Entanglement entropy of a three-spin-interacting spin chain with a time-reversal-breaking impurity at one boundary

We investigate the effect of a time-reversal-breaking impurity term (of strength ${\lambda }_d$) on both the equilibrium and nonequilibrium critical properties of entanglement entropy (EE) in a three-spin-interacting transverse Ising model, which can be mapped to a $p$-wave superconducting chain with next-nearest-neighbor hopping and interaction. Importantly, we find that the logarithmic scaling of the EE with block size remains unaffected by the application of the impurity term, although, the coefficient (i.e., central charge) varies logarithmically with the impurity strength for a lower range of ${\lambda }_d$ and eventually saturates with an exponential damping factor $[\sim exp(-{\lambda }_d)]$ for the phase boundaries shared with the phase containing two Majorana edge modes. On the other hand, it receives a linear correction in term of ${\lambda }_d$ for an another phase boundary. Finally, we focus to study the effect of the impurity in the time evolution of the EE for the critical quenching case where the impurity term is applied only to the final Hamiltonian. Interestingly, it has been shown that for all the phase boundaries, contrary to the equilibrium case, the saturation value of the EE increases logarithmically with the strength of impurity in a certain regime of ${\lambda }_d$ and finally, for higher values of ${\lambda }_d$, it increases very slowly dictated by an exponential damping factor. The impurity-induced behavior of EE might bear some deep underlying connection to thermalization.

Figure 1

Zero-temperature phase diagram of the model Hamiltonian (2) for $h=1$. The line ${\lambda }_2=1+{\lambda }_1$ represents the phase boundary between upper three-spin dominated phase (with two zero-energy Majorana end modes at each edge) and ferromagnetic phase with one Majorana at each end. There are two more phase boundaries, given by ${\lambda }_2=1-{\lambda }_1$ ($a\text{−}e\text{−}d$ line) and ${\lambda }_2=-1$ ($b\text{−}d$ line). The paramagnetic region denoted by $n=0$ does not have any zero-energy Majorana mode. In terms of topology both the three-spin dominated phases have two zero-energy Majorana modes at each end.

Figure 2

The derivative of EE with ${\lambda }_2$ shows peak, dip or kink at the phase boundaries pointing toward the fact that it can be used as an indicator of QPTs. For ${\lambda }_d=0$, the derivative shows a dip at $n=2-n=0$ phase boundary while the peaks are observed for $n=1-n=0$ and $n=1-n=2$ phase boundaries. These dip and peak in the derivative of EE that appear in the phase boundaries separating $n=2$ phase with others change to a kink like structure for finite ${\lambda }_d$. This qualitative change in behavior of the derivative is clearly evident from the variation of EE with ${\lambda }_2$ as shown in the inset (we set ${\lambda }_1=1$); the value of EE reduces inside the $n=2$ phase when ${\lambda }_d$ becomes finite. The dip or peak height of the derivative at these kinks is maximum for an infinitesimal impurity strength and decreases with increasing ${\lambda }_d$; inset suggests similar feature that at $n=2-n=0$ and $n=1-n=2$ phase boundaries the EE shows a peak that decreases with ${\lambda }_d$. In contrast, at $n=0-n=1$ phase boundary, the peak height of the derivative enhances with ${\lambda }_d$ as peak height of EE increases. We choose the system size to be $N=100$ and block length to be $l=30$.

Figure 3

Plot shows that peak or dip, observed in the derivative of EE for a clean system with ${\lambda }_d=0.1$, becomes more sharper with block size $l$. We choose $n=1-n=2$ phase boundary. We see that $\frac{\mathrm{\Delta }S({\lambda }_2=2)}{\mathrm{\Delta }{\lambda }_2}$ matches well with $(m/l)logl+b$ (depicted by blue dashed lines) where $m$ and $b$ is found to be 13.31 and $-2.36$. Inset shows the variation of the derivative with $l$ by varying ${\lambda }_2$ and ${\lambda }_1$ is kept fixed at unity.

Figure 4

The equilibrium EE, $S_l$ as function of block length $l$ for different values of ${\lambda }_d$ is plotted for (a) $n=2-n=0$ (with ${\lambda }_2=-1$), (b) $n=1-n=0$ (with ${\lambda }_2=0$), (c) $n=2-n=1$ (with ${\lambda }_2=-2$), and (d) $n=2-n=1$ (with ${\lambda }_2=2.1$) phase boundaries. For (a) and (b), the EE becomes maximum for ${\lambda }_d=4.8$ as shown by the solid brown line appeared at the top of the plots. The dotted long-dashed sky line with ${\lambda }_d=3.2$, long-dashed red line with ${\lambda }_d=2.4$, short-dashed blue line with ${\lambda }_d=1.6$, dotted green line with ${\lambda }_d=0.8$, and solid black line with ${\lambda }_d=0$ appear in a decreasing order. The order is reversed for (c) and (d) compared to (a) and (b). The value of EE with the impurity strength ${\lambda }_d$ increases for (a) and (b), whereas decreases for (c) and (d). The inset shows the variation of the effective central charge $c_{\lambda }$ as a function of ${\lambda }_d$. The behavior exhibited by $c_{\lambda }$ is opposite to that of the EE in all the above cases. Additionally, insets of (a), (c), and (d) show that $c_{\lambda }$ varies logarithmically with ${\lambda }_d$ for ${\lambda }_d^l<{\lambda }_d<{\lambda }_d^u$; $c_{\lambda }=\alpha log{\lambda }_d+\beta$ (indicated by blue solid lines). On the other hand, for (b) it is linear throughout the range of ${\lambda }_d$; $c_{\lambda }=\alpha {\lambda }_d+\beta$ (drawn using blue solid lines). In this case, $\beta$ equals to the central charge ($c_0=1/12$) for the clean system. Numerically, the extrapolated value of $c_{\lambda }$ at ${\lambda }_d=0$ is $\beta =c_0=0.082$ (close to $1/12$).

Figure 5

(a) and (b) show the variation of $c_{\lambda }$ with ${\lambda }_d$ as shown in Figs. 4 and 4, respectively. Here, it has been seen that both the plots can be fitted with $c_{\lambda }=a+bexp(-d{\lambda }_d)$ for ${\lambda }_d>{\lambda }_d^u$ as shown by blue solid lines. It indeed indicates that the effective central charge eventually saturates for strong impurity limit with an exponential damping. The values of fitting parameters are provided inside the plots.

Figure 6

Evolution of the EE as a function of time for different values of the impurity strength (${\lambda }_d$). It can be observed that the EE gets affected by the ${\lambda }_d$ after a certain value of ${\lambda }_d$ when the system is quenched to $n=0-n=1$ phase boundary (with ${\lambda }_1=1$ and ${\lambda }_2=0$) from $n=0$ phase with ${\lambda }_1=0.5$. The different curves from bottom to top are given by, solid red line (${\lambda }_d=0.8$), dashed black line (${\lambda }_d=1.2$), long-dashed-dotted pink line (${\lambda }_d=1.6$), double-dotted gray line (${\lambda }_d=2.0$), dashed red line (${\lambda }_d=2.4$), long-dashed black line (${\lambda }_d=2.8$), dashed green line (${\lambda }_d=4.0$), dashed orange line (${\lambda }_d=6.0$). Inset: Shows that saturation value of the EE is almost independent ${\lambda }_d$ upto some value, say ${\lambda }_d^{*}$. At the same time, main plot depicts that the saturation value of the EE increases logarithmically with ${\lambda }_d$ after ${\lambda }_d^{*}$. Here, block length, $l=20$ and $N=300$.

Figure 7

Variation of the EE with time after quenching the system to $n=0-n=2$ boundary (${\lambda }_1=1$ and ${\lambda }_2=-1$) from $n=0$ phase with ${\lambda }_2=-0.5$ and ${\lambda }_1=1$. The various curves from bottom to top are given as solid red line (${\lambda }_d=2.0$), dashed black line (${\lambda }_d=2.4$), long-dashed-dotted green line (${\lambda }_d=2.8$), double-dotted gray line (${\lambda }_d=3.4$), dashed red line (${\lambda }_d=3.8$), solid blue line (${\lambda }_d=5.0$), long-dashed black line (${\lambda }_d=6.0$). Here, block length, $l=20$ and $N=300$. Inset plot shows that saturation value of the EE is weakly dependent on ${\lambda }_d$ and decreases by a small amount with increasing the value of ${\lambda }_d$ up to a certain value, say ${\lambda }_d^{*}$. On the other hand, the main plot depicts that the saturation value of EE increases with ${\lambda }_d$ in a nonlinear fashion after ${\lambda }_d^{*}$.

Figure 8

Time evolution of the EE is affected by the impurity strength as the system is quenched to $n=1-n=2$ boundary (with ${\lambda }_1=1$ and ${\lambda }_2=2$) from $n=2$ phase with ${\lambda }_1=0.5$ and ${\lambda }_2=2$. The different curves from bottom to top are denoted by solid black line (${\lambda }_d=2.0$), solid pink line (${\lambda }_d=2.4$), solid gray line (${\lambda }_d=2.8$), long-dashed red line (${\lambda }_d=3.4$), long-dashed blue line (${\lambda }_d=3.8$), long-dashed black line (${\lambda }_d=5.0$), long-dashed green line (${\lambda }_d=6.0$). Here, block length, $l=20$ and $N=300$. Inset: It can be seen that the saturation value of the EE decreases slowly when ${\lambda }_d$ increases up to a certain value ${\lambda }_d^{*}$. Main plot: Similar to the previous cases, after ${\lambda }_d^{*}$, the saturation value of the EE increases with ${\lambda }_d$ in a nonlinear fashion.

Figure 9

We have plotted the saturation value of the EE as a function of $ln{\lambda }_d$ for three different cases studied in Figs. 6, 7, and 8. This plot shows that saturation value of the EE increases linearly with $ln{\lambda }_d$ after ${\lambda }_d^{*}$. The blue dashed lines represent the fitted curve: $S_{\mathrm{sat}}=\alpha ln{\lambda }_d+\beta$.

Figure 10

Variation of $S_{\mathrm{sat}}$ as a function of ${\lambda }_d$ when the quenching is performed up to (a) $n=1-n=2$ and (b) $n=1-n=0$ phase boundaries. After a logarithmic increase, $S_{\mathrm{sat}}$ follows the relation $c_{\lambda }=a+bexp(-{\lambda }_d)$ (depicted by blue solid lines).