Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks

We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of electroencephalogram or functional magnetic resonance imaging measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson’s correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, $Z_2$, mean field superfluid to Mott insulator, and a Berzinskii-Kosterlitz-Thouless crossover.

Figure 1

Sketch of the mutual information complex network. A chain of $L$ quantum bits for the transverse Ising model, the “fruit fly” of quantum many body physics. (a) Links originating from site 3 and site 6 for the mutual information complex network ${\mathcal{I}}_{ij}$, corresponding to phases and critical point in (b). In this weighted network, the height of the links in our sketch denotes their relative strength; note descending vertical axes from left to right. The entire complex network is far too dense to depict, so we show just two representative sites. (b) Sketch of ferromagnetic phase, critical point, and paramagnetic phase. The sinusoidal potential corresponds to an optical lattice for ultracold atoms or molecules. In the ferromagnetic limit the ground state is the $Z_2$ symmetric superposition between all spin up and all spin down, indicated by two rows of arrows.

Figure 2

Complex network measures on the mutual information. (a) Transverse quantum Ising model describing quantum spins (qubits). The clustering coefficient $C$ and density $D$ serve as order parameters for the ferromagnetic phase. The average disparity $Y$ identifies the short range correlations of the paramagnetic ground state. The Pearson correlation coefficient $R$ develops a cusp near the critical point $g_c=1$, identifying a structured nature to correlations near criticality. (b) Bose-Hubbard model describing massive particles for commensurate lattice filling, with BKT crossover occurring in the limit $L\to \infty$ at a ratio of tunneling $J$ to interaction $U$ of $(J/U{)}_{\mathrm{BKT}}=0.305$; for smaller system sizes, the effective critical point [30] can be as small as $(J/U{)}_{\mathrm{BKT}}\simeq 0.2$. The density and clustering coefficient grow as spatial correlations develop in the superfluid phase. The average disparity is high in the Mott insulator phase where correlations are short-ranged. Critical and crossover behavior is most evident in derivatives of these measures; see Fig. 3 and Table 1. Note: all network measures have been self-normalized to unity for display on a single plot.

Figure 3

Finite-size scaling for the Bose-Hubbard model and transverse Ising model. (a) BHM quantum phase diagram for fixed $L=40$ showing superfluid and Mott Insulating phases with mean field phase transition along the Mott lobe and a BKT transition at its tip. (b) BHM BKT transition at unit filling. Scaling in $1/L$ places the critical point at $(J/U{)}_{\mathrm{BKT}}=0.314$ (clustering $C$, dashed red), 0.281 (average disparity $Y$, solid blue), and 0.289 (density $D$, black dashed), respectively. Compare to the best value to date [29, 32] of 0.305, or the Luttinger liquid prediction of 0.328. (c) Approaching the BHM mean field superfluid to Mott insulator transition for fixed $(J/U)=0.1$. Maximum disparity, central extrema in clustering, and minimum density scale towards the commensurate-incommensurate phase boundary and lie on top of each other. (d) Scaling of multiple measures and their derivatives for the TIM; see also Table 1.