Gauge-fixed Wannier wave functions for fractional topological insulators

We propose an improved scheme to construct many-body trial wave functions for fractional Chern insulators (FCI), using one-dimensional localized Wannier basis. The procedure borrows from the original scheme on a continuum cylinder, but is adapted to finite-size lattice systems with periodic boundaries. It fixes several issues of the continuum description that made the overlap with the exact ground states insignificant. The constructed lattice states are translationally invariant, and have the correct degeneracy as well as the correct relative and total momenta. Our prescription preserves the (possible) inversion symmetry of the lattice model, and is isotropic in the limit of flat Berry curvature. By relaxing the maximally localized hybrid Wannier orbital prescription, we can form an orthonormal basis of states which, upon gauge fixing, can be used in lieu of the Landau orbitals. We find that the exact ground states of several known FCI models at $\nu =1/3$ filling are well captured by the lattice states constructed from the Laughlin wave function. The overlap is higher than $0.99$ in some models when the Hilbert space dimension is as large as $3\times {10}^4$ in each total momentum sector.

Figure 1

Single-particle orbitals in the lowest Landau level (LLL) and the Chern band. In the left panel, we show the Landau orbitals in the LLL on a torus. The two fundamental cycles are marked by ${\mathbf{L}}_1$ and ${\mathbf{L}}_2$, and the twist angle is labeled by $\theta$. The Landau orbitals are plane waves in the ${\widehat{e}}_y$ direction that are localized in the direction perpendicular to propagation. In the right panel, we show a 1D Wannier state localized in the ${\mathbf{b}}_1$ direction and in the lowest band of the kagome lattice model. The size of the spheres depicts the weights of the Wannier state on each lattice site.

Figure 2

The principal Brillouin zone for $C=+1$. Plotted are the flows of the Wannier center position ${\chi }^{X,k_y}=X-\arg \left[W_x\left(k_y\right)\right]/\left(2\pi \right)$ [Eq. (35)] as a function of $k_y$, color coded according to the unit-cell index $X$. The movement of the Wannier center in each unit cell is monotonic in the principal Brillouin zone $C{k}_y+{\delta }_y\in [0\cdots N_y)$ (marked by the gray shade), but not monotonic in the Brillouin zone $k_y\in [0\cdots N_y)$.

Figure 3

The loop $(0,k_y)(0,k_y+1)(k_x,k_y+1)(k_x,k_y)(0,k_y)$ in the $N_x\times N_y$ lattice Brillouin zone. The Wilson loop around this rectangular contour is defined as $W_{5pt2pt}(k_x,k_y)$. The explicit expression in terms of the Berry connections is given in Eq. (54). Notice that at $k_x=N_x$, the connections at the two vertical edges cancel each other due to periodic boundary, leading to $W_{5pt2pt}(N_x,k_y)=W_x\left(k_y\right)/W_x(k_y+1)$.

Figure 4

Flow of $\arg \left[{\lambda }_x\left(\kappa \right)\right]$ in the interval $(-2\pi /N_x,0]$. The solid dots represents the $N_y$ values of $\arg \left[{\lambda }_x\left(\kappa \right)\right]$ for the $N_y$ Wannier centers in each unit cell. Here, we show the case of $C=+1$: $\arg \left[{\lambda }_x\left(\kappa \right)\right]$ is a monotonically decreasing function of $\kappa$ in the pBZ. (a) When $k_y$ and $k_y+1$ can be put in the principal Brillouin zone simultaneously, we have $\arg [{\lambda }_x\left(k_y\right)/{\lambda }_x(k_y+1)]\in (0,2\pi /N_x)$. (b) When going from $k_y$ to $k_y+1$ crosses the boundary of the principal Brillouin zone, we have $\arg [e^{i2\pi /N_x}{\lambda }_x\left(k_y\right)/{\lambda }_x(k_y+1)]\in (0,2\pi /N_x)$. Summarizing both cases, we have $\arg \left[{\mu }_x\left(k_y\right)\right]\in (0,2\pi /N_x)$, for ${\mu }_x\left(k_y\right)$ defined in Eq. (55).

Figure 5

The shift parameter ${\delta }_y$ fixed by inversion for $C=+1$. The three panels correspond to the three cases in Eq. (94). Compared with Fig. 2, here we highlight the discrete nature of the finite-size system. The wave number $k_y$ takes $N_y$ discrete values in each Brillouin zone. The solid dots represents the $N_y$ Wannier centers in each unit cell, colored coded by the unit-cell index $X$. The gray shade shows the principal Brillouin zone $C{k}_y+{\delta }_y\in [0\cdots N_y)$.

Figure 6

Overlap between variational states and the exact ground states. The size of the lattice is labeled by $N_x\times N_y$ at top. The plot includes the checkerboard lattice model (purple crosses), the Haldane model (red hexagons), the kagome lattice model (green triangles), the ruby lattice model (blue Y shapes), and the two-orbital model (cyan squares). The Wannier construction is performed using the $|X,k_y\rangle$ basis. At $6\times 3$, the three states in a multiplet are in the same momentum sector, and thus there is one overlap value for each model. At other lattice sizes, two out of the three states have identical overlap due to inversion symmetry. Also shown is the overlap between the FQH ground state of the Coulomb interaction at filling $\nu =1/3$ and the Laughlin model state, denoted by black circles and labeled by the number of flux $N_{\phi }$. As a baseline, we also plot the top ${10}^{-6}$ and ${10}^{-12}$ percentiles (gray lines) of the overlap values between random unit vectors as a function of the Hilbert space dimension. The overlap axis range is set to $[0,1]$ to contrast with Fig. 7.

Figure 7

Overlaps between variational states and the exact ground states using the prescription in Ref. 15. Inversion breaking for the checkerboard lattice model and the Haldane model is clearly visible at $4\times 3$, and present (although not discernible in this figure) at other lattice sizes. Inversion breaking for these models is also present at $7\times 3$ and $6\times 4$, even though obscured by the finite resolution of the plot. The inversion symmetry of the two-orbital model at $6\times 4$ is also broken.

Figure 8

Overlaps between the Laughlin FCI states and the exact diagonalization states, as a function of the shift parameter ${\delta }_y$ in the Wannier-LLL mapping. Please refer to the legend and caption of Fig. 6 for the annotation of scatter symbols. The left (right) panel shows the calculations on a $6\times 4$ ($6\times 3$) lattice. For clarity, for each model at each ${\delta }_y$ we show the average overlap of the threefold states. The two-orbital model is not included in the $6\times 3$ calculation, as it does not have a gapped topological ground state at this lattice size. The values of ${\delta }_y$ that produce significant overlaps are in full agreement with Eq. (94).

Figure 9

Overlaps between the Laughlin FCI states and the exact diagonalization states on a $6\times 4$ lattice, as a function of the twist angle $\theta$ of the continuum torus for the FQH calculations. For clarity, for each model at each ${\delta }_y$ we show the average overlap of the threefold states. For each model, a peak in overlap is clearly visible at the value of $\theta$ given in Table , in accordance with the discussion in Sec. 3c.

Figure 10

Overlap optimization for the Laughlin lattice states by tuning the single-particle gauge. The calculations are performed on a $6\times 4$ lattice. Please refer to the legend and caption of Fig. 6 for the annotation of scatter symbols. Shown in (a) is the overlap upper bound (110) obtained by removing the relative phase at each component between the exact ground state and the Laughlin lattice state, against the actual overlap. Shown in (b) is the supremum (the least upper bound) of the overlap, obtained by a brute-force optimization of the single-particle gauge of the Bloch states, against the actual overlap.

Figure 11

Particle entanglement spectrum of the kagome lattice model of $N=8$ particles on the $N_x\times N_y=6\times 4$ lattice. Shown in (a) and (b) are the spectra of the Laughlin FCI state and the exact ground state, respectively, with $N_A=3$. Shown in (c) and (d) are the corresponding spectra with $N_A=4$. In each momentum sector, we calculate the number of admissible configurations $n$, and color the lowest $n$ entanglement energy levels in red and the rest in blue. The structure shown in the left panel exhibits the defining characteristic of the Laughlin state.

Figure 12

Single-particle orbitals in the lowest Landau level (LLL) and the Chern band in the alternative gauge choice. In the left panel, we show the Landau orbitals in the LLL on a torus. The two fundamental cycles are marked by ${\mathbf{L}}_1$ and ${\mathbf{L}}_2$, and the twist angle is labeled by $\theta$. The Landau orbitals are plane waves in the ${\widehat{e}}_v$ direction that are localized in the direction perpendicular to propagation. In the right panel, we show a 1D Wannier state localized in the ${\mathbf{b}}_2$ direction and in the lowest band of the kagome lattice model. The size of the spheres depicts the weights of the Wannier state on each lattice site.

Figure 13

Isotropy of the Wannier construction, measured by the overlap between the two sets of Laughlin lattice states constructed using the $|X,k_y\rangle$ and $|Y,k_x\rangle$ Wannier bases. Please refer to the caption and legend of Fig. 6 for the annotation of scatter groups and lines. Inversion symmetry symmetry can be observed similar to Fig. 6.