Quantum phase transitions in antiferromagnetic planar cubic lattices

Motivated by its relation to an $\mathcal{NP}$-hard problem, we analyze the ground-state properties of antiferromagnetic Ising-spin networks embedded on planar cubic lattices, under the action of homogeneous transverse and longitudinal magnetic fields. This model exhibits a quantum phase transition at critical values of the magnetic field, which can be identified by the entanglement behavior, as well as by a majorization analysis. The scaling of the entanglement in the critical region is in agreement with the area law, indicating that even simple systems can support large amounts of quantum correlations. We study the scaling behavior of low-lying energy gaps for a restricted set of geometries, and find that even in this simplified case, it is impossible to predict the asymptotic behavior, with the data allowing equally good fits to exponential and power-law decays. We can therefore draw no conclusion as to the algorithmic complexity of a quantum adiabatic ground-state search for the system.

Figure 1

Expected phase diagram in the $(B,\Gamma )$ plane for the systems under study.

Figure 2

“Ladder on a circle” for $N=16$ (left) and $N=18$ (right). For $N=16$ the system is not frustrated, while for $N=18$ it exhibits frustration.

Figure 3

Phase diagram of the nonfrustrated “ladder on a circle,” for $N=16$ spins. The energy gap between the ground and second excited states reveals the appearance of two well-separated phases. The critical line numerically fits the law $\Gamma =1.93949-0.367302B+0.237182B^2-0.106107B^3$.

Figure 4

Entanglement entropy of the reduced density matrix for the particles on the inner (outer) ring, for $N=16$. The peak indicates the crossing of the critical region in the $(B,\Gamma )$ plane. The critical line numerically fits the law $\Gamma =1.93949-0.367302B+0.237182B^2-0.106107B^3$.

Figure 5

Maximum entanglement entropy per site between two rings of a nonfrustrated ladder configuration as a function of the number of lattice sites, up to $N=24$, along the line $B=1$. For large $N$ the quantity $S_{\max }\left({\rho }_{N∕2}\right)∕N$ tends to a constant, which reveals an asymptotic linear scaling for the entropy peak.

Figure 6

Majorization arrow for $B=1$ when decreasing $\Gamma$, for $N=16$. There is a change in the behavior of the cumulants at the critical point ${\Gamma }_c\sim 1.8$. For simplicity only the first five cumulants are plotted.

Figure 7

Majorization analysis of the eigenvalues of the density matrix ${\rho }_{N∕2}$ when tracing out one of the rings, when decreasing $\Gamma$, for $N=16$ and $B=1$. There is a change in the behavior of the cumulants at the critical point ${\Gamma }_c\sim 1.8$. For simplicity only the first five cumulants are plotted.

Figure 8

Pentagonal (left) and tetragonal (right) geometries for cubic planar lattices of $N=20$ and 16 spins, respectively.

Figure 9

Majorization cumulants for the pentagonal lattice of $N=20$ spins, for $B=1$ when decreasing $\Gamma$. A change in their behavior is observed at the critical point ${\Gamma }_c\sim 0.7$. For simplicity only the first five cumulants are plotted.

Figure 10

Majorization cumulants for the tetragonal lattice of $N=16$ spins, for $B=1$ when decreasing $\Gamma$. A change in their behavior is observed at the critical point ${\Gamma }_c\sim 1.6$. For simplicity only the first five cumulants are plotted.

Figure 11

Majorization analysis of the eigenvalues of the density matrix ${\rho }_{N∕2}$ when tracing out half of the system, when decreasing $\Gamma$, for $N=20$ and $B=1$ on the pentagonal lattice. There is again a change in the behavior of the cumulants at the critical point ${\Gamma }_c\sim 0.7$. For simplicity only the first five cumulants are plotted.

Figure 12

Majorization analysis of the eigenvalues of the density matrix ${\rho }_{N∕2}$ when tracing out half of the system, when decreasing $\Gamma$, for $N=16$ and $B=1$ on the tetragonal lattice. There is a change in the behavior of the cumulants at the critical point ${\Gamma }_c\sim 1.6$. For simplicity only the first five cumulants are plotted. Unavoidable numerical noise is present for the first cumulant close to $\Gamma =1$.

Figure 13

Here we plot various parameters calculated for all 1249 possible planar cubic lattices with $N=18$, and averaged. The data were taken along the line $B=0.5$, and each parameter exhibits a signature of a quantum phase transition. The parameters plotted include the first and second energy gaps ${\Delta }_{12},{\Delta }_{13}$, each of which displays a clear minimum. The entropy of the reduced density matrix of a single spin (chosen at random) $S\left[\rho \left(1\right)\right]$ and a block of $N∕2$ connected spins (again chosen at random) divided by $N,S\left[\rho (N∕2)\right]∕N$. Each of these quantities shows a peak at the critical region, between $\Gamma =1$ and 1.4. Finally we have plotted the lowest cumulants of a majorization analysis based on the ordering of the ground state in the $z$ basis, $c{1}_z$, and based on an ordering of the eigenvalues of the $N∕2$-particle reduced density matrix $c{1}_{\rho }$. The first of these shows a sudden increase, while the second exhibits a minimum in the vicinity of the QPT.