Level Spacing Distribution Revisited - Theoretical Group Atomic ...

Level Spacing Distribution Revisited 

Karol ˙Zyczkowski 

in collaboration with 

Tomasz Tkocz and Marek Ku´s (Warsaw) 

Institute of Physics, Jagiellonian University, Cracow, Poland 

and 

Center for Theoretical Physics, Polish Academy of Sciences 

IF UJ, Kraków, March 14, 2011 

K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 1 / 20

Quantum Chaos & Unitary Dynamics 

’Quantum chaology’: analogues of classically chaotic systems 

Quantum analogues of classically chaotic dynamical systems can be 

described by random matrices 

a) autonomous systems – Hamiltonians: 

Gaussian ensembles of random Hermitian matrices, (GOE, GUE, GSE) 

b) periodic systems – evolution operators: 

Dyson circular ensembles of random unitary matrices, (COE, CUE, CSE) 


Random Matrices & Universality 

Universality classes 

Depending on the symmetry properties of the system one uses ensembles 

form 

orthogonal (β = 1); 

unitary (β = 2) and 

symplectic (β = 4) ensembles. 

The exponent β determines the level repulsion, P(s) ∼ s β for s → 0 where 

s stands for the (normalised) level spacing, si = φi+1 − φi. 

see e.g. F. Haake, Quantum Signatures of Chaos 


Gaussian Ensembles of Hermitian matrices H 

Hermitian random matrix H = H † 

consists of independent Gaussian entries 

a) orthogonal ensemble, β = 1 - real random numbers, 

b) unitary ensemble, β = 2 - complex numbers (real at the diagonal!) 

c) symplectic ensemble, β = 4 - quaternions (2 × 2 matrices) 

leading to 2N × 2N matrix with each eigenvalue occurring twice. 

Different normalization conditions, we use the one implied by 

the normal distribution 〈Hij〉 = 0 and σ2 = 〈H2 ij 〉 = 1 

GUE ⇒ Unitary invariance, P(H) = P(UHU † ) 

leads to joint probability distribution (jpd) of eigenvalues xi 

β P 

− 

Pβ(x1, . . .,xN) = CNe 2 j x2 j 

 

|xj − xk| β 

j

Wigner Semicircle Law 

Spectral density P(x) 

can be obtained by integrating out all eigenvalues but one from jpd. 

For all three Gaussian ensembles of Hermitian random matrices one 

obtains (asymptotically, for N → ∞) the Wigner Semicircle Law (1955) 

P(x) = 1 

2 − x2 2π 

where x denotes a normalized eigenvalue, xi = λi/ √ N 



Extremal eigenvalues & Tracy–Widom Law 

Statistics of extremal cases - the largest eigenvalue xmax 

The normalized largest eigenvalue 

s := (xmax − 2 √ N)N −1/6 

of a GUE random matrix is (asymptotically) distributed according to the 

Tracy-Widom law (1994) 

F2(s) = det(1 − K) , 

where K is the integral operator with the Airy kernel 

K(x, y) = Ai(x)Ai′ (y) − Ai ′ (x)Ai(y) 

. 

x − y 

The scaling behaviour of the finite size effect (as 1/N 1/6 ) 

is due to Bowick & Brezin (1991) and Forrester (1991). 


Tracy–Widom distributions 

Tracy–Widom distributions Fβ(s) 

Distributions Fβ(s) and the largest eigenvalue 

of random GUE matrices 

(image by A. Edelman) 


Some applications of Tracy–Widom Law 

Combinatorics: length of the longest increasing permutation 

Baik, Deift, Johansson (1999) 

Statistics: largest eigenvalues of correlation matrices C = XX T 

Johnstone (2001) 

Growth processes: fluctuations around the limiting shape 

Johansson (2000) 

Superconductors: fluctuations of excitation gap in a quantum dot 

Vavilov, Brouer, Ambegaokar, Benakker (2001) 

Random Tilings & Aztec Diamond: fluctuations around 

the Arctic Circle Johansson (2002) 


Aztec Diamond & domino lattice 

Aztec diamond of order n – a region of 2n(n + 1) unit squares tiled with 

n(n + 1) dominos (i.e. union of two squares) 

The Aztec diamond 

of order n = 3 

K(3) = 2 6 = 64 

There exists exactly K = 2 n(n+1)/2 different domino tilings 

Elkies, Kuperberg, Larsen, Propp (1992) 

How a random Aztec Diamond looks like?? 


Random Domino & Arctic Circle Theorem 

Arctic Circle theorem - randomness inside the circle 

& ’Frozen Phase’ in four corners Jockush, Propp, Shor (1995) 

n = 50, K = 2 1275 ≈ 10 383 

Fluctuations around the Arctic Circle are described by 

the Tracy–Widom distribution F2 Johansson (2002) 


Random matrices & Finite N Effects 

Analogy 

Infinite matrices ↔ semiclassical → 0 limit ↔ geometric optics 

diffraction and Airy discs 

Finite N Random Matrices ↔ Quantum Theory > 0 ↔ Wave 

Optics 


Level spacing distribution P(s) 

Nearest neighbour spacing s 

si = xi+1−xi 

∆ where ∆ is the mean spacing 

a) Gaussian ensembles for N = 2 ⇒ Wigner surmise 

β = 1 GOE (orthogonal) P1(s) = π 

2s exp(−π 4s2 ) 

β = 2 GUE (unitary) P2(s) = 32 

π 2s 2 exp(− 4 

π s2 ) 

β = 4 GSE (symplectic) P4(s) = 218 

3 6 π 3s 4 exp(− 64 

9π s2 ) 

These distributions derived for N = 2 work well also for Gaussian 

ensembles in the asymptotic case, N → ∞. 

Random unitary matrices & Circular ensembles of Dyson 

Uniform density of phases along the unit circle, P(φ) = 1/2π. 

Phase spacing, si = N 

2π [φi+1 − φi] since ∆ = 2π/N. 

For large matrices the level spacing distributions for Gaussian ensembles 

(Hermitian matrices) and circular ensembles (unitary matrices) coincide. 


Composite systems & tensor products 

Consider tensor product of random matrices V = U1 ⊗ U2 

of size N, where Uj are random matrices according to the Haar measure 

(CUE). Eig(V) = {ψkm} = (αk + βm)|mod2π, 

where {αk} N k=1 and {βm} N m=1 denote spectra of U1 and U2. 

What is the (asymptotic) spacing distribution PCUE⊗CUE(s) ? 

Two qubits & random local gates 

Analytical results P2⊗2(s) for CUE(2) ⊗ CUE(2), 

(case by case analysis & direct integration) 

P2⊗2(s) = Θ(2 − s) 1 

 

2π (s − 2)sin2 ( πs 

4 ) π(s − 2) − 2 sin( πs 

2 ) 

+ 1 

128π2 

2(36 + π2 (s − 4) 2 ) + 128 cos( πs 

4 ) 

+ (56 + π 2 (s − 4) 2 )cos( πs 

2 ) 


Level spacing for unitary matrices of order N = 4 

Comparison of spacing distribution P(s) for 

a) Poisson CPE(4), b) CUE(2) ⊗ CUE(2), c) CUE(4), 

Poisson ensemble, β = 0, CPE(4), P (4) 

0 (s) = (1 − s/4)3 , 

Large matrices, N → ∞ asymptotics P (∞) 

0 (s) = e −s , 


Extremal spacings distributions 

for random unitary matrices 

consider a) Minimal spacing smin = minj{sj} N j=1 

and b) Maximal spacing smax = maxj{sj} N j=1 

Minimal spacing distribution for N = 4 random unitary matrices 

Two qubits & random local gates 

Analytical results P2⊗2(t) for CUE(2) ⊗ CUE(2) case, where t = smin 

P2⊗2(t) = 1 

 

2π(1−t) 

4π 

4−cos( πt 

2 )−3 sin( πt 

2 

)+8 sin(πt)−3 sin(3πt 

2 ) 

 

CUE, β = 2, CUE(4), P (2) 

4 (t) = ... explicit result to long to reproduce it 

here... 

Poisson ensemble, β = 0, CPE(4), P (0) 

4 (t) = 3(1 − t)2 . 


Minimal spacing P(smin) for N = 4 unitary matrices 

Comparison of spacing distribution P(smin) for 

a) Poisson CPE(4), b) CUE(2) ⊗ CUE(2), c) CUE(4). 

mean values: 〈smin〉CPE4 = 1/4, 〈smin〉CUE2⊗CUE2 ≈ 0.4, 

〈smin〉CUE4 ≈ 0.54 


Average minimal spacing 〈smin〉 for large unitary 

matrices 

Approximation of independent spacings 

Assume spacings sj described by the distribution Pβ(s) are independent. 

Minimal spacing 

Since for small spacings Pβ(s) ∼ s β so the integrated distribution 

I(s) = s 

0 P(s′ )ds ′ behaves as Iβ(s) ∼ s 1+β 

Matrix of order N yields N spacings sj. The minimal spacing smin occurs 

for such an argument that Iβ(smin) ≈ 1/N. 

Thus (smin) 1+β ≈ 1/N =⇒ smin ≈ N −β−1 


Average maximal spacing 〈smax〉 



Mean maximal spacing for COE 

Since for large spacings Pβ(s) ∼ s 1 exp(−s 2 ) so the integrated 

distribution 

I1(s) = s 

0 P(s′ )ds ′ behaves as I1(s) ∼ −exp(−s 2 ) 

Matrix of order N yields N spacings sj. The maximal spacing smax occurs 

for such an argument that 1 − I1(smax) ≈ 1/N. 

Thus exp[−(smax) 2 ] ≈ 1/N =⇒ smax ≈ √ lnN 

However, 


Average maximal spacing 〈smax〉 



Mean maximal spacing for COE 

Since for large spacings Pβ(s) ∼ s 1 exp(−s 2 ) so the integrated 

distribution 

I1(s) = s 

0 P(s′ )ds ′ behaves as I1(s) ∼ −exp(−s 2 ) 

Matrix of order N yields N spacings sj. The maximal spacing smax occurs 

for such an argument that 1 − I1(smax) ≈ 1/N. 

Thus exp[−(smax) 2 ] ≈ 1/N =⇒ smax ≈ √ lnN 

However, these results (and many more) appeared recently (October 2010) 

in a preprint arXiv:1010.1294 ”Extreme gaps between eigenvalues of 

random matrices” by Ben Arous and Bourgade. 


Concluding Remarks 

Random Matrices: 

a) offer a useful tool applicable in several branches of science 

including physics ! 

b) display (asymptotically) universal properties, which depend on 

the symmetry with respect to orthogonal / unitary / symplectic 

transformations 

Limit N → ∞ corresponds to the semiclassical limit of quantum 

theory or geometric optics as a limiting case of wave optics! 

Problem of extremal eigenvalues is described by Tracy–Widom 

distributions. 

The analogous problem of extremal spacings is studied only recently 

Preliminary results on spectral statistics of local unitary evolution 

operators (with tensor product structure).

Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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