Rigidity of the Laughlin Liquid

Lieb, Élliott H.; Rougerie, Nicolas; Yngvason, Jakob

doi:10.1007/s10955-018-2082-1

Cited by 22 publications

(25 citation statements)

References 51 publications

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“…In the next section we state our main results precisely and discuss them further (they were announced in the short paper [42]). The rest of the paper is devoted to their proofs.…”

Section: Introductionmentioning

confidence: 88%

“…A remarkable consequence is that the Laughlin state stays an approximate ground state in any radial increasing potential, however steep. We refer to [59,60] again and to the companion paper [42] for further discussion of the significance of this.…”

Section: 2mentioning

confidence: 99%

“…By the previous lemma, we can assume that the density of such vanishing points is bounded below uniformly, and combining this with a bound on the gradient of the potential we obtain an upper bound on the potential outside the disk. We can now proceed in two different ways (the second one was sketched in [42]):…”

Section: Lemma 45 (Maximal Configurations Have No Vacancies)mentioning

confidence: 99%

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Local incompressibility estimates for the Laughlin phase

Lieb

Rougerie

Yngvason

2018

Commun. Math. Phys.

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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.

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“…In the next section we state our main results precisely and discuss them further (they were announced in the short paper [42]). The rest of the paper is devoted to their proofs.…”

Section: Introductionmentioning

confidence: 88%

Section: 2mentioning

confidence: 99%

Section: Lemma 45 (Maximal Configurations Have No Vacancies)mentioning

confidence: 99%

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Local incompressibility estimates for the Laughlin phase

Lieb

Rougerie

Yngvason

2018

Commun. Math. Phys.

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“…is an exact ground state (L 2 -normalized by the constant in front). One can then argue, and prove to some extent [20,21,29,24], that such functions and natural variants are extremely robust, in particular to the addition of the external potential V .…”

Section: Projected Hamiltonians and Densities In Quantum Hall Physicsmentioning

confidence: 99%

“…As noted by many people since long, and emphasized in particular in [11,12,5], a holomorphic representation of states is not limited to the lowest Landau level, where it has proved to be important for deriving some basic properties, e.g. [20,21,24,25,29]. In fact, there is a natural unitary correspondence between states in different Landau levels, in particular between higher levels and the lowest one.…”

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confidence: 99%

Holomorphic quantum Hall states in higher Landau levels

Rougerie

Yngvason

2020

Journal of Mathematical Physics

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Eigenstates of the planar magnetic Laplacian with homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems, in particular we represent effective Hamiltonians and particle densities in higher Landau levels by corresponding quantities in the lowest Landau level.

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Fluctuations of Two Dimensional Coulomb Gases

2018

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We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the support of the equilibrium measure. This can be stated in terms of convergence of the random electrostatic potential to a Gaussian Free Field.Our result is the first to be valid at arbitrary temperature and at the mesoscopic scales, and we recover previous results of Ameur-Hendenmalm-Makarov and Rider-Virág concerning the determinantal case, with weaker assumptions near the boundary. We also prove moderate deviations upper bounds, or rigidity estimates, for the linear statistics and a convergence result for those corresponding to energy-minimizers.The method relies on a change of variables, a perturbative expansion of the energy, and the comparison of partition functions deduced from our previous work. Near the boundary, we use recent quantitative stability estimates on the solutions to the obstacle problem obtained by Serra and the second author.

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Rigidity of the Laughlin Liquid

Cited by 22 publications

References 51 publications

Local incompressibility estimates for the Laughlin phase

Local incompressibility estimates for the Laughlin phase

Holomorphic quantum Hall states in higher Landau levels

Fluctuations of Two Dimensional Coulomb Gases

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