Quantum geometry in condensed matter

Describing natural phenomena with geometric language is one of the most important achievements of modern physics. In the context of general relativity, Einstein treats spacetime as a dynamic geometric structure in which the distribution of matter determines the curvature of spacetime, and the geometry of spacetime in turn determines the motion of matter, thus revealing the deep connection between spacetime geometry and the motion of celestial bodies. Geometric theories have not only been verified on the macroscopic cosmic scale, but also provide profound insights and accurate predictions on the microscopic particle scale. In the context of condensed matter physics, the quantum geometry of the electron wave functions controls the motion of electrons and thus affects the fundamental physical properties of matter. Quantum geometry has become an important theoretical tool in modern condensed matter physics. It plays a key role in revealing novel transport phenomena, resolving the microscopic mechanism of superconductivity, and searching for novel topological states, and has a profound impact on the development of condensed matter physics.

National Science Review recently published online a review article "Quantum Geometry in Condensed Matter" co-authored by a team led by chair professor Hai-Zhou Lu of Southern University of Science and Technology and academician X. C. Xie of Peking University.

It is pointed out in the review that the quantum geometry of the electron wave functions in condensed matter is characterized by a tensor. The real part of this quantum geometry tensor is the symmetric Fubini-Study metric (also known as the quantum metric), which describes the amplitude distance between two quantum states. The imaginary part is the antisymmetric Berry curvature tensor (up to a coefficient of -1/2), which describes the phase distance between two quantum states. This quantum geometry tensor plays a key role in understanding the electromagnetic field dependence of nonlinear transport, the superconducting transition temperature of flat-band superconductors, and the stability of fractional Chern insulators.

Quantum geometry leads to both longitudinal and transverse nonlinear transport. According to electromagnetic field dependence, the transverse transport can be further divided into the electric nonlinear Hall effect and the magnetic nonlinear Hall effect. The former arises from either the Berry curvature dipole or the quantum metric dipole, depending on the symmetry, while the latter results from the anomalous orbital polarizability, a quantum geometric quantity derived from Berry curvature (Fig. 1).

Quantum geometry enables in the flat-band superconductors a superconducting transition temperature much higher than the BCS expectation. Although the electrons in flat bands are quite inert (the effective mass tends to infinity), the Cooper pairs formed from such electrons can remarkably have a finite effective mass because of the inter-band quantum metric effect. The resulting geometric supercurrent greatly exceeds the BCS expectation, revealing the reason for the enhancement of the superconducting transition temperature (Fig. 2).

Quantum geometry plays a key role in the stabilization of fractional Chern insulating states. The topological Bloch bands with strong correlation highly mimic the flat Landau levels in the fractional quantum Hall effect, giving rise to fractionally quantized Chen insulating states. The homogeneity of quantum geometry distribution in the Brillouin zone stabilizes the fractional Chern insulating states by isolating them from other strongly correlated states such as the charge density wave (Fig. 3).

In summary, it is emphasized in the review that the interplay between quantum geometry and disorder in nonlinear transport, the relation between quantum geometry and pairing symmetry in flat-band superconductors, and the influence of quantum geometry on other strongly correlated phases in fractional Chern insulators are important outstanding problems in the field of quantum geometry and had better be urgently settled.