Frustrated Quantum Magnetism on Complex Networks: What Sets the Total Spin

Abstract

Consider equal antiferromagnetic Heisenberg interactions between qubits forming a complex, nonbipartite network. We ask the question: How does the network topology determine the net magnetization of the ground state and to what extent is it tunable? By examining over 75000 networks of different families with tunable structural properties, we demonstrate that (i) heterogeneity in the number of neighbors is essential for a nonzero total spin, and (ii) apart from the number of neighbors, the key determinant is the presence of (disassortative) hubs, as opposed to the frustration level. In fact, one can vary the magnetization throughout its range by embedding such hubs. We also discuss simple, exactly solvable networks where such tunability leads to both abrupt and continuous transitions, with quantum effects giving rise to a diverging susceptibility. Our findings can be realized on emerging platforms and pose a number of fundamental questions, strongly motivating wider exploration of quantum many-body phenomena on complex networks.

Figures (15)

• Figure 1: (a) Distribution of total spin in the ground state of random graphs with $N = 30$ qubits and $N_e = 45$ bonds with equal antiferromagnetic Heisenberg coupling. Dashed curve shows the average magnetization, $S^z_{\text{total}}$, for Ising spins (max-cut bipartitions). (b) Ensemble average of $S_{\text{total}}$ vs average degree $\bar{k} = 2 N_e / N$. (c) Correlation with various graph metrics. Points inside the gray band are statistically insignificant.
• Figure 2: Sample graph (top), degree distribution (middle), and $S_{\text{total}}$ distribution (bottom) for random regular, random, and scale-free (Barabási-Albert) graphs with $N = 30$ and $\bar{k} \approx 4$. The power-law tail in (f) gives rise to hubs [circled in (c)] that want to polarize their neighborhoods (inset). Dashed curves in (g-i) show the distribution of average $S^z_{\rm total}$ for Ising spins.
• Figure 3: (a) Average total spin and (b) heterogeneity, given by the standard deviation $\Delta k$ of the degree distribution, for random graphs without short loops and $N = 30$. (c) $\bar{S}_{\text{total}}$ for Barabási-Albert graphs with $N = 30$, $\bar{k} = 3.8$, and tunable number of triangles, measured by the mean clustering coefficient $C \in [0,1]$WattsStrogatz1998. Dashed curve shows $\bar{S}^z_{\text{total}}$ for Ising spins.
• Figure 4: (a) Most disassortative and (b) most assortative networks with the same set of degrees. Node colors show a bipolar ground state of Ising spins. (b) Variation of $\bar{S}_{\text{total}}$ as the assortativity Newman2002 is tuned for random graphs with $N = 30$ and $\bar{k} = 4$. Dashed curve shows $\bar{S}^z_{\rm total}$ for Ising spins.
• Figure 5: (a) Graph with embedded hubs and (b) graph with no hub from the "copy model" described in text for $m=2$. Node colors show a ground state for Ising spins and bond colors show ground-state correlations for our Heisenberg model. (c)--(e) Variation of total-spin distribution with the structure parameter $p$ for $N = 30$ and $m=2$. Red dots show the average excitation gap for a given $S_{\rm total}$. (f) Variation of heterogeneity $\Delta k$, frustration index $N_f$, and assortativity $A$. (g) $\bar{S}_{\rm total}$ as a function of $p$ and $\bar{k} = 2m - m(m+1)/N$ for $N = 30$.