Non-hermitian Diffusion Maciej A. Nowak Mark Kac Complex Systems Research Center, Marian Smoluchowski Institute of Physics, Jagiellonian University, Krak´ ow, Poland July 13th, 2015 UPON 2015, Barcelona Supported in part by the grant DEC-2011/02/A/ST1/00119 of National Centre of Science.
Outline Why to bother about nonhermiticity Diffusion Redux Diffusion of hermitian matrices - ”Dysonian way” Diffusion of hermitian matrices - ”Burgulent way” Unraveling the diffusion of nonhermitian matrices [Burda, Grela, MAN, Tarnowski and Warcho� l, Phys. Rev. Lett. 113 (2014) 104102, Nucl. Phys. B897 (2015) 421] Example Prospects and open problems Maciej A. Nowak Shock waves in Ginibre ensemble
Nonhermitian operators Nonhermitian quantum mechanics (resonances, complex potentials...) Euclidean Quantum Field Theory (finite density QCD) � T Statistics (lagged correlators) C i , j (∆) = 1 t =1 X i , t X j , t +∆ T Complexity (directed graphs/networks, non-backtracking operators for sparse systems) Maciej A. Nowak Shock waves in Ginibre ensemble
Diffusion Redux Wiener process X t = X 0 + B t , where dB t = B t + dt − B t = N (0 , dt ) dt � F ( B t ) � = 1 d 2 � F xx ( B t ) � Proof: F ( B t + dt ) = F ( B t ) + F x ( B t ) dB t + 1 2 F xx ( B t ) dB 2 t < dB 2 Diffusion : < dB t > = 0 , t > = dt Ito trick: dF ( B t ) ≡ F x ( B t ) dB t + 1 2 F xx ( B t ) dt Heat equation (Smoluchowski-Fokker-Planck eq.) ∂ t ρ ( x , t ) = 1 � 2 ∂ xx ρ ( x , t ), where < F ( B t ) > ≡ F ( x ) ρ ( x , t ) dx 2 π t e − x 2 1 Example: For ρ ( x , 0) = δ ( x ), ρ ( x , t ) = √ 2 t Maciej A. Nowak Shock waves in Ginibre ensemble
”Dysonian way” ([Dyson; 1962]) 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 Wiener process: H τ = H 0 + B τ , where τ = t N Perturbation calculus for H τ + d τ = H τ + dH τ yields | <ψ i | dH τ | ψ j > | 2 λ i ( H τ + d τ ) = λ i + < ψ i | dH τ | ψ i > + 1 � i � = j 2 λ i − λ j Ito calculus: d λ i ≡ dB i N + 1 dt � √ N i � = j λ i − λ j Eigenvalues interact! Maciej A. Nowak Shock waves in Ginibre ensemble
Diffusion of N by N hermitian matrices H = H † Gaussian Unitary Ensemble (GUE) � x ii if i = j H ij = x ij + iy ij if i < j √ 2 where all x ij , y ij drawn from standard Gaussians, so < ( dH ij ) 2 > = 1 < dH ij > = 0 , N dt Probability distribution ∂ t P ( H , t ) = LP ( H , t ), where � � ∂ 2 ∂ 2 ∂ 2 1 1 L = � ∂ 2 x kk + � ∂ 2 x ij + 2 N k 2 N i < j ∂ 2 y ij � < F ( H ) > t = [ dH ] P ( H , t ) F ( H ) Maciej A. Nowak Shock waves in Ginibre ensemble
”Burgulent way” We define d N ( z , t ) = det( z 1 N − H ) Integrable, exact eq. (for any N and for any initial conditions) ∂ t < d N ( z , t ) > t = − 1 2 N ∂ zz < d N ( z , t ) > t [Blaizot,MAN,Warcho� l; 2008-2013] Large N limit: lim N →∞ 1 N ∂ z ln < d N > = 1 N ∂ z < ln d N > = 1 N ∂ z < tr ln( z 1 N − H ) > ≡ g ( z , t ) (motivated by the Cole-Hopf transformation) � � �� N � Green’s function g ( z , t ) = 1 1 = 1 1 tr k =1 N z 1 N − H N z − λ k ”Heat equation” becomes inviscid complex Burgers equation ∂ t g + g ∂ z g = 0 (case of Voiculescu eq. ∂ t g + R ( g ) ∂ z g = 0) Spectrum from Sochocki Plemelj eq. 1 1 ′ ) λ − λ ′ ± i ǫ = P . V . λ − λ ′ ∓ i πδ ( λ − λ ρ ( λ, t ) = − 1 π lim ǫ → 0 ℑ g ( z ) | z = λ + i ǫ Shock phenomena at the edges of the spectrum Maciej A. Nowak Shock waves in Ginibre ensemble
”Burgelent way” - cont. ”Eulerian” solution of Burgers equation (on complex plane) reads g ( z , t ) = g 0 ( z − tg ( z , t )), so for simplest initial condition H 0 = 0, g ( z , 0) = g 0 ( z ) = 1 z , problem downgrades to the solution of the quadratic equation, i.e. reads √ z 2 − 4 t ). g ( z , t ) = 1 2 t ( z − Spectral density comes from the imaginary part of the Green’s √ 4 t 2 − λ 2 1 function, i.e. ρ ( λ, t ) = 2 π t (Diffusing Wigner’s semicircle) Diffusing Wigner’s semicircle (from Burgers equation) is a counterpart of the diffusing Gaussian (from the heat equation) in the world of large matrices. 1 Finite N effects appear as a spectral viscosity ν s ∼ 2 N , leading to universal spectral fluctuations in the vicinity of shock waves Maciej A. Nowak Shock waves in Ginibre ensemble
Non-hermitian case - large N - electrostatic analogy Analytic methods break down, since spectra are complex ρ ( z , t ) = 1 �� i δ (2) ( z − λ i ( t )) � . N Electrostatic potential � 1 N tr ln[ | z − X | 2 + | w | 2 ] � w , t ) ≡ lim N →∞ φ ( z , ¯ z , w , ¯ � 1 � = lim N →∞ N ln D N where w ) = det ( Q ⊗ 1 N − X ) with D N ( z , ¯ z , w , ¯ � z � X � � − ¯ w 0 X = Q = X † ¯ 0 w z Electric field g = ∂ z φ ∂ 2 φ Gauss law ρ ( z , t ) = 1 z g | w =0 = 1 π ∂ ¯ z | w =0 π ∂ z ∂ ¯ | w | 2 Proof: δ (2) ( z ) = lim w → 0 1 ( | z | 2 + | w | 2 ) 2 π Maciej A. Nowak Shock waves in Ginibre ensemble
Hidden variable Historically, | w | was treated as an infinitesimal regulator only [Brown;1986],[Sommers et al.;1988]. We promote w to full, complex-valued dynamical variable. Then, ”orthogonal direction” w unravels the eigenvector correlator O ( z , t ) = 1 1 k O kk δ (2) ( z − λ k ( t )) w φ | w =0 = �� � π ∂ w φ∂ ¯ , where N 2 O ij = < L i | L j >< R j | R i > and | L i > ( | R i > ) are left (right) eigenvectors of X . Maciej A. Nowak Shock waves in Ginibre ensemble
Approach to nonhermitian matrices We supersede d N ( z ) = det( z 1 N − H ) by the determinant D N ( z , ¯ z , w , ¯ w ) = det ( Q ⊗ 1 N − X ) For nonhermitian matrices X , we have left and right k ¯ eigenvectors X = � k λ k | R k >< L k | = � λ k | L k >< R k | where X | R k > = λ k | R k > and < L k | X = λ k < L k | < L j | R k > = δ jk , but < L i | L j > � = 0 and < R i | R j > � = 0. D N = det[ U − 1 ( Q ⊗ 1 N − X ) U ] = � � z 1 N − Λ − ¯ w < L | L > det z 1 N − ¯ w < R | R > ¯ Λ Spectrum (Λ) entangled with overlap of eigenvectors O ij ≡ < L i | l j >< R j | R i > . Maciej A. Nowak Shock waves in Ginibre ensemble
Random walk for the Ginibre ensemble We define random walk of X ij = x ij + iy ij , where 1 1 2 N dB y 2 N dB x dx ij = ij and dy ij = ij . √ √ � DX P ( X , t ) det( Q − X ) We consider < D N ( z , w , t ) > = Using Grassmannian integration tricks and the evolution 1 � ( ∂ 2 x ij + ∂ 2 equation ∂ t P ( X , τ ) = y ij ) P ( X , t ) we arrive at 4 N exact 2d diffusion equation ∂ t < D N ( z , w , t ) > = 1 N ∂ w ¯ w < D N ( z , w , t ) > Solution reads < D N ( z , | w | , t ) > = � ∞ � − N q 2 + | w | 2 � � � 2 Nq | w | 2 N 0 q exp I 0 D N ( z , q , t = 0) dq t t t z − X † 0 ) + | w | 2 ). where D N ( z , | w | , t = 0) = det(( z − X 0 )(¯ Maciej A. Nowak Shock waves in Ginibre ensemble
”Burgulent way” - nonhermitian case, N = ∞ limit Let define v ≡ | ∂ w φ | and | w | ≡ r . Note that v 2 controls eigevectors and g controls the complex spectrum The hermitian-case Burgers equation ∂ t g + g ∂ z g = 0 is now superimposed by the system = ∂ t v v ∂ r v ∂ z v 2 ∂ t g = Evolution of overlaps ( v ) prior to the evolution of spectra Shock phenomenon in eigenvector sector ”Missed ” complex plane ( w ) is relevant - quaternion ( Q ) description. Maciej A. Nowak Shock waves in Ginibre ensemble
Examples 1 X 0 = 0 π t 2 ( t − | z | 2 )Θ( √ t − | z | ) 1 O ( z , t ) = [Chalker-Mehlig;1998],[Janik et al.;1998] π t Θ( √ t − | z | ) 1 ρ ( z , t ) = [Ginibre; 1964] 2 X 0 = diag ( a , a , ..., − a , − a , ... ) Maciej A. Nowak Shock waves in Ginibre ensemble
The spiric example Initial condition X 0 = diag ( a , .. a , − a , ... − a ). Maciej A. Nowak Shock waves in Ginibre ensemble
Evolution of singularities in ( z , w ) space. Ginibre versus spiric example Maciej A. Nowak Shock waves in Ginibre ensemble
The spiric example cont. Spectral density (left) and eigenvector correlator (right) snapshots ρ ( x ) O ( x )/ N 0.4 0.15 0.3 0.10 0.2 0.05 0.1 2 x 2 x - 2 - 1 1 - 2 - 1 1 Maciej A. Nowak Shock waves in Ginibre ensemble
Conclusions and open problems Formalism of Dysonian dynamics for non-hermitian RMM, involving coevolution of eigenvalues and eigenvectors , for arbitrary N Conjecture, that above presented scenario, based on Ginibre ensemble, is generic for all non-hermitian RMM - paramount role of eigenvectors Unexpected similarity between hermitian and non-hermitian RMM based on ”Burgulence” concepts Verification in various application of hermitian and non-hermitian random matrix models Unexplored mathematics Maciej A. Nowak Shock waves in Ginibre ensemble
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