Isoperimetric Inequalities in Quantum Geometry

TL;DR

Isoperimetric inequalities in quantum geometry establishes a quantum analogue of the classical isoperimetric problem by relating the macroscopic quantum geometric quantities $dFS$ and $gammaB$ for closed loops in Hilbert space. It derives a strong inequality $(|gammaB|-pi)^2 + dFS^2 >= pi^2$ for two-band systems and a general weak inequality $dFS >= gammaB$ for $M$-band systems, with circles on the Bloch sphere saturating the strong bound. These results extend to complex projective spaces $CP^{M-1}$, and the authors show how the bounds constrain physically measurable quantities such as Wannier function spread, quantum speed limits, electron-phonon coupling, and geometric superfluid weight. Together, the work provides a unifying geometric framework that connects quantum distance and Berry phase to tangible condensed-matter phenomena, offering a fresh perspective on quantum geometry beyond symmetry-protected settings.

Abstract

We reveal strong and weak inequalities relating two fundamental macroscopic quantum geometric quantities, the quantum distance and Berry phase, for closed paths in the Hilbert space of wavefunctions. We recount the role of quantum geometry in various quantum problems and show that our findings place new bounds on important physical quantities.

Figures (2)

• Figure 1: Visual of the 2D isoperimetric inequalities. Right: equilateral triangle, square, hexagon, and circle of equal radius (red lines). As the number of sides, $N$, of a regular polygon increases, the inverse isoperimetric quotient (shown at the top) saturates to $1$, revealing that the planar isoperimetric problem is satisfied by the circle. Left: the same visual but for spherical polygons on a sphere with their radial lengths fixed at the polar angle of ${\pi}/{4}$. The spherical isoperimetric problem is again solved by a shape with the highest symmetry -- a closed path of constant polar angle.
• Figure 2: Illustration of Bloch sphere parameterization and quantum isoperimetric inequalities. (a) Any two-band state can be expressed as $\ket{z_1} = (1,z_1)^{T}/\sqrt{1+|{z_1}|^2}$ using one complex number $z_1=e^{i\phi}\tan(\theta/2)$, where $\phi$ is the azimuthal angle and $\theta$ is the polar angle of the Bloch sphere. $\theta=0$ ($z_1=0$) and $\pi$ ($z_1=\infty$) denote the the north and south poles of the Bloch sphere, respectively. (b) Representatives of trivial, geometric, and topological loops. (c) Geometric implication of the weak quantum isoperimetric inequality derived from the strong one.