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

Bauerschmidt, Roland; Bourgade, Paul; Nikula, Miika; Yau, Horng‐Tzer

doi:10.4310/atmp.2019.v23.n4.a1

Cited by 52 publications

(64 citation statements)

References 61 publications

(164 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, it is possible to prove a central limit theorem (see [40] for α = 1, and e.g. [41,42] for general α) for linear statistic associated to smooth test functions. In this case the gaussian fluctuations are of order one.…”

Section: Discussionmentioning

confidence: 99%

Entanglement spectroscopy of chiral edge modes in the quantum Hall effect

Estienne

Stéphan

2020

Phys. Rev. B

View full text Add to dashboard Cite

We investigate the entanglement entropy in the Integer Quantum Hall effect in the presence of an edge, performing an exact calculation directly from the microscopic two-dimensional wavefunction. The edge contribution is shown to coincide exactly with that of a chiral Dirac fermion, and this correspondence holds for an arbitrary collection of intervals. In particular for a single interval the celebrated conformal formula is recovered with left and right central charges c+c = 1. Using Monte-Carlo techniques we establish that this behavior persists for strongly interacting systems such as Laughlin liquids. This illustrates how entanglement entropy is not only capable of detecting the presence of massless degrees of freedom, but also of pinpointing their position in real space, as well as elucidating their nature.

show abstract

Section: Discussionmentioning

confidence: 99%

Entanglement spectroscopy of chiral edge modes in the quantum Hall effect

Estienne

Stéphan

2020

Phys. Rev. B

View full text Add to dashboard Cite

show abstract

“…is imposed in order to extremise the free energy. Equation (11) has the limiting density ρ c (ζ ) = lim N→∞ 1 N R N,1 (ζ ) as its solution. Applying the Laplace operator ∆ = ∂∂ to (11) and using that its Green's function is the logarithm, we obtain that the crossover density ρ c satisfies (12) below.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For Gaussian potential Q(z) = |z| 2 for example, this gives the circular law, with a constant density on the unit disc. Note that (16) also can be obtained from (15) by requiring a saddle point condition as in (11). We emphasize that the standard choice of scaling (9) makes the droplet and density ρ independent of the inverse temperature β .…”

Section: Theoremmentioning

confidence: 99%

The High Temperature Crossover for General 2D Coulomb Gases

Akemann

Byun

2019

J Stat Phys

View full text Add to dashboard Cite

We consider N particles in the plane influenced by a general external potential that are subject to the Coulomb interaction in two dimensions at inverse temperature β . At large temperature, when scaling β = 2c/N with some fixed constant c > 0, in the large-N limit we observe a crossover from Ginibre's circular law or its generalization to the density of non-interacting particles at β = 0. Using several different methods we derive a partial differential equation of generalized Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential and present the analytic solution for the density in case of a Gaussian plus logarithmic potential.

show abstract

“…random variables where the variance grows like n. Furthermore in many cases, one can also show a central limit theorem proving that X(f ) − E(X(f )) converges to a Gaussian distribution. See for e.g., [4,3,10,11,12,20,38,39,42]. However in dimensions bigger than two, there is a change in behavior of X(f ) and the variance grows polynomially in the system size.…”

Section: 2mentioning

confidence: 99%

“…Rigidity for Coulomb type systems. Hyperuniformity of the two-dimensional Coulomb gas has recently been established in [4,3,42]. But as far as we know, before Chatterjee's results, no such results establishing rigidity for systems in dimensions three and higher were available, although very precise information about the partition function in such contexts had recently been obtained in [52,41].…”

Section: Introductionmentioning

confidence: 99%

Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions

Ganguly

Sarkar

2019

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Stochastic point processes with Coulomb interactions arise in various natural examples of statistical mechanics, random matrices and optimization problems. Often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties, that is, the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i.i.d. random points. A well known conjecture from physics appearing in the works of Jancovici, Lebowitz, Manificat, Martin, and Yalcin (see [43,46,37]), states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i.i.d. random points. In a recent beautiful work [18], Chatterjee gave the first proof of such a result in dimension three for a Coulomb type system, known as the hierarchical Coulomb gas, inspired by Dyson's hierarchical model of the Ising ferromagnet [22,23]. However the case of dimensions greater than three had remained open. In this paper, we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three, for the hierarchical model. Using similar methods, we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two. A key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems, and could be of independent interest.

show abstract

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

Cited by 52 publications

References 61 publications

Entanglement spectroscopy of chiral edge modes in the quantum Hall effect

Entanglement spectroscopy of chiral edge modes in the quantum Hall effect

The High Temperature Crossover for General 2D Coulomb Gases

Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions

Contact Info

Product

Resources

About