Quantum Geometry: Revisiting electronic scales in quantum matter (2504.07173v1)

Abstract: Electronic properties of solids are often well understood via the low energy dispersion of Bloch bands, motivating single band approximations in many metals and semiconductors. However, a closer look reveals new length and time scales introduced by quantum dipole fluctuations due to interband mixing, which are reflected in the momentum space textures of the electronic wavefunctions. This structure is usually referred to as quantum geometry. The scales introduced by geometry not only qualitatively modify the linear and nonlinear responses of a material but can also have a vital role in determining the many-body ground state. This review explores how quantum geometry impacts properties of materials and outlines recent experimental advances that have begun to explore quantum geometric effects in various condensed matter platforms. We discuss the separation of scales that can allow us to estimate the significance of quantum geometry in various response functions.

• The paper demonstrates that quantum dipole fluctuations redefine electronic length scales in quantum materials.
• The study applies the quantum geometric tensor to link interband transitions with conventional band theory, enhancing experimental interpretations.
• The research shows that incorporating quantum geometry shifts electron transport and many-body ground state behaviors in new directions.

Quantum Geometry: Revisiting Electronic Scales in Quantum Matter

Introduction

The paper focuses on the impact of quantum geometry on the electronic properties of solids, particularly metals and semiconductors. Traditional analyses consider only low-energy dispersion within Bloch bands; however, this paper emphasizes the need to incorporate quantum dipole fluctuations arising from interband mixing, a concept central to quantum geometry. This review explores how quantum geometry not only alters material responses but also influences many-body ground states, with recent experiments supporting these claims.

The Role of Quantum Geometry

Quantum geometry can be understood through the paper of Riemannian structures on the manifold of electronic wavefunctions, introduced by the quantum geometric tensor (QGT). This tensor elucidates both geometric phases and a loss of state norm, which are crucial in defining electronic properties. The quantum metric, in particular, highlights interband transitions, marking a deviation from conventional band theories.

Quantum dipole fluctuations, emerging from these geometric transitions, are instrumental in defining novel length scales within metals. These scales often exceed lattice parameters and redefine interactions and transport phenomena within various materials.

Figure 1: Dipole fluctuations leading to electron position uncertainty, with a length scale characterized by the quantum geometry of the ground state.

Quantum Geometry in Band Theory

Quantum geometry introduces fundamental shifts in band theory, traditionally focused on single-band approximations. The QGT bridges Bloch state characteristics with broader electronic behaviors, embedding quantum geometric phenomena into band structures and transitions.

The Kohn metric and single-band quantum metrics compare interband effects against intraband transitions, driving the realization that quantum effects play a central role beyond semiclassical predictions.

Figure 2: Visualization of scale separation in tight-binding approximations and the emergence of geometric scales.

Practical Findings and Separations of Scale

The paper extends the notion of scale separation, identifying diverse dynamics inherent to quantum materials at various timescales. Intermediate timescales highlight quantum geometric contributions, distinguishing them from those defined by longer mean free paths. Sum rules provide a theoretical foundation for understanding interband transitions, deconvoluting observed spectral weights to infer physical properties across different conditions.

Experimental methodologies align with theoretical predictions, helping measure quantum geometry's impact on observable material characteristics.

Conclusion

Quantum geometry offers a framework to understand nuances of electronic structures by introducing enhanced interpretive tools beyond traditional band theory. These insights into interband transitions and dipole fluctuations position quantum geometry as crucial in comprehending the compound responses and states of quantum materials. Extending these foundational principles to practical applications promises significant advancements in material sciences and condensed matter physics.