On the Averages of Operators in Finite Fermion Systems

Masao Nomura

It is shown that known reduction formulas for partial averages of two different kinds, the fixed-seniority and the fixed-isospin averages, are transcribed into each other by replacement of arguments. The average here is that of a general k-body operator in a finite space of fermions (bosons). Description of the formulas in a common form in terms of eigenvalues of Casimir operators is also exploited.

J. I. Horváth, I.K. Gyémánt

J.B. French, Jerry Paul Draayer

Siew Ann Cheong, Christopher L. Henley

The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors ${\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})$ and ${\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction $\overline{n}$, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.

J. B. French, J. P. Draayer

Dmitri A. Ivanov

It is known that computing the permanent of the matrix 1+A, where A is a finite-rank matrix, requires a number of operations polynomial in the matrix size. Motivated by the boson-sampling proposal of restricted quantum computation, I extend this result to a generalization of the matrix permanent: An expectation value in a product of a large number of identical bosonic states with a bounded number of bosons. This result complements earlier studies on the computational complexity in boson sampling and related setups. The proposed technique based on the Gaussian averaging is equally applicable to bosonic and fermionic systems. This also allows us to improve an earlier polynomial complexity estimate for the fermionic version of the same problem.