Nonclassicality and criticality in symmetry-protected magnetic phases

Quantum and global discord in a spin-1 Heisenberg chain subject to single-ion anisotropy (uniaxial field) are studied using exact diagonalization and the density matrix renormalization group. We find that these measures of quantum nonclassicality are able to detect the quantum phase transitions confining the symmetry-protected Haldane phase and show critical scaling with universal exponents. Moreover, in the case of thermal states, we find that quantum discord can increase with increasing temperature.

Figure 1

Lowest three energy levels of Hamiltonian Eq. (12) with periodic boundary conditions as a function of $U$ for $L=5$. For odd length chains, the ground state changes suddenly due to energy level crossings. Here we see the first crossing at $U\simeq -1.6$ and a second crossing at $U\simeq 0.9$.

Figure 2

Nearest-neighbor symmetric QD, ${\mathcal{D}}_2$, for the reduced state of the two central spins of an open-ended chain of length $L=8$ (red), 16 (green), 32 (blue), 64 (gray), 128 (black), and 256 (orange) going from bottom to top. Notice that the curves for $L=32$, 64, 128, and 256 are almost indistinguishable except near $U\sim -0.3$.

Figure 3

(a) Derivative of the nearest-neighbor symmetric QD against $U$ for $L=8$ (red), 16 (green), 32 (blue), 64 (gray), 128 (black), and 256 (orange). (b) Finite size scaling of the value of $U$ where the derivative is maximum against inverse chain length.

Figure 4

Second derivative of the symmetric QD with respect to $U$ in the critical region of the Gaussian QPT. The point markers are the numerically calculated values $L=16$ (downward green triangles), 32 (blue diamonds), 64 (gray circles), 128 (black squares), and 256 (upward orange triangles). The solid lines are quadratic functions of best fit for each data set. The vertical red dashed line at $U=0.9667$ is approximately where the curves cross each other. Inset: Finite size scaling of the second derivative of ${\mathcal{D}}_2$ using the estimate for the critical point $U=0.9667$ and the fitted value $\nu =1.6\pm 0.1$ [see Eq. (13)].

Figure 5

(a) GQD of the thermal ground state for $L=2$, 4, 6, and 8 (going from bottom to top). (b) GQD for the thermal ground state when $L=3$, 5, and 7 from bottom to top. See text for a discussion.

Figure 6

(a) GQD against $T$ for a closed ring of six spins with $U=0$ (topmost), 2 (middle curve), and $-2$ (bottom). The remaining panels are the symmetric QD for the reduced state of two (b) nearest-neighbor, (c) next-nearest-neighbor, and (d) next-next-nearest neighbor spins, from a six-spin closed ring.