Detecting quantum phase transition via magic resource in the $XY$ spin model

Quantum phase transition in the $XY$ spin model with three-spin interaction is investigated using magic resource (non-stabilizerness), which is crucial in universal fault-tolerant quantum computation. The magic quantifier we employ here is defined straightforwardly via characteristic functions of quantum states, which are well defined for all dimensional quantum systems (in sharp contrast to those defined by discrete Wigner functions) and can be easily calculated. We show that the magic quantifier of both the reduced single-site spins and two-site spins of the system ground state increase to their maximum around the critical points for quantum phase transition. This indicates that the magic resource can be used to detect the critical phenomena in the $XY$ spin model and reveals a connection between quantum phase transition in many-body systems and quantum resource in stabilizer quantum computation.

Figure 1

Single-spin magic quantifier $M\left({\rho }_i\right)$ and its first-order derivative $dM\left({\rho }_i\right)/d\alpha$ versus the three-spin interaction parameter $\alpha$ for $N=20001, \lambda =0$. We see that the magic quantifier $M\left({\rho }_i\right)$ achieves its maximum around the critical point ${\alpha }_c=0.5$, while the derivative $dM\left({\rho }_i\right)/d\alpha$ changes dramatically around the critical point ${\alpha }_c=0.5$.

Figure 2

Single-spin magic quantifier $M\left({\rho }_i\right)$ and its derivative $dM\left({\rho }_i\right)/d\lambda$ versus the external magnetic-field parameter $\lambda$ for $N=20001, \alpha =0$. We see their distinctive behaviors around the critical point ${\lambda }_c=1$.

Figure 3

Two-spin magic quantifier $M\left({\rho }_{i,i+1}\right)$ and its derivative $dM\left({\rho }_{i,i+1}\right)/d\alpha$ versus the three-spin interaction parameter $\alpha$ for $N=20001, \lambda =0$. Both exhibit apparent singular behaviors around the critical point ${\alpha }_c=0.5$.

Figure 4

Two-spin magic quantifier $M\left({\rho }_{i,i+1}\right)$ and its derivative $dM\left({\rho }_{i,i+1}\right)/d\lambda$ versus the magnetic-field parameter $\lambda$ for $N=20001, \alpha =0$. Both exhibit apparent singular behavior around the critical point ${\lambda }_c=1$. It is interesting to note that they also exhibit special behavior at $\lambda \approx 0.64,$ which might indicate some unusual feature therein awaiting for further investigation.