Work inspired by studies of Rajamani Narayanan and Herbert Neuberger Spectral shock waves in QCD Maciej A. Nowak (In collaboration with Jean-Paul Blaizot and Piotr Warcho� l) Mark Kac Complex Systems Research Center, Marian Smoluchowski Institute of Physics, Jagiellonian University, Krak´ ow, Poland Large N Gauge Theories, Galileo Galilei Institute, Florence May 9th, 2011
Motivation Shock waves Catastrophes ”Hard edge” Outline Diffusion of large (huge) matrices Non-linear Smoluchowski-Fokker-Planck equations and shock waves Finite N as viscosity in the spectral flow – Burgers equations Order-disorder phase transition in large N YM theory, colored catastrophes and universality Shock waves in large N SYM? Chiral shock waves Summary Maciej A. Nowak Spectral shock waves
Motivation Shock waves Simplest diffusion – additive Brownian walk of huge matrices Catastrophes Surfing the shock wave ”Hard edge” Motivation Matricial ( N × N , N ∼ ∞ ) analogue of classical probability calculus (physics, telecommunication, life science itd) Large N QFT in 0+0 dimensions (on one space-time point) Building in the dynamics: systems evolve as a function of some exterior parameters (time, length of the wire, area of the surface, temperature ...) Finding out universality windows where this simplified dynamics is shared by non-trivial theories Maciej A. Nowak Spectral shock waves
Motivation Shock waves Simplest diffusion – additive Brownian walk of huge matrices Catastrophes Surfing the shock wave ”Hard edge” Two probability calculi MATRICIAL (FRV for N = ∞ ) CLASSICAL spectral measure of probability density matrix-valued ensemble distribution � < ... > = ... P ( H ) dH � < ... > = ... p ( x ) dx � � 1 Fourier transform F ( k ) of Resolvent G ( z ) = Tr z − H pdf generates moments R-transform generates ln F ( k ) of Fourier tr. additive cumulants generates additive cumulants G [ R ( z ) + 1 / z ] = z Gaussian – Non-vanishing Wigner semicircle – second cumulant only, Non-vanishing second ln F ( k ) = c 2 k 2 cumulant only, R ( z ) = C 2 z Maciej A. Nowak Spectral shock waves
Motivation Shock waves Simplest diffusion – additive Brownian walk of huge matrices Catastrophes Surfing the shock wave ”Hard edge” Spectral observables in RMT (FRV) P ( H ) dH = e − N Tr V ( H ) dH = � N i =1 dx i e − N � i V ( x i ) � i < j ( x i − x j ) 2 Jacobian (Vandermonde determinant) triggers interactions between eigenvalues All nontrivial correlations in the spectral functions reflect this interaction � � One-point function G ( z ) = 1 1 1 = � Tr z k +1 m k , where k N z − H m k = 1 � Tr H k � � dxx k ρ ( x ) = N Note that − 1 π ℑ G ( z ) | z = x + i ǫ = ρ ( x ) Maciej A. Nowak Spectral shock waves
Motivation Shock waves Simplest diffusion – additive Brownian walk of huge matrices Catastrophes Surfing the shock wave ”Hard edge” Inviscid Burgers equation After considerable and fruitless efforts to develop a Newtonian theory of ensembles, we discovered that the correct procedure is quite different and much simpler...... from F.J. Dyson, J. Math. Phys. 3 (1962) 1192 H ij → H ij + δ H ij with < δ H ij = 0 > and < ( δ H ij ) 2 > = (1 + δ ij ) δ t For eigenvalues x i , random walk undergoes in the ”electric � � 1 field” (Dyson) < δ x i > ≡ E ( x i ) δ t = � δ t and i � = j x j − x i < ( δ x i ) 2 > = δ t Resulting SFP equation for the resolvent in the limit N = ∞ and τ = Nt reads ∂ τ G ( z , τ ) + G ( z , τ ) ∂ z G ( z , τ ) = 0 Non-linear, inviscid complex Burgers equation, very different comparing to Fick equation for the ”classical” diffusion ∂ τ p ( x , τ ) = 1 2 ∂ xx p ( x , τ ) Maciej A. Nowak Spectral shock waves
Motivation Shock waves Simplest diffusion – additive Brownian walk of huge matrices Catastrophes Surfing the shock wave ”Hard edge” Inviscid Burgers equation - details SFP eq: ∂ t P ( { x j } , t ) = 1 i ∂ 2 � ii P ( { x j } , t ) − � i ∂ i ( E ( x i ) P ( { x j } , t )) 2 Integrating, normalizing densities to 1 and rescaling the time τ = Nt we get dy ρ ( y ) � ∂ τ ρ ( x ) + ∂ x ρ ( x ) P . V . x − y = dy ρ c ( x , y ) 2 N ∂ 2 1 � xx ρ ( x ) + P . V . x − y r.h.s. tends to zero in the large N limit x ± i ǫ = P . V . 1 1 x ∓ i πδ ( x ) Note that contrary to Dyson we consider free diffusion and not Ornstein-Uhlenbeck process, since we focus on non-equilibrium phenomena. Maciej A. Nowak Spectral shock waves
Motivation Shock waves Simplest diffusion – additive Brownian walk of huge matrices Catastrophes Surfing the shock wave ”Hard edge” Dolphins wisdom - surfing the shock wave Tracing the singularities of the flow allows to understand the pattern of the evolution of the complex system without explicit solutions of the complicated hydrodynamic equations... UK Daily Mail, July 11th 2007 Maciej A. Nowak Spectral shock waves
Motivation Complex Burgers Equation Shock waves Physical manifestation - finite N YM Catastrophes Caustics ”Hard edge” Complex Burgers Equation Burgers equation ∂ τ G + G ∂ z G = 0 Complex characteristics G 0 ( z ) = G ( τ = 0 , z ) = 1 G ( z , τ ) = G 0 ( ξ [ z , τ )]) z ξ = z − G 0 ( ξ ) τ ( ξ = x − vt ), so solution reads G ( z , τ ) = G 0 ( z − τ G ( z , τ )) Shock wave when d ξ dz = ∞ √ z 2 − 4 τ ), 1 Since explicit solution reads G ( z , τ ) = 2 πτ ( z − √ 1 4 τ − x 2 , shock waves appear at the edges i.e. ρ ( x , τ ) = of the spectrum ( x = ± 2 √ τ ). 2 πτ But we can infer the same information from the condition dz / d ξ = 0, since ξ c = ±√ τ , so z c = ξ c + G 0 ( ξ c ) τ = ± 2 √ τ Maciej A. Nowak Spectral shock waves
Motivation Complex Burgers Equation Shock waves Physical manifestation - finite N YM Catastrophes Caustics ”Hard edge” Universal preshock – relaxing N = ∞ condition � 1 � � G ( z , τ ) = 1 1 � = ∂ z N Tr ln( z − H ( τ )) = Tr N z − H ( τ ) � 1 � ∂ z N ln det( z − H ( τ )) We define f ( z , τ ) = 1 N ∂ z ln < det( z − H ( τ )) > Note that f and G coincide only when N = ∞ (cumulant expansion) Remarkably f fulfills for any N an exact equation 1 ∂ τ f + f ∂ z f = − ν∂ zz f ν = 2 N Exact viscid Burgers equation with negative (!) viscosity Positive viscosity smoothens the shocks, negative is ”roughening” them ± x = 2 √ τ + ν 2 / 3 s and f N ( x , τ ) ∼ ± 1 √ τ + ν 1 / 3 ξ N ( s , τ ), where s ξ N ∼ ∂ s ln Ai ( 2 √ τ ) Preshock: ”soft edge” (Airy) universality Maciej A. Nowak Spectral shock waves
Motivation Complex Burgers Equation Shock waves Physical manifestation - finite N YM Catastrophes Caustics ”Hard edge” Multiplicative matricial random walk classically: y i +1 − y i = y i η ( η -noise) matricially: product of < � k (1 + H k ) > in general has complex spectra. But we can impose the constraint of unitarity < � k exp iH k > , then eigenvalues are complex, but always confined to the unit circle ( x = e i θ ) � π − π d θ ρ ( τ,θ ) Resolvent G ( z , τ ) = z − e i θ . Related function F ( z = e i θ , τ ) = i ( zG ( z , τ ) − 1 2 ) = i ( 1 2 + � ∞ n =1 w n ( τ ) e − in θ ) Maciej A. Nowak Spectral shock waves
Motivation Complex Burgers Equation Shock waves Physical manifestation - finite N YM Catastrophes Caustics ”Hard edge” Diffusion of unitary matrices: Burgers equation for F ( z = e i θ , τ ) Durhuus, Olesen, Migdal, Makeenko, Kostov, Matytsin, Gross, Gopakumar, Douglas, Rossi, Kazakov, Voiculescu, Pandey, Shukla, Janik, Wieczorek, Neuberger, Biane... Collision of two shock waves, since they propagate on the circle Universal preshock - expansion at the singularity for finite N Universal, wild oscillations anticipating the shock – Pearcey universality Maciej A. Nowak Spectral shock waves
Motivation Complex Burgers Equation Shock waves Physical manifestation - finite N YM Catastrophes Caustics ”Hard edge” Three phases If we encounter branch singularity ( θ − θ c ) µ on the complex plane, then for large n , w n = | n | − µ − 1 e − n ∆ ℜ e in θ ∗ , where θ c = θ ∗ + i ∆ Gapped phase Closure of the gap Gappless phase τ < 4 τ = 4 τ > 4 real singularities inflection point, so complex singularities, µ = 1 / 2 µ = 1 / 3 µ = 1 / 2 moments oscillate in Durhuus-Olesen moments decay time phase transition exponentially modulo power law different power law modulo power law Photos by Jean Guichard (La Jument lighthouse, Brittany) Maciej A. Nowak Spectral shock waves
Motivation Complex Burgers Equation Shock waves Physical manifestation - finite N YM Catastrophes Caustics ”Hard edge” Central limit theorem Nontrivial evolution from order ( ρ ( θ, 0) = δ ( θ )) to disorder 1 ( ρ ( θ, ∞ ) = 2 π ) (Haar measure), unravelled due to τ = Nt Gapped phase: laminar ”flow” Critical point: inflection point Gapless phase: Inverse spectral cascade L. Da Vinci, Florence (?), ca 1506 Maciej A. Nowak Spectral shock waves
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