Coulomb Blockade of Anyons in Quantum Antidots

Abstract

Coulomb interaction turns anyonic quasiparticles of a primary quantum Hall liquid with filling factor ν = 1/(2m + 1) into hard-core anyons. We have developed a model of coherent transport of such quasiparticles in systems of multiple antidots by extending the Wigner-Jordan description of 1D abelian anyons to tunneling problems. We show that the anyonic exchange statistics manifests itself in tunneling conductance even in the absence of quasiparticle exchanges. In particular, it can be seen as a non-vanishing resonant peak associated with quasiparticle tunneling through a line of three antidots. [5,6]. The situation with fractional statistics is so far less certain even in the case of the abelian statistics, which is the subject of this work. Although the recent experiments [7] demonstrating unusual flux periodicity of conductance of a quasiparticle interferometer can be interpreted as a manifestation of the fractional statistics [8,9], this interpretation is not universally accepted [10,11]. There is a number of theoretical proposals (see, e.g., [12,13]) suggesting tunnel structures where the statistics manifests itself through noise properties. Partly due to complexity of noise measurements, such experiments have not been performed successfully up to now. In this work, we show that coherent quasiparticle dynamics in multiantidot structures should provide clear signatures of the exchange statistics in dc transport. Most notably, in tunneling through a line of three antidots, fractional statistics leads to a non-vanishing peak of the tunnel conductance which would vanish for integer statistics.These effects rely on the ability of quantum antidots to localize individual quasiparticles of the QH liq- uids [4,14,15]. The resulting transport phenomena in antidots are very similar to those associated with the Coulomb blockade [16] in tunneling of individual electrons in dots. For instance, similarly to a quantum dot [19], the linear conductance of one antidot shows periodic oscillations with each period corresponding to the addition of one quasiparticle [4,14,15,17,18]. Recently, we have developed a theory of such Coulomb-blockade-type tunneling for a double-antidot system [20], where quasiparticle exchange statistics does not affect the transport. The goal of this work is to extend this theory to antidot structures where the statistics does affect the conductance. The two simplest structures with this property consist of three antidots and have quasi-1D geometries with either periodic or open boundary conditions (Fig. 1). A technical issue that needed to be resolved to calculate the tunnel conductance is that the anyonic field operators defined through the Wigner-Jordan transformation [21,22,23,24], are not fully sufficient in the situations of tunneling. As we show below, to obtain correct matrix elements for anyon tunneling, one needs to keep track of the appropriate boundary conditions of the wavefunctions which are not accounted for in the field operators.Specifically, we consider the antidots coupled by tunneli...

One-dimensional impenetrable anyons in thermal equilibrium: II. Determinant representation for the dynamic correlation functions

“…The anyonic model considered in this work can be viewed as a generalization to an arbitrary statistics parameter κ of the model of impenetrable bosons obtained from the Bose gas with repulsive δ-function interaction [1,2] in the limit of infinitely large coupling constant (for other anyonic extensions of well known models see [3,4,5]). This model, which we call the Lieb-Liniger gas of anyons, was formulated in this form by Kundu [6], clarified in [7,8] and further studied in [9,10,11,12,13,14]. In the bosonic case, the first step in the analysis of the correlation functions is the derivation of the Fredholm determinant representation for these functions [15].…”

“…The order of these operators adopted in Eq. (17) (leading to the phase +iπκ): the particle with the first coordinate z 1 created first, then z 2 , etc., is convenient [7] for the subsequent calculation of the form factors. In this paper, we limit our discussion to the case of infinitely strong interaction, c → ∞, which corresponds to impenetrable anyons.…”

“…In contrast to particles of integer statistics, wavefunctions of the anyons satisfy different quasi-periodic boundary conditions in their different arguments, the difference resulting from the statistical phase shift 2πκ [7,8]. In general, the quasi-periodic boundary conditions also include the external phase shift η (we will consider η=2π× rational), so that the boundary conditions on the wavefunctions (20) are:…”

“…Since these phase shifts are canceled in Eq. (36), the expression under the integrals over z j is periodic in each of the variable [7]. This feature is the necessary consistency condition for the Hilbert spaces of anyon wavefunctions with different numbers of particles, and is important in what follows for the appropriate calculation of the form factors (36).…”

Junctions of anyonic Luttinger wires

“…The reason for this generalization is twofold. On one hand the study of 1D anyonic model is attracting a renewed interest [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] , mainly motivated by possible experiments with cold atoms 60 . On the other hand, the transport of wires joined with a quantum Hall island is driven by anyonic excitations 12 .…”