Quantum discord length is enhanced while entanglement length is not by introducing disorder in a spin chain

Classical correlation functions of ground states typically decay exponentially and polynomially, respectively, for gapped and gapless short-range quantum spin systems. In such systems, entanglement decays exponentially even at the quantum critical points. However, quantum discord, an information-theoretic quantum correlation measure, survives long lattice distances. We investigate the effects of quenched disorder on quantum correlation lengths of quenched averaged entanglement and quantum discord, in the anisotropic $XY$ and $XYZ$ spin glass and random field chains. We find that there is virtually neither reduction nor enhancement in entanglement length while quantum discord length increases significantly with the introduction of the quenched disorder.

Figure 1

Entanglement length in quantum $XY$ spin glass vs the homogeneous $XY$ model. In each panel, the blue squares denote the quenched averaged concurrence, $\langle C_{i,j}\rangle$, between sites $i$ and $j$, plotted against the lattice distance $r=|i-j|$, for the quantum $XY$ spin glass for $\langle J\rangle /h=$ 0.5 (a), 0.9 (b), 1.1 (c), and 1.5 (d). The red circles denote corresponding homogeneous systems. The dotted and solid lines show the exponential fits for disordered and homogeneous systems, respectively. The vertical axis denotes the concurrences, while the horizontal ones denote the lattice distances. Here $N=50$ and $\gamma =0.5$. For the disordered case, the number of realizations of the random coupling is taken to be ${10}^4$, for the quenched averaging. The lines are exponential fits and the obtained data at integer values of $r$. The vertical axis are measured in ebits, while the horizontal axis are in lattice length units.

Figure 2

All considerations here are the same as in Fig. 1, except that QD is considered instead of concurrence and that the former is measured in ebits.

Figure 3

Scalings of QD length for the 1D quantum $XY$ spin glass. We plot the quenched averaged QD on the vertical axis against lattice distance on the horizonal one for different values of $N$. In the main figure, the vertical axis is in bits, while the horizontal one is in lattice length units. In the inset, the unit of the horizontal and vertical axes are, respectively, logarithms of the total numbers of lattice sites and of lattice length. The logarithms are with base $e$.

Figure 4

Entanglement length in random field quantum $XY$ spin chain vs homogeneous $XY$ chain for $J/\langle h\rangle =$ (a) 0.67, (b) 0.91, (c) 1.1, and (d) 2.0. All considerations, except for the model considered and for the fact that the different panels are now for different values of $J/\langle h\rangle$, remain the same as in Fig. 1.

Figure 5

All considerations here are the same as in Fig. 4, except that QD is considered instead of concurrence and that the former is measured in ebits.

Figure 6

The homogeneous and disordered $XYZ$ models. Scaling of entanglement as a function of distance for the homogeneous and disordered spin chains with $N=24$ and $\gamma =0.5$ for $\langle J\rangle /h$ = (a) 0.5 and (b) 1.5. The vertical axes denote the concurrences, while the horizontal ones denote the lattice distances. The up and down triangles correspond to the homogeneous system with $\mathrm{\Delta }=0.1$ and $\mathrm{\Delta }=0.5$, respectively. The circles and the squares correspond to the disordered system for $\mathrm{\Delta }=0.1$ and $\mathrm{\Delta }=0.5$, respectively. The solid and dotted lines show exponential fits for different cases. For the disordered case, the number of random realizations taken is 8000.

Figure 7

All considerations here are the same as in Fig. 6, except that QD is considered instead of concurrence and that the vertical axes are measured in bits.