Entanglement in non-critical inhomogeneous quantum chains

TL;DR

This paper investigates entanglement properties in inhomogeneous quantum Ising chains, revealing how inhomogeneity affects ground states, entanglement entropy, and topological phases through analytical and RG techniques.

Contribution

It provides a detailed analysis of entanglement in inhomogeneous critical and non-critical Ising chains, including mappings to Majorana fermions and topological phase characterization.

Findings

Ground state forms concentric singlets in strong inhomogeneity

Entanglement entropy scales linearly in weak inhomogeneity

Identification of trivial and topological phases in non-critical regime

Abstract

We study an inhomogeneous critical Ising chain in a transverse field whose couplings decay exponentially from the center. In the strong inhomogeneity limit we apply Fisher's renormalization group to show that the ground state is formed by concentric singlets similar to those of the rainbow state of the XX model. In the weak inhomogeneity limit we map the model to a massless Majorana fermion living in a hyperbolic spacetime, where, using CFT techniques, we derive the entanglement entropy that is violated linearly. We also study an inhomogeneous non-critical Ising model that for weak inhomogeneity is mapped to a massive Majorana fermion, while for strong inhomogeneity regime it exhibits trivial and non-trivial topological phases and a separation between regions with high and low entanglement. We also present the entanglement Hamiltonian of the model.