Isoperimetric Inequalities in Quantum Geometry (2503.16604v1)

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.

Isoperimetric Inequalities in Quantum Geometry

The paper "Isoperimetric Inequalities in Quantum Geometry" by Praveen Pai and Fan Zhang investigates the interconnection between classical isoperimetric inequalities and quantum geometric states within the field of wavefunction behavior in Hilbert space. Addressing the fundamental quantum geometric quantities—quantum distance and Berry phase—the authors advance our understanding of path-dependent properties of wavefunctions, providing both strong and weak inequalities concerning these macroscopic quantities.

In classical geometry, the isoperimetric problem considers the shape that maximizes area while conserving perimeter. Classical results indicate the circle as a solution in two-dimensional Euclidean space, with rigorous solutions being extended to higher-dimensional manifolds. This geometric construct is mirrored in the paper's quantum interpretation, framing quantum state evolution within a geometric landscape. By mapping the two-band quantum isoperimetric problem to spherical geometry, the authors establish a strong inequality that binds quantum distance and Berry phase for closed paths in Hilbert space. The strong quantum isoperimetric inequality (QII) formula is expressed as:

$$(|\gamma_{\text{B}}| - \pi)^2 + d_{\text{FS}}^2 \geq \pi^2,$$

where $\gamma_{\text{B}}$ and $d_{\text{FS}}$ represent the Berry phase and Fubini-Study quantum distance, respectively.

Complementarily, a weak inequality posits that quantum distance is a lower bound for Berry phase:

$d_{\text{FS}} \geq \gamma_{\text{B}},$

which notably holds without asserting symmetries and can be generalized for multiple-band systems $(M \geq 2)$. The implications of these inequalities are manifold.

Implications and Applications:

1. Wannier Function Spread: The gauge-invariant part of Wannier function spread is constrained by quantum geometry, effectuating a tethering to both quantum distance and Berry phase squared. The paper illuminates that in one-dimensional systems, the Wannier spread is inherently bounded by these geometric and topological properties.
2. Quantum Speed Limit: By linking the evolution of quantum states to energy uncertainty, the paper specifies a quantum speed limit that now incorporates Berry phase as a limiter. This progression presents a quantifiable timeframe on the evolution path, impacting how quantum systems are temporally regulated, especially in adiabatic processes.
3. Electron-Phonon Coupling: The coupling constant $\lambda$, pivotal in determining superconductivity transition temperatures, is significantly informed by quantum geometric contributions. The weak QII confines the quantum metric's effect on this coupling in two-dimensional systems, providing a bridge between Fermi surface properties and superconducting characteristics.
4. Geometric Superfluid Weight: In multiband superconductors, the superfluid weight $D_s$ is affected by topological invariants, such as the Chern number. This paper demonstrates that even without traditional interband contributions, gauge-independence of quantum distance enhances the bound set by Berry phase, with potential attributions to superfluid properties in topological flat bands.

The theoretical framework advanced represents a significant stride in reconciling classical notions in geometry with quantum mechanics, providing new perspectives to explore quantum paths within Hilbert spaces. The investigation suggests future avenues in further broadening these inequalities to more complex quantum systems and manifold structures. These insights could also foster innovation in materials science, particularly in fields like superconductivity and electronic properties modulation, which are heavily reliant on understanding quantum geometry. Considerations for future research could include expanding the analysis to encompass higher order topological systems and exploring non-Euclidean influences on state evolution paths.