Band structure engineering of ideal fractional Chern insulators

As lattice analogs of fractional quantum Hall systems, fractional Chern insulators (FCIs) exhibit enigmatic physical properties resulting from the intricate interplay between single-body and many-body physics. In particular, the design of ideal Chern band structures as hosts for FCIs necessitates the joint consideration of energy, topology, and quantum geometry of the Chern band. We devise an analytical optimization scheme that generates prototypical FCI models satisfying the criteria of band flatness, homogeneous Berry curvature, and isotropic quantum geometry. This is accomplished by adopting a holomorphic coordinate representation of the Bloch states spanning the basis of the Chern band. The resultant FCI models not only exhibit extensive tunability despite having only few adjustable parameters but are also amenable to analytically controlled truncation schemes to accommodate any desired constraint on the maximum hopping range or density-density interaction terms. Together, our approach provides a starting point for engineering ideal FCI models that are robust in the face of specifications imposed by analytical, numerical, or experimental implementation.

Figure 1

Cartoon picture of the role of the lattice in affecting the geometry of a FQH fluid, which in turn determines the shape of correlation functions. The left figure depicts the rotationally-symmetric case of the conventional FQH effect, with magnetic length $l_B$ far exceeding the lattice constant $a_0$. The right figure depicts the opposite FCI limit, where the lattice constant is equal or commensurate with the magnetic length; as shown here, a $C_4$-symmetric distortion of the correlation functions should generically be expected upon placing the FQH fluid on the square lattice.

Figure 2

The real parts of illustrative elliptic functions used in our construction of ideal FCI Bloch bands. (Top left) The Weierstrass elliptic function $W(z+\pi (1+i\left)\right)$ [Eq. (32)] with a double pole at the origin. (Top right) Its antiderivative, the Weierstrass Zeta function $\zeta (z+\pi (1+i\left)\right)$ with a single pole at the origin. Note that it is not doubly periodic, as forbidden for meromorphic functions with only one pole; however, linear superpositions of pairs of such Weierstrass zeta functions can be doubly periodic. (Bottom left) $W^{\prime }(z+\pi (1+i))$ with a third order pole at the origin. (Bottom right) The function ${(W^{\prime }\left(z\right)/W\left(z\right))}^2$ used in the Chern number 4 model in Eq. (50); it inherits its set of two double poles from the poles of $W^{\prime }\left(z\right)$ and the zeros of $W\left(z\right)$.

Figure 3

Comparison of our truncated ${\varphi }_1=0.88\frac{W^{\prime }}{W}$ and ${\varphi }_2=4.3W$ model [Eqs. (51) and (52)] with a well-known optimized FCI model with Chern number 3 based on pyrochlore slabs (Ref. [82]). In both the left panel (band dispersion) and right panel (Berry curvature), our model (bluish-green) is seen to possess much better uniformities than that from Ref. [82] (yellow). Numerically, we have $f\approx 40$ and $\langle {\left(\mathrm{\Delta }F\right)}^2\rangle =1.562\times {10}^{-3}$.

Figure 4

Illustration of the truncation procedure results for the $C=3$ ideal FCI model in Sec. 5b. The top, middle, and bottom rows depict the Berry curvature, Chern band dispersion, and satisfaction of the ideal droplet condition for our untruncated ideal FCI model (right-most column “Weierstrass”) as well as real-space truncations to second-, fifth-, seventh-, and ninth-neighbor hopping (left and center columns). Inset labels quantify the Berry curvature deviation $\mathrm{\Delta }F=(1/4{\pi }^2)\int d^k{(F_{xy}-3/2\pi )}^2$, band flatness ratio, and deviation from the ideal isotropic droplet condition ${\mathrm{\Delta }}_{\mathrm{iso}}=\int d^k(\mathrm{Tr}g-F_{xy})$. These quantities are systematically plotted as a function of real-space truncation length in the bottom plot. Notably, Berry curvature fluctuations are suppressed for shorter-ranged truncations, however at the price of breaking FCI isotropy.

Figure 5

(Left) The branch cuts and three singularities of Eq. (B4). There is also an additional singularity when $g_x=\pm k_y$ since $W\left(x\right)$ diverges at $x=0$. (Right) $\sqrt{\mathrm{Tr}{h}_{ij}^{†}{h}_{ij}}$ for $|i-j|=0$ to 4. Terms beyond NNs and NNNs are more than one order of magnitude smaller than the onsite term.