Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands

We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low-energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the ground-state manifold. We extract the modular transformation matrices $\mathcal{S}$ ($\mathcal{U}$) which contain the information of mutual (self) statistics, quantum dimensions, and the fusion rule of quasiparticles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.

Figure 1

(a) Haldane model on a $2\times 4\times 6$ HC lattice with lattice vectors ${\vec{w}}_1,{\vec{w}}_2$. The arrow directions on red dotted lines present the signs of the phases $\pm \varphi$ in the NNN hopping terms. The NNNN hoppings are represented by the blue dashed lines. The two ways to bipartition the system along dashed lines are labeled as cut I and cut II. (b) Checkerboard lattice with basis vectors ${\vec{w}}_1,{\vec{w}}_2$. The arrow directions present the signs of the phases $\pm \varphi$ in the NN hopping terms. The two ways to partition the system along the dashed lines are labeled as cut I and cut II, respectively.

Figure 2

Surface and contour plots of Renyi $n=2$ entanglement entropy ($-S$) of wave function $|{\Phi }_{c_1,\varphi }\rangle$ on a $2\times 4\times 4$ HC lattice filled with eight hard-core bosons. The two MESs are labeled by arrows.

Figure 3

Surface and contour plots of Renyi $n=2$ entanglement entropy ($-S$) of wave function $|{\Psi }_{c_1,c_2,{\varphi }_2,{\varphi }_3}\rangle$ on a $2\times 4\times 6$ HC lattice filled with eight fermions. (a) The entropy ($-S$) vs ${\varphi }_2$ and ${\varphi }_3$ by setting $c_1=c_2=c_3=\frac{1}{\sqrt{3}}$. The three MESs are labeled by arrows. (b) The entropy ($-S$) vs $c_1$ and $c_2$ by setting ${\varphi }_2=0.546\pi$, ${\varphi }_3=0.286\pi$ [the phase parameters for the MES labeled by the red arrow in (a)].

Figure 4

The entropy of wave function $|\Psi \rangle$ on a $2\times 4\times 5$ checkerboard lattice with five interacting hard-core bosons by setting $c_1=c_2=c_3=c_4=1/2$, which is the optimized values for MESs. Here we only show the entropy smaller than $2.872$. The calculation is for a bipartition system along the cut II direction.

Figure 5

Eigenvalues ${\lambda }_i$ of the reduced density matrix of two MESs (square) and two maximal entangled states (diamond) for a (a) $2\times 4\times 4$ and (b) $2\times 3\times 6$ HC lattice with hard-core bosons at $\nu =1/2$. The number near the dots shows the degeneracy. Crossed blue (purple) dots stand for the combination of $\left\{{\lambda }_i^{\text{min1(2)}}\right\}$ as described in the text.

Figure 6

(a)–(f) Disorder effect on MESs on a $2\times 4\times 4$ HC lattice filled with eight hard-core bosons. (a) Energy spectrum (the two lowest eigenvalues are labeled by a diamond and cross) and (b) particle entanglement spectrum (PES) for tracing out five particles. There are 352 states below the PES gap for $W<0.6$, in good agreement with the counting of quasihole excitations in the FQH state. (c)–(f) Contour plot of entropy for (c) $W=0.1$, (d) $W=0.4$, (e) $W=0.8$, and (f) $W=1.0$. (g)–(l) Anisotropic interaction effect on MESs on a $2\times 4\times 6$ HC lattice filled with eight fermions. (g) Energy spectrum (the three lowest eigenvalues are labeled by a blue square, red diamond, and green cross) and (h) PES for tracing out six particles. There are 228 states below the PES gap for $V_1^a\ge 0$. (i)–(l) Contour plot of entropy for (i) $V_1^a=0.5$, (j) $V_1^a=0.0$, (k) $V_1^a=-0.5$, and (l) $V_1^a=-1.0$. Bipartitions are all along the cut I direction.