Method for calculating operator traces

Abstract

A new method is presented for the calculation of operator traces in reasonably large ($N\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{10}^{4}$) model spaces. The method is based on the approximation of a multiparticle vector $|s〉$, which simultaneously satisfies the conditions $〈s|{O}_{i}|s〉=\mathrm{Tr}{O}_{i}$ for a given set of commuting operators ${{O}_{i}}$, including the Hamiltonian $H$. An algorithm is presented for finding such an approximation to $|s〉$, which we call $|\overline{s}〉$. To illustrate the method we apply it to the calculation of $\mathrm{Tr}{H}^{n}$ and $\mathrm{Tr}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}}^{2} {H}^{n})$, $n=0 \mathrm{to} 8$, in the model space of five nucleons in the $\mathrm{sd}$ shell ($^{21}\mathrm{Ne}$; $T=\frac{1}{2}$) using the Chung-Wildenthal matrix elements. The calculated results agree with exact diagonalization results to within a few percent. The method is economical in that it requires the storage of only the array $|\overline{s}〉$ and the relatively small arrays for the operators ${{O}_{i}}$. It is also particularly appropriate for bit representation of both operators and vectors in terms of a Fock-space basis. An important characteristic of this method is the ease with which submanifolds of a particular symmetry and/or configuration may be projected out for study. This feature was used to calculate configuration traces for higher moments in the $^{21}\mathrm{Ne}$ five-nucleon problem as well as to project out the $T=\frac{1}{2}$ manifold. The results in this case demonstrate the inadequacy of low moment truncations in configuration level density calculations.NUCLEAR STRUCTURE Calculated level density of $^{21}\mathrm{Ne}$ using spectral distribution expansion; a new method for evaluating operator moments; results compared with values obtained from exact diagonalization.