Entanglement in a simple quantum phase transition
Tobias J. Osborne, Michael A. Nielsen
TL;DR
The paper investigates entanglement structure in the 1D XY model and the transverse Ising model, solving exactly via Jordan-Wigner to obtain one- and two-site density matrices. It analyzes ground-state entanglement (single-site von Neumann entropy and two-site concurrence) and how it peaks or changes at the quantum critical point, with symmetry breaking and degeneracy shaping the results, and extends to thermal entanglement showing a quantum critical region where entanglement persists at finite temperature. The findings indicate that criticality is associated with distributed, scale-spanning entanglement and highlight constraints from entanglement sharing, suggesting avenues toward universal measures of entanglement in critical quantum systems and future work on dynamics and broader parameter regimes.
Abstract
What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems which can be solved. An example of such a system is the 1D infinite-lattice anisotropic XY model. This model is exactly solvable using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest neighbour entanglement (though not the nearest-neighbour entanglement) is a maximum at the critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a transition in the behaviour of the entanglement between a single site and the remainder of the lattice.