Geometric phases and quantum phase transitions in open systems

The relationship is established between quantum phase transitions and complex geometric phases for open quantum systems governed by a non-Hermitian effective Hamiltonian with accidental crossing of the eigenvalues. In particular, the geometric phase associated with the ground state of the one-dimensional dissipative Ising model in a transverse magnetic field is evaluated, and it is demonstrated that the related quantum phase transition is of the first order.

Figure 1

(Color online) Left panel $(\epsilon =0)$: Clear step function behavior of the geometric phase at the diabolic point $r=z=0$. Right panel: $\mathrm{Re}\gamma$ near the exceptional point $(\epsilon =0.5)$.

Figure 2

(Color online) Real part of the overall geometric phase ${\gamma }_g$ (left) and imaginary part of the overall geometric phase ${\gamma }_g$ (right) versus $\delta$ and $h$.

Figure 3

(Color online) Real part of derivative of the overall geometric phase $\partial {\gamma }_g∕\partial h$ (left) and its imaginary part (right) versus $\delta$ and $h$.