Particle diagrams and embedded many-body random matrix theory

We present a method which uses Feynman-like diagrams to calculate the statistical quantities of embedded many-body random matrix problems. The method provides a promising alternative to existing techniques and offers many important simplifications. We use it here to find the fourth, sixth, and eighth moments of the level density of an $m$-body system with $k$ fermions or bosons interacting through a random Hermitian potential $\left(k\le m\right)$ in the limit where the number of possible single-particle states is taken to infinity. All share the same transition, starting immediately after $2k=m$, from moments arising from a semicircular level density to Gaussian moments. The results also reveal a striking feature; the domain of the $2n$th moment is naturally divided into $n$ subdomains specified by the points $2k=m,3k=m,...,nk=m$.

Figure 1

(a) illustrates the particle diagram implied by $A_{\mu \nu \rho \sigma }=\langle \mu |a_{\mathbf{j}}^{†}{a}_{\mathit{i}}|\sigma \rangle \langle \rho |a_{\mathit{i}}^{†}{a}_{\mathbf{j}}|\nu \rangle$ of (2) and (5), and (b) illustrates the particle diagram implied by the factor $A_{\sigma \mu \nu \rho }$ in (5). Each bond between compound states represents a set of single-particle states shared by both of the compound states.

Figure 2

(a) illustrates the particle diagram for the term $6A_{ptqq}\left(A_{twvu}{A}_{upwv}\right)$. This is almost identical to Fig. 1 but with the equivalent of Figs. 1 and 1 juxtaposed on the same diagram. The additional tail adds a factor $\left(\begin{array}{c}m\\ k\end{array}\right)\left(\begin{array}{c}l-m+k\\ k\end{array}\right)$. (b) illustrates the particle diagram of $3A_{putq}{A}_{qwvt}{A}_{upwv}$, where all the bonds implied by the expression are illustrated in a single three-dimensional (3D) triangular prism. Using the diagram it becomes simpler to identify the single-particle states which must overlap maximally in order to maximize the argument of the whole sum.

Figure 3

Plot of the factors $C_4=\left(\begin{array}{c}m-k\\ k\end{array}\right)/\left(\begin{array}{c}m\\ k\end{array}\right)$, $C_6=\left(\begin{array}{c}m-2k\\ k\end{array}\right)/\left(\begin{array}{c}m\\ k\end{array}\right)$, and $C_8=\left(\begin{array}{c}m-3k\\ k\end{array}\right)/\left(\genfrac{}{}{0pt}{}{m}{k}\right)$ for $m=15$. The term with coefficient 1 for the fourth moment is $C_4$ (10), for the sixth moment is $C_4{C}_6$ (17) and for the eighth moment is $C_4{C}_6{C}_8$ (18). These coefficients illustrate the increasing degree to which the domain of the $2n$th moment is partitioned for increasing $n$. For $k/m>\frac{1}{2}$ the moments are those of a semicircle and for $k=0$ they are those of a Gaussian.