Complex-network description of thermal quantum states in the Ising spin chain

We use network analysis to describe and characterize an archetypal quantum system—an Ising spin chain in a transverse magnetic field. We analyze weighted networks for this quantum system, with link weights given by various measures of spin-spin correlations such as the von Neumann and Rényi mutual information, concurrence, and negativity. We analytically calculate the spin-spin correlations in the system at an arbitrary temperature by mapping the Ising spin chain to fermions, as well as numerically calculate the correlations in the ground state using matrix product state methods, and then analyze the resulting networks using a variety of network measures. We demonstrate that the network measures show some traits of complex networks already in this spin chain, arguably the simplest quantum many-body system. The network measures give insight into the phase diagram not easily captured by more typical quantities, such as the order parameter or correlation length. For example, the network structure varies with transverse field and temperature, and the structure in the quantum critical fan is different from the ordered and disordered phases.

Figure 1

Examples of weighted networks illustrating the meanings of the different network measures we use, namely, the local measures, density, disparity, and betweenness centrality for a node, and the global measures, clustering coefficient, average geodesic distance, and diameter for a network. In the examples above, the thickness of a link is proportional to its weight. Node $A$ has a large density and betweenness centrality, and a small disparity. Node $B$ has a small density and disparity, and a large betweenness centrality. Node $C$ has a small density and betweenness centrality, and a large disparity. The network in (a) has a smaller diameter, average geodesic distance, and clustering coefficient than the network in (b).

Figure 2

von Neumann mutual information versus distance for the TIM at different temperatures and transverse field couplings. The mutual information is nearly uniform when $\{h,k_{B}T\}\ll J$, as shown in the plot at the lower-left corner. The mutual information algebraically decays near the quantum critical point, $k_{B}T\ll J\approx h$, as shown in the log-log plot in the lower middle. The mutual information exponentially decays in all other regimes, as shown in the remaining four log-linear plots.

Figure 3

Network measures of the von Neumann mutual information network for the TIM in the thermodynamic limit as a function of temperature and magnetic field. (a) Density of links connected to a node in the network. (b) Disparity of a node. (c) Normalized betweenness centrality $\frac{B_i}{N^2}$ of a node $i$. (d) Clustering coefficient of the network. (e) Normalized average geodesic distance $\frac{\overline{D}}{N}$ between nodes. (f) Normalized diameter $\frac{D_{\max }}{N}$ of the network. The distance between two nodes across a link is defined as the inverse of the mutual information between them. The gradients of all network measures are observed to have an extremum at the quantum phase transition at $T=0,h=J$.

Figure 4

Network measures of the von Neumann mutual information network for the TIM at zero temperature as a function of inverse system size and magnetic field. (a)–(f) plot the same quantities as in Fig. 3, with inverse system size instead of temperature on the vertical axis.

Figure 5

(a) The strength $\lambda =Nd$ and (b) disparity $Y$ of the von Neumann mutual information network for different network sizes $N$. Solid lines: $h=0$, and dashed lines: $h=J$ for system sizes $N=6$ (large circles), $N=10$ (diamonds), $N=20$ (small squares), $N=50$ (large squares), and $N=10000$ (small circles). At $T=0$, the strength diverges as $\lambda \approx O\left(N\right)$, and therefore the density is nonzero. For all $T>0$, the strength converges to a finite value as $N$ is increased, and therefore the density for a network in the thermodynamic limit is $d=0$. Unlike the density, all other network measures converge to a finite function of temperature as the network grows in size.

Figure 6

Network measures of the Rényi mutual information network as a function of Rényi order $q$ for the TIM in the thermodynamic limit at different magnetic fields and temperatures. (a)–(f) plot the same quantities as Fig. 3 versus Rényi order $q$ at $h=0,T=0$ (black solid circles), $h=J,T=0$ (blue solid diamonds), $h=1.5J,T=0$ (red solid squares), $h=0,k_{B}T=J$ (black open circles), $h=J,k_{B}T=J$ (blue open diamonds), and $h=1.5J,k_{B}T=J$ (red open squares).

Figure 7

Network measures of the Rényi mutual information network as a function of Rényi order $q$ for the TIM at zero temperature at different magnetic fields and system sizes. (a)–(f) plot the same quantities as Fig. 3 versus Rényi order $q$ at $h=0,L=10$ (black solid circles), $h=J,L=10$ (blue solid diamonds), $h=1.5J,L=10$ (red solid squares), $h=0,L=90$ (black open circles), $h=J,L=90$ (blue open diamonds), and $h=1.5J,L=90$ (red open squares).

Figure 8

Network measures of the spin-spin correlation networks for the TIM. Left panels: Density of links connected to a node in the adjacency network for different spin correlations. Right panels: Disparity of a node in these adjacency networks.

Figure 9

Network measures of the concurrence network for the TIM. The top panels show (a) the strength of a node ($=N\times$ density) in the network, and the normalized diameter $\frac{D_{\max }}{N}$, at a finite temperature (vertical axis) for a system in the thermodynamic limit. The bottom panels show the same measures at zero temperature for systems of lengths ranging from 10 to 50 sites (vertical axis). Above and to the left of the dotted line in all four panels, the concurrence is zero for all the links in the network, the strength is zero, and the diameter of the network is infinite. The gradient of the density is observed to have a maximum at the quantum phase transition at $T=0,h=J$ in the thermodynamic limit.

Figure 10

Network measures of the Rényi mutual information network as a function of system size $L$ for the TIM at $T=0$ at different magnetic fields and Rényi order $q$. (a)–(f) plot the same quantities as Fig. 3 versus system size $L$ at $h=0,q=1$ (black solid circles), $h=J,q=1$ (blue solid diamonds), $h=1.5J,q=1$ (red solid squares), $h=0,q=10$ (black open circles), $h=J,q=10$ (blue open diamonds), and $h=1.5J,q=10$ (red open squares).