Global entanglement and quantum criticality in spin chains

The entanglement of quantum $XY$ spin chains of arbitrary length is investigated via a recently developed global measure suitable for generic quantum many-body systems. This entanglement is determined over the phase diagram and found to exhibit rich structure. In particular, the field derivative of the entanglement density becomes singular along the critical line. The form of this singularity is dictated by the universality class controlling the quantum phase transition.

Figure 1

(Color online) Entanglement density ${\mathcal{E}}_N$ (upper) and phase diagram (lower) vs $(r,h)$ for the $XY$ model with $N={10}^4$ spins, which is essentially in the thermodynamic limit. There are three phases: $O$: ordered oscillatory, for $r^2+h^2<1$ and $r\ne 0$; $F$: ordered ferromagnetic, between $r^2+h^2>1$ and $h<1$; $P$: paramagnetic, for $h>1$. As is apparent, there is a sharp rise in the entanglement across the line $h=1$, which signifies a quantum phase transition. The arc $h^2+r^2=1$, along which the entanglement density is zero (see also Fig. 2), separates phases $O$ and $F$. Along $r=0$ lies the $XX$ model, which belongs to a different universality class from the anisotropic $XY$ model.

Figure 2

(Color online) Upper panel: Entanglement density and its $h$ derivative (inset) for the ground state of three systems at $N=\infty$. Solid blue line: Ising $(r=1)$ limit; dashed black line: anisotropic $(r=1∕2)$ $XY$ model; dash-dotted red line: $(r=0)$ $XX$ model. For the sake of clarity, the $XY$-case curves are shifted to the right by 0.5, indicated by the green arrow. For the $r=1∕2$, at $h^2+r^2=1$ the entanglement density vanishes, which is a general property for the anisotropic $XY$ model. Note that whilst the entanglement itself has a nonsingular maximum at $h\approx 1.1$ (Ising), $h\approx 1.04$ ($XY$ $r=1∕2$), $h=0$ $\left(XX\right)$, respectively, it has a singularity at the quantum critical point at $h=1$, as revealed by the divergence of its derivative. Lower panel: The solutions of the single rotation parameter $\xi$ for the ansatz state vs $h$.