Non-hermitian Diffusion Maciej A. Nowak Mark Kac Complex Systems - - PowerPoint PPT Presentation

Non-hermitian Diffusion

Maciej A. Nowak

Mark Kac Complex Systems Research Center, Marian Smoluchowski Institute of Physics, Jagiellonian University, Krak´

• w, 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) Statistics (lagged correlators) Ci,j(∆) = 1

T

T

t=1 Xi,tXj,t+∆

Complexity (directed graphs/networks, non-backtracking

• perators for sparse systems)

Maciej A. Nowak Shock waves in Ginibre ensemble

Diffusion Redux

Wiener process Xt = X0 + Bt, where dBt = Bt+dt − Bt = N(0, dt)

d dt F(Bt) = 1 2 Fxx(Bt)

Proof: F(Bt+dt) = F(Bt) + Fx(Bt)dBt + 1

2Fxx(Bt)dB2 t

Diffusion : < dBt >= 0, < dB2

t >= dt

Ito trick: dF(Bt) ≡ Fx(Bt)dBt + 1

2Fxx(Bt)dt

Heat equation (Smoluchowski-Fokker-Planck eq.) ∂tρ(x, t) = 1

2∂xxρ(x, t), where < F(Bt) >≡

• F(x)ρ(x, t)dx

Example: For ρ(x, 0) = δ(x), ρ(x, t) =

1 √ 2πt e− x2

2t 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: Hτ = H0 + Bτ, where τ = t

N

Perturbation calculus for Hτ+dτ = Hτ + dHτ yields λi(Hτ+dτ) = λi+ < ψi|dHτ|ψi > + 1

2

• i=j

|<ψi|dHτ|ψj>|2 λi−λj

Ito calculus: dλi ≡ dBi

√ N + 1 N

• i=j

dt λi−λj

Eigenvalues interact!

Maciej A. Nowak Shock waves in Ginibre ensemble

Diffusion of N by N hermitian matrices H = H†

Gaussian Unitary Ensemble (GUE) Hij =

• xii

if i = j

xij+iyij √ 2

if i < j where all xij, yij drawn from standard Gaussians, so < dHij >= 0, < (dHij)2 >= 1

N dt

Probability distribution ∂tP(H, t) = LP(H, t), where L =

1 2N

• k

∂2 ∂2xkk + 1 2N

• i<j
• ∂2

∂2xij + ∂2 ∂2yij

• < F(H) >t=
• [dH]P(H, t)F(H)

Maciej A. Nowak Shock waves in Ginibre ensemble

”Burgulent way”

We define dN(z, t) = det(z1N − H) Integrable, exact eq. (for any N and for any initial conditions) ∂t < dN(z, t) >t= − 1

2N ∂zz < dN(z, t) >t

[Blaizot,MAN,Warcho l; 2008-2013] Large N limit: limN→∞ 1

N ∂z ln < dN > = 1 N ∂z< ln dN > = 1 N ∂z < tr ln(z1N − H) >≡ g(z, t) (motivated by the

Cole-Hopf transformation) Green’s function g(z, t) = 1

N

• tr

1 z1N−H

• = 1

N

N

k=1 1 z−λk

• ”Heat equation” becomes inviscid complex Burgers equation

∂tg + g∂zg = 0 (case of Voiculescu eq. ∂tg + R(g)∂zg = 0) Spectrum from Sochocki Plemelj eq.

1 λ−λ′±iǫ = P.V. 1 λ−λ′ ∓ iπδ(λ − λ

′)

ρ(λ, t) = − 1

π limǫ→0 ℑg(z)|z=λ+iǫ

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) = g0(z − tg(z, t)), so for simplest initial condition H0 = 0, g(z, 0) = g0(z) = 1

z , problem downgrades

to the solution of the quadratic equation, i.e. reads g(z, t) = 1

2t (z −

√ z2 − 4t). Spectral density comes from the imaginary part of the Green’s function, i.e. ρ(λ, t) =

1 2πt

√ 4t2 − λ2 (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. Finite N effects appear as a spectral viscosity νs ∼

1 2N , 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

N

• i δ(2)(z − λi(t))
• .

Electrostatic potential φ(z, ¯ z, w, ¯ w, t) ≡ limN→∞ 1

N tr ln[|z − X|2 + |w|2]

• = limN→∞

1

N ln DN

• where

DN(z, ¯ z, w, ¯ w) = det(Q ⊗ 1N − X) with Q = z − ¯ w w ¯ z

• X =

X X †

• Electric field g = ∂zφ

Gauss law ρ(z, t) = 1

π∂¯ zg|w=0 = 1 π ∂2φ ∂z∂¯ z |w=0

Proof: δ(2)(z) = limw→0 1

π |w|2 (|z|2+|w|2)2

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

π∂wφ∂ ¯ wφ|w=0 = 1 N2

• k Okkδ(2)(z − λk(t))
• , where

Oij =< Li|Lj >< Rj|Ri > and |Li > (|Ri >) are left (right) eigenvectors of X.

Maciej A. Nowak Shock waves in Ginibre ensemble

Approach to nonhermitian matrices

We supersede dN(z) = det(z1N − H) by the determinant DN(z, ¯ z, w, ¯ w) = det(Q ⊗ 1N − X) For nonhermitian matrices X, we have left and right eigenvectors X =

k λk|Rk >< Lk| = k ¯

λk|Lk >< Rk| where X|Rk >= λk|Rk > and < Lk|X = λk < Lk| < Lj|Rk >= δjk, but < Li|Lj >= 0 and < Ri|Rj >= 0. DN = det[U−1(Q ⊗ 1N − X)U] = det

• z1N − Λ

− ¯ w < L|L > w < R|R > ¯ z1N − ¯ Λ

• Spectrum (Λ) entangled with overlap of eigenvectors

Oij ≡< Li|lj >< Rj|Ri >.

Maciej A. Nowak Shock waves in Ginibre ensemble

Random walk for the Ginibre ensemble

We define random walk of Xij = xij + iyij, where dxij =

1 √ 2N dBx ij and dyij = 1 √ 2N dBy ij .

We consider < DN(z, w, t) >=

• DX P(X, t) det(Q − X)

Using Grassmannian integration tricks and the evolution equation ∂tP(X, τ) =

1 4N

(∂2

xij + ∂2 yij)P(X, t) we arrive at

exact 2d diffusion equation ∂t < DN(z, w, t) >= 1

N ∂w ¯ w < DN(z, w, t) >

Solution reads < DN(z, |w|, t) >=

2N t

∞

0 q exp

• −N q2+|w|2

t

• I0
• 2Nq|w|

t

• DN(z, q, t = 0)dq

where DN(z, |w|, t = 0) = det((z − X0)(¯ z − X †

0) + |w|2).

Maciej A. Nowak Shock waves in Ginibre ensemble

”Burgulent way” - nonhermitian case, N = ∞ limit

Let define v ≡ |∂wφ| and |w| ≡ r. Note that v2 controls eigevectors and g controls the complex spectrum The hermitian-case Burgers equation ∂tg + g∂zg = 0 is now superimposed by the system ∂tv = v∂rv ∂tg = ∂zv2 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 X0 = 0

O(z, t) =

1 πt2 (t − |z|2)Θ(√t − |z|)

[Chalker-Mehlig;1998],[Janik et al.;1998] ρ(z, t) =

1 πt Θ(√t − |z|)

[Ginibre; 1964]

2 X0 = diag(a, a, ..., −a, −a, ...) Maciej A. Nowak Shock waves in Ginibre ensemble

The spiric example

Initial condition X0 = 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

• 2
• 1

1 2 x 0.1 0.2 0.3 0.4 ρ(x)

• 2
• 1

1 2 x 0.05 0.10 0.15 O(x)/N

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

References

• E. Gudowska-Nowak, R.A. Janik, J. Jurkiewicz, MAN, Nucl.Phys.

B670 (2003) 479-507 J.-P. Blaizot, MAN, Phys. Rev. Lett. 101, (2008) 102001 J.-P. Blaizot, MAN, Phys. Rev. E82 (2010) 051115 J.-P. Blaizot, MAN, P. Warcho l, Phys. Rev. E87 (2013) 052134 J.-P. Blaizot, MAN, P. Warcho l, Phys. Lett. B724 (2013) 170 J.-P. Blaizot, MAN, P. Warcho l, Phys. Rev. E89 (2014) 042130 J-.P. Blaizot, J. Grela, MAN, P. Warcho l, arXiv 1405.5244

• Z. Burda, J. Grela, MAN, W. Tarnowski, P. Warcho

l, Phys. Rev.

• Lett. 113 (2014) 104102
• Z. Burda, J. Grela, MAN, W. Tarnowski, P. Warcho

l, Nucl. Phys. B897 (2015) 421

Maciej A. Nowak Shock waves in Ginibre ensemble