Fate of the cluster state on the square lattice in a magnetic field

The cluster state represents a highly entangled state which is one central object for measurement-based quantum computing. Here we study the robustness of the cluster state on the two-dimensional square lattice at zero temperature in the presence of external magnetic fields by means of different types of high-order series expansions and variational techniques using infinite projected entangled pair states. The phase diagram displays a first-order phase transition line ending in two critical end points. Furthermore, it contains a characteristic self-dual line in parameter space allowing many precise statements. The self-duality is shown to exist on any lattice topology.

Figure 1

(a) Eigenvalues $k_{\mu }$ of stabilizer operator $K_{\mu }$ on the square lattice with spin $1/2$ degree of freedom on each lattice site. Since $K_{\mu }$ forms a superlattice since it commutes with each of the starlike five spin interaction. This superlattice is again defined on a square lattice. (b) Effect of ${\sigma }^{\alpha }$ operators on the effective superlattice. In conclusion the mainly investigated perturbation with a combined $h_x$,$h_z$ field can create or destroy (depending on the initial state) either 0, 1, 2, or 4 clusterons.

Figure 2

(a) Tensor network diagram for the five-body operator $e^{\delta \tau J{K}_{\mu }}$. This is given in terms of five tensors, $T_z$ (repeated four times) and $T_x$. (b) Nonzero components of tensors $T_z$ and $T_x$. (c) Tensor network diagram for the one-body operator $e^{\delta \tau {\sum }_{\alpha }{h}_{\alpha }{\sigma }_i^{\alpha }}$. This is given in terms of one tensor $T_h$. (d) Nonzero components of tensor $T_h$.

Figure 3

(a) The contraction of four $T_z$ tensors together with $T_x$ and $T_h$ produces a tensor $R$ with double indices, which can be considered as single indices after grouping them. (b) The evolution operator $U\left(\delta \tau \right)$ can be understood as an iPEPO that can be constructed just from tensor $R$.

Figure 4

At step $k$, (a) the iPEPS for state $|{\Psi }_k\rangle$ is defined by tensors ${\Gamma }_A,{\Gamma }_B$ and positive diagonal matrices ${\lambda }_1,...,{\lambda }_4$. The iPEPO for operator $U\left(\delta \tau \right)$ is applied. (b) iPEPS for the evolved state $|{\tilde{\Psi }}_{k+1}\rangle$. Matrices ${\tilde{\Gamma }}_A,{\tilde{\Gamma }}_B$ are obtained by contracting ${\Gamma }_A$ and ${\Gamma }_B$ with $R$, whereas matrices $\widetilde{{\lambda }_1},...,{\tilde{\lambda }}_4$ are obtained by doing the tensor product of ${\lambda }_1,...,{\lambda }_4$ with the $4\times 4$ identity operator. (c) Quasicanonical form for the iPEPS of state $|{\tilde{\Psi }}_{k+1}\rangle$. (d) iPEPS for state $|{\Psi }_{k+1}\rangle$.

Figure 5

One-particle gap $\Delta$ (upper panel) and ground-state energy per site $e_0$ (lower panel) as a function of the magnetic field $h$. The field direction is parametrized by $h_x=h\cos \left(\theta \right)$ and $h_z=h\sin \left(\theta \right)$ and is chosen to be $\theta =\frac{5}{128}\pi$. Results of series expansions are shown as lines while circles correspond to iPEPS data with bond dimension $D=2,3,4$ (energy differences are negligible in the scale of the plot). The bare series obtained in order 9 (order 8) for $e_0$ ($\Delta$) is plotted as solid red lines. Different dlogPadé (Padé) extrapolations of the highest order are shown as dashed (dotted) lines (deviant extrapolations are omitted in the plot for the sake of clarity). Notice that these extrapolations almost collapse in most of the plot. It can be clearly seen that $e_0^{\mathrm{iPEPS}}$ is well below $e_0^{\mathrm{SE}}$ for a field value for which the one-particle gap is still finite, i.e., $h^{\mathrm{crit}}>h^{*}$ (black vertical lines) holds. We therefore detect a first-order phase transition for $h\approx 0.86$ illustrated as the dashed vertical line at $h^{*}$.

Figure 6

Phase diagram of the perturbed cluster Hamiltonian as a function of $X$ and $Y$ obtained by the series expansion plus the iPEPS approach. The corners of the triangle correspond to the three limits: (i) the pure cluster Hamiltonian $(J,0,0)$ (left corner), (ii) the pure $h_x$ field $(0,0,h_x)$ (right corner), and (iii) the pure $h_z$ field $(0,h_z,0)$ (upper corner). The self-dual line is illustrated by the vertical red solid line cutting the triangle in the middle. The whole spectrum and therefore also the phase diagram on both sides of the self-dual line are fully symmetric. First-order phase transition lines are drawn as orange solid lines. The two critical end points are marked by blue circles [coordinates of the left point $(X_{\mathrm{left}}\approx 0.604,Y_{\mathrm{left}}\approx 0.174)$ and coordinates of the right point $(X_{\mathrm{right}}\approx 0.810,Y_{\mathrm{right}}=Y_{\mathrm{left}})$].

Figure 7

The gap $\Delta /J$ as a function of $h_z$ is shown along the self-dual line $h_x=J$ setting $J=1$. The solid line corresponds to the bare series and dashed lines represent different Padé extrapolations. We remark that dlogPadé extrapolations reveal no poles along the real axis. (Inset) Comparing the iPEPS data with the series expansion for the ground-state energy per site $e_0$ in the high $h_z$-field case. In this limit $e_0$ behaves linearly with the field $e_0/J=-h_z$. The good agreement between the different limits indicates that there is no phase transition along the full self-dual line.

Figure 8

Direct comparison between the series expansion for the fidelity and the iPEPS results. We contrast the results in the limit $h_z=0$ (upper panel) and $h_x=0$ (lower panel). One clearly sees the deviation of both approaches near the first-order transition (red dashed line). We ascribe this fact to the insensitivity of the SE approach to first-order effects. The agreement along the $h_z$ axis remains very good, considering the Padé extrapolations of the bare series. Notice also the collapse of the different Padé extrapolations with the bare series in the scale of the plot.

Figure 9

Phase diagram including the useful region for quantum computing. The shaded gray (green) area and the gray inset indicate the usable region for measurement-based quantum computing (MBQC) according to the fidelity threshold at $d⩾0.986$ [43]. (Inset) Comparing the extrapolations for the series expansion of the fidelity per site (black line) with the iPEPS results (green dots). Clearly there is no agreement beyond the first-order transition line ($h_x>h^{*}$). This is an artifact of the series expansion technique, which is not sensitive to first-order effects. Departing from the phase boundary, both fidelities are in very good agreement.