Luttinger’s theorem at the center of topology

The laws of physics are based on universal principles, often associated with mathematical theorems. Identifying physical phenomena that escape these fundamental rules leads to a paradigm shift, and usually to major discoveries. In a paper published in Physical Review Letters, Lucila Peralta Gavensky, Nathan Goldman (Faculty of Science) and Subir Sachdev (Harvard), reveal a fundamental link between the violation of two major rules of solid-state physics: Luttinger’s theorem and the classification rule for insulating materials.

In 1960, Luttinger proposed a fundamental and universal relationship that links the total number of particles a physical system can contain to its behavior under low-energy excitations. Luttinger’s theorem (which dictates the total number of particles a physical system can contain) obviously applies to systems of independent particles, but it is also valid for systems that exhibit strong correlations between particles. Surprisingly, there are exotic phases of matter for which Luttinger’s theorem does not hold. The violation of Luttinger’s theorem and its consequences for the behavior of quantum matter are currently the focus of intense research in condensed matter physics.

In this context, it was demonstrated that a large class of topological insulators can be labeled by a unique integer, called the Ishikawa-Matsuyama invariant, which perfectly describes the material’s conduction properties. This result is remarkable in that it offers a simple and direct method for classifying insulating states despite the presence of strong interactions between particles. Very recently, however, theorists have identified exotic insulator models that escape this classification: corrections to the Ishikawa-Matsuyama invariant are therefore necessary in very special systems.

In a paper published in Physical Review Letters, Lucila Peralta Gavensky and Nathan Goldman (Faculty of Science, ULB), in collaboration with Subir Sachdev (Harvard) , reveal that the violation of Luttinger’s theorem and the classification of insulating materials are linked by a fundamental relationship. In essence, these authors demonstrate that the Ishikawa-Matsuyama invariant perfectly characterizes correlated insulators, provided that Luttinger’s theorem is satisfied. Otherwise, this topological invariant is insufficient to characterize correlated insulators, and the authors provide explicit mathematical expressions to make the necessary corrections in terms of the relevant physical quantities.

This important connection between Luttinger’s theorem and the topological classification of quantum matter sheds new light on the emergence of exotic phenomena in strongly correlated quantum matter.

https://actus.ulb.be/fr/actus/­recherche/­le-theorem­e-de-lutti­nger-au-ce­ntre-de-la­-matiere-t­opologique