Patterns of genuine multipartite entanglement in frustrated quantum spin systems

We investigate the behavior of genuine multipartite entanglement of paradigmatic frustrated quantum spin systems. We consider six different spin models, whose frustration ranges from being very high to very low. We find that the highly frustrated quantum spin systems are near-maximally genuine multipartite entangled, while those with low frustration do not possess a similar definite behavior with regard to their genuine multipartite entanglement.

Figure 1

Illustration of the definition of frustration degree in small spin systems. The first row contains a recollection of the general definition. The quantum Hamiltonian is first “Isingized,” in the sense that we replace each of the terms of the Hamiltonian, irrespective of whether it is one-body, two-body, three-body, etc., by an Ising interaction term to obtain the “Isingized Hamiltonian.” The values and signs of the interaction and field strengths are kept unchanged. This Hamiltonian is then transformed into the corresponding classical Hamiltonian by replacing the Pauli-${\sigma }_z$ matrices with the classical Ising variables (that take values $\pm 1$). We then find the ground-state configurations of this classical Hamiltonian. For every such ground state, we find out which term in the classical Hamiltonian has a positive (negative) contribution to the ground-state energy, $H_{gs}$, for that ground state. These “positive” (“negative”) terms of the Hamiltonian tend to increase (decrease) the $H_{gs}$ and therefore are incorporated into the “frustrated” (“nonfrustrated”) parts of the Hamiltonian, for a given ground state. The ratio of the positive and the modulus of the negative contributions, averaged over all possible ground-state configurations, form the frustration degree of the quantum Hamiltonian. In the second row, we consider, as an example, a frustrated AFM 3-spin periodic linear quantum system, with six degenerate ground-state configurations (each of the three schematic diagrams contain two possible configurations), both for the quantum as well as of the corresponding classical Hamiltonian. The presence of frustration is evident from the lack of a configuration of the classical (Ising) spins leading to a ground-state energy of $H_{gs}=-3J$. The contributions from each spin-pair interaction is shown next to the interaction line (light dashed). Each of these ground states are shown to have one positive-energy contributing (frustrating, shown in bold blue) term and two negative-energy contributing [nonfrustrating, shown in red (nonbold font)] terms. The ratio of the positive and (modulus of) negative terms, averaged over the six possible ground states, is the frustration degree ($=$$\frac{1}{2}$), for the frustrated 3-spin quantum system. We next, in the third row, consider a 4-spin nonfrustrated AFM periodic linear quantum system with two degenerate ground states. The absence of frustration is evident in the figure from the presence of a classical spin configuration leading to the ground-state energy $H_{gs}=-4J$. We observe that there is no positive-energy contributing (frustrating) term for both the ground-state configurations. The corresponding frustration degree is thereby zero.

Figure 2

GGM ($\mathcal{G}$) for the cooled state of the Ising gas. The horizontal axis represents the total number of spins, $N$ = $2m$. We observe that $\mathcal{G}$ increases and finally converges to $\frac{1}{2}$ as $N$ is increased. The vertical axis is dimensionless, while the horizontal one is measured in spins.

Figure 3

GGM ($\mathcal{G}$) for the cooled state of the RVB gas, with respect to the total number of spins, $N=2m$. We observe that the $\mathcal{G}$ is an increasing function of $N$. Dimensions are the same as mentioned in Fig. 2.

Figure 4

The GGM ($\mathcal{G}$) of the cooled state for the MG model, with respect to the total number of spins, $N=2m$. We observe that $\mathcal{G}$ increases with $N$ and converges to $\mathcal{G}\approx \frac{3}{8}$ for large $N$. The dimensions of the axes are the same as mentioned in Fig. 2.

Figure 5

The maximum eigenvalues of ${\rho }_{12}$ (red, dark) and ${\rho }_{23}$ (green, light) of the MG model for $m=40$. In grayscale, the maximum eigenvalue of ${\rho }_{12}$ can be identified as the surface with a dome-shaped protrusion at the center of the base. The other surface corresponds to the maximum eigenvalue of ${\rho }_{23}$. The $x$ and $y$ axes represent $p$ and $q$, respectively. The $z$ axis represents the maximum eigenvalue of ${\rho }_{12}$ and ${\rho }_{23}$ for different values of $p$ and $q$. The eigenvalues are equal along the circle, $p^2+q^2=1$, which correspond to the points of maximum GGM. All axes are dimensionless.

Figure 6

GGM ($\mathcal{G}$) of the Ising ring cooled state with respect to the total number of spins, $N=2m$. We observe that $\mathcal{G}$ increases with $N$ and is maximally multipartite entangled ($\mathcal{G}=0.5$) at the high $N$ limit. The dimensions of the axes are the same as mentioned in Fig. 2.