Entanglement Entropy and Spectra of the One-dimensional Kugel-Khomskii Model

TL;DR

This study investigates the entanglement properties of the one-dimensional Kugel-Khomskii model, revealing new features in entanglement spectra, including robust gaps and breakdown of the virtual edge picture, and refines the phase diagram using entanglement analysis.

Contribution

The paper introduces novel insights into the entanglement spectra of the Kugel-Khomskii model, highlighting features not seen in previous studies and challenging existing theoretical frameworks.

Findings

Robust entanglement spectrum gaps in both gapped and gapless phases.

Breakdown of the virtual edge picture for this model.

Qualitative differences in momentum-space entanglement spectra compared to Heisenberg chain.

Abstract

We study the quantum entanglement of the spin and orbital degrees of freedom in the one- dimensional Kugel-Khomskii model, which includes both gapless and gapped phases, using analytical techniques and exact diagonalization with up to 16 sites. We compute the entanglement entropy, and the entanglement spectra using a variety of partitions or "cuts" of the Hilbert space, including two distinct real-space cuts and a momentum-space cut. Our results show the Kugel-Khomski model possesses a number of new features not previously encountered in studies of the entanglement spectra. Notably, we find robust gaps in the entanglement spectra for both gapped and gapless phases with the orbital partition, and show these are not connected to each other. We observe the counting of the low-lying entanglement eigenvalues shows that the "virtual edge" picture which equates the low-energy Hamiltonian of a virtual edge, here one gapless leg of a two-leg ladder, to the "low-energy" entanglement Hamiltonian breaks down for this model, even though the equivalence has been shown to hold for similar cut in a large class of closely related models. In addition, we show that a momentum space cut in the gapless phase leads to qualitative differences in the entanglement spectrum when compared with the same cut in the gapless spin-1/2 Heisenberg spin chain. We emphasize the new information content in the entanglement spectra compared to the entanglement entropy, and using quantum entanglement present a refined phase diagram of the model. Using analytical arguments, exploiting various symmetries of the model, and applying arguments of adiabatic continuity from two exactly solvable points of the model, we are also able to prove several results regarding the structure of the low-lying entanglement eigenvalues.