Numerical study of a transition between ${\mathbb{Z}}_2$ topologically ordered phases

Distinguishing different topologically ordered phases and characterizing phase transitions between them is a difficult task due to the absence of local order parameters. In this paper, we use a combination of analytical and numerical approaches to distinguish two such phases and characterize a phase transition between them. The “toric code” and “double semion” models are simple lattice models exhibiting ${\mathbb{Z}}_2$ topological order. Although both models express the same topological ground state degeneracies and entanglement entropies, they are distinct phases of matter because their emergent quasiparticles obey different statistics. For a 1D model, we tune a phase transition between these two phases and obtain an exact solution to the entire phase diagram, finding a second-order Ising$\times$Ising transition. We then use exact diagonalization to study the 2D case and find indications of a first-order transition. We show that the quasiparticle statistics provides a robust indicator of the distinct topological orders throughout the whole phase diagram.

Figure 1

(a) indicates the mapping from the spin to the loop space. (b) shows the terms in the Hamiltonian. The dots indicate Hermitian conjugates of the ones shown here and also equivalent terms obtained by rotating the lattice.

Figure 2

The ladder geometry used for the quasi-1D systems. In (a), we show the domain wall representation of loops in the even sector of the Hilbert space; note the use of auxiliary $\uparrow$ spins off the lattice. (b) shows a domain wall representation for loops in the odd sector of the Hilbert space; here we need auxiliary $\uparrow$ spins above the chain and $\downarrow$ spins below it.

Figure 3

The spectrum of the 1D ladder model in the even sector at various points in the phase diagram. For $0\le \eta <\mu$ $\left(0\le \mu <\eta \right)$ and $0\le \lambda <\mu \left(\eta \right)$ we are in the DSem (TC) phase, and the spectrum is fully gapped. For any $0\le \lambda <0.5$ there is a continuous phase transition involving two critical Ising theories as we tune $\mu$ to be greater than $\eta$ (vertical red line). For $\mu \ne \eta$ and when $\lambda =0.5$ (horizontal thick line), there is a critical point in the 2D critical Ising universality class, and the spectrum has a single Dirac point. For $\lambda >0.5$, ${\sigma }^z$ loops are short, and the auxiliary Ising spins ${\tau }^z$ are ordered. At $\lambda =0.5$, $\mu =\eta =0.25$ which is the intersection of two lines of critical points (horizontal and vertical), the spectrum shows a quadratic dispersion.

Figure 4

Derivative of ground state energy density as the system is tuned from the TC to DSem phase. The increase in steepness at $\mu =\eta =0.5$ as we go to large system sizes $(N$ = number of spins) indicates a possible first-order transition.

Figure 5

Spectral decomposition $A$ of the ground state wave function at the transition points (dashed lines) for different system sizes. The left column corresponds to 48 spins and the right column corresponds to 75 spins. The top row shows the behavior of $A(\omega ,\lambda )$ at the known second-order transition from the TC phase to the polarized phase at $\lambda \approx 0.18$. The bottom row shows $A(\omega ,\mu )$ at the transition from the TC to the DSem phase where a transition occurs at $\mu =0.5$.

Figure 6

Variation of the topological entanglement entropy (TEE) as the system is tuned from the TC fixed point to the DSem fixed point (along $\lambda =0$; red circles, top $x$ axis), and from the TC fixed point to the polarized fixed point (along $\mu =0$; blue triangles, bottom $x$ axis). The inset shows the TEE at all points in the phase diagram with dark red regions indicating higher TEE. This indicates that the bottom part of the phase diagram is topologically ordered. The size of the system considered is 48 spins.

Figure 7

Nontrivial bipartitions of the torus. We consider the partition on the right and trace over one-half of the torus to obtain the reduced density matrix. The yellow lines indicate the long loops around the torus characterizing the possible winding sectors.

Figure 8

Variation of the $U$ matrix as the system is tuned from the TC phase to the DSem phase. The angle ${\alpha }_j$ plotted is related to the $U$ matrix by $U_{jj}=e^{i{\alpha }_j}$. We see a transition from the fermionic statistics indicative of the TC phase to the semionic statistics indicative of the DSem phase. The region near the transition point is ambiguous due to an additional degeneracy which comes into play. The size of the system considered is 48 spins.

Figure 9

Duality mapping between the loops in the topological models and domain walls in the dual ${\tau }_p^z$ Ising spins which live on plaquettes.