Quantum correlations in quenched disordered spin models: Enhanced order from disorder by thermal fluctuations

We investigate the behavior of quantum correlations of paradigmatic quenched disordered quantum spin models, viz., the $XY$ spin glass and random-field $XY$ models. We show that quenched averaged quantum correlations can exhibit the order-from-disorder phenomenon for finite-size systems as well as in the thermodynamic limit. Moreover, we find that the order-from-disorder can become more pronounced in the presence of temperature by suitable tuning of the system parameters. The effects are found for entanglement measures as well as for information-theoretic quantum correlation ones, although the former show them more prominently. We also observe that the equivalence between the quenched averages and their self-averaged cousins—for classical and quantum correlations—is related to the quantum critical point in the corresponding ordered system.

Figure 1

Density plot of the difference between self-averaged and quenched averaged observables, i.e., of ${\mathcal{S}}^{\mathcal{Q}}=\frac{1}{N}{\sum }_{i=1}^N\mathcal{Q}\left({\rho }_{i,i+1}\right)-\int \int \cdots \int \mathcal{Q}\left(\left\{{\lambda }_i\right\}\right)d\left\{{\lambda }_i\right\}$ versus $\langle \lambda \rangle$ on the horizontal axis, and the disorder strength $\overline{\sigma }$ on the vertical axis, for the quantum $XY$ spin glass Hamiltonian. Here $\langle \lambda \rangle =\frac{\langle J_i\rangle }{h}$ and $\overline{\sigma }=\frac{\sigma }{h}$. We choose $N={10}^3$ and $\gamma =0.7$. The observables are chosen as (a) $m^z$, (b) $T^{xx}$, (c) $T^{yy}$, (d) $T^{zz}$, as defined in Sec. 2a, and (e) concurrence. The white region indicates ${\mathcal{S}}^{\mathcal{Q}}=0$, which implies that the self-averaged and quenched averaged observables are equivalent. We set ${\mathcal{S}}^{\mathcal{Q}}$ to be 0 when ${\mathcal{S}}^{\mathcal{Q}}<0.001$.

Figure 2

Order-from-disorder for concurrence with different system sizes. Comparison between the concurrences of the nearest-neighbor density matrices of the zero-temperature states of the ordered $XY$ model [(red) circles, dashed line] and those of the $XY$ spin glass system [(blue) triangles, solid line] versus $\lambda$ for different system sizes. Here, $\gamma =0.4$ and $\overline{\sigma }=0.3$. (a) $N=8$, (b) $N=12$, (c) $N=20$, and (d) $N=100$. The horizontal axis is dimensionless, while the vertical axis is in ebits. Note that the horizontal axis represents $\frac{J}{h}$ for the ordered-system curve, while it represents $\frac{\langle J\rangle }{h}$ for the disordered-system curve.

Figure 3

Order-from-disorder for concurrence for different $\gamma$ values with $\overline{\sigma }=0.3$ and $N={10}^3$. (a) $\gamma =0.1$, (b) $\gamma =0.4$, (c) $\gamma =0.7$, and (d) $\gamma =1.0$. All other considerations are the same as in Fig. 2.

Figure 4

Order-from-disorder for quantum discord with different system sizes. The vertical axis is measured in bits. All other considerations are the same as in Fig. 2.

Figure 5

Order-from-disorder for quantum discord for different $\gamma$'s and for $N={10}^3$. The vertical axis is measured in bits. All other considerations are the same as in Fig. 3.

Figure 6

(a, b) Density plot of thermal enhancement scores for concurrence versus $\lambda$ on the abscissa and $h\beta$ on the ordinate, for (a) $\gamma =0.1$ and (b) $\gamma =0.7$, for the $XY$ spin glass model. We set $N={10}^3$ and $\overline{\sigma }=0.3$. (c) ${\mathrm{\Delta }}_{\beta ,\lambda }^C$ versus $h\beta$ for fixed $\lambda =0.8$ with $\gamma =0.1$ [(blue) circles, dashed line] and $\gamma =0.7$ [(black) pluses, solid line]. The data are fitted with the best polynomial fit. Other parameters are the same as in (a) and (b). Clearly this shows nonmonotonicity of the entanglement enhancement scores with temperature for the $XY$ spin glass model. (d–f) Quantum discord. All other considerations in (d) and (e) are the same as in (a) and (b), respectively. (f) ${\mathrm{\Delta }}_{\beta ,\lambda }^D$ versus $h\beta$ for $\lambda =0.1$ with $\gamma =0.1$ [(blue) circles, dashed line] and $\gamma =0.7$ [(black) pluses, solid line]. Nonmonotonicity of the enhancement score for quantum discord is observed only for lower values of $\gamma$. Horizontal axes are dimensionless in all panels. Vertical axes are also dimensionless in (a), (b), (d), and (e). ${\mathrm{\Delta }}_{\beta ,\lambda }^C$ is measured in ebits, while ${\mathrm{\Delta }}_{\beta ,\lambda }^D$ is measured in bits.

Figure 7

Behavior of the total enhancement score for concurrence (a) and quantum discord (b) for the thermal state of the quantum $XY$ spin glass. The data are fitted with the best polynomial fit. Two values of anisotropy constants are chosen: $\gamma =0.1$ [(blue) circles, dashed line] and $\gamma =0.7$ [(black) pluses, solid line]. Here, $\lambda =0.5,\overline{\sigma }=0.3$, and $N={10}^3$. Horizontal axes are dimensionless in both panels, while the vertical axis is measured in ebits in (a) and in bits in (b).

Figure 8

Order-from-disorder for concurrence in the random-field $XY\text{model}$ in the zero-temperature state. The (red) circles correspond to the ordered system, while the (blue) triangles correspond to the disordered one. Here, $\tilde{\sigma }=\frac{\sigma }{J}=0.3$ and $N={10}^3$. Each panel is for a different value of $\gamma$. Horizontal axes are dimensionless, while vertical axes are measured in ebits. All other considerations are the same as in Fig. 3.

Figure 9

Order-from-disorder for quantum discord. Vertical axes are measured in bits. All other considerations are the same as in Fig. 8.

Figure 10

The same as Fig. 6, but for the random-field $XY$ model. For both the bottom panels [(c) and (f)], we have chosen $\mu =0.5.$

Figure 11

The same as Fig. 7, but for the random-field $XY$ model. We choose $\mu =0.5$.