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SU(N) Wigner-Racah algebra for the matrix of second moments of embedded Gaussian unitary ensemble of random matrices

Recently Pluhar and Weidenmueller [Ann. Phys. (N.Y.) Vol 297, 344 (2002)] showed that the eigenvectors of the matrix of second moments of embedded Gaussian unitary ensemble of random matrices generated by k-body interactions (EGUE(k)) for m fermions in N single particle states are SU(N) Wigner coefficients and derived also an expression for the eigenvalues. Going beyond this work, we will show that the eigenvalues of this matrix are square of a SU(N) Racah coefficient and thus the matrix of second moments of EGUE(k) is solved completely by SU(N) Wigner-Racah algebra.


I. INTRODUCTION
Interacting finite quantum systems such as nuclei, atoms, quantum dots, nanometer-scale metallic grains etc. are governed by hamiltonians of low (compared to the number of particles in the system) particle rank. Therefore, for many purposes, the random matrix models appropriate for these systems are embedded random matrix ensembles of k-body interactions originally introduced, via nuclear shell model calculations, by French and Wong and Bohigas and Flores [1]. For a system of m spinless fermions in N single particle states (we will use fermions throughout this paper and turn briefly to bosons at the end) the embedded Gaussian unitary ensemble of random matrices of k-body interactions [EGUE(k)] is generated by defining the hamiltonian H, which is given to be k-body, to be GUE in the k-particle spaces and then propagating it to the m particle spaces by using the geometry (direct product structure) of the m-particle spaces. Just as EGUE(k), the EGOE(k) and other embedded ensembles are defined [2]. With m particle space dimension given by N m = N m , one has the unitary groups SU(N), U(N k ) and U(N m ) with EGUE(k) invariant under U(N k ) and the embedding in m-particle spaces is defined by SU(N); note that a GUE in m particle spaces is invariant under U(N m ) but not the EGUE(k), k < m.
Very early, using the so called binary correlation approximation, Mon and French [3] and later French et al [4] derived some analytical properties of embedded ensembles valid in the dilute limit (defined by (N, m, k) → ∞, m/N → 0 and k/m → 0). However only recently rigorous analytical results, valid for any (N, m, k) are derived for these ensembles by Benet et al. [5] and very soon Pluhar and Weidenmüller (hereafter called PW) demonstrated [6] that these results indeed follow from considerations based on the SU(N) embedding algebra. With all the m-particle matrix elements being linear combinations of the k-particle matrix elements (see Eq. (9) ahead), the joint distribution for the matrix elements will be a multivariate Gaussian. Thus all the information about EGUE(k) is in the covariance matrix or the matrix of second moments (Eq. (10) ahead). PW have shown that the eigenvectors of this matrix are SU(N) Wigner (or Clebsch-Gordon (CG)) coefficients and derived the expression for their eigenvalues using a duality relation for EGUE(k). The purpose of this paper is to show that the eigenvalues can be written as SU(N) Racah coefficients and thus the matrix of second moments is solved completely by SU(N) Wigner-Racah algebra.
To this end results for SU(N) Racah coefficients given in [7] and [8,9] are used. We will start with some basic results given in PW paper.

II. BASIC DEFINITIONS AND RESULTS
Let us begin with m particles in N single particle states (unfortunately in PW ℓ is used in place of N but to keep the notations same as in our earlier papers [2,10], we use N). The single particle (sp) creation operator a † i for any i-th sp state transforms as the irreducible representation {1} of U(N) and similarly a product of r creation operators transform, as we have fermions, as the irrep {1 r } in Young tableaux notation. Let us add that a U(N) irrep {λ 1 , λ 2 , · · · , λ N } defines the corresponding also other equivalent ways of defining SU(N) irreps given a U(N) irrep [6]). This U(N) ↔ SU(N) correspondence is used throughout and therefore we use U(N) and SU(N) interchangeably. In PW, {1 r } is denoted by f r and we will follow this notation from now on. With v r denoting irreps (and other multiplicity labels) of the groups in a subgroup chain of U(N) that supply the labels needed for a complete specification of any m-particle state (for the purpose of the present paper the subgroup chain need not be specified), the operator r i=1 a † i and a normalized r-particle creation operator A † (f r v r ) behave as the SU(N) tensors T frvr and 1 √ r! T frvr respectively. Using the composition formula, where can be written as a product of k and s = m − k particle states as, Some properties of the CG coefficients, used later in simplifications, are In (3), φ is a function that defines the phase for the 1 ↔ 2 interchange in the CG coefficients, d(f ) is the dimension of the irrep f and f is the irrep conjugate to f .
and it also contains a phase factor as given in Eq. (53) of [7] (this is also seen easily by comparing the second and third equalities in (3) with the corresponding relations for the standard CG coefficients for angular momentum [11]). Similarly with Note that g 0 = {0} for SU(N) and also g ν = g ν . The function φ in Eq. (3) is of the where F is some function with F (λ) an integer and F (λ) = F (λ); for SU(N) irreps that appear in this paper these results are valid as can be seen from Eq. (60) of [7]) except that there can be an overall N dependent factor which will not change any of the final results.
It is useful to note that the tensors B's in (4) multiplied by k! are, to within a phase factor, same as the tensors defined in Eq. (48) of [7]. Wigner-Eckart theorem decomposes the matrix elements of B k (g ν ω ν ) in m-particle spaces into a reduced matrix element || || and a CG coefficient, Two important properties of B k (g ν ω ν ) are, In (6), k denotes trace over the k-particle spaces. The first equality in Eq. (6) easily follows from the fact that here g ν = {0} as traces are scalars with respect to SU(N) and then applying the fifth equality in Eq. (3). Similarly the second equality in Eq. (6) follows from the fact that only g ν = g µ will give a scalar. Now we will turn to EGOE(k).
Consider a k-body hamiltonian where V vav b (k) are matrix elements of H(k) in k-particle space and form a GUE, i.e.
V vav b (k) are independent Gaussian variables with zero center and variance given by In Eq. (8) the overline indicates ensemble average and λ 2 is ensemble averaged variance of H(k) in k-particle space. The m-particle matrix elements of H(k) are, with (9) Eq. (9) is obtained easily by substituting the definition (7) for H(k), then inserting complete set of states between A † and A operators and applying Eq. (2). The EGUE(k) in m ≥ k spaces is defined by Eqs. (7)- (9). Now it is clear that for any analysis of EGUE(k) all one need to know is the covariance between any two m-particle matrix elements H v 1 m v 2 m (k) and this defines the matrix of second moments, As stressed by PW, most important step in EGUE(k) analysis is to derive a "generalized eigenvalue expansion" of A defined by A ij = k C ik E k C jk with E k the eigenvalues and C jk the eigenvectors such that E k are positive and C's hermitian. To this end, it is useful to consider the unitary decomposition of H(k) in terms of the The expansion coefficients W gν ων (k) are easily given by 12) and this follows by using the definition (11) and Eq. (6). Most significant property of the W coefficients is that they are independent Gaussian variables with zero center and variance given by (derived using Eqs. (11), (4), (7) and (8) in that order and using the orthonormal properties of the CG coefficients),

III. MATRIX OF SECOND MOMENTS
Firstly we will derive an expression for the covariance H v 1 m (k) in terms of SU(N) CG coefficients and Racah coefficients and then turn to the eigenvalues and eigenvectors of A, the matrix of second moments. Applying Eqs. (10), (11) and (5) in that order gives, Eqs. (55) and (56) of [7] together with Eqs. (3) (see also the remark just after Eq. (4)) allows one to write the reduced matrix element in Eq. (14) as a SU(N) Racah or U-coefficient, Before going further let us define, for a given (m, N), a function Λ ν (k), Now, using Eq. (16) and substituting the expression given by Eq. (61) of [7] for the U-coefficient in (15), it is seen that Combining (14) with (18) yield an expression for the covariance between H matrix elements in m-particle spaces, Eq. (19) will be useful in deriving expressions for the moments of H spectrum, i.e.
H p m . However the disadvantage of (19) is that it is not in a proper form to give the eigenvalues and eigenvectors of the matrix A in Eq. (10). In order obtain them, the CG coefficients in Eq. (14) should be changed to C Then Eqs. (20) and (3) will give, Finally Eqs. (21) and (15) combined with (14) produce the generalized eigenvalue expansion of the matrix of second moments A, Obviously the quantity in the curly brackets in Eq. (22) gives the eigenvalues of A and the C's are eigenvectors.

APPLICATIONS
In order to proceed further, it is useful to consider 6j symbols of SU(N) and they are defined by (see Eq. (3.2.18) of [9]), In (23) λ's are SU(N) irreps and the four couplings involved in the U-coefficient are assumed to be multiplicity free (for the applications in the present paper this assumption is always valid). Symmetry properties of the 6j-symbol appearing on the r.h.s of (23) are well known [8,9]. In the present analysis, the Biedenharn-Elliott sum rule extended to SU(N) [8,9] plays a central role. This sum rule relates a product of three Racah coefficients (weighted appropriately by dimension factors and phase factors with the irreps in the Racah coefficients appearing in some particular order) with sum over a common irrep label to a product of two Racah coefficients. After converting the Racah coefficients in (22) into 6j symbols of SU(N) using Eq. (23) and then applying the symmetry properties of the 6j symbols, it is seen that the sum in the square brackets in Eq. (22) is exactly in the required form. Applying the Biedenharn-Elliott sum rule, the sum then simplifies to Here µ = 0, 1, . . . , m − k. Now the eigenvalues of the matrix A, in terms of the U(N)

Racah coefficients is given by
with degeneracy d(g µ ) (see Eq. (16)). Eq. (24) is the central result of this paper.
With this, the matrix A is completely specified by the U(N) Wigner and Racah coefficients. Now substituting the formula (Eq. (61)) of [7]) for the U-coefficients in Eq. (24) produces the result of PW, where, Note that the function Λ µ (k) is defined by Eq. (17).
Information about EGUE(k) is contained in the ensemble averaged moments M p = H p m and the bivariate moments Σ pq = H p m H q m . In deriving the formulas for the lower order moments, we will show the usefulness of Eq. (19). Obviously, ensemble averaged centroid is zero and the variance is The second equality follows from (25) and the final result follows by applying (3); (27) is in λ 2 /N k units and this factor is dropped as all the quantities we consider from now on are all scaled with respect to { H 2 m } 1/2 . As the third moment is zero, we will turn to the fourth moment, The second equality in Eq. (28) follows by applying Eqs. (11) and (13). In the third equality, it is easy to recognize the first term. The second term follows by applying Eq. (19) and (25) to the two pieces in the corresponding term in the second equality.
The final result follows by applying the orthonormality of the CG coefficients. Eqs.
(27,28) will give the excess (γ 2 ) parameter of the density of eigenvalues of EGUE(k), Eqs. (27,29,31,33) are also given by Benet et al [5]; this paper neither gives details of the derivations nor uses SU(N) Racah coefficients. Also in this work,Σ 11 ,Σ 22 and γ 2 + 1 are denoted by S, R and Q respectively while we have followed Ref. [3]. The two different forms given by Eqs. (19,25) for the covariances of m-particle H matrix elements, give in a simple manner the formulas for the low order moments M p that define the state density and the bivariate moments Σ pq that give information about fluctuations.