Some Recent Advances in Density Matrix Theory

J. Cooper

In recent years the theory of line broadening has been considerably advanced through the generalized impact approximation, which has as its starting point an expression for the quantum-mechanical autocorrelation function. In this paper the results for isolated lines in the impact approximation are rederived using a density matrix formulation. The method presented here is not particularly rigorous; however, it assists in the understanding of the approximations made in the autocorrelation function approach, and in some cases possible methods of extending the theory become much more apparent. For example, in this formulation, the broadening of an electric quadrupole transition is obvious.

K. Husimi, Yoshihiro Kitano, Toshiyuki Nishiyama

Karl Blum

Stefano Scopa, Lorenzo Piroli, Pasquale Calabrese

Abstract We provide a thorough characterisation of the zero-temperature one-particle density matrix of trapped interacting anyonic gases in one dimension, exploiting recent advances in the field theory description of spatially inhomogeneous quantum systems. We first revisit homogeneous anyonic gases with point-wise interactions. In the harmonic Luttinger liquid expansion of the one-particle density matrix for finite interaction strength, the non-universal field amplitudes were not yet known. We extract them from the Bethe Ansatz formula for the field form factors, providing an exact asymptotic expansion of this correlation function, thus extending the available results in the Tonks–Girardeau limit. Next, we analyse trapped gases with non-trivial density profiles. By applying recent analytic and numerical techniques for inhomogeneous Luttinger liquids, we provide exact expansions for the one-particle density matrix. We present our results for different confining potentials, highlighting the main differences with respect to bosonic gases.

Yangyi Lu, Jiali Gao

On the basis of recent advancements in the Hamiltonian matrix density functional for multiple electronic eigenstates, this study delves into the mathematical foundation of the multistate density functional theory (MSDFT). We extend a number of physical concepts at the core of Kohn-Sham DFT, such as density representability, to the matrix density functional. In this work, we establish the existence of the universal matrix functional for many states as a proper generalization of the Lieb universal functional for the ground state. Consequently, the variation principle of MSDFT can be rigorously defined within an appropriate domain of matrix densities, thereby providing a solid framework for DFT of both the ground state and excited states. We further show that the analytical structure of the Hamiltonian matrix functional is considerably constrained by the subspace symmetry and invariance properties, requiring and ensuring that all elements of the Hamiltonian matrix functional are variationally optimized in a coherent manner until the Hamiltonian matrix within the subspace spanned by the lowest eigenstates is obtained. This work solidifies the theoretical foundation to treat multiple electronic states using density functional theory.

Karl Blum