Random Matrix Theory with U(N) Racah Algebra for Transition Strengths

0.15 MB·English

by  V. K. B. Kota

5
1
0 Random Matrix Theory with U(N) Racah Algebra
2

n for Transition Strengths
a
J

0 V.K.B. Kota
3
Physical Research Laboratory, Ahmedabad 380 009, India

]
E-mail: [email protected]
h
p
- Abstract.
h
t Forfinitequantummany-particlesystems,agivensystem,inducedbyatransitionoperator,
a
makestransitionsfromitsstatestothestatesofthesamesystemortothoseofanothersystem.
m
Examples are electromagnetic transitions (thentheinitial and finalsystems aresame), nuclear
[ beta and double beta decay (then the initial and final systems are different), particle addition

toorremovalfromagivensystemandsoon. Workingtowardsdevelopingacompletestatistical
1
theory for transition strength densities (transition strengths multiplied bythedensityof states
v
at the initial and final energies), we have started a program to derive formulas for the lower
0
7 orderbivariatemomentsofthestrengthdensitiesgeneratedbyavarietyoftransitionoperators.
6 Inthispaperresults arepresentedforatransition operator thatremovesk0 numberofparticle
7 by considering m spinless fermions in N single particle states. The Hamiltonian that is k-
0 bodyisrepresentedbyEGUE(k)[embeddedGaussian unitaryensembleofk-bodyinteractions]
. and similarly the transition operator by an appropriate independent EGUE. Employing the
1
0 embedding U(N) algebra, finite-N formulas for moments up to order four are derived and
5 they show that in general the smoothed (with respect to energy) bivariate transition strength
1 densities takebivariate Gaussian form. Extension of these results to particle addition operator
: and betadecay typeoperators are discussed.
v
i
X
r
a
1. Introduction
Let us begin with the statement from the preface to the proceedings of the meeting held
in 2006 on ”Applications of Random Matrices in Physics”: [1] Random matrices are widely
and successfully used in physics for almost 60-70 years, beginning with the works of Wigner
and Dyson. Initially proposed to describe statistics of levels in complex nuclei, the Random
Matrix Theory has grown far beyond nuclear physics, and also far beyond just level statistics.
It is constantly developing into new areas of physics and mathematics, and now constitutes
a part of the general culture and curriculum of a theoretical physicist. Besides applications
in all branches of quantum physics, RMT is being used in disciplines such as Econophysics,
Wireless communication, information theory, multivariate statistics, number theory, neural and
biological networks and so on. The focus in this article is on the frontier topic of statistical
properties of isolated finite many-particle quantum systems [2]. Examples for these systems are
atoms,atomicnuclei, mesoscopicsystems(quantumdots,smallmetallicgrains), interactingspin
systems modeling quantum computing core, ultra-cold atoms and so on. A route to investigate
statistical properties is to employ the classical GOE or GUE or GSE random matrix ensembles
with various deformations. For these, as Wigner states: the assumption is that the Hamiltonian
which governs the behavior of a complicated system is a random real symmetric or complex
Hermitian or Quaternion real matrix, with no special properties except for its symmetric or
Hermitian or Quaternion real nature. However, for most of the isolated finite many-particle
quantum systems, their constituents predominantly interact via two-particle interactions and
the classical random matrix ensembles are too unspecific to account for this most important
feature. One refinement which retains the basic stochastic approach but allows for this feature
consists in the use of embedded random matrix ensembles [3, 4, 5, 6, 7, 8].
Therefore, it is more appropriate to represent an isolated finite interacting quantum system,
say with m particles (fermions or bosons) in N single particle (sp) states by random matrix
models generated by random k-body (note that k < m and most often we have k = 2)
interactions and propagate the information in the interaction to many particle spaces. Thus
we have random interaction matrix models for m-particle systems. In the simplest version,
the k-particle Hamiltonian (H) of a spinless fermion (or boson) system is represented by
GOE/GUE/GSE and then the m particle H matrix is generated using the m-particle Hilbert
spacegeometry. ThekeyelementhereistherecognitionthattheU(N)Liealgebratransportsthe
information in the two-particle spaces to many-particle spaces. As a GOE/GUE/GSE random
matrix ensemblein two-particle spaces is embeddedinthe m-particle H matrix, these ensembles
are generically called embedded ensembles (EE). With GUE embedding we have EGUE and in
this paper EGUE is used throughout. For EE, a general formulation for deriving analytical
results is to use the Wigner-Racah algebra of the embedding Lie algebra [8]. The focus in the
present paper is on transition strengths and they are not yet studied in any detail using EE [8].
Also, as emphasized in [4, 9], there are many open questions in the random matrix theory for
transition strengths in finite interacting quantum many-particle systems.
For finite quantum many-particle systems, induced by a transition operator, a given system
makes transitions from its states to the states of the same system or to the states of another
system. Examples are electromagnetic transitions (then the initial and final systems are same),
nuclear beta and double beta decay (then the initial and final systems are different), particle
addition to or removal from a given system and so on. Given a state with energy E and say
i
it is connected to a state E by a transition operator O, then the transition strengths are
f
|hE |O |E i|2 and these will determine for example the life time of a state. It is important to
f i
recognize that the transition strengths probe the structure of the eigenfunctions of a quantum
many-body system and thus they are very important from the point of view of experiments
probing the structure of a system. Also, they are needed in many applications (for example,
beta decay transition strengths are essential for nucleosynthesis studies). In the statistical
theories, it is more useful to deal with the corresponding transition strength density (this will
take into account degeneracies in the eigenvalues) defined by
I (E ,E ) = I(E ) |hE |O | E i|2 I(E ) . (1)
O i f f f i i
In Eq. (1), I(E) are state densities normalized to the dimension of the m particle spaces. Note
that E and E belong to the same m particle system or different systems depending on the
i f
nature of O, the transition operator.
Working towards developing a complete statistical theory for transition strength densities
(transition strengths multiplied by the density of states at the initial and final energies) for
isolated finite many-particle quantum systems, we have started a program to derive formulas for
the lower order bivariate moments of the strength densities generated by a variety of transition
operators. Ingeneral,theHamiltonianmayhavemanysymmetrieswiththefermions(orbosons)
carrying other degrees of freedom such as spin, orbital angular momentum, isospin and so on.
Alsowemayhaveinthesystemdifferenttypesoffermions(orbosons)andforexampleinatomic
nuclei we have protons and neutrons. In addition, a transition operator may preserve particle
numberandother quantumnumbersoritmay changethem. Amongallthesevarioussituations,
in our program we have considered five different systems: (i) a system of m spinless fermions
andatransitionoperatorthatpreservestheparticlenumber;(ii)asystemofmspinlessfermions
and a transition operator that removes say k number of particles from the m fermion system;
0
(iii) same as (ii) but for particles addition operator; (iv) a system with two types of spinless
fermions with the transition operator changing k number of particles of one type to k number
0 0
of other particles as in nuclear beta and double beta decay; (v) same as (i)-(iii) but for spinless
boson systems. In [10], results are presented for (i). In the present paper results are presented
for (ii) and their extensions to (iii) and (iv) are briefly discussed; Ref. [11] gives results from a
first study for (iv). Let us add that (ii) and (iii) are important for example in nuclear physics
as one and two particle removal and addition to a nucleus are important experimental probes of
the structure of the atomic nucleus; see for example [12, 13]. Thus, the results in Sections 3-5
have applications in nuclear physics. Now we will give a preview.
Section 2gives somebasicresultsforEGUE(k) forspinlessfermionsystemsasderived in[14].
Using these results, formulas for the lower order bivariate moments of the transition strength
densities for the situation (ii) above are derived and they are presented in detail in Section 3.
Using these, results in the asymptotic limit are derived and they are presented in Section 4.
Extension of the results in Section 3 to the situations (iii) and (iv) above are briefly discussed
in Section 5. Finally, in Section 6 gives conclusions and future outlook.
2. Basic EGUE(k) results for a spinless fermion system
Let us consider m spinless fermions in N degenerate sp states with the Hamiltonian Hˆ a k-body
operator,
Hˆ = V (k) A†(k)A (k) , V (k) = k,i | Hˆ |k,j . (2)
ij i j ij
Xi,j D E
†
Here A (k) is a k particle (normalized) creation operator and A (k) is the corresponding
i i
annihilation operator (a hermitian conjugate). Also, i and j are k-particle indices. The k
and m particle space dimensions are N and N respectively. Representing the V matrix,
k m
defined by the matrix elements V , by GUE we have EGUE(k) for the H matrix in m-particle
ij
(cid:0) (cid:1) (cid:0) (cid:1)
spaces. Note that, for V a GUE, the real and imaginary parts of V are independent zero
ij
centered Gaussian random variables with variance satisfying,
V (k)V (k) = V2 δ δ . (3)
ab cd H ad bc
Here the ’over-line’ indicates ensemble average. From now on we will drop the hat over H and
denote when needed H by H(k). In physical systems in general k = 2 but in some systems such
as atomic nuclei and BEC it is possible to have k = 3 and even k = 4 [15, 16, 17].
The U(N) algebra that generates the embedding, as shown in [14], gives formulas for the
lower order moments of the one-point function, the eigenvalue density I(E) = hhδ(H −E)ii and
alsoforthetwo-pointfunctionintheeigenvalues. UsedhereistheU(N)tensorialdecomposition
of the H(k) operator giving ν = 0,1,...,k irreducible parts Bν,ων(k) and then,
k
H(k) = W (k) Bν,ων(k) . (4)
ν,ων
ν=X0;ων∈ν
Note that ω are labels of the irreducible representations (irreps) of the subalgebras of U(N)
ν
and their explicit structure will not play any role in the present work. With the GUE(k)
representation for the H(k) operator, the expansion coefficients W’s will be independent zero
centered Gaussian random variables with
W (k) W (k) = V2 δ δ . (5)
ν1,ων1 ν2,ων2 H ν1,ν2 ων1ων2
For deriving formulas for the various moments, the first step is to apply the Wigner-Eckart
theorem for the matrix elements of Bν,ων(k). Given the m-fermion states |f v i, we have with
m i
respect to the U(N) algebra, f = {1m}, the antisymmetric irrep in Young tableaux notation
m
and v are additional labels. Note that the ν label used for B’s corresponds to the Young
i
tableaux {2ν1N−2ν}. Now, Wigner-Eckart theorem for U(N) ⊃ G (here G is some subalgebra
of U(N) giving v and ω labels) gives
hf v | Bν,ων(k) |f v i= hf || Bν(k) || f i Cν,ων . (6)
m f m i m m
fmvf,fmvi
Here, h−−||−−||−−i is the reduced matrix element and C−−−− is a U(N) ⊃ G Clebsch-
−−−−
Gordan(C-G)coefficient [notethat wearenotmakingadistinction between U(N)andSU(N)].
Also,if |f virepresentsastateforrnumberoffermions,then f v representsthecorresponding
r r
state for r number of holes or N−r number of fermions (see [14] for details). In Young tableaux
(cid:12) (cid:11)
notation fm ={1N−m}. Definition of Bν,ων(k) and the U(N)(cid:12)Wigner-Racah algebra will give,
|hf || Bν(k) ||f i|2 = Λν(N,m,m−k) ,
m m
(7)
m′−µ N′−m′+r−µ
Λµ(N′,m′,r)= .
r r
(cid:18) (cid:19)(cid:18) (cid:19)
The Λν(N,m,k) are nothing but, apart from a N and m dependent factor, a U(N) Racah
coefficient [14]. This and the various properties of the U(N) Wigner and Racah coefficients give
two formulas for the ensemble average of a product any two m particle matrix elements of H,
hf v | H(k) |f v i hf v |H(k) | f v i
m 1 m 2 m 3 m 4
k
(8)
= V2 Λν(N,m,m−k) Cν,ων Cν,ων ,
H fmv1,fmv2 fmv3,fmv4
ν=X0;ων
and also
hf v |H(k) | f v i hf v | H(k) | f v i
m 1 m 2 m 3 m 4
m−k
(9)
= V2 Λν(N,m,k) Cν,ων Cν,ων
H fmv1,fmv4 fmv3,fmv2
ν=X0;ων
Eq. (9) follows by applying a Racah transform to the product of the two C-G coefficients
appearing in Eq. (8). Let us mention two important properties of the U(N) C-G coefficients
that are quite useful,
−1/2
N N
Cν,ων = δ , C0,0 = δ . (10)
fmvi,fmvi s m ν,0 fmvi,fmvj m vi,vj
Xvi (cid:18) (cid:19) (cid:18) (cid:19)
From now on we will use the symbol f only in the C-G coefficients, Racah coefficients and
m
the reduced matrix elements. However, for the matrix elements of an operator we will use m
implying totally antisymmetric state for fermions. An important by-product of Eqs. (9) and
(10) is
hmv | H(k) | mv i hmv |H(k) | mv i = h[H(k)]2im δ (11)
i j j k vi,vk
Xvj
and we will use this in Section 3.
Starting with Eq. (4) and using Eqs. (5), (9) and (10) will immediately give the formula,
−1
N
h[H(k)]2im = hmv | [H(k)]2 |mv i = V2 Λ0(N,m,k) . (12)
m i i H
(cid:18) (cid:19) Xvi
Similarly, for hH4im first the ensemble average is decomposed into 3 terms as,
hh[H(k)]4iim = hmv |[H(k)]4 | mv i
i i
Xvi
= hmv |H(k) | mv i hmv |H(k) | mv i hmv |H(k) | mv ihmv |H(k) | mv i
i j j p p l l i
vi,vXj,vp,vlh
+hmv |H(k) |mv i hmv |H(k) | mv i hmv |H(k) | mv ihmv |H(k) |mv i
i j l i j p p l
+hmv | H(k) |mv i hmv | H(k) |mv i hmv | H(k) |mv ihmv |H(k) |mv i .
i j p l j p l i
i (13)
Note that the trace H4 m = N H4 m. It is easy to see that the first two terms simplify to
m
give 2[hH2im]2 and the third term is simplified by applyingEq. (8) to the firstensemble average
(cid:10)(cid:10) (cid:11)(cid:11) (cid:0) (cid:1)(cid:10) (cid:11)
and Eq. (9) to the second ensemble average. Then, the final result is
−1 min(k,m−k)
2 N
h[H(k)]4im = 2 hH2im +V4 Λν(N,m,k)Λν(N,m,m−k)d(N :ν) ; (14)
H m
h i (cid:18) (cid:19) Xν=0
2 2
N N
d(N :ν)= − . (15)
ν ν −1
(cid:18) (cid:19) (cid:18) (cid:19)
Now, we will derive the formulas for the moments of the transition strength densities generated
by a transition operator O that removes k number of particles from a m-particle system.
0
3. Lower-order moments of transition strength densities: results for particle
removal operators
Particle removal(or addition)operatorsareofgreat interestinnuclear physics. For exampleone
particle (proton or neutron) removal from a target nucleus gives information about the single
particle levels in the target and similarly, two-particle removal gives information about pairing
force. Let us begin with a particle removal operator O and say it removes k numberof particles
0
when acting on a m fermion state. Then the general form of O is,
O = V A (k ) . (16)
α0 α0 0
α0
X
Here, A (k ) is a k particle annihilation operator and α are indices for a k particle state.
α0 0 0 0 0
Note that A (k )transforms as {f } ={1N−k0} with respectto U(N) and A† (k ) transforms
α0 0 k0 α0 0
as {f }. It important to recognize that the O matrices will be rectangular matrices connecting
k0
m particle states to m−k particle states. In the defining space, the matrix will be a 1×d
0 0
matrix with matrix elements given by V . Note that α takes d values and d = N . We will
α0 0 0 0 k0
represent O by EGUE implying that the defining space matrix elements V are zero centered
α0 (cid:0) (cid:1)
independentGaussian random variables [also they are independentof the V (k) variables in Eq.
ij
(2) and therefore also independent of the W variables in Eq. (4)] with variance satisfying
V V† = V2 δ . (17)
α β O αβ
In many particle spaces the O matrix will be a d ×d matrix connecting d = N number of
1 2 1 m
m-particle states to d = N number of (m−k )-particle states. Using Eqs. (16) and (17),
2 m−k0 0 (cid:0) (cid:1)
(cid:0) (cid:1)
we have
m N −m
hO†Oim = V2 , hOO†im = V2 . (18)
O k O k
(cid:18) 0(cid:19) (cid:18) 0 (cid:19)
Similarly, Eq. (11) gives the relations,
hO†OHpim = hO†Oim hHpim , hO†HpOim = hO†Oim hHpim−k0 . (19)
Another useful result follows by introducing complete set of states between the O† and O
operators in Eq. (18) and applying the Wigner-Eckart theorem,
N −k
m || A†(k ) ||m−k hm−k || A(k ) ||mi = 0 . (20)
0 0 0 0
m−k
(cid:18) 0(cid:19)
D E
Following the procedure used in Section 2, it is possible to derive formulas for the lower order
moments of the transition strength densities generated by O defined by Eq. (16). The bivariate
moments are defined by
M = hO†HQOHPim . (21)
PQ
and we will consider the moments P +Q = 2 and 4 (the P +Q = 3 moments are zero as we are
using independent EGUE representations for O and H matrices).
Firstly, Eqs. (19) gives,
M =hO†Oim hH2im , M = hO†Oim hH2im−k0 ,
20 02 (22)
M =hO†Oim hH4im , M = hO†Oim hH4im−k0 .
40 04
Now, Eq. (18) along with Eqs. (12) and (14) will give the formulas for M , M , M and M .
20 02 40 04
Formula for the first non-trivial moment M =hO†HOHim is derived by introducing complete
11
set of states between O† and H, H and O and O and H in the trace giving,
N M (m) = N hO†HOHim = hm,v |O† |m−k ,v ihm−k ,v |O | m,v i
m 11 m 1 0 2 0 3 4
v1,v2,v3,v4
(cid:0) (cid:1) (cid:0) (cid:1) X
× hm−k ,v |H |m−k ,v ihm,v |H | m,v i.
0 2 0 3 4 1
(23)
Using Eq. (16) and applying Eq. (17) along with Eqs. (4) - (7) and the Wigner-Eckart theorem
will give,
k
M (m)= V2V2 N −1 N−k0 [Λν(N,m−k ,m−k −k)Λν(N,m,m−k)]1/2
11 O H m m−k0 0 0
ν=0 (24)
× C(cid:0)fk0(cid:1),α (cid:0) (cid:1)CXfk0,α Cν,ων Cν,ων .
fmv1,fm−k0v2 fm−k0v3,fmv4 fm−k0v2,fm−k0v3 fmv4,fmv1
v1,v2,vX3,v4;α;ων
Simplifying the four C-G coefficients will give finally,
k
M (m) = V2V2 N −1 N−k0 Z (N,m,k ,k,ν) ;
11 O H m m−k0 11 0
ν=0
Z (N,m,k ,k,ν)(cid:0)=(cid:1) N(cid:0) d(N(cid:1):Xν)Λν(N,m,m−k)Λν(N,m−k ,m−k −k) 1/2 (25)
11 0 k0 0 0
h i
(cid:0) (cid:1)
×(−1)φ(fm,fm−k0,fk0)+φ(fm−k0,fm−k0,ν) U(fmfm−k0fmfm−k0; fk0ν).
Here φ is a phase factor and it is a function of the U(N) irreps. It is shown elsewhere (V.K.B.
Kota and Manan Vyas, in preparation) that (−1)φ(fm,fm−k0,fk0)+φ(fm−k0,fm−k0,ν) U(−−−) will
be positive where U(−−) is a U(N) U-coefficient. Therefore we need only U2 and the formula
for this is given by [18],
N+1 2 m−ν N−ν−p (N −2ν +1)
2 ν p−ν m−p
U(f , f , f , f ; f ν) = . (26)
m p m p m−p (cid:0) (cid:1)N(cid:0)−m+(cid:1)p(cid:0)2 N (cid:1)(N +1)
p m−p
(cid:2) (cid:3)
(cid:0) (cid:1) (cid:0) (cid:1)
Turning to the fourth order moments, we need M , M and M . As O† 6= O, here
13 31 22
M 6= M [similarly M 6= M and M 6= M as seen from Eq. (22)]. Following the
13 31 40 04 20 02
procedure used for deriving the formula for M (m), we have for M (m)
11 31
M (m) = hO†HOH3im = 2 hH2im M (m)
31 11
+ V2 V2 N −1
O H m
(cid:0) (cid:1) v1,v2,v3,v4,v5,vX6:α,ν1,ων1,ν2,ων2
(27)
†
m,v |A (k ) |m−k ,v hm−k ,v |A (k ) |m,v i
1 α 0 0 2 0 3 α 0 4
D E
×hm−k0,v2 | Bν1,ων1(k) | m−k0,v3ihm,v5 | Bν1,ων1(k) |m,v6i
× hm,v4 |Bν2,ων2(k) | m,v5ihm,v6 |Bν2,ων2(k) | m,v1i .
Now, applying the Wigner-Eckart theorem, using the results in Section 2 and simplifying the
resulting C-G coefficients will give,
M (m) = hO†HOH3im = 2hH2im M (m)
31 11
min(k,m−k) (28)
+V2V2 N −1 N−k0 Λν(N,m,k) Z (N,m,k ,k,ν) .
O H m m−k0 11 0
ν=0
(cid:0) (cid:1) (cid:0) (cid:1) X
The function Z is defined in Eq. (25). Following the same procedure as above, the formula
11
for M is,
13
M (m) = hO†H3OHim = 2 hH2im−k0 M (m)
13 11
(29)
min(k,m−k0−k)
+V2V2 N −1 N−k0 Λν(N,m−k ,k) Z (N,m,k ,k,ν) .
O H m m−k0 0 11 0
ν=0
(cid:0) (cid:1) (cid:0) (cid:1) X
Derivation of the formula for M is more involved. Leaving details to a long paper under
22
preparation, the final result (with ρ a multiplicity label) is
M (m)= hO†H2OH2im = hO†Oim hH2im hH2im−k0
22
2
k
−1
+V2 V2 N N N−k0 Z (N,m,k ,k,ν)
O H m k0 m−k0 11 0
( )
n(cid:0) (cid:1)(cid:0) (cid:1)o (cid:0) (cid:1) Xν=0
k k 2k
+V2 V2 N −1 N−k0 N d(N : ν) (30)
O H(cid:0)m(cid:1) (cid:0)m−k0(cid:1) νX1=0νX2=0Xν=0s(cid:18)k0(cid:19)
× hm || [Bν1(k)Bν2(k)]ν:ρ || mi hm−k || [Bν1(k)Bν2(k)]ν:ρ || m−k i
0 0
ρ
X
× (−1)φ(fm,fm−k0,k0)+φ(fm−k0,fm−k0,ν) U(fmfm−k0fmfm−k0; fk0ν).
The moments M can be converted into reduced (scale free) cumulants k that gives
PQ PQ
information about the shape of the bivariate transition strength density. For our purpose the
first non-trivial cumulants are the fourth order cumulants and they are given by,
k = µ −3, k = µ −3,
40 40 04 04
k = µ −3ξ, k = µ −3ξ,
31 31 13 13
k22 = µ22−2ξ2−1 ; (31)
−1
P/2 Q/2
µ = M M M and M = M /M .
PQ 20 02 PQ PQ PQ 00
(cid:26) (cid:27)
h i h i
f f f f
The k , P +Q = 4 follow from Eqs. (12), (14), (18), (22),(25), (28), (29) and (30). Numerical
PQ
results for some typical values of (N,m,k,k ) are shown in Table 1. These results show that
0
in general |kPQ| <∼ 0.3 indicating that the bivariate strength density will be close to a bivariate
Gaussian. For further confirming this result, we will derive asymptotic results for k .
PQ
4. Asymptotic formulas for bivariate moments and approach to bivariate Gaussian
form
Here we will consider the asymptotic limit defined by N → ∞ with m, k and k fixed and k,
0
k << m. Note that in the dilute limit (or true asymptotic limit) we also have m → ∞ and
0
m/N → 0 with k and k fixed. Using the formulas given in Sections 2 and 3 first we can show
0
that in the asymptotic limit (asymp),
N −1 N−k0 Z (N,m,k ,k,k) = N d(N :k)Λk(N,m,m−k)Λk(N,m−k ,m−k −k) 1/2
m m−k0 11 0 k0 0 0
h i
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
× N −1 N−k0 U(f ,f ,f ,f ; f ,k) as−y→mp m N m−k ,
m m−k0 m m−k0 m m−k0 k0 k k k0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:12) asymp (cid:12) (cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)
h[H(k)]2im = Λ0(cid:12)(N,m,k) −→ m N , (cid:12)
k k
N −1Λk(N,m,m−k)Λk(N,m,k(cid:0))d(cid:1)((cid:0)N(cid:1):k) as−y→mp m m−k N 2 .
m k k k
(32)
(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)
Table 1. Bivariate correlation coefficient (ξ) and fourth order bivariate cumulants k with
rs
r ≥ s and r + s = 4 for various values of number of sp states (N), number of fermions (m),
Hamiltonianbodyrank(k)andtherank(k )oftheparticleremovaltransitionoperator. Results
0
are obtained using the formulas given in Section 3. Note that for the M that is needed for k ,
22 22
we have used Eq. (30) with the third term replaced by the corresponding asymptotic formula
given by Eq. (37) as a formula for the reduced matrix elements in Eq. (30) is not available.
N m k k ξ k k k k k
0 40 04 31 13 22
20 10 2 1 0.82 −0.54 −0.55 −0.44 −0.45 −0.21
30 10 2 1 0.85 −0.48 −0.50 −0.41 −0.43 −0.26
60 10 2 1 0.88 −0.42 −0.46 −0.37 −0.40 −0.30
80 10 2 1 0.88 −0.41 −0.45 −0.36 −0.39 −0.31
50 12 2 1 0.89 −0.38 −0.40 −.034 −0.36 −0.25
15 2 1 0.91 −0.33 −0.35 −0.30 −0.31 −0.19
20 2 1 0.92 −0.29 −0.29 −0.26 −0.27 −0.13
25 2 1 0.92 −0.27 −0.27 −0.25 −0.25 −0.08
24 8 2 1 0.82 −0.56 −0.61 −0.46 −0.49 −0.31
2 2 0.66 −0.56 −0.67 −0.37 −0.43 −0.22
40 15 2 1 0.90 −0.36 −0.37 −0.32 −0.33 −0.18
2 2 0.80 −0.36 −0.38 −0.29 −0.31 −0.12
60 20 2 1 0.93 −0.27 −0.27 −0.25 −0.25 −0.14
3 1 0.89 −0.51 −0.53 −0.46 −0.47 −0.30
3 2 0.79 −0.51 −0.54 −0.40 −0.43 −0.22
Starting with ξ, it should be clear that in the asymptotic limit only the term with ν = k in Eq.
(25) will survive. Then, applying the first two relations in Eq. (32) will give
M (m)
11
ξ(m) =
1/2
M (m) M (m)M (m)
00 20 02
h i (33)
m−k m 1f/2 f
asymp
−→ k0 k .
(cid:0)m (cid:1)m−(cid:0)k0(cid:1)1/2
k0 k
Similarly, for k and k only the te(cid:0)rm(cid:1)s(cid:0)with (cid:1)ν = k in Eq. (14) will survive and then applying
40 04
the second and third relations in Eq. (32) will give,
m−k
M (m) asymp
k (m) = 40 −3 −→ k −1 ,
40 2 m
M (m) (cid:0) k (cid:1)
f20
h i (cid:0) (cid:1) (34)
fM (m) asymp m−k0−k
k (m) = 04 −3 −→ k −1.
04 2 m−k0
Mf02(m) (cid:0) k (cid:1)
h i (cid:0) (cid:1)
For M31, the first term in Eq. (28)fis trivial and in the sum in the second term only the ν = k
term will survive in the asymptotic limit. Now, applying Eqs. (32) and (7) will give the result
for k (m),
31
m−k m−k
asymp
k (m) −→ k k0 −ξ(m)= ξ(m)k (m) . (35)
31 40
(cid:0) m(cid:1)(cid:0) m(cid:1)−k
m 0
k0 s k k
(cid:18) (cid:19)(cid:18) (cid:19)
(cid:0) (cid:1)
Similarly k (m) is given by,
13
m−k0−k m−k m 1/2
asymp
k (m) −→ k k0 k −ξ(m)= ξ(m)k (m) . (36)
31 04
(cid:0) m (cid:1)(cid:0)m−k0(cid:1)3(cid:0)/2(cid:1)
k0 k
(cid:0) (cid:1) (cid:0) (cid:1)
Finally, in M only the third term in Eq. (30) is complicated. This is simplified using its
22
relation, valid in the asymptotic limit, to ξ(m) as described in [10]. Following this we have for
k ,
22
m m−k 2 m−2k m−k
asymp
k (m) −→ −2[ξ(m)]2 + k k0 + k0 k
22 (cid:0)m−(cid:1)k(cid:0)0 m(cid:1)2 (cid:0) m (cid:1)m(cid:0)−k0 (cid:1)
k k0 k0 k
(37)
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
m−2k
m m−k
≈ −2[ξ(m)]2 + k0 + .
m m−k0 k k
k0(cid:0) k(cid:1) (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
In the dilute limit with m → ∞ an(cid:0)d(cid:1)m(cid:0)/N →(cid:1) 0 and expanding the binomials in Eqs. (33) to
(37), it is seen that to order 1/m the cumulants k , r+s = 4 will be −k2/m (independent of
rs
k ) and the correlation coefficient ξ(m)→ 1−(kk )/2m. Thus, the cumulants will tend to zero
0 0
giving bivariate Gaussian form. However, as ξ → 1 as m → ∞, in practice it is necessary to
add the k , r+s= 4 corrections to the bivariate Gaussian. This is same as the result seen for
rs
t-body transition operators before in [10, 19].
5. Extensions to particle addition operators and beta decay type operators
Firstly, particle addition operator is O+ = V A†(k ) and acting on a m-particle state it
α α α 0
will generate m+k particle states. It is easy to see that the formulation in Section 3 will apply
0
directly to O+ operator by appropriately chanPging everywhere m−k by m+k giving formulas
0 0
for ξ and M , r+s = 4. Explicit formulas are not given here due to lack of space.
rs
Nowwewillturntobetadecay typeoperatorandforthisconsiderasystemwithm fermions
1
in N sp states and m fermions in N sp states with H preserving (m ,m ). Then, the H
1 2 2 1 2
operator, assumed to be k-body, is given by,
H(k) = V (i,j) A†(i)A (i)A†(j)A (j) ,
αa:βb α β a b
i+Xj=kαX,β∈iaX,b∈j (38)
V (i,j) = hi,α :j,a | H | i,β :j,bi .
αa:βb
HereweareusingGreek labels α,β,... todenote themany particle states generated byfermions
occupying the orbit with N sp states and the Roman labels a,b,... for the many particle states
1
generated by the fermions occupying the orbit with N sp states. For each (i,j) pair with
2
i + j = k, we have a matrix V(i,j) in the k-particle space and we assume that the V(i,j)
matrices are represented by independent GUE’s with their matrix elements being zero centered
with variance,
Vαa:βb(i,j) Vα′a′:β′b′(i′,j′) = VH2(i,j) δii′δjj′δαβ′δab′δβα′δba′ . (39)