Guest Charge and Potential Fluctuations in Two-Dimensional Classical Coulomb Systems

Jancovici, B.; Šamaj, L.

doi:10.1007/s10955-008-9515-1

Cited by 8 publications

(7 citation statements)

References 25 publications

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“…An example is a recent proof of a second-moment sum rule for correlations near a guest charge in a two-dimensional one-component plasma [5]. Here the use of symmetry properties of the Ursell functions leads to a short proof [6], whereas in deriving the analogous sum rule for the three-dimensional case a detailed analysis of the statistical ensemble properties has to be carried out [7]. Sometimes the analogy between the two-and three-dimensional cases gets lost altogether, as seems to be the case for a higher-order sum rule of the two-dimensional one-component plasma [8].…”

Section: Introductionmentioning

confidence: 99%

Sum rules for correlation functions of ionic mixtures in arbitrary dimensiond⩾ 2

Suttorp¹

2008

J. Phys. A: Math. Theor.

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Abstract. The correlations in classical multi-component ionic mixtures with spatial dimension d ≥ 2 are studied by using a restricted grand-canonical ensemble and the associated hierarchy equations for the correlation functions. Sum rules for the first few moments of the two-particle correlation function are derived and their dependence on d is established. By varying d continuously near d = 2 it is shown how the sum rules for the two-dimensional mixture are related to those for mixtures at higher d.

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Section: Introductionmentioning

confidence: 99%

Sum rules for correlation functions of ionic mixtures in arbitrary dimensiond⩾ 2

Suttorp¹

2008

J. Phys. A: Math. Theor.

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“…This result goes back to [1]. The second moment is more involved [3,29,30]. Curiously, the second moment vanishes at ν = 1 3 and a = 1.…”

Section: Resultsmentioning

confidence: 64%

Collective field theory for quantum Hall states

2015

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We develop a collective field theory for fractional quantum Hall (FQH) states. We show that in the leading approximation for a large number of particles, the properties of Laughlin states are captured by a Gaussian free field theory with a background charge. Gradient corrections to the Gaussian field theory arise from the covariant ultraviolet regularization of the theory, which produces the gravitational anomaly. These corrections are described by a theory closely related to the Liouville theory of quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies.PACS numbers: 73.43. Cd, 73.43.Lp, Introduction Since the work of Laughlin [1], a common approach to analyzing the physics of the fractional quantum Hall effect (FQHE) starts with a trial ground state wave function for N electrons. Despite its success, this approach is an impractical framework for studying the collective behavior of a large number of electrons ( N ∼ 10 6 , in samples exhibiting the QHE). As a result, some subtle properties of QHE states, such as the gravitational anomaly [2-10], were computed only recently.The effects of quantum anomalies are essential in the physics of the QHE. Although anomalies originate at short distances on the order of the magnetic length, they control the large-scale properties of the state, such as transport. It was recently shown in [10] that, like the Hall conductance, transport coefficients determined by the gravitational anomaly are expected to be quantized on QH plateaus. For this reason it is important to formulate the theory of QH effect in a fashion which makes the quantum anomalies manifest. The field theory approach seems the most appropriate for this purpose.In this paper, we develop a field theory for Laughlin states. This approach naturally captures universal features of the QHE, and emphasizes geometric aspects of QH-states. We demonstrate how the field theory encompasses recent developments in the field [2-10] and obtain some properties of quasi-hole excitations. Preliminary treatment of this approach appears in [3].The field theory framework uncovers a connection between the QHE and random geometry, specifically 2D Liouville quantum gravity. Since its introduction, the Laughlin wave function has been a practical model wave function mainly because of the plasma analogy. This analogy to a 2D statistical mechanical system allowed the most salient features of the state -uniform density and fractional quasi-hole charge -to be easily captured by a saddle point approach to the partition function of the equivalent plasma.Every analysis to date has stopped at the saddle point. as a result subtle features of the theory such as the gravitational anomaly were missed. We show how the Laughlin wave function maps to a full quantum field theory. This approach allows to go beyond the saddle point and

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“…In the case of the second-moment, such generalization was made in 2D by using the BGY hierarchy in Ref. [17] and in higher spatial dimensions in Ref. [29].…”

Section: Discussionmentioning

confidence: 99%

“…The rederivation of this result and its generalization to a 2D mixture of several species by using the BGY hierarchy was done in Ref. [17]. The extension of the analysis to ionic mixtures in higher spatial dimensions was the subject of Ref.…”

Section: Introductionmentioning

confidence: 99%

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Fourth Moment of the Charge Density Induced Around A Guest Charge in Two-Dimensional Jellium

Šamaj

2019

Preprint

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The model under consideration is the classical two-dimensional one-component plasma (jellium) of pointlike particles with charge e, interacting pairwisely via the logarithmic Coulomb potential and immersed in a uniform neutralizing background charge density. The system is in thermal equilibrium at the inverse temperature β, its thermodynamics depends only on the coupling constant Γ = βe 2 . We put into an infinite (homogeneous and translationally invariant) plasma a guest particle of charge Ze with Z > −2/Γ in order to prevent from the collapse of the jellium charges onto it. The guest particle induces a screening cloud (the excess charge density) in the plasma. The zeroth and second moments of this screening cloud were derived previously for any fluid value of Γ . In this paper, we propose a formula for the fourth moment of the screening cloud. The derivation is based on the assumption that the fourth moment is, similarly as the second moment, analytic in Z around Z = 0. An exact treatment of the limit Z → ∞ shows that it is a finite (cube) polynomial in Z. The Γ -dependence of the polynomial coefficients is determined uniquely by considering the limits Z → 0 and Z → ∞, and the compressibility sum rule for Z = 1. The formula for the fourth moment of screening cloud is checked in the leading and first correction orders of the Debye-Hückel limit Γ → 0 and at the exactly solvable free-fermion point Γ = 2. Sufficient conditions for sign oscillations of the induced charge density which follow from the second-moment and fourth-moment sum rules are discussed.

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Guest Charge and Potential Fluctuations in Two-Dimensional Classical Coulomb Systems

Cited by 8 publications

References 25 publications

Sum rules for correlation functions of ionic mixtures in arbitrary dimensiond⩾ 2

Sum rules for correlation functions of ionic mixtures in arbitrary dimensiond⩾ 2

Collective field theory for quantum Hall states

Fourth Moment of the Charge Density Induced Around A Guest Charge in Two-Dimensional Jellium

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