Coherence dynamics of spin systems in critical environment with topological characterization

Abstract

The coherence dynamics of the central systems are investigated in the spin-chain environment with topological characterization. For the single- and two-spin coherence, their critical behaviors can detect the topological quantum phase transitions (TQPTs) in weaker coupling regions. Their noncritical behaviors show periodic oscillations or keep constant. For the quantum coherence of a symmetric W state, the bipartite coherence is dominant due to its polygamy. The critical behaviors of global coherence ($\textrm{QC}_{\mathrm{a:bc}}$ and $\textrm{QC}_{\mathrm{b:c}}$) and multipartite monogamy can also detect the TQPTs. In the strong coupling regions ($g_{a(b,c)} \gg 1$), the coherence dynamics of single-spin, two-spin, and bipartite block of three-spin system can be characterized by Gaussian envelope, but the envelope line of three-spin coherence can be only described by negative logarithmic function. Finally, the resistance to the topological spin-chain environment becomes stronger from the single- to three-spin coherence.

Appendices

Coherence dynamics and TQPTs induced by the anisotropy parameters

Some references have reported that the topological phase transition points are $\gamma _{c1}=(-\sqrt{5}+1)/2$ and $\gamma _{c2}=(\sqrt{5}+1)/2$ as the anisotropy of the nearest-neighbor spins $\gamma$ is the driving parameter, and $\delta _{c1}\approx -1.2747$ and $\delta _{c2} \approx 0.5604$ as the anisotropy of next-to-nearest-neighbor spins $\delta$ is the driving parameter [64,65,66, 69]. Based on these results, we have displayed the coherence dynamics of the single-spin system as a function of $\gamma$ and $\delta$ in Fig. 9a and b, respectively. It is found that the single-spin coherence tends to undergo a rapid decay near each TQPT, but there is a surge at the critical point $\delta _{c2}$. The more detailed information can see Fig. 9c and d, which display the single-spin coherence as a function of the parameters $\gamma$ and $\delta$ at $t=10$. One can observe that the quantum coherence is sensitive to the parameters $\gamma$ and $\delta$. Although the quantum coherence declines sharply around the TQPTs, it always increases sharply at the phase transition points, especially at the critical point $\delta _{c2}$. This situation is similar to single-spin and two-spin coherence at the critical point $\alpha _{c3}=(\sqrt{5}-1)/2$, and Cheng et al. also proved that non-Markovianity decreases to zero on each side of the critical point $\left( \delta _{c2}\right)$ and exhibits a local maximum on the point exactly in their Fig. 5b [64]. We all know that the non-Markovianity can describe the memory effects and the backflow of information from the surrounding environment. Hence, we can understand that the backflow of coherence information from the spin-chain environment prevents the decay of single-spin coherence.

Coherence distribution of a tripartite system

We briefly illustrate the coherence distribution of a tripartite system in this section [21]. A geometric picture of different coherences is displayed in Fig. 10, and the distances between any two states represent the different coherences. It is found that the three-spin coherence, also known as absolute coherence ${\textrm{QC}}_{\textrm{A}}$, can be decomposed in two ways: one is the global coherence ${\textrm{QC}}_{\textrm{G}}$ and local coherence ${\textrm{QC}}_{\textrm{L}}$, the other is ${\textrm{QC}}_{\mathrm{a:bc}}$ and $\textrm{QC}_{\textrm{A}}^{{\mathrm{a:bc}}}$. Here, $\textrm{QC}_{a:bc}$ describes the coherence between qubit a and the bipartite block bc. ${\textrm{QC}}_{\mathrm {b:c}}$ measures the coherence in the bipartite block bc. They both constitute the coherence distribution of ${\textrm{QC}}_{\textrm{G}}$. ${\textrm{QC}}_{\textrm{A}}^{\mathrm {a:bc}}$ evaluates the coherence between the $\rho _a\otimes \rho _{bc}$ and $[\pi \left( \rho \right) ]_d$, and it can be decomposed into ${\textrm{QC}}_{\mathrm {b:c}}$ and ${\textrm{QC}}_{\textrm{L}}$. Based on the geometric diagram of the coherence distribution in Fig. 10, there are four triangles, and each triangle satisfies the trade-off relation. They can be expressed as

$$\begin{aligned}{} & {} {\textrm{QC}}_{\textrm{A}} \le {\textrm{QC}}_{{\textrm{G}}} + {\textrm{QC}}_{\textrm{L}}, \ {\textrm{QC}}_{\textrm{A}} \le {\textrm{QC}}_{{\mathrm {a:bc}}}+{\textrm{QC}}_{\textrm{A}}^{{\mathrm {a:bc}}}, \nonumber \\{} & {} {\textrm{QC}}_{{\textrm{G}}} \le {\textrm{QC}}_{{\mathrm {a:bc}}} + {\textrm{QC}}_{\mathrm {b:c}}, \ {\textrm{QC}}_{\textrm{A}}^{{\mathrm {a:bc}}} \le {\textrm{QC}}_{\mathrm {b:c}} + {\textrm{QC}}_{\textrm{L}}. \end{aligned}$$

(B1)

In our work, there is no local coherence ${\textrm{QC}}_{\textrm{L}}$ for the symmetric W state due to the $\pi \left( \rho \right) =[\pi \left( \rho \right) ]_d$, and the absolute coherence ${\textrm{QC}}_{\textrm{A}}$ is equal to the global coherence ${\textrm{QC}}_{{\textrm{G}}}$. Thus, we have investigated the physical properties of absolute coherence ${\textrm{QC}}_{\textrm{A}}$ and its coherence distribution (${\textrm{QC}}_{{\mathrm {a:bc}}}$ and ${\textrm{QC}}_{\mathrm {b:c}}$) in the topologically critical environment. By the way, the geometric diagram in Fig. 10 is not the only coherence distribution. For example, the three-spin coherence can also be decomposed into the $\textrm{QC}_{ab:c}$ and $\textrm{QC}_{a:b}$ or $\textrm{QC}_{ac:b}$ and $\textrm{QC}_{a:c}$.

Fig. 10

A schematic diagram of coherence distribution for a tripartite system