Quantum geometry is universally bounded from below by Wilson-loop windings. In this Letter, we define an isolated set of bands to be Wilson-loop ideal, if their quantum metric saturates the Wilson-loop lower bound. The definition naturally incorporates the known Chern-ideal and Euler-ideal bands, and allows us to define other types of ideal bands, such as Kane-Mele ${\mathbb{Z}}_2$-ideal and inversion-fragile-ideal bands. In particular, we find that in the case of zero total Chern number, an isolated Wilson-loop-ideal set of two bands with nonsingular non-Abelian Berry curvature and nontrivial normal Wilson-loop winding always admits a Chern-ideal gauge, without the need of a global good quantum number (such as spin). This enables the direct construction of new topologically ordered states, such as fractional topological insulator wave functions. We further propose a general framework of constructing monotonic flows that achieve Wilson-loop-ideal states starting from nonideal bands through band mixing, where Wilson-loop-ideal states are not energy eigenstates but have smooth projectors similar to isolated bands. We apply the constructed flows to the realistic model of 3.89° twisted bilayer ${\mathrm{MoTe}}_2$, a moiré Rashba model, and another moiré time-reversal-breaking model, and numerically find Chern-ideal, ${\mathbb{Z}}_2$-ideal, and inversion-fragile states, respectively, with relative error in the integrated quantum metric below $5\times {10}^{-3}$. Our exact-diagonalization calculations on the numerically ideal states demonstrate the potential of our general definition of Wilson-loop-ideal bands and general procedure of constructing Wilson-loop-ideal states for future study of novel correlated physics.

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• Anyons
• Topological phases of matter
• Topological materials