Local and intrinsic quantum coherence in critical systems

Local and intrinsic quantum coherence, as two contributions of total quantum coherence, have been introduced by Radhakrishnan et al. [C. Radhakrishnan, M. Parthasarathy, S. Jambulingam, and T. Byrnes, Phys. Rev. Lett. 116, 150504 (2016)]. In the present work we study the property of local, intrinsic, and total quantum coherence in critical systems such as the $XY$ model, extended $XY$ model, and Ashkin-Teller model to analyze their capability in identifying the second-order, infinite-order quantum phase transitions, as well as the topological quantum phase transitions. It is shown that not only the total coherence, but also the local and intrinsic coherence can spotlight the critical points efficiently. Moreover, the long-range properties of total and local coherence, and the short-range properties of intrinsic coherence are analyzed. The total and local coherence for spin-pairs farther than nearest neighbors can highlight the critical points while the intrinsic coherence is absent for long-distance spin-pairs.

Figure 1

(a) The energy spectra as a function of $\gamma$ with parameters $\alpha =1,\delta =1$, and $\lambda =-0.5$. (b) Trajectories of winding vectors in $Y–Z$ plane with parameters $\alpha =1,\delta =1,\lambda =-0.5$, and $\gamma =-1,0,2$.

Figure 2

The scheme of Ashkin-Teller model.

Figure 3

(a) The intrinsic coherence ${\mathcal{C}}_I$ as functions of $\lambda$ with $\gamma =0.6,0.8$. The inset shows the intrinsic coherence susceptibility as functions of $\lambda$ with $\gamma =0.6,0.8$. (b) The dependence of the local and total coherence, i.e., ${\mathcal{C}}_L$ and ${\mathcal{C}}_T$, and $\lambda$ with $\gamma =0.6,0.8$. (c) The dependence of the local and total coherence susceptibility with respect to $\lambda$, i.e., $\chi \left({\mathcal{C}}_L\right)$ and $\chi \left({\mathcal{C}}_T\right)$, and $\lambda$ with $\gamma =0.8$. (d) The dependence of the local and total coherence susceptibility with respect to $\lambda$, i.e., $\chi \left({\mathcal{C}}_L\right)$ and $\chi \left({\mathcal{C}}_T\right)$, and $\lambda$ with $\gamma =0.6$.

Figure 4

(a) The intrinsic coherence, local coherence, and total coherence ${\mathcal{C}}_I, {\mathcal{C}}_L$, and ${\mathcal{C}}_T$ for the reduced density operators of spin-pairs with long distance $R=3$ as functions of $\lambda$ with $\gamma =0.6$. (b) Total quantum coherence ${\mathcal{C}}_T$ for the reduced density operators of spin-pairs with $R=1,3,5,7$ as functions of $\lambda$. (c) The relation of the value of total quantum coherence at the critical point ${\lambda }_c=1$ and distance of spin-pairs $R$. The expression of fitting curve is $C_T{|}_{\lambda ={\lambda }_c}=0.3862R^{-0.4627}$

Figure 5

Total quantum coherence as a function of $\gamma$ with parameters (a) $\alpha =1,\delta =1,\lambda =-0.5$ and (c) as a function of $\alpha$ with parameters $\gamma =1,\delta =-1,\lambda =1$. (b) and (d) display the dependence of winding numbers on $\gamma$ with the same parameters of (a) and on $\alpha$ with the same parameters of (c).

Figure 6

(a) The total, intrinsic, and local quantum coherence as a function of $\alpha$ with parameters $\gamma =1,\delta =-1,\lambda =1$ for the reduced density operator of nearest spin-pairs, i.e., $R=1$, and the inset shows the dependence of intrinsic quantum coherence susceptibility on $\alpha$. (b) The total, intrinsic and local quantum coherence as a function of $\alpha$ with the same parameters in (a) except for the distance of spin-pairs $R=3$. (c) The dependence of total quantum coherence on $\alpha$ with several different $R$. (d) The relation of the value of total quantum coherence at the critical point ${\alpha }_c=0$ and distance of spin-pairs $R$. The expression of fitting curve is $C_T{|}_{\alpha ={\alpha }_c}=0.4853R^{-0.5171}$.

Figure 7

(a) The total quantum coherence $C_T$ as functions of $\alpha$ close to the critical point ${\alpha }_c=2$ for several values of $R$ without duality transformation. (b) The relation of $C_T{|}_{\alpha =2}$ and the distances of spin-pairs $R$ without duality transformation. (c) $C_T$ as functions of $\alpha$ near the critical point ${\alpha }_c=2$ for several values of $R$ in the dual lattice. (d) The relation of $C_T{|}_{\alpha =2}$ and the distances of spin-pairs $R$ in the dual lattice, which can be fitted perfectly by $C_T{|}_{\alpha =2}=0.3696R^{-1.371}$.

Figure 8

The total quantum coherence ${\mathcal{C}}_T$ (a) and coherence susceptibility with respect to $\mathrm{\Delta }\chi \left({\mathcal{C}}_T\right)$ (b) as functions of $\mathrm{\Delta }$ with various values of $\beta$.

Figure 9

(a) The energy spectra as a function of $\alpha$ with parameters $\gamma =1,\delta =-1,\lambda =1$. (b, c) Trajectories of winding vectors in $Y–Z$ plane for parameters $\gamma =1,\delta =-1,\lambda =1$ with various values of $\alpha$. For $\alpha =-2.5,-1.5,-0.5,0.5,1.5,2.5$, the winding numbers are $\nu =2,0,0,1,-1,-2$, respectively.

Figure 10

The dependence of total quantum coherence $C_T$ and $\alpha$ around (a) $\alpha =-0.85$ and (b) $\alpha =\left(-\sqrt{5}-1\right)/2$.