Geometric Phases and Criticality in Spin-Chain Systems

A relation between geometric phases and criticality of spin chains is established. As a result, we show how geometric phases can be exploited as a tool to detect regions of criticality without having to undergo a quantum phase transition. We analytically evaluate the geometric phase that corresponds to the ground and excited states of the anisotropic $XY$ model in the presence of a transverse magnetic field when the direction of the anisotropy is adiabatically rotated. It is demonstrated that the resulting phase is resilient against the main sources of errors. A physical realization with ultracold atoms in optical lattices is presented.

Figure 1

(a) The regions of criticality are presented as a function of the Hamiltonian parameters $\lambda$, $\gamma \cos 2\varphi$, and $\gamma \sin 2\varphi$. The geometric phase corresponding to the ground state (b) and the relative one between the ground and excited state (c) as a function of the path parameters $\lambda$ and $\gamma$. Values of the geometric phase corresponding to the loops $C_1$, $C_2$, and $C_3$ in (a) are also indicated.