Communications in Mathematical Physics

, Volume 365, Issue 2, pp 431–470 | Cite as

Local incompressibility estimates for the Laughlin phase

  • Elliott H. Lieb
  • Nicolas Rougerie
  • Jakob YngvasonEmail author
Open Access


We prove sharp density upper bounds on optimal length-scales for the ground states of classical 2D Coulomb systems and generalizations thereof. Our method is new, based on an auxiliary Thomas–Fermi-like variational model. Moreover, we deduce density upper bounds for the related low-temperature Gibbs states. Our motivation comes from fractional quantum Hall physics, more precisely, the perturbation of the Laughlin state by external potentials or impurities. These give rise to a class of many-body wave-functions that have the form of a product of the Laughlin state and an analytic function of many variables. This class is related via Laughlin’s plasma analogy to Gibbs states of the generalized classical Coulomb systems we consider. Our main result shows that the perturbation of the Laughlin state cannot increase the particle density anywhere, with implications for the response of FQHE systems to external perturbations.



Open access funding provided by University of Vienna. We received financial support from the French ANR Project ANR-13-JS01-0005-01 (N. Rougerie) and the US NSF Grant PHY-1265118 (E. H. Lieb).


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Nicolas Rougerie
    • 2
  • Jakob Yngvason
    • 3
    • 4
    Email author
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.LPMMC (UMR 5493)Université Grenoble 1 & CNRSGrenobleFrance
  3. 3.Fakultät für PhysikUniversität WienViennaAustria
  4. 4.Erwin Schrödinger Institute for Mathematical PhysicsViennaAustria

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