Entangled unique coherent line in the ground-state phase diagram of the spin-1/2 $XX$ chain model with three-spin interaction

Entangled spin coherent states are a type of quantum states that involve two or more spin systems that are correlated in a nonclassical way. These states can improve metrology and information processing, as they can surpass the standard quantum limit, which is the ultimate bound for precision measurements using coherent states. However, finding entangled coherent states in physical systems is challenging because they require precise control and manipulation of the interactions between the modes. In this work we show that entangled unique coherent states can be found in the ground state of the spin-1/2 $XX$ chain model with three-spin interaction, which is an exactly solvable model in quantum magnetism. We use the spin squeezing parameter, the $l_1$-norm of coherence, and the entanglement entropy as tools to detect and characterize these unique coherent states. We find that these unique coherent states exist in a gapless spin liquid phase, where they form a line that separates two regions with different degrees of squeezing. We call this line the entangled unique coherent line, as it corresponds to the almost maximum entanglement between two halves of the system. We also study the critical scaling of the spin squeezing parameter and the entanglement entropy versus the system size.

Figure 1

(a) Schematic picture of the ground-state phase diagram of the spin-1/2 $XX$ chain model with three-spin interaction [44]. (b) Same as in (a) but from the viewpoint of the spin squeezing parameter and the entanglement entropy.

Figure 2

The SSP as a function of the magnetic field for three different chain sizes $N=500$, 700, and 1000 (a) in the absence of TSI, $\alpha =0$, and (b) in the presence of TSI, $\alpha =2$. (c) Density plot of the SSP for a chain size $N=200$. (d) The $l_1$-norm of coherence as a function of the magnetic field for a chain size $N=1000$ and $\alpha =2$. The inset shows the local minima at exactly the coherent point for three chains lengths $N=500$, 700, and 1000.

Figure 3

The EE as a function of the magnetic field for three different chain sizes $N=500$, 700, and 1000 (a) in the absence of TSI, $\alpha =0$, and (b) in the presence of TSI, $\alpha =2$. (c) Density plot of the EE parameter for a chain size $N=200$.

Figure 4

(a) The SSP with respect to the size of the system on the critical line separating SL-I and SL-II phases for $\alpha =0$ and $h_c=0$. (b) Linear fit of the form $Y=mX+b$ applied to the function $3S_A(N/2)/log\left(N\right)=c_{\text{eff}}+m/log\left(N\right)$ on the critical line separating SL-I and SL-II phases and also in these gapless phases for for $\alpha =2$. Chain sizes are considered up to $N=1000$. Logarithms are calculated using base 2.

Figure 5

Concurrence $\mathcal{C}$ as a function of the magnetic field between (a) $1N$, (b) $2N$, and (c) $3N$ pairs of spin. (d) Plot of the one-tangle as a function of the magnetic field, for chain sizes $N=500$, 700, and 1000 in the presence of TSI, $\alpha =2$.