Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order

We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin-$1∕2$ particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper order parameter. We demonstrate that our conclusions are robust under the addition of random perturbations, not only in the topological phase, but also in the spin-polarized phase and even at the critical point.

Figure 1

(Color online) A square lattice with $32$ spins. The spin degrees of freedom are placed on the vertices. The red dashed lines $t_1^x,t_2^x$ are the incontractible loops around the torus. The product $B7\times B8\times B11$ denotes the loop operator drawn in red. All the spins on the vertices crossed by a loop are flipped. The region $A\cup B\cup C$ is a ring containing eight spins, used in computing $S_{\mathrm{top}}$. For the lattice of $32$ spins, the ring has diameter $R=2$ and width $r=1$.

Figure 2

(Color online) Fidelity between the time-dependent solution of the Schrödinger equation and the adiabatic state, for different values of the total evolution time: $T=20$,40,60. (a) The unperturbed model for $L18$. The evolution is adiabatic for $T=60$. Note that the drop in adiabaticity is a precursor of the QPT. (b) $L8$: fidelity in both the ideal and perturbed $(P=1)$ cases. The perturbed model is indistinguishable from the ideal one.

Figure 3

(Color online) QPT detectors for $L8,L18,L32$, for the unperturbed and perturbed model. All graphs show strong resilience of the model and its QPT against perturbations: (a) Second derivative of $E\left(\tau \right)$, diverging for ${\tau }_c\sim 0.7$. The QPT is thus second order. (b) Derivative of the von Neumann entropy, measuring the entanglement of a plaquette with the rest of the lattice. Its divergence at criticality also signals a second order QPT. The perturbation has no effect for $P=20$ (triangles indistinguishable from circles) but is visible for $P=40$. (c) Ground state fidelity $\mathcal{F}\left(\tau \right)$: the fidelity drop at the critical point signals a QPT, associated with a drastic change in the properties of the ground state. (d) Overlap between the perturbed and the ideal ground state. The clearly visible susceptibility to the perturbation at the critical point also signals the QPT.

Figure 4

(Color online) Expectation value of Wilson loop operators of increasing size for $L32$. The expectation value of the loop operators starts to increase at ${\tau }_c$, more steeply so for the largest loops, indicating that this observable can be used to detect the TOQPT for large systems.

Figure 5

(Color online) von Neumann entropy for a plaquette of spins and $S_{\mathrm{top}}$ for $L32$ with an Ising Hamiltonian in a transverse field. Note the different vertical scales. For a system without topological order, $S_{\mathrm{top}}$ is always $\sim 0$ (the small bump is a finite-size effect).

Figure 6

(Color online) $S_{\mathrm{top}}$ for $L18$ and $L32$, and von Neumann entropy for $L32$, for the ideal and perturbed model. $S_v$ assumes the value $l-1=3$ in the entire TO phase, where $l-1$ is the exact value of $S_v$ for the pure Kitaev model $(\tau =1)$ and $l=4$ is the length of the border of a plaquette. $S_{\mathrm{top}}$ is zero in the spin-polarized phase and quickly reaches unity in the TO phase. Also the topological character of the QPT and of the phases is resilient to perturbations.

Figure 7

(Color online) First derivative of $S_{\mathrm{top}}$ for $L18$ and $L32$, for the ideal and perturbed model, diverging at ${\tau }_c$ thus signaling the TOQPT; (inset) derivative and full width at half maximum of $S_{\mathrm{top}}$ at ${\tau }_c$ as a function of perturbation strength $P$. $S_{\mathrm{top}}$ remains robust up to $P\sim 25$.