Unifying geometric entanglement and geometric phase in a quantum phase transition

Geometric measure of entanglement and geometric phase have recently been used to analyze quantum phase transition in the $XY$ spin chain. We unify these two approaches by showing that the geometric entanglement and the geometric phase are respectively the real and imaginary parts of a complex-valued geometric entanglement, which can be investigated in typical quantum interferometry experiments. We argue that the singular behavior of the complex-value geometric entanglement at a quantum critical point is a characteristic of any quantum phase transition, by showing that the underlying mechanism is the occurrence of level crossings associated with the underlying Hamiltonian.

Figure 1

Derivative of the ground-state GE density $\varepsilon (r,h)$ of the $XY$ model as a function of the magnetic field $h$ in the thermodynamic limit. Solid curve: Ising limit ($r=1$); dashed curve: anisotropic ($r=0.5$) $XY$ model; and dot-dashed curve: anisotropic ($r=0.05$) $XY$ model. The inset corresponds to the Ising limit for different finite lattice sizes $N=11,18,31,64$.

Figure 2

Top panel: Derivative of the GP density, GP per particle, corresponding to cyclic evolution of the $XY$ ground state given in Eq. (8), as a function of the magnetic field $h$ in the thermodynamic limit. Lower panel: $h$ derivative of the difference between GP densities corresponding to the evolutions given in Eqs. (8) and (9), plotted as a function of magnetic field $h$ in the thermodynamic limit. Solid curve: Ising limit ($r=1$); dashed curve: anisotropic ($r=0.5$) $XY$ model; dot-dashed: anisotropic ($r=0.05$) $XY$ model. The insets correspond to the Ising limits for different finite lattice sizes $N=11,18,31,64$.

Figure 3

Paths $C_{\Psi }$ and $C_{\Phi }$ of many-body state $\Psi$ and its closest product state $\Phi$, respectively. $G$ is a geodesic in state space projecting $\Psi$ onto $\Phi$.