Quantum phase transitions exposed by rotating the spins

Using the exactly solvable $XY$ spin chain as an example, we consider nonadiabatic variation of the Hamiltonian along an isospectral trajectory. We suggest that quantum phase transitions (QPTs) can be revealed by the nonadiabatic geometric or Aharonov-Anandan phase, accumulated in a cycle of the state that starts from a particular initial state. On the other hand, starting as the ground state of the instantaneous Hamiltonian, the state does not return to the initial state if the variation of the Hamiltonian is not adiabatic, but the survival probability can indicate QPTs and display revival phenomena.

Figure 1

SP as a function of the parameters $\lambda$ and $\gamma$, for $\Omega =0.01$ and $N=2001$.

Figure 2

SP as a function of $\gamma$ for various values of $\lambda$. $\Omega =0.01,N=2001$.

Figure 3

SP for $\lambda =0$. (a) $\gamma$ dependence of the SP for $N=2001$ and $\Omega =0.002,0.005,0.01,0.02$. (b) $\gamma$ dependence of the SP for $\Omega =0.01$ and $N=100001$, 10001, 2001, 501. The insets enlarge the regimes close to the quantum critical points.

Figure 4

Ising criticality (with $\gamma =1$). (a) $\lambda$ dependence of SP for $N=2001$ and $\Omega =0.002,0.005,0.01,0.02,0.05$. (b) $\lambda$ dependence of SP for $\Omega =0.01$ and $N=200001$, 50001, 10001, 2001, 201.

Figure 5

Three kinds of revival phenomena of the SP with different revival time. (a) $T_{\text{rev}}\approx \pi /|\Omega -{\Lambda }_{\lambda +\Omega }^{k_0}|$. (b) $T_{\text{rev}}\approx \pi /\Omega$ for $\Omega =0.02$, which is close to adiabaticity. $T_{\text{rev}}\approx N/2$ for $\Omega =100$. The inset in (b) shows the revival of the average magnetization $m_z\left(t\right)$ with $T_{\text{rev}}\approx N/2$ for $N=201$. The revival of $S\left(t\right)$ under the same parameter values as for $m_z\left(t\right)$ is shown in (a).