Identifying non-Abelian topological ordered state and transition by momentum polarization

Using a method called momentum polarization, we study the quasiparticle topological spin and the edge-state chiral central charge of non-Abelian topological ordered states described by Gutzwiller-projected wave functions. Our results verify that the fractional Chern insulator state obtained by Gutzwiller projection of two partons in bands of Chern number 2 is described by SU(2)${}_2$ Chern-Simons theory coupled to fermions, rather than the pure $\text{SU}{\left(2\right)}_2$ Chern-Simons theory. In addition, by introducing an adiabatic deformation between one Chern-number-2 band and two Chern-number-1 bands, we show that the topological order in the Gutzwiller-projected state does not always agree with the expectation of topological field theory. Even if the parton mean-field state is adiabatically deformed, the Gutzwiller projection can introduce a topological phase transition between Abelian and non-Abelian topologically ordered states. Our approach applies to more general topologically ordered states described by Gutzwiller-projected wave functions.

Figure 1

An illustration of the hopping Hamiltonian in Eq. (2). The two orbitals on each lattice site are shown as different layers and colored in black and blue, respectively. The hopping is $+1$ ($-1$) along the solid (dashed) lines, and $i/\sqrt{2}$ ($-i/\sqrt{2}$) along (against) the red arrows.

Figure 2

The partial translation operator $T_y^L$ translates the left half of the cylinder by one lattice constant along the $\widehat{y}$ direction. The red arrows indicate the chiral edge modes. The topological sector $a$ is determined by the type of quasiparticle threaded through the cylinder, denoted by the large blue arrow.

Figure 3

The value of $-L_{y}arg\left({\lambda }_1\right)$ versus $L_y^2$ for the identity sector $a=1$. The intercept at $L_y^2=0$ of the linear fitting gives $-2\pi p_1=0.7513\pm 0.046$. We set $L_x=8$ and $l=1$ for all calculations.

Figure 4

An illustration of the hopping Hamiltonian in Eq. (11). The two orbitals on each lattice site are shown in different layers and colored in black and blue, respectively. The hoppings along the solid (dashed) lines are $+\sqrt{2}$ ($-\sqrt{2}$), and along (against) the red arrows are $i/\sqrt{2}$ ($-i/\sqrt{2}$). It is straightforward to separate the system into two uncoupled zigzag subsystems with odd and even values of $x_i+I$.

Figure 5

Illustration of a $C=1$ Chern insulator model on a two-dimensional square lattice. The nearest-neighbor hopping amplitudes are $\sqrt{2}$ along the square edges and $-$$\sqrt{2}$ along the dashed lines. The next-nearest-neighbor hoppings are along the square diagonal with amplitude $+i/\sqrt{2}$ along ($-i/\sqrt{2}$ against) the arrow. The two lattice sites in the unit cell are marked as $A$ and $B$.

Figure 6

The topological spin $h$ for the semion (non-Abelian quasiparticle) sector versus various values of $\Theta \in [0,\pi /4]$ for the projected Chern insulator in Eq. (10) from momentum polarization calculations. The red dashed line and the blue dotted line are the theoretical values of $h$ for the $\text{SU}{\left(2\right)}_1\times \text{SU}{\left(2\right)}_1$ CS theory ($h_s=1/4$) and $\nu =2$ fermions coupled to an $\text{SU}\left(2\right)$ gauge field ($h_{\sigma }=5/16$), respectively.

Figure 7

A histogram of the number of sampled configurations versus the parton number around its average $N_1-{\overline{N}}_1$ in one of the decomposed $C=1$ bands. While the $N_1={\overline{N}}_1$ central peak contains more than $98\%$ of the configurations for $\Theta =0.05\pi$ (red), the spread for $\Theta =0.125\pi$ (black) is much wider and the percentage of the $N_1={\overline{N}}_1$ configurations is only $15\%$, suggesting that $N_1$ is no longer a good quantum number. The results are obtained on system size $L_x=L_y=28$ with periodic boundary conditions.

Figure 8

The mean squared deviation $\sqrt{\langle {(N_1-{\overline{N}}_1)}^2\rangle /{\overline{N}}_1}$ versus $\Theta$ for system sizes $L_x=L_y=8,12,16,20,24,28$.