Modification of the Porter-Thomas distribution by rank … · Modi cation of the Porter-Thomas...

Introduction Rank-one formalism Probability dist κ2 < 1 κ2 > 1 Large-window dist Numerics Conclusion

Modification of the Porter-Thomas distribution by rank-oneinteraction

Eugene Bogomolny

University Paris-Sud, CNRSLaboratoire de Physique Theorique et Modeles Statistiques, Orsay

France

XII Brunel-Bielefeld Workshop on Random Matrix Theory, Uxbridge, 10.12.2016

Eugene Bogomolny, LPTMS Orsay France Modification of the Porter-Thomas distribution by rank-one interaction

Outlook

1 Introduction

2 Rank-one formalism

3 Probability dist

4 κ2 < 1

5 κ2 > 1

6 Large-window dist

7 Numerics

8 Conclusion

Porter–Thomas distribution = everything is Gaussian

C. E. Porter and R. G. Thomas Phys. Rev. 104 483 (1956)

Fluctuations of nuclear reaction widths

Time-invariant systems

P1(x) =1

√2πlx

exp(−

)Nuclear physics : x = reduced resonance width. Random matrices : x = N|Ψ|2Time-non-invariant systems

P2(x) =1

)l = mean value of x . Standard normalisation : 〈x〉 = 1 −→ l = 1

Old experiments agreed well with the PT distribution

Recent experiments P. E. Koehler et al

Reduced neutron widths in the nuclear data ensemble :experiment and theory do not agreePhys. Rev. C 84, 034312 (2011)Neutron resonance data exclude random matrix theoryFortschritte Phys. 61, 80 (2013)

P1(x) =1

√2πlx

exp(−

P2(x) =1

P1(x) =1

√2πlx

exp(−

P2(x) =1

Modification of the model

Within standard (invariant) random matrix ensembles : PT distribution = theorem

Modifications required

A. Volya, H. A. Weidenmuller, and V. Zelevinsky PRL 115, 052501 (2015)Neutron resonance widths and the Porter-Thomas distribution

Realistic model of nuclear s-wave resonances : Mij = G(β)ij + Z δi1δj1

G(β)ij = standard random matrix, β = 1→ GOE, β = 2→ GUE

Z δi1δj1 = interaction which couples resonances to decay channels

Distribution (GOE or GUE) : P(Gij ) ∼ exp(− β

4σ2 Tr (G G†))

β = 1, 2

Density = Wigner semicircle law : ρW (E) = 12πσ2

√4σ2N − E2

Z may be complex. ReZ is due to coupling to neutron channel and is ”immanent tothe theory”. ImZ is related with hypothetical non-statistical gamma decays

Here only Hermitian matrix M is considered, Z = real . To get a nontrivial limit

σ√N

remains constant when N →∞

Numerically : distribution of xα = N|Ψ1(α)|2 does deviate from the PT law

4σ2 Tr (G G†))

β = 1, 2

√4σ2N − E2

σ√N

4σ2 Tr (G G†))

β = 1, 2

√4σ2N − E2

σ√N

Known results

Real Z −→ local spectral statistics is independent on Z (after unfolding)

For GUE there is a direct proof using the Itzykson-Zuber integralE. Brezin and S. Hikami, Nucl. Phys. B 479 697 (1996)

For GOE ?

Imaginary Z = iγ

σ√N−→ there exists an exact solution (using non-linear sigma model)

for the distribution of imaginary parts of eigenvalues, E = e + i2

Γ, y = πΓ/ρW (e),

GUE, Y. V. Fyodorov and H.-J. Sommers, JETP Lett. 63, 1036 (1996)

P(y) = −d

(e−yg sinh y

), g =

2(γ + γ−1)

GOE, H.-J. Sommers, Y. V. Fyodorov, and M. Titov, J. Phys. A 32, L77 (1999)

P(y) =1

−1(1− λ2)e2λy (g − λ)F (λ, y)dλ

F (λ, y) =

∫ ∞g

e−yp1 dp1

(λ− p1)2√

(p21 − 1)(p1 − g)

e−yp2 dp2

(λ− p2)2√

(p22 − 1)(g − p2)

Rank-one formalism Mij = G(β)ij + Z δi1δj1

Two N × N Hermitian matrices G and M differ by a rank-one interaction

Mij = Gij + v∗i vj , vj =√Z(1, 0, . . . , 0)

Eigenvalues and eigenfunctions for Hermitian matrices assumed to be orthogonal

N∑j=1

GijΦj (α) = eαΦi (α),N∑j=1

MijΨj (α) = EαΨi (α)

Expansions : Ψj (α) =∑Nβ=1 CαβΦj (β), Φj (α) =

∑Nβ=1 C

−1αβΨj (β)

Consequences

Cαβ =aαb∗β

Eα − eβ, bβ =

N∑j=1

vjΦj (β), aα =∑β

Cαβbβ

Quantisation condition ∑β

|bβ |2

Eα − eβ= 1, Eα, eα

Complementary relations

C−1αβ =

bαa∗αEβ − eα

|aβ |2

eα − Eβ= −1

Rank-one formalism Mij = G(β)ij + Z δi1δj1

Two N × N Hermitian matrices G and M differ by a rank-one interaction

Mij = Gij + v∗i vj , vj =√Z(1, 0, . . . , 0)

Eigenvalues and eigenfunctions for Hermitian matrices assumed to be orthogonal

N∑j=1

GijΦj (α) = eαΦi (α),N∑j=1

MijΨj (α) = EαΨi (α)

Expansions : Ψj (α) =∑Nβ=1 CαβΦj (β), Φj (α) =

∑Nβ=1 C

−1αβΨj (β)

Consequences

Cαβ =aαb∗β

Eα − eβ, bβ =

N∑j=1

vjΦj (β), aα =∑β

Cαβbβ

Quantisation condition ∑β

|bβ |2

Eα − eβ= 1, Eα, eα

Complementary relations

C−1αβ =

bαa∗αEβ − eα

|aβ |2

eα − Eβ= −1

Schematic picture

Quantisation conditions :∑β

|bβ |2

Eα − eβ= 1,

|aβ |2

eα − Eβ= −1

New and old eigenvalues are interlacinge1 ≤ e2 ≤ eN , eα ≤ Eα ≤ eα+1, eN ≤ EN (for Z > 0)

Main relations

Solve for numerators

|bα|2 =

∏γ(Eγ − eα)∏γ 6=α(eγ − eα)

, |aα|2 = −∏γ(eγ − Eα)∏

γ 6=α(Eγ − Eα)

Theorem (xm, ym are known)

N∑m=1

xm − yn= 1, −→ bm =

∏n(xm − yn)∏

s 6=m(xm − xs)

Proof (one among many others)

Consider the function

fn(x) =

∏r 6=n(x − yr )∏s(x − xs)

∏r (x − yr )

(x − yn)∏

s(x − xs)

Asymptotically fn(x)→ 1/x −→ integral over a large contour encircling all polesequals 1Rewriting this integral as the sum over all finite poles gives

1 =∑m

∏r (xm − yr )

(xm − yn)∏

s 6=m(xm − xs)Q.E.D.

Many different relations. E.g.∑β |bβ |2 =

∑β |aβ |2 =

∑β(Eβ − eβ)

Initial probability distribution

Eigenvalues eα and eigenfunctions Φ1(α) of matrix Gβ are distributed as instandard random matrix ensembles

P({eα}, {rα}) ∼∏α<γ

|eγ − eα|β∏α

rβ/2−1α δ

(∑α

rα − 1

)exp(−V ({eα}))

rα = |Φ1(α)|2 (the same is valid for other components as well)

V ({eα}) = confinement term. For standard Gaussian ensembles

V ({eα}) =β

Mean density of matrix eigenvalues = Wigner semicircle law

ρW (E) =1

2πσ2

√4Nσ2 − E2

Main question

How new eigenvectors Ψ1(α) and new eigenvalues Eα are distributed ?

Two steps of changing variables : (eα,Φ1(α)) −→ (eα,Eα) −→ (Ψ1(α),Eα)

Initial probability distribution

Eigenvalues eα and eigenfunctions Φ1(α) of matrix Gβ are distributed as instandard random matrix ensembles

P({eα}, {rα}) ∼∏α<γ

rβ/2−1α δ

(∑α

rα − 1

)exp(−V ({eα}))

rα = |Φ1(α)|2 (the same is valid for other components as well)

V ({eα}) = confinement term. For standard Gaussian ensembles

V ({eα}) =β

Mean density of matrix eigenvalues = Wigner semicircle law

ρW (E) =1

2πσ2

√4Nσ2 − E2

Main question

How new eigenvectors Ψ1(α) and new eigenvalues Eα are distributed ?

Two steps of changing variables : (eα,Φ1(α)) −→ (eα,Eα) −→ (Ψ1(α),Eα)

Joint distribution of old and new eigenvalues

P({eα}, {rα})∏α

deαdrα ∼∏α<γ

rβ/2−1α δ(

rα − 1)∏α

deαdrα

bα =∑N

j=1 vjΦj (β), vj =√Z(1, 0, . . . , 0) −→ |bα|2 = Z |Φ1(α)|2

Expression of rα through old and new eigenvalues,

rα ≡ |Φ1(α)|2 =1

Z|bα|2 =

Change of variables : N quantities rα −→ N quantities Eα

Derivatives : ∂rα/∂Eγ = rα/(Eγ − eα)

Cauchy determinant : det( 1xm−yn

) =∏

i<j (xi−xj )∏

i<j (yj−yi )∏i,j (xi−yj )

Jacobian

(∂rα

∂Eγ

rα) det

Eγ − eα

)= (∏α

∏α<β(Eα − Eβ)(eβ − eα)∏

α,β(Eα − eβ)

∏α<γ

(Eγ − Eα)

(eα − eγ),

rα =1

(Eα − eα)

Joint distribution of old and new eigenvalues

P({eα}, {rα})∏α

deαdrα ∼∏α<γ

rβ/2−1α δ(

rα − 1)∏α

deαdrα

bα =∑N

j=1 vjΦj (β), vj =√Z(1, 0, . . . , 0) −→ |bα|2 = Z |Φ1(α)|2

Expression of rα through old and new eigenvalues,

rα ≡ |Φ1(α)|2 =1

Z|bα|2 =

Change of variables : N quantities rα −→ N quantities Eα

Derivatives : ∂rα/∂Eγ = rα/(Eγ − eα)

Cauchy determinant : det( 1xm−yn

) =∏

i<j (xi−xj )∏

i<j (yj−yi )∏i,j (xi−yj )

Jacobian

(∂rα

∂Eγ

rα) det

Eγ − eα

)= (∏α

∏α<β(Eα − Eβ)(eβ − eα)∏

α,β(Eα − eβ)

∏α<γ

(Eγ − Eα)

(eα − eγ),

rα =1

(Eα − eα)

Change from old eigenvalues to new eigenfunctions

I. L. Aleiner and K. A. Matveev Shifts of random energy levels by a local perturbation,PRL 80, 814 (1998)

P({eα}, {Eα}) ∼∏γ>α(eγ − eα)(Eγ − Eα)∏γ,α |eγ − Eα|β/2−1

(∑α

(Eα − eα)− Z

)∏α

deαdEα

Ψ1(α) =∑β CαβΦ1(β) = 1√

∑β Cαβbβ = aα√

∑β|bβ |2

Eα−eβ= aα√

Expression of rα through old and new eigenvalues

zα = |Ψ1(α)|2 =1

Z|aα|2 =

∏γ(Eα − eγ)∏

γ 6=α(Eα − Eγ)

N quantities rα −→ N quantities eα. Derivatives : ∂zα∂eγ

= − zαEα−eγ

Jacobian

(∂zα

∂eβ

∏α<γ

(eγ − eα)

(Eα − Eγ)

Identities∏α,γ

(Eα − eγ) =∏α

zα( ∏α<γ

(Eα − Eγ))2,

zα =1

(Eα − eα)

Change from old eigenvalues to new eigenfunctions

I. L. Aleiner and K. A. Matveev Shifts of random energy levels by a local perturbation,PRL 80, 814 (1998)

P({eα}, {Eα}) ∼∏γ>α(eγ − eα)(Eγ − Eα)∏γ,α |eγ − Eα|β/2−1

(∑α

(Eα − eα)− Z

)∏α

deαdEα

Ψ1(α) =∑β CαβΦ1(β) = 1√

∑β Cαβbβ = aα√

∑β|bβ |2

Eα−eβ= aα√

Expression of rα through old and new eigenvalues

zα = |Ψ1(α)|2 =1

Z|aα|2 =

∏γ(Eα − eγ)∏

γ 6=α(Eα − Eγ)

N quantities rα −→ N quantities eα. Derivatives : ∂zα∂eγ

= − zαEα−eγ

Jacobian

(∂zα

∂eβ

∏α<γ

(eγ − eα)

(Eα − Eγ)

Identities∏α,γ

(Eα − eγ) =∏α

zα( ∏α<γ

(Eα − Eγ))2,

zα =1

(Eα − eα)

Distribution of new eigenvalues and new eigenfunctions

New joint distribution = the old one

∏α<γ

rβ/2−1α δ(

rα − 1)∏α

deαdrα =

=∏α<γ

|Eγ − Eα|β∏α

zβ/2−1α δ(

zα − 1)∏α

dEαdzα

Symmetry : Gij = Mij − v∗i vj eα ↔ −Eα P({eα}, {Eα}) is symmetric

Symmetry is valid only without the confinement term : exp(−βTrGG†/4σ2)

TrGG† = Tr (Mij − Zδi1δj1)2, Mij =∑α EαΨi (α)Ψ∗j (α), M11 =

∑α Eα|Ψ1(α)|2∑

α e2α =

∑α E2

α − 2Z∑α Eαzα + Z2

Full joint distribution of new eigenvalues Eα and new eigenvectors, zα ≡ |Ψ1(α)|2

P({Eα}, {zα}) ∼∏α<β

|Eβ − Eα|β∏α

zβ/2−1α δ(

zα − 1)

× exp[−

(∑α

E2α − 2Z

Eαzα)]

∏α<γ

rβ/2−1α δ(

rα − 1)∏α

deαdrα =

=∏α<γ

zβ/2−1α δ(

zα − 1)∏α

dEαdzα

∑α Eα|Ψ1(α)|2∑

α e2α =

∑α E2

P({Eα}, {zα}) ∼∏α<β

zβ/2−1α δ(

zα − 1)

× exp[−

(∑α

E2α − 2Z

Eαzα)]

∏α<γ

rβ/2−1α δ(

rα − 1)∏α

deαdrα =

=∏α<γ

zβ/2−1α δ(

zα − 1)∏α

dEαdzα

∑α Eα|Ψ1(α)|2∑

α e2α =

∑α E2

P({Eα}, {zα}) ∼∏α<β

zβ/2−1α δ(

zα − 1)

× exp[−

(∑α

E2α − 2Z

Eαzα)]

Mean level density

L. A. Pastur, On the spectrum of random matrices, TMP, 10, 67 (1972)

Mean Green function

G(E) =1

N〈Tr (E −M)−1〉

The Pastur equation

G(E) =1

E − Z − σ2NG(E)+

N − 1

E − σ2NG(E)

Solution in 2 lowest orders in N−1

G(E) = G0(E)

ZG0(E)

N(1− ZG0(E))

)G0(E) = the mean Green function for standard RM ensembles

G0(E) =1

E − σ2NG0(E), G0(E) =

E −√E2 − 4σ2N

Formation of a collective state (= outlier)

Mean level density

ρ(E) =1

πTrG(E−i0), G0(E) =

E −√E2 − 4σ2N

2σ2N−→ ρ0(E) =

2πσ2N

√4Nσ2 − E2

For perturbed problem E = 2σ√N cosφ

ρ(φ) =( N

2κ(2 cosφ− κ)

π(κ2 − 2κ cosφ+ 1)

)sin2 φ

Total level number :∫ π

0 ρ(φ)dφ =

{N κ2 < 1N − 1 + 1/κ2 κ2 > 1

Collective state (outlier) appears at κ2 > 1

Ec − Z − σ2NG0(Ec) = 0 −→ Ec = σ√N(κ+

Close to Ec G(E) ≈ G0(E) + δG(E) (a = 2σ(1− κ−2))

δG(E) =1

κ2 − 1(E−Ec)−σ2NδG(E)]−1 −→ δρ(E) =

√a2 − (E − Ec)2

∫δρ(E)dE = 1− κ−2. Total :

∫ρ(E)dE = N

Exact result : E − Ec = Gaussian variable with mean= 0, variance= 2(κ2−1)σ2

Mean level density

ρ(E) =1

E −√E2 − 4σ2N

2σ2N−→ ρ0(E) =

2πσ2N

√4Nσ2 − E2

ρ(φ) =( N

2κ(2 cosφ− κ)

π(κ2 − 2κ cosφ+ 1)

)sin2 φ

0 ρ(φ)dφ =

{N κ2 < 1N − 1 + 1/κ2 κ2 > 1

Ec − Z − σ2NG0(Ec) = 0 −→ Ec = σ√N(κ+

δG(E) =1

κ2 − 1(E−Ec)−σ2NδG(E)]−1 −→ δρ(E) =

√a2 − (E − Ec)2

∫ρ(E)dE = N

Mean level density

ρ(E) =1

E −√E2 − 4σ2N

2σ2N−→ ρ0(E) =

2πσ2N

√4Nσ2 − E2

ρ(φ) =( N

2κ(2 cosφ− κ)

π(κ2 − 2κ cosφ+ 1)

)sin2 φ

0 ρ(φ)dφ =

{N κ2 < 1N − 1 + 1/κ2 κ2 > 1

Ec − Z − σ2NG0(Ec) = 0 −→ Ec = σ√N(κ+

δG(E) =1

κ2 − 1(E−Ec)−σ2NδG(E)]−1 −→ δρ(E) =

√a2 − (E − Ec)2

∫ρ(E)dE = N

Mean level density

ρ(E) =1

E −√E2 − 4σ2N

2σ2N−→ ρ0(E) =

2πσ2N

√4Nσ2 − E2

ρ(φ) =( N

2κ(2 cosφ− κ)

π(κ2 − 2κ cosφ+ 1)

)sin2 φ

0 ρ(φ)dφ =

{N κ2 < 1N − 1 + 1/κ2 κ2 > 1

Ec − Z − σ2NG0(Ec) = 0 −→ Ec = σ√N(κ+

δG(E) =1

κ2 − 1(E−Ec)−σ2NδG(E)]−1 −→ δρ(E) =

√a2 − (E − Ec)2

∫ρ(E)dE = N

Case κ2 < 1

zα = |Ψ1(α)|2 for all energies Eα are O(N−1)Condition

∑α zα = 1 can be taken into account by the Lagrange multiplier

δ(∑α

zα − 1) −→ exp(−µ(∑α

zα − 1))

zα = independent, each zα ≡ z(Eα) has PDF

P(z,E) =(µ−

2σ2E)β/2

(πz)β/2−1 exp

(−(µ−

2σ2E)z

)µ has to be calculated from normalisation :

∑α zα = 1∑

∫ ∞0

z P(z,Eα) dz = 1,∑α

E − Eα=

σ2, E =

2µσ2

The sum is mean unperturbed Green function, G0(E)

G0(E) =1

N〈Tr (E − G)−1〉 =

E −√E2 − 4σ2N

2σ2N−→ µ = βN

κ2 + 1

Distribution of perturbed problem = local PT distribution

Pβ(x) =1

(2πx)1−β/2(l(E))β/2exp

), l(E) =

(κ2 + 1−

σ√NE

Case κ2 > 1

When κ2 > 1 there exists one collective state. zc = |Ψ1(Ec)|2 = O(1)

Other zα are O(N−1) and their distribution is the local PT distribution

P(z,E) =(µ−

2σ2E)β/2

(πz)β/2−1 exp

(−(µ−

2σ2E)z

)The only difference is that their normalisation is

∑N−1α=1 zα = 1− zc −→

µ(zc) =βN

(κ2(1− zc) +

1− zc

Calculation of Ec and zcJoint distribution of E ≡ Ec and r ≡ zc

P(E , r) ∼∏α

|E − Eα|β(µ(r)−β√Nκ

2σEα)−β/2

× exp[−

4σ2E2 +

β√Nκ

2σEr − (1− r)µ(r)

]∼ e−Nβ F (E ,r)

F (E , r) = −1

ln(E − Eα) +1

4σ2NE2 +

ln(ν(r)−

2σ√NEα)

2σ√NEr − (1− r)ν(r), ν(r) =

(κ2(1− r) +

1− r

Case κ2 > 1

When κ2 > 1 there exists one collective state. zc = |Ψ1(Ec)|2 = O(1)

Other zα are O(N−1) and their distribution is the local PT distribution

P(z,E) =(µ−

2σ2E)β/2

(πz)β/2−1 exp

(−(µ−

2σ2E)z

)The only difference is that their normalisation is

∑N−1α=1 zα = 1− zc −→

µ(zc) =βN

(κ2(1− zc) +

1− zc

)Calculation of Ec and zc

Joint distribution of E ≡ Ec and r ≡ zc

P(E , r) ∼∏α

|E − Eα|β(µ(r)−β√Nκ

2σEα)−β/2

× exp[−

4σ2E2 +

β√Nκ

2σEr − (1− r)µ(r)

]∼ e−Nβ F (E ,r)

F (E , r) = −1

ln(E − Eα) +1

4σ2NE2 +

ln(ν(r)−

2σ√NEα)

2σ√NEr − (1− r)ν(r), ν(r) =

(κ2(1− r) +

1− r

)Eugene Bogomolny, LPTMS Orsay France Modification of the Porter-Thomas distribution by rank-one interaction

Final answer

Saddle-point equations ∂F (E ,r)∂E

= 0 ∂F (E ,r)∂r

= 0 plus mean Green function −→

Ec = σ√N(κ+ 1

), zc = 1− 1

For both κ2 < 1 and κ2 > 1

µ = βNκ2 + 1

Local Porter-Thomas distribution

For −2σ√N ≤ E ≤ 2σ

√N and all κ

Pβ(x ,E) =1

l(E) ≡ N〈|Ψ1(E)|2〉 =

(κ2 + 1−

σ√NE

N∑Eα

〈|Ψ1(Eα)|2〉 ≈∫ 2σ

−2σ√NρW (E)

[∫ ∞0

z Pβ(z,Eα) dz

{1 κ2 < 11/κ2 κ2 > 1

Cf. H. A. Weidenmuler, PRL 105, 232501 (2010) :l(E) is associated with the existence of a virtual or weakly bound state near threshold

Final answer

= 0 ∂F (E ,r)∂r

Ec = σ√N(κ+ 1

), zc = 1− 1

µ = βNκ2 + 1

For −2σ√N ≤ E ≤ 2σ

√N and all κ

Pβ(x ,E) =1

l(E) ≡ N〈|Ψ1(E)|2〉 =

(κ2 + 1−

σ√NE

N∑Eα

〈|Ψ1(Eα)|2〉 ≈∫ 2σ

−2σ√NρW (E)

[∫ ∞0

z Pβ(z,Eα) dz

{1 κ2 < 11/κ2 κ2 > 1

Final answer

= 0 ∂F (E ,r)∂r

Ec = σ√N(κ+ 1

), zc = 1− 1

µ = βNκ2 + 1

For −2σ√N ≤ E ≤ 2σ

√N and all κ

Pβ(x ,E) =1

l(E) ≡ N〈|Ψ1(E)|2〉 =

(κ2 + 1−

σ√NE

N∑Eα

〈|Ψ1(Eα)|2〉 ≈∫ 2σ

−2σ√NρW (E)

[∫ ∞0

z Pβ(z,Eα) dz

{1 κ2 < 11/κ2 κ2 > 1

Final answer

= 0 ∂F (E ,r)∂r

Ec = σ√N(κ+ 1

), zc = 1− 1

µ = βNκ2 + 1

For −2σ√N ≤ E ≤ 2σ

√N and all κ

Pβ(x ,E) =1

l(E) ≡ N〈|Ψ1(E)|2〉 =

(κ2 + 1−

σ√NE

N∑Eα

〈|Ψ1(Eα)|2〉 ≈∫ 2σ

−2σ√NρW (E)

[∫ ∞0

z Pβ(z,Eα) dz

{1 κ2 < 11/κ2 κ2 > 1

Another approach based on the averaged Green function

The exact Schur complement formula (G(E) ≡ (E −M)−1)

G11(E) = (E − Z −∑j,k 6=1

M1j GjkMk1)−1

Gjk = the Green function of (N − 1)× (N − 1) matrix obtained from matrix M by

removing line 1 and row 1 = standard (N − 1)× (N − 1) random matrix, HFor Gβ matrices M1jMk1 −→

N→∞σ2δjk

ThereforeG11(E) −→

N→∞(E − Z − σ2G0(E))−1

G0 = Tr (E − H)−1. For N →∞

G0(E) ≈E −√E2 − 4σ2N

Eigenfunction expansion :

Gij (E) =∑α

Ψi (α)Ψ∗j (α)

E − Eα−→ 〈|Ψ1(E)|2〉 ≈

ImG11(E − i0)

πρW (E)

After some algebra : l(E) ≡ N〈|Ψ1(E)|2〉 =(κ2 + 1− κ

σ√NE)−1

Generalisation for finite rank-interaction is straightforward but (i) outlier interactionrequires more careful calculations and (ii) the Gaussian distribution cannot be proved

Large-window distribution

Local Gaussian distribution only for Ψ1(E) in a small energy window |δE | � σ√N

If E1 < Eα < E2

〈zqα〉[E1,E2] =cβ(q)

∫ E2

ρW (E)

(κ2 + 1−

σ√NE

dE , δN =

∫ E2

ρW (E)dE

ρW (E) = Wigner’s spectral density, cβ(q) = the Gaussian moments

c1(q) = 2qΓ(q+1/2)√π

, c2(q) = Γ(q + 1)

The full distribution = weighted integral of local PT distributions

Pβ(x) =1

∫ E2

ρW (E)

)dE , l(E) =

κ2 + 1− κ

If all states are included, E1 = −2σ√N, E2 = 2σ

For β = 1 (GOE)

P1(x) =

∫ π

0dφ sin2 φ

√κ2 + 1− 2κ cosφ e−

(κ2−2κ cosφ+1)x

For β = 2 (GUE)

P2(x) =I1(2κx)

κxe−(κ2+1)x

Numerics : mean values of N〈(Ψ1(E ))2〉 for different κ

0 0.5 1 1.5 2

Red circles are mean values for energies in the interval [−√N/2,

√N/2]

Blue diamonds are the same but for energies in the interval [√N/2, 3

√N/2]

Solid black lines = large-window theoretical predictionsDashed black line is the small-window predictionsN = 1000, σ = 1, and each point is averaged over 50 random realisations

Numerics : distribution of x =√

NΨ1(E ) for β = 1 (GOE)

-3 -2 -1 0 1 2 3x

)Left : states with energies in [−

√N/2,

√N/2] (lower curve)

and [√N/2, 3

√N/2] (upper curve) for κ = 0.6

Right : the same but for κ = 1.5

Solid lines are the Gaussian fits with zero mean whose variance agrees well with thetheoretical prediction

N ReΨ1(E ) (left) and

x =√

N ImΨ1(E ) (right) for β = 2 (GUE) and κ = 0.6

-3 -2 -1 0 1 2 3x

Energies in the interval [−√N/2,

√N/2]

-3 -2 -1 0 1 2 3x

Energies in the interval [√N/2, 3

√N/2]

NΨ1(E ) for all states, κ = .8, β = 1

-4 -2 0 2 4x

Blue dashed line is the PT distribution (Gaussian) : P(x) = e−x2/2/√

2πBlack solid line is theoretical prediction

P(x) =

∫ π

0dφ sin2 φ

√κ2 + 1− 2κ cosφ e−

(κ2−2κ cosφ+1)x2

Conclusion

Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v.

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

When ensemble of standard random matrices with Gaussian distribution isperturbed by a rank-one perturbation, the distribution of x = N|Ψ1(E)|2 has thesame functional form but with

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

When κ2 > 1 there exists one collective state whose mean energy is

Ec = σ√N(κ+ κ−1) and 〈|Ψ(c)

1 |2〉 = 1− κ−2

When all eigenfunctions in a large energy interval are considered their distributionis not Gaussian but is given by an integral over Gaussian functionsIn the limit N →∞ all other components of eigenfunctions (except Ψ1(E)) remaindistributed according to the usual PT distributionExperimentally one has to fit a width distribution for a resonance in a small energywindow to the local PT formula thus finding the l(E) from the fitTaking into account together different resonances with different energies is not asensitive way to investigate this phenomenonIntroduction of rank-one interaction does not change local spectral statisticsExperimentally for resonances ∆3(L) at small distances = RM prediction butdeviates from it at L ≈ 40− 70 d as for dynamical systems

Conclusion

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

1 |2〉 = 1− κ−2

Conclusion

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

1 |2〉 = 1− κ−2

Conclusion

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

1 |2〉 = 1− κ−2

When all eigenfunctions in a large energy interval are considered their distributionis not Gaussian but is given by an integral over Gaussian functions

In the limit N →∞ all other components of eigenfunctions (except Ψ1(E)) remaindistributed according to the usual PT distributionExperimentally one has to fit a width distribution for a resonance in a small energywindow to the local PT formula thus finding the l(E) from the fitTaking into account together different resonances with different energies is not asensitive way to investigate this phenomenonIntroduction of rank-one interaction does not change local spectral statisticsExperimentally for resonances ∆3(L) at small distances = RM prediction butdeviates from it at L ≈ 40− 70 d as for dynamical systems

Conclusion

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

1 |2〉 = 1− κ−2

When all eigenfunctions in a large energy interval are considered their distributionis not Gaussian but is given by an integral over Gaussian functionsIn the limit N →∞ all other components of eigenfunctions (except Ψ1(E)) remaindistributed according to the usual PT distribution

Experimentally one has to fit a width distribution for a resonance in a small energywindow to the local PT formula thus finding the l(E) from the fitTaking into account together different resonances with different energies is not asensitive way to investigate this phenomenonIntroduction of rank-one interaction does not change local spectral statisticsExperimentally for resonances ∆3(L) at small distances = RM prediction butdeviates from it at L ≈ 40− 70 d as for dynamical systems

Conclusion

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

1 |2〉 = 1− κ−2

When all eigenfunctions in a large energy interval are considered their distributionis not Gaussian but is given by an integral over Gaussian functionsIn the limit N →∞ all other components of eigenfunctions (except Ψ1(E)) remaindistributed according to the usual PT distributionExperimentally one has to fit a width distribution for a resonance in a small energywindow to the local PT formula thus finding the l(E) from the fitTaking into account together different resonances with different energies is not asensitive way to investigate this phenomenon

Introduction of rank-one interaction does not change local spectral statisticsExperimentally for resonances ∆3(L) at small distances = RM prediction butdeviates from it at L ≈ 40− 70 d as for dynamical systems

Conclusion

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

1 |2〉 = 1− κ−2

When all eigenfunctions in a large energy interval are considered their distributionis not Gaussian but is given by an integral over Gaussian functionsIn the limit N →∞ all other components of eigenfunctions (except Ψ1(E)) remaindistributed according to the usual PT distributionExperimentally one has to fit a width distribution for a resonance in a small energywindow to the local PT formula thus finding the l(E) from the fitTaking into account together different resonances with different energies is not asensitive way to investigate this phenomenonIntroduction of rank-one interaction does not change local spectral statistics

Experimentally for resonances ∆3(L) at small distances = RM prediction butdeviates from it at L ≈ 40− 70 d as for dynamical systems

Conclusion

P1(x) =1

√2πlx

exp(−

), P2(x) =

), x = N|Ψ|2, l = 1

l(E) ≡ 〈N |Ψ1(E)|2〉 =(κ2 + 1−

σ√NE)−1

1 |2〉 = 1− κ−2

Modification of the Porter-Thomas distribution by rank … · Modi cation of the Porter-Thomas...

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