Local incompressibility estimates for the Laughlin phase

Abstract

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.

Rigidity of the Laughlin Liquid

“…They hold essentially on length scales O(N 1/4 ), much smaller than the full extent of the liquid, which is of order N 1/2 , and are related to earlier partial results proved in [37,38]. The full proofs are somewhat involved and presented elsewhere [25], but we sketch the main arguments below and discuss physical applications.…”

“…We now turn to sketching the proof of Theorem 1. Details are given in the longer paper [25]. It is convenient to change variables and consider the scaled N -particle probability density…”

The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem

“…Still for the two-dimensional Coulomb gas at any temperature, a local density [32,36] was recently proved, together with abnormally small charge fluctuations in the sense of rigidity [32], see (1.8). Other recent results in this direction include [3,[35][36][37]41].…”

Systems of Points with Coulomb Interactions

“…[La1,La2,La3], and for recent mathematical progress using this correspondence, cf. [RSY2,RY,LRY]. For β = 2 it also arises as the wave-function density of the ground state for the system of N non-interacting fermions confined to a plane with a perpendicular magnetic field [Fo1,Chap.…”