A universal conformal field theory approach to the chiral persistent currents in the mesoscopic fractional quantum Hall states

Abstract

We propose a general and compact scheme for the computation of the periods and amplitudes of the chiral persistent currents, magnetizations and magnetic susceptibilities in mesoscopic fractional quantum Hall disk samples threaded by Aharonov-Bohm magnetic field. This universal approach uses the effective conformal field theory for the edge states in the quantum Hall effect to derive explicit formulas for the corresponding partition functions in presence of flux. We point out the crucial role of a special invariance condition for the partition function, following from the Bloch-Byers-Yang theorem, which represents the Laughlin spectral flow. As an example we apply this procedure to the Z k parafermion Hall states and show that they have universal non-Fermi liquid behavior without anomalous oscillations. For the analysis of the high-temperature asymptotics of the persistent currents in the parafermion states we derive the modular S -matrices constructed from the S matrices for the u(1) sector and that for the neutral parafermion sector which is realized as a diagonal affine coset.

Thermoelectric Characteristics of $${\mathbb Z}_k$$ Z k Parafermion Coulomb Islands

“…where v F is the Fermi velocity at the edge and L is the circumference of the edge. The CFT disk partition function in presence of AB flux φ = eB.A/h, threading the disk, is modified by simply shifting the chemical potential [12] ζ…”

Thermopower and thermoelectric power factor ofZkparafermion quantum dots

“…where K l (τ, ζ ; m) is the u(1) partition function for the charged part which is completely determined by the filling factor ν H and coincides with that for a chiral Luttinger liquid with a compactification radius [23] R c = 1/m, in the notation of [21,16]…”

“…The modular parameter ζ used in the definition of the rational CFT partition functions is related to the chemical potential µ by [16,10]…”

“…[10] to the other Read-Rezayi states in order to compute the thermopower of a Coulomb blockaded island in these FQH states in terms of the Grandcanonical averages of the edge states' Hamiltonian and particle number operators. To this end we will use the structure of the Grand-canonical partition functions for general FQH states on a disk properly modified in presence of Aharonov-Bohm (AB) flux [16]. We calculate numerically and plot the thermopower, the electric and thermal conductances and the power factors for these states with odd or even number of bulk quasiparticles.…”

“…(12), according to Eqs (15). and(16) Z φ (τ, µ) = K 1+3(φ +µ) (τ, 0; 15)ch 1,1 (rτ) + K 6+3(φ +µ) (τ, 0; 15)ch 1,2 (rτ) + K −4+3(φ +µ) (τ, 0; 15)ch 2,2 (rτ) (24) where the K functions are defined in Eq. (13), r = v n /v c and the neutral partition functions are [15] ch 1,1 (τ) = ch(Λ 0 + Λ 1 ; τ) = q 1 30…”