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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.

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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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We study free electrons on an infinite half-filled chain, starting in the ground state with a bond defect. We find a logarithmic increase of the entanglement entropy after the defect is removed, followed by a slow relaxation towards the value of the homogeneous chain. The coefficients depend continuously on the defect strength.

The properties of the reduced density matrix describing an interval of N sites in an infinite chain of free electrons are investigated. A commuting operator is found for arbitrary filling and also for open chains. For a half filled periodic chain it is used to determine the eigenfunctions for the dominant eigenvalues analytically in the continuum limit. Relations to the critical six-vertex model are discussed.Reduced density matrices for some portion of a larger system play a central role in the density-matrix renormalization group (DMRG) method [1][2][3]. For this reason, they have been investigated for a number of model systems in the last years [4][5][6][7][8]. It was found that for free electrons or bosons the reduced density matrices have exponential form exp(−H) and thus look like thermal operators. It was also realized that one can base the determination of H on the one-particle correlation functions in the state one is studying [7,[9][10][11]. However, for critical lattice systems, which are particularly interesting in view of DMRG applications, the properties of H have only been studied numerically so far. For free electrons hopping on a chain one can see in this way the typical finite-size effects on the spectra and the concentration of the eigenfunctions near the boundaries. Nevertheless, a more explicit solution, which also makes contact with some results found in field theory [12] in calculations of entanglement entropies, would be desirable.In the following we present such a solution by looking at the hopping model in a somewhat different way. We show that there exists a relatively simple operator T which commutes with the reduced density matrix for the ground state and thus has the same eigenfunctions. This holds for arbitrary filling and also for a subsystem at the end of a semi-infinite chain. The case of a half-filled periodic chain is then studied in detail. Working with T and taking a proper continuum limit allows to determine the general character of the single-particle eigenstates. For the low-lying states it is found that the eigenvalues of H and T even coincide up to a scale factor. We also discuss the connection with the critical six-vertex model. This allows to derive the spectrum of H by a conformal mapping. It also allows to view the density matrix as a particular transfer matrix and the commutation relation as a special case of similar relations which occur in the treatment of integrable two-dimensional models.