Classical correlation and quantum discord in critical systems

We discuss the behavior of quantum and classical pairwise correlations in critical systems, with the quantumness of the correlations measured by the quantum discord. We analytically derive these correlations for general real density matrices displaying $Z_2$ symmetry. As an illustration, we analyze both the $XXZ$ and the transverse field Ising models. Finite size as well as infinite chains are investigated and the quantum criticality is discussed. Moreover, we identify the spin functions that govern the correlations. As a further example, we also consider correlations in the Hartree-Fock ground state of the Lipkin-Meshkov-Glick model. It is then shown that both classical correlation and quantum discord exhibit signatures of the quantum phase transitions.

Figure 1

(Color online) Quantum and classical correlations for nearest-neighbor spins in the $XXZ$ chain for $L\to \infty$.

Figure 2

(Color online) Values of the parameters $\left|c_1\right|=\left|c_2\right|$ and $\left|c_3\right|$ as a function of $\Delta$. The parameter $\left|c_3\right|$ governs the correlations for $\Delta <-1$, while $\left|c_1\right|=\left|c_2\right|$ is dominant in the gapless disordered phase $-1<\Delta <1$. In the ferromagnetic phase, $\left|c_1\right|=\left|c_2\right|=0$ and $\left|c_3\right|=1$, which implies the fixed value $C\left(\rho \right)=1$ and $Q\left(\rho \right)=0$ for any $\Delta >1$.

Figure 3

(Color online) Classical and quantum correlations for nearest-neighbor spins in the transverse field Ising model for a chain with 1024 sites.

Figure 4

(Color online) First derivative of the classical correlation for nearest-neighbor spins with respect to $g$ in the transverse field Ising chain for different lattice sizes $L$. The derivative of $C$ has a pronounced minimum at $g_{min}$, which tends to the QCP $g=1$ as $L\to \infty$. Inset: $dC/dg$ taken at $g_{min}$ exhibits a logarithmic divergence fitted by $\left(dC/dg\right){\mid }_{g_{min}}=-0.29161-0.22471\text{log}L$.

Figure 5

(Color online) First derivative of the quantum correlation for nearest-neighbor spins with respect to $g$ in the transverse field Ising chain for different lattice sizes $L$. Inset: $dQ/dg$ presents an inflexion point that tends to the QCP $g=1$ as $L\to \infty$.

Figure 6

(Color online) Second derivative of the quantum correlations for nearest-neighbor spins with respect to $g$ in the transverse field Ising chain for different lattice sizes $L$. Observe that the second derivative of $Q$ has a pronounced maximum at $g_{max}$, which tends to the QCP $g=1$ as $L\to \infty$. Inset: $d^{2}C/d{g}^2$ taken at $g_{max}$ exhibits a quadratic logarithmic divergence fitted by $\left(d^{2}Q/d{g}^2\right){\mid }_{g_{max}}=1.268-0.94712\text{log}L+0.15176{\text{log}}^{2}L$.

Figure 7

(Color online) Spin functions $c_1=G_{xx}^{i,i+1}$ and $c_4=G_z^i$ as a function of $g$. These functions govern the behavior of correlations, exhibiting a crossing exactly at the QCP. The plot is for a chain with $L=1024$ sites.

Figure 8

(Color online) Classical and quantum correlations between modes $\left(+m\right)$ and $\left(-m\right)$ in the HF ground state of the Lipkin model. Inset: derivative of correlations is nonanalytic at $\lambda =1$.