Post on 29-Jan-2021
Santosh KumarShiv Nadar UniversityDadri, India
DISTRIBUTION OF SCATTERING MATRIX ELEMENTS IN QUANTUM CHAOTIC SCATTERING
Joint work with
A. Nock*, H.-J. Sommers, T. Guhr Universität Duisburg-Essen, Duisburg.*Queen Mary University of London, London
B. Dietz, M. Miski-Oglu, A. Richter, F. Schäfer Institut für Kernphysik, Technische Universität Darmstadt
IX Brunel-Bielefeld Workshop on Random Matrix TheoryBielefeld 2013
Distribution of S-matrix element in Fourier Space
OUTLINE
Introduction
Brief sketch of the derivation
Results and comparison with simulations and experiments
Conclusion
SCATTERING
Scattering of light waves by clouds Scattering of radio waves
Rutherford-Geiger-Marsden experiment
Scattering of laser beam
Electron scattering inside aquantum dot
A candidate scattering event at LHC leading to the discovery of Higgs Boson
Deviation of a wave or a particle from its trajectory because of some localized non-uniformity in the medium.
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SCATTERING: GENERIC SET-UP
States before the scattering event
Interaction region(Scattering event) States after the
scattering event
Channels of reaction (Characterized by total
energy E and other quantum numbers)
Scattering matrix (S-matrix) connects states existing asymptotically before and after the scattering event
(Relates incoming and outgoing waves)
B1(k)A2(k)
�=
S11 S12S21 S22
� A1(k)B2(k)
� 1(x) = A1(k)e
ikx + B1(k)e�ikx
2(x) = A2(k)eikx + B2(k)e
�ikx
SS† = S†S = 1
SCATTERING MATRIX
H
Relates incoming and outgoing waves
2 channel example:
S- matrix is unitary owing to the flux conservation,
N(� 1) bound states : hl|mi = �lmM channels : ha,E1|b, E2i = �ab �(E1 � E2)
HAMILTONIAN FORMULATION
H =X
lm
|liHlmhm|+X
c
ZdE|c, EiEhc, E|
+
X
l,c
✓|li
ZdEWlchc, E|+ herm. conj.
◆
Hab
d
e
f
c
Schematic view of the general scattering problem: Different channels of reaction (labeled a,b,...) are
connected via a compact interaction region described by a Hamiltonian H.
Schematic view of the general scattering problem: Different channels of reaction (labeled a,b,...) are
connected via a compact interaction region described by a Hamiltonian H.
GaAs based quantum dot(Source: http://www.newton.ac.uk/reports/9798/dqc1.gif)
• C. Mahaux and H. A. Weidenmüller, Shell Model Approach to Nuclear Reactions (North Holland, Amsterdam, 1969)
SCATTERING MATRIX
The resolvent G(E) is given by:
The N-component coupling vectors are assumed to satisfy the orthogonality
Sab(E) = �ab � i2⇡W †aG(E)Wb
G(E) =
E1N �H + i⇡
MX
c=1
WcW†c
!�1
W †cWd =�c⇡�cd; c, d = 1, · · · ,M
• C. Mahaux and H. A. Weidenmüller, Shell Model Approach to Nuclear Reactions (North Holland, Amsterdam, 1969)
STATISTICAL DESCRIPTION
Scattering process is quite often of chaotic nature.
Complicated dependence on
‣ Parameters of incoming waves (e.g., energy)
‣ The scattering region (e.g., the form or strength of the scattering potential.)
Scattering description of S-matrix is needed!
S-matrix, itself, is treated as a stochastic quantity and is described by the Poisson kernel.
(Based on the assumption of minimal information content)
M: Dimension of the S-matrix (Number of channels): Average S-matrixβ: Symmetry class
Dependence on the parameters not obvious!
P (S) / |det�1M � ShS†i
�|�(�M+2��)
MEXICO APPROACH
• P. Mello, P. Pereyra and T. H. Seligman, Ann. Phys. (NY) 161, 254 (1985)• H. U. Baranger and P. Mello , Phys. Rev. Lett. 73,142 (1994); Europh. Lett. 33, 465 (1996)• P. Mello and H. Baranger , Physica A 220, 15 (1995)
HEIDELBERG APPROACHIntroduces stochasticity on the level of the Hamiltonian describing the scattering center.
Random Matrix Universality: Universal and generic features can be extracted by modeling the Hamiltonian describing the scattering center using appropriate ensemble of Random matrices
N: Dimension of the matrices Hv: Energy scale
β: Symmetry class
β=1 Time reversal invariant “spinless” systems (Real-Symmetric H) β=2 Time reversal noninvariant systems (Hermitian H)
• D. Agassi, H. A. Weidenmüller and Z. Mantzouranis, Phys. Rep. 22, 145 (1975)
P(H) / exp✓��N
4v2trH2
◆
KNOWN RESULTS
• Two-point correlation function (β=1) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer Phys. Rep. 129, 367 (1985)
• Up to fourth moment (β=1) E. D. Davis and D. Boose Phys. Lett. B 211, 379 (1988); Z. Phys. A 332, 427 (1989)
• Two-point correlation function (β=2) D. V. Savin, Y. V. Fyodorov, H.-J. Sommers Acta Phys. Pol. A 109, 53-64 (2006)
• Two-point correlation function (β=2, Arbitrary UN invariant distribution for H) S. Mandt and M. R. Zirnbauer, J. Phys. A: Math. Theor. 43, 025201 (2010)
KNOWN RESULTS
• Two-point correlation function (β=1) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer Phys. Rep. 129, 367 (1985)
• Up to fourth moment (β=1) E. D. Davis and D. Boose Phys. Lett. B 211, 379 (1988); Z. Phys. A 332, 427 (1989)
• Two-point correlation function (β=2) D. V. Savin, Y. V. Fyodorov, H.-J. Sommers Acta Phys. Pol. A 109, 53-64 (2006)
• Two-point correlation function (β=2, Arbitrary UN invariant distribution for H) S. Mandt and M. R. Zirnbauer, J. Phys. A: Math. Theor. 43, 025201 (2010)
‣ Distribution of diagonal elements (β=1, 2): Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers J. Phys. A: Math. Gen 38, 10731 (2005)
KNOWN RESULTS
• Two-point correlation function (β=1) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer Phys. Rep. 129, 367 (1985)
• Up to fourth moment (β=1) E. D. Davis and D. Boose Phys. Lett. B 211, 379 (1988); Z. Phys. A 332, 427 (1989)
• Two-point correlation function (β=2) D. V. Savin, Y. V. Fyodorov, H.-J. Sommers Acta Phys. Pol. A 109, 53-64 (2006)
• Two-point correlation function (β=2, Arbitrary UN invariant distribution for H) S. Mandt and M. R. Zirnbauer, J. Phys. A: Math. Theor. 43, 025201 (2010)
‣ Distribution of diagonal elements (β=1, 2): Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers J. Phys. A: Math. Gen 38, 10731 (2005)
The problem of finding distribution of off-diagonal S-matrix elements remained unsolved!
SOME OBSERVATIONS
Already in 1975 numerical simulations revealed that
The distributions, in general, exhibit Non - Gaussian behavior (expected because of Unitarity constraint).
The real and imaginary parts show different deviations from the Gaussian behavior for β=1.
Similar conclusions were arrived at from the data obtained from experiments on microwave resonators.
• J.W.Tepel, Z.Physik A 273, 59 (1975). • J.Richert, M. H. Simbel, and H. A. Weidenmüller,Z.Physik A 273, 195 (1975).• B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, and H. A.
Weidenmüller, Phys. Rev. E 81, 036205 (2010).
SOME OBSERVATIONS
Already in 1975 numerical simulations revealed that
The distributions, in general, exhibit Non - Gaussian behavior (expected because of Unitarity constraint).
The real and imaginary parts show different deviations from the Gaussian behavior for β=1.
Similar conclusions were arrived at from the data obtained from experiments on microwave resonators.
• J.W.Tepel, Z.Physik A 273, 59 (1975). • J.Richert, M. H. Simbel, and H. A. Weidenmüller,Z.Physik A 273, 195 (1975).• B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, and H. A.
Weidenmüller, Phys. Rev. E 81, 036205 (2010).
Ps(xs) =
Zd[H]P(H)�(xs � }s(Sab)); s = 1, 2
GOAL
Distribution of off-diagonal S-matrix elements
Real part of Sab:
Imaginary part of Sab:
}1(Sab) =Sab + S⇤ab
2= �i⇡(W †aGWb �W
†bG
†Wa)
}2(Sab) =Sab � S⇤ab
2i= �⇡(W †aGWb +W
†bG
†Wa)
Rs(k) =
Zd[H]P(H) exp(�ik}s(Sab)); s = 1, 2
Ps(xs) =1
2⇡
Z 1
�1dkRs(k) exp(ikxs)
CHARACTERISTIC FUNCTIONS
The characteristic function also serves as the moment generating function
The distributions can be obtained as the Fourier transform of Rs(k):
Rs(k) =
Zd[H]P(H) exp(�ik}s(Sab)); s = 1, 2
W =
WaWb
�As =
0 (�i)sG
isG† 0
�
Rs(k) =
Zd[H]P(H) exp(�ik⇡W †AsW )
P(H) / exp⇣� �N
4v2trH2
⌘
CHARACTERISTIC FUNCTIONS
Introduce a 2N-dimensional vector W and a 2N×2N dimensional matrix As
H appears in the denominator of G: Ensemble averaging nontrivial !
G =
E1N �H + i⇡
MX
c=1
WcW†c
!�1
SUPERMATHEMATICSAnticommuting (Grassmann or Fermionic) variables:
Any function of the anticommuting variables is a finite polynomial,e.g., exp(α)=1+α
“Complex conjugate”
Conventions:
•F. A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987)• K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997)
↵1↵2 = �↵2↵1↵2 = 0
↵ ↵⇤
(↵⇤)⇤ = �↵ (↵�)⇤ = ↵⇤�⇤(↵⇤↵)⇤ = (↵⇤)⇤↵⇤ = �↵↵⇤ = ↵⇤↵
SUPERMATHEMATICSIntegrals (Berezin Integrals):
In contrast, for the ordinary complex variables
Superintegral:
•F. A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987)• K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997)
Zd↵ = 0,
Zd↵↵ =
1p2⇡Z Z
d↵⇤ d↵ exp(iq ↵⇤↵) =q
2⇡i
Z Z Z Zdz⇤dz d↵⇤d↵ exp(iq(z⇤z + ↵⇤↵)) = 1
Z Zdz⇤ dz exp(iq z⇤z) =
2⇡i
q
str� = tra� trb
sdet� =det(a� µb�1⌫)
det b=
det a
det(b� ⌫a�1µ)
SUPERMATHEMATICS
Supervectors:
Supermatrices:
DefinitionsSupertrace:
Superdeterminant:
=
z⇣
� † = [z† ⇣†]
� =
a µ⌫ b
��T =
aT ⌫T
�µT bT�
�† =
a† ⌫†
�µ† b†�
z =
zazb
�⇣ =
⇣a⇣b
�A�1s =
0 (�i)s(G�1)†
isG�1 0
�W =
WaWb
�
Zd[⇣] exp
✓i
4⇡k⇣†A�1s ⇣
◆= det
A�1si8⇡2k
!
SUPERMATHEMATICSMultivariate Gaussian Integrals:
Using vectors with commuting entries:
Using vectors with anticommuting entries:
Zd[z] exp
✓i
4⇡kz†A�1s z
◆exp
✓i
2
(W †z + z†W )
◆= det
�1
A�1si8⇡2k
!exp(�ik⇡W †AsW )
Zd[z]
Zd[⇣] exp
✓i
4⇡k(z†A�1s z + ⇣
†A�1s ⇣)
◆exp
✓i
2
(W †z + z†W )
◆= exp(�ik⇡W †AsW )
SUPERMATHEMATICS
Combining the above integral results we obtain
Zd[z]
Zd[⇣] exp
✓i
4⇡k(z†A�1s z + ⇣
†A�1s ⇣)
◆exp
✓i
2
(W †z + z†W )
◆= exp(�ik⇡W †AsW )
Rs(k) =
Zd[H]P(H) exp(�ik⇡W †AsW )
SUPERMATHEMATICS
Combining the above integral results we obtain
The exponential on RHS is exactly the factor in our expression for Rs(k)
Rs(k) =
Zd[ ] exp
⇣ i2
(W† + †W)⌘Z
d[H]P(H) exp⇣ i4⇡k
†A�1s ⌘
W =
2
664
WaWb00
3
775 =
2
664
zazb⇣a⇣b
3
775 A�1s =
2
666664
0 (�i)s(G�1)†
isG�1 00
00 (�i)s(G�1)†
isG�1 0
3
777775
G�1 = E1N �H + i⇡MX
c=1
WcW†c
Rs(k) =
Zd[ ] exp
⇣ i2
(W† + †W)⌘Z
d[H]P(H) exp⇣ i4⇡k
†A�1s ⌘
W =
2
664
WaWb00
3
775 =
2
664
zazb⇣a⇣b
3
775 A�1s =
2
666664
0 (�i)s(G�1)†
isG�1 00
00 (�i)s(G�1)†
isG�1 0
3
777775
G�1 = E1N �H + i⇡MX
c=1
WcW†c
H is now linear in the exponent containing the supervectors
Rs(k) =
Zd[ ] exp
⇣ i2
(W† + †W)⌘Z
d[H]P(H) exp⇣ i4⇡k
†A�1s ⌘
W =
2
664
WaWb00
3
775 =
2
664
zazb⇣a⇣b
3
775 A�1s =
2
666664
0 (�i)s(G�1)†
isG�1 00
00 (�i)s(G�1)†
isG�1 0
3
777775
G�1 = E1N �H + i⇡MX
c=1
WcW†c
H is now linear in the exponent containing the supervectors
As-1 is not block diagonal!
!⌅+ 00 2�⌅
�
�
† ! †⌅± =
0 ±(�i)s1N
�is1N 0
�
Ψ and Ψ† can be treated as independent complex quantities and therefore admit independent transformations.
Jacobian: (-1)N 2-2N for β= 1 and (-1)N for β= 2
The choice of Ξ± ensures proper convergence requirements for the supermatrix introduced later
Rs(k) = (�1)NZ
d[ ] exp⇣ i2
(U†s + †W)
⌘Zd[H]P(H) exp
⇣ i4⇡k
†A�1 ⌘
A�1 = diag[�(G�1)†, G�1,�(G�1)†,�G�1]
β = 2 (HERMITIAN H)
Rs(k) = (�1)NZ
d[ ] exp⇣ i2
(U†s + †W)
⌘Zd[H]P(H) exp
⇣ i4⇡k
†A�1 ⌘
=
2
664
zazb⇣a⇣b
3
775
A�1 = diag[�(G�1)†, G�1,�(G�1)†,�G�1]
β = 2 (HERMITIAN H)
Rs(k) = (�1)NZ
d[ ] exp⇣ i2
(U†s + †W)
⌘Zd[H]P(H) exp
⇣ i4⇡k
†A�1 ⌘
z†aHza � z†bHzb + ⇣
†aH⇣a + ⇣
†bH⇣b
= tr(HD)
D = zaz†a � zbz
†b � ⇣a⇣
†a � ⇣b⇣
†b
H-part in exponent involving the supervectors:
where
A�1 = diag[�(G�1)†, G�1,�(G�1)†,�G�1]
β = 2 (HERMITIAN H)
Rs(k) = (�1)NZ
d[ ] exp�i †Vs
� Zd[H]P(H) exp
⇣ i4⇡k
†A�1 ⌘
A�1 = diag(�(G�1)†, G�1,�(G�1)†,�G�1)⌦ 12
β= 1 (REAL SYMMETRIC H)
=
2
66666666664
xa
ya
xb
yb
⇣a
⇣
⇤a
⇣b
⇣
⇤b
3
77777777775
Rs(k) = (�1)NZ
d[ ] exp�i †Vs
� Zd[H]P(H) exp
⇣ i4⇡k
†A�1 ⌘
A�1 = diag(�(G�1)†, G�1,�(G�1)†,�G�1)⌦ 12
β= 1 (REAL SYMMETRIC H)
Rs(k) = (�1)NZ
d[ ] exp�i †Vs
� Zd[H]P(H) exp
⇣ i4⇡k
†A�1 ⌘
xTaHxa + yTa Hya � xTb Hxb � yTb Hyb + ⇣†aH⇣a � ⇣Ta H⇣⇤a + ⇣
†bH⇣b � ⇣
Tb H⇣
⇤b
= tr(HD)
D = xaxTa + yay
Ta � xbxTb � ybyTb � ⇣a⇣†a + ⇣⇤a⇣Ta � ⇣b⇣
†b + ⇣
⇤b ⇣
Tb
A�1 = diag(�(G�1)†, G�1,�(G�1)†,�G�1)⌦ 12H-part in exponent involving the supervectors:
where
β= 1 (REAL SYMMETRIC H)
Zd[H]P(H) exp
⇣ i4⇡k
trHD⌘= exp
⇣� 1
4rtrD2
⌘
= exp
⇣� 1
4rstr (K1/2BK1/2)2
⌘
r =4�⇡2k2N
v2
Bmn =NX
j=1
( m)j( †n)j ; m,n = 1, 2, .., 8/�
K = diag(1,�1, 1, 1)⌦ 12/�
ENSEMBLE AVERAGING
where
exp
⇣� 1
4rstr (K1/2BK1/2)2
⌘=
Zd[�] exp
�� r str�2 + i str�K1/2BK1/2
�
=
Zd[�] exp
�� r str�2 + i †K1/2(� ⌦ 1N )K1/2
�
HUBBARD-STRATONOVICH TRANSFORMATION
σ is an 8/β-dimensional supermatrix having same structure as B, and K = K ⌦ 12/�
Rs(k) = (�1)NZ
d[�] exp(�r str�2)Z
d[ ] exp⇣i †K1/2⌃K1/2 + i †Vs
⌘
Rs(k) = (�1)NZ
d[�] exp(�r str�2)Z
d[ ] exphi †K1/2⌃K1/2 +
i
2
(U†s + †W)
i
⌃ =⇣� � E
4⇡k18/�
⌘⌦ 1N +
i
4kL⌦
MX
c=1
WcW†c
L = diag(1,�1, 1,�1)⌦ 12/�
β=1
β=2
Rs(k) = (�1)NZ
d[�] exp(�r str�2)Z
d[ ] exp⇣i †K1/2⌃K1/2 + i †Vs
⌘
Rs(k) = (�1)NZ
d[�] exp(�r str�2)Z
d[ ] exphi †K1/2⌃K1/2 +
i
2
(U†s + †W)
i
⌃ =⇣� � E
4⇡k18/�
⌘⌦ 1N +
i
4kL⌦
MX
c=1
WcW†c
L = diag(1,�1, 1,�1)⌦ 12/�
Integral over the supervector can now be performed
β=1
β=2
Rs(k) =
Zd[�] exp
⇣� r str�2 � �
2
str ln⌃� i4
Fs⌘
⌃ =⇣� � E
4⇡k18/�
⌘⌦ 1N +
i
4kL⌦
MX
c=1
WcW†c
L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�
Fs =
(VTs L
�1/2⌃�1L�1/2Vs, � = 1
U†sL�1/2⌃�1L�1/2W, � = 2
REPRESENTATION IN SUPERMATRIX SPACE
β=1 32 independent integration variables β=2 16 independent integration variables
Rs(k) =
Zd[�] exp
⇣� r str�2 � �
2
str ln⌃� i4
Fs⌘
⌃ =⇣� � E
4⇡k18/�
⌘⌦ 1N +
i
4kL⌦
MX
c=1
WcW†c
L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�
Fs =
(VTs L
�1/2⌃�1L�1/2Vs, � = 1
U†sL�1/2⌃�1L�1/2W, � = 2
REPRESENTATION IN SUPERMATRIX SPACE
β=1 32 independent integration variables β=2 16 independent integration variables
Drastic reduction in the number of integration variables!
Rs(k) =
Zd[�] exp
⇣� r str�2 � �
2
str ln⌃� i4
Fs⌘
⌃ =⇣� � E
4⇡k18/�
⌘⌦ 1N +
i
4kL⌦
MX
c=1
WcW†c
L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�
Fs =
(VTs L
�1/2⌃�1L�1/2Vs, � = 1
U†sL�1/2⌃�1L�1/2W, � = 2
REPRESENTATION IN SUPERMATRIX SPACE
β=1 32 independent integration variables β=2 16 independent integration variables
Drastic reduction in the number of integration variables!Form similar to that of generating function for correlations
Rs(k) =
Zd[�] exp
⇣� r str�2 � �
2
str ln⌃� i4
Fs⌘
⌃ =⇣� � E
4⇡k18/�
⌘⌦ 1N +
i
4kL⌦
MX
c=1
WcW†c
L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�
Fs =
(VTs L
�1/2⌃�1L�1/2Vs, � = 1
U†sL�1/2⌃�1L�1/2W, � = 2
REPRESENTATION IN SUPERMATRIX SPACE
β=1 32 independent integration variables β=2 16 independent integration variables
Drastic reduction in the number of integration variables!Form similar to that of generating function for correlations
(Verbaarschot, Weidenmüller, Zirnbauer) apart from the Fs part
Rs(k) =
Zd[�] exp(�L� �L)
L = N 4�⇡2k2
v2str�2 +N
�
2str ln
⇣� � E
4⇡k18/�
⌘
�L =MX
c=1
str ln⇣18/� +
i�c4⇡k
⇣� � E
4⇡k18/�
⌘�1L⌘+
i
4Fs
�0 =1
8⇡k
�E ± i
p4v2 � E2
�
SADDLE POINT ANALYSIS
We are interested in N >> M limit. We fix M and let N → ∞
Fs is a linear combination of matrix elements of multiplied with γc, where c=a, b.
Saddle point equation:
Scalar solution:
8�⇡2k2
v2� +
�
2
⇣� � E
4⇡k18/�
⌘�1= 0
�G =1
8⇡k
�E18/� �
p4v2 � E2 Q
�
Q = �i T�1LT ; strQ = 0; Q2 = �18/�
L = diag(1,�1, 1,�1)⌦ 12/�
MANIFOLD OF SOLUTIONS
The dominant part of the free energy is invariant under the application of T
β=1:T belongs to Lie supergroup UOSP(2,2/4) Q belongs to the coset superspace UOSP(2,2/4)/(UOSP(2/2)×UOSP(2/2))
β=2:T belongs to Lie supergroup U(1,1/2) Q belongs to the coset superspace U(1,1/2)/(U(1/1)×U(1/1))
•K. B. Efetov, Adv. Phys. 32, 53 (1983) • J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985)• Y. V. Fyodorov, and H.-J. Sommers, J. Math. Phys. 38,1918 (1997)
� = �G + ��
SEPARATING “GOLDSTONE” AND “MASSIVE” MODES
Expand up to the second power in δσ. The integrals involving Goldstone and Massive modes factorize. Symbolically:
The part involving Massive modes are Gaussian integrals and yields unity.
Z(�) =
Z(�G)
Z(��)
• L. Schäfer and F. Wegner, Z. Phys. B 38, 113 (1980)• J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985)
Rs(k) =
Zdµ(�G)e
�iFs/4MY
c=1
sdet��2
⇣18/� +
i�c4⇡k
��1E L⌘
�G =1
8⇡k
�E18/� �
p4v2 � E2 Q
� �E = �G �E
4⇡k18/�
NONLINEAR SIGMA MODEL
Parametrization of Q:β=1: Eight commuting variables Eight anticommuting variables
β=2: Four commuting variables, Four anticommuting variables
• K. B. Efetov, Adv. Phys. 32, 53 (1983) • J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985)• Y. V. Fyodorov, and H.-J. Sommers, J. Math. Phys. 38,1918 (1997)
Rs(k) = 1�Z 1
1d�1
Z 1
�1d�2
k2
4(�1 � �2)2FU(�1,�2)
�t1at
1b + t
2at
2b
�J0
⇣kq
t1at1b
⌘
Ps(xs) =@
2f(xs)
@x
2s
,
f(x) = x⇥(x) +
Z 1
1d�1
Z 1
�1d�2
FU(�1,�2)4⇡(�1 � �2)2
t
1at
1b + t
2at
2b�
t
1at
1b � x2
�1/2⇥(t1at
1b � x2)
FU =MY
c=1
gc + �2gc + �1
gc =v2 + �2c
�cp4v2 � E2
=2
Tc� 1
tjc =
q|�2j � 1|
(gc + �j), j = 1, 2
RESULTS (β=2)
Identical results for real (s=1) and imaginary (s=2) parts
Characteristic Function
Distribution
Ps(xs) = �(xs) +@f
(s)1
@xs+
@
2f
(s)2
@x
2s
+@
3f
(s)3
@x
3s
+@
4f
(s)4
@x
4s
FO =MY
c=1
gc + �0(gc + �1)1/2(gc + �2)1/2
J = (1� �20)|�1 � �2|
2(�21 � 1)1/2(�22 � 1)1/2(�1 � �0)2(�2 � �0)2
Rs(k) = 1 +1
8⇡
Z 1
�1d�0
Z 1
1d�1
Z 1
1d�2
Z 2⇡
0d J (�0,�1,�2)FO(�0,�1,�2)
4X
n=1
(s)n kn
RESULTS (β=1)Different results for real (s=1) and imaginary (s=2) parts
Characteristic Function
Distribution
EXPERIMENTS WITH MICROWAVE RESONATORS
Equivalence in mathematical structure of the time-independent Schrödinger and Hemholtz equations (two-dimensions)
The shape of microwave cavity is such that the dynamics of the corresponding classical billiard is chaotic
Not only moduli, but both real and imaginary parts of the S-matrix elements can be measured
(r2 + k2) = 0 (r2 + k2)Ez = 0
COMPARISON WITH EXPERIMENTAL DATA (β=1)
Characteristic functions for the real and imaginary parts of S12 for the frequency range 10-11 GHz
Characteristic functions for the real and imaginary parts of S12 for the frequency range 24-25 GHz
COMPARISON WITH EXPERIMENTAL DATA (β=1)
Distributions for the real and imaginary parts of S12 for the frequency range 18-19 GHz
Distributions for the real and imaginary parts of S12 for the frequency range 24-25 GHz
COMPARISON WITH NUMERICAL SIMULATIONS (β=1)
COMPARISON WITH NUMERICAL SIMULATIONS (β=2)
CONCLUSION
We solved a long-standing problem of finding the exact results (in the N→∞ limit) for distributions of off-diagonal S- matrix elements.
We accomplished this task using a novel route to the nonlinear sigma model based on the characteristic function.
We validated our results with experimental data obtained with chaotic microwave billiards, and thus presented a new confirmation of the random matrix universality conjecture.
• S. Kumar, A. Nock, H.-J. Sommers, T. Guhr, B. Dietz, M. Miski-Oglu, A. Richter, and F. Schäfer, Phys. Rev. Lett. 111, 030403 (2013)• A. Nock, S. Kumar, H.-J. Sommers, T. Guhr, Ann. Phys. (In press); Preprint: arXiv:1307.4739
Thank You!