Luttinger’s theorem holds the key to classifying correlated quantum matter

In 1960, Joaquin Luttinger put forth a universal statement linking a system’s total particle capacity to its behavior during low-energy excitations. Although Luttinger’s theorem is typically confirmed in systems of independent particles, it intriguingly extends to correlated quantum matter with robust particle interactions.

Despite its general validity, Luttinger’s theorem faces surprising failures in specific instances within strongly correlated phases of matter. The repercussions of this breakdown on the behavior of quantum matter have become a focal point in condensed matter physics research.

Parallel to these investigations, considerable efforts have focused on categorizing and characterizing correlated insulating states of matter. Notably, a significant stride was made when it was revealed that a wide array of topological insulators could be identified by a single integer – the Ishikawa-Matsuyama invariant – effectively encapsulating their transport properties.

This breakthrough marked a milestone by providing a straightforward method for classifying insulating states in the presence of strong interactions. However, recent theoretical developments have uncovered exotic models of correlated insulators that defy this elegant classification, necessitating corrections to the Ishikawa-Matsuyama invariant under specific conditions.

In a recent article published in Physical Review Letters, Lucila Peralta Gavensky and Nathan Goldman (ULB), in collaboration with Subir Sachdev (Harvard), unveil a profound connection between the failure of Luttinger’s theorem and the classification of insulating states of matter. The authors demonstrate that the Ishikawa-Matsuyama invariant becomes the complete descriptor for correlated insulators when Luttinger’s theorem holds.

Conversely, when Luttinger’s theorem is violated, this topological invariant proves inadequate for labeling correlated phases. The authors go on to provide explicit expressions for the necessary corrections, expressed in terms of pertinent physical quantities.

This crucial link between Luttinger’s theorem and the topological classification of quantum matter provides insights into the emergence of exotic phenomena within strongly correlated quantum systems.

Source: Université libre de Bruxelles

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