also includes Lie groups in ordinary space Paris, April 2012

Crossover Transitions and Dyson’s Brownian Motion in Superspace Paris, April 2012

Crossover Transitions random matrix H drawn from GOE, GUE, GSE (random) matrix H (0) with arbitrary P (0) ( H (0) ) interpolating ensemble with α “strength of chaos” H ( α ) = H (0) + αH correlations on local scale of mean level spacing D depend on ξ p = x p /D and λ = α/D fictitious time t = α 2 / 2 and locally τ = t/D 2 = λ 2 / 2 Brownian motion transports initial H (0) into chaos, τ → ∞ Dyson, French, Pandey, Mehta, ... Paris, April 2012

Dyson’s Brownian Motion in Superspace generator for k –level correlations of initial condition � P (0) ( H (0) ) sdet − 1 � � 1 ⊗ H (0) − s ⊗ 1 Z (0) d [ H (0) ] k ( s ) = generator for crossover correlations, r = x + J ∂ diffusion ∆ r Z k ( r, t ) = ∂tZ k ( r, t ) � Γ k ( s, r, t ) Z (0) Z k ( r, t ) = k ( s ) B k/k ( s ) d [ s ] convolution � � � − 1 t str ( u − 1 su − r ) 2 Γ k ( s, r, t ) = exp dµ ( u ) kernel where u ∈ U( k/k ) or u ∈ UOSp(2 k/ 2 k ) diagonalizes hierarchic equations of French et al. (1988) TG (1996) Paris, April 2012

Special Case: GUE plus External Field for P (0) ∼ δ H (0) = diag ( E (0) 1 , . . . , E (0) − → N ) fixed √ transition to GUE, H ( t ) = H (0) + 2 tH , find immediately R k ( x 1 , . . . , x k ) = det [ C N ( x p , x q )] p,q =1 ,...,k � � � + ∞ � + ∞ − ( s p 1 + x p ) 2 + ( is q 2 + x q ) 2 C N ( x p , x q ) = 1 ds p 1 ds q 2 exp 2 t s p 1 − is q 2 2 t 2 t −∞ −∞ N � is q 2 − E (0) n p 1 − E (0) s − n n =1 cf. Brézin, Hikami (1996) average over H (0) destroys determinant structure Paris, April 2012

General Case on Local Scale local scale, unfolding ξ = x/D , j = J/D , τ = t/D 2 and ρ = ξ + j = r/D , s ′ = s/D z (0) N →∞ Z (0) k ( s ′ ) = lim z k ( ρ, τ ) = lim N →∞ Z k ( r, t ) k ( s ) and generator for local crossover correlations obeys same equations: ∂ diffusion ∆ ρ z k ( ρ, τ ) = ∂τ z k ( ρ, τ ) � Γ k ( s ′ , ρ, τ ) z (0) k ( s ′ ) B k/k ( s ′ ) d [ s ′ ] convolution z k ( ρ, τ ) = diffusion process to chaos is scale invariant ! TG (1996) Paris, April 2012

Some Applications Paris, April 2012

Symmetry Breaking � H 1 � 0 ensemble H ( α ) = + αH , all from GUE’s 0 H 2 ω = ξ 2 − ξ 1 λ = α/D = 0 , 0 . 3 , 0 . 9 , ∞ Y 2 ( ω, λ ) X 2 ( ω, λ ) = 1 − Y 2 ( ω, λ ) statistical enhancement ω TG, Weidenmüller (90’s) Paris, April 2012

Chaotic Billiard with Random Scatterers unitary, L scatterers, strength α , generator of k -level correlations � � sdet L (1 + ασ ) � − str σ 2 + str σ ( x + J ) Z k ( x + J ) = exp d [ σ ] sdet N σ GUE statistics for all λ in bulk, transition to Poisson for tail states TG, Stöckmann (2004) Paris, April 2012

Chiral Random Matrix Theory and Yet Another Group Integral Paris, April 2012

Chiral Random Matrix Theory Dirac operator has chiral symmetry, in chiral basis it is � � � � i � D c 0 0 W i � D = − → , W random matrix ( i � D c ) † W † 0 0 compare chRMT with lattice gauge calculations hard edge microscopic limit (zoom) chiral condensate spectral density Verbaarschot (1993) Paris, April 2012