We review the properties of reduced density matrices for free fermionic or bosonic many-particle systems in their ground state. Their basic feature is that they have a thermal form and thus lead to a quasi-thermodynamic problem with a certain free-particle Hamiltonian. We discuss the derivation of this result, the character of the Hamiltonian and its eigenstates, the single-particle spectra and the full spectra, the resulting entanglement and in particular the entanglement entropy. This is done for various one-and two-dimensional situations, including also the evolution after global or local quenches.
We compute the single-particle spectral density, susceptibility near the Kohn anomaly, and pair propagator for a one~ensional interacting-electron gas. With an attractive interaction, the pair propagator is divergent in the zero-temperature limit and the Kohn singularity is removed. For repulsive interactions, the Kohn singularity is stronger than the free-particle case and the pair propagator is finite. The low-temperature behavior of the interacting system is not consistent with the usual Ginzburg-Landau functional because the frequency, temperature, and momentum dependences are characterized by power-law behavior with the exponent dependent on the interaction strength.Similarly, the energy dependence of the single-particle spectral density obeys a power law whose exponent depends on the interaction and exhibits no quasiparticle character. Our calculations are exact for the Luttinger or Tomonaga model of the one-dimensional interacting system.
We consider noninteracting fermions on a lattice and give a general result for the reduced density matrices corresponding to parts of the system. This allows to calculate their spectra, which are essential in the densitymatrix renormalization group method, by diagonalizing small matrices. We discuss these spectra and their typical features for various fermionic quantum chains and for the two-dimensional tight-binding model.
͑3͒The quantities ⌳ k 2 are the eigenvalues of the matrices (AÀB)(A¿B) and (A¿B)(AÀB), the corresponding eigenvectors being ki ϭg ki ϩh ki and ki ϭg ki Ϫh ki ,respectively.Consider now the ground state ͉⌽ 0 ͘ of the Hamiltonian ͑1͒ for an even number of sites L. Due to the structure of H,
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