Reduced density matrices and entanglement entropy in free lattice models

TL;DR

This work shows that reduced density matrices for free lattice models adopt a Boltzmann-like form governed by an effective quadratic Hamiltonian, allowing entanglement properties to be read off from single-particle spectra. It develops and compares three practical methods to obtain these spectra, analyzes 1D and 2D geometries, and connects spectral features to entanglement measures such as the entropy and the entanglement spectrum. The paper further investigates how these spectra and entropies evolve under global, local, and periodic quenches, revealing light-cone spreading, plateau formation, and regime-dependent growth, with clear implications for DMRG efficiency and quantum quench physics. Overall, it provides a coherent framework linking spectral structure, interface geometry, and dynamical entanglement in non-interacting lattice systems.

Abstract

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.

Figures (18)

• Figure 1: Left: Density matrices for a quantum chain as two-dimensional partition functions. Far left: Expression for $\rho$. Half left: Expression for $\rho_1$. The matrices are defined by the variables along the thick lines. Right: Two-dimensional system built from four quadrants with corresponding corner transfer matrices $A,B,C,D$. The arrows indicate the direction of transfer. After Ref. Greifswald08.
• Figure 2: Level spacing as a function of the parameter $k$.
• Figure 3: Density-matrix spectra for one-half of a transverse Ising chain with $N=20$ sites in its ground state. Left: All ten single-particle eigenvalues $\varepsilon_l$. Right: The largest total eigenvalues $w_n$. Reprinted with permission from Chung/Peschel01. © 2001 by the APS.
• Figure 4: Size dependence of the density-matrix spectra in a critical system. Shown are results for segments of different lengths in an infinite hopping model. Left: Single-particle eigenvalues $\varepsilon_l$. Right: Total eigenvalues $w_n$. After Ref. Greifswald08.
• Figure 5: Single-particle spectra for different ground states. Left: Variation with the filling. Right: Variation with the number of equal-size Fermi seas at half filling. All results are for a segment of $L= 20$ sites in an infinite hopping model.