Effect of surface curvature on magnetic moment and persistent currents in two-dimensional quantum rings and dots

Bulaev, D. V.; Geyler, V. A.; Margulis, V. A.

doi:10.1103/physrevb.69.195313

Cited by 50 publications

(50 citation statements)

References 39 publications

(58 reference statements)

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“…The above expressions (5.1)-(5.3) for the energy levels can also be extracted from the recent work [7]. It is worthwhile to emphasize that the discrete spectrum is absent for |b| < 1/2.…”

Section: Spectrum and Density Of Statesmentioning

confidence: 95%

Aharonov-Bohm effect on the Poincaré disk

Lisovyy

2007

Journal of Mathematical Physics

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“…The above expressions (5.1)-(5.3) for the energy levels can also be extracted from the recent work [7]. It is worthwhile to emphasize that the discrete spectrum is absent for |b| < 1/2.…”

Section: Spectrum and Density Of Statesmentioning

confidence: 95%

Aharonov-Bohm effect on the Poincaré disk

Lisovyy

2007

Journal of Mathematical Physics

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“…Such curvature effects have gained renewed attention in the last decade, mainly because of the technological progress that has enabled the fabrication of low-dimensional nanostructures with complex geometry [6,7,8,9,10,11,12,13,14]. From the theoretical perspective, many intriguing phenomena pertinent to electronic states [15,16,17,18,19,20,21,22], electron diffusion [23], and electron transport [24,25,26,27] have been suggested. In particular, the correlation between surface curvature and spin-orbit interaction [28,29] as well as with the external magnetic field [30,31,32] has been recently considered as a fascinating subject.…”

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confidence: 99%

Geometry-driven shift in the Tomonaga-Luttinger exponent of deformed cylinders

2009

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We demonstrate the effects of geometric perturbation on the Tomonaga-Luttinger liquid (TLL) states in a long, thin, hollow cylinder whose radius varies periodically. The variation in the surface curvature inherent to the system gives rise to a significant increase in the power-law exponent of the single-particle density of states. The increase in the TLL exponent is caused by a curvature-induced potential that attracts low-energy electrons to region that has large curvature. PACS numbers: 73.21.Hb, 71.10.Pm, 03.65.Ge Studying the quantum mechanics of a particle confined to curved surfaces has been a problem for more than fifty years. The difficulty arises from operator-ordering ambiguities [1], which permit multiple consistent quantizations for a curved system. The conventional method used to resolve the ambiguities is the confining-potential approach [2,3]. In this approach, the motion of a particle on a curved surface (or, more generally, a curved space) is regarded as being confined by a strong force acting normal to the surface. Because of the confinement, quantum excitation energies in the normal direction are raised far beyond those in the tangential direction. Hence, we can safely ignore the particle motion normal to the surface, which leads to an effective Hamiltonian for propagation along the curved surface with no ambiguity.It is well known that the effective Hamiltonian involves an effective scalar potential whose magnitude depends on the local surface curvature [2,3,4,5]. As a result, quantum particles confined to a thin, curved layer behave differently from those on a flat plane, even in the absence of any external field (except for the confining force). Such curvature effects have gained renewed attention in the last decade, mainly because of the technological progress that has enabled the fabrication of low-dimensional nanostructures with complex geometry [6,7,8,9,10,11,12,13,14]. From the theoretical perspective, many intriguing phenomena pertinent to electronic states [15,16,17,18,19,20,21,22], electron diffusion [23], and electron transport [24,25,26,27] have been suggested. In particular, the correlation between surface curvature and spin-orbit interaction [28,29] as well as with the external magnetic field [30,31,32] has been recently considered as a fascinating subject.Most of the previous works focused on noninteracting electron systems, though few have focused on interacting electrons [33] and their collective excitations. However, in a low-dimensional system, Coulombic interactions may drastically change the quantum nature of the system. Particularly noteworthy are one-dimensional systems, where the Fermi-liquid theory breaks down so that the system is in a Tomonaga-Luttinger liquid (TLL) state [34]. In a TLL state, many physical quantities exhibit a power-law dependence stemming from the absence of single-particle excitations near the Fermi energy; this situation naturally raises the question as to how geometric perturbation affects the TLL behaviors of quasi onedimensional curved sys...

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“…Notice that this potential is bounded and thus the spectrum of the corresponding quantum Hamiltonian contains an absolutely continuous part, the interval [E 0 (a), ∞[ . The spectral analysis for this case is covered by paper [6]. It turns out that E 0 (a) ∼ a 2 as a → ∞, and so the continuous part disappears in the flat limit.…”

Section: The Hamiltonianmentioning

confidence: 99%

On the harmonic oscillator on the Lobachevsky plane

Šťovı́ček

Tušek

2007

Russ. J. Math. Phys.

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We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential V (̺) = (a 2 ω 2 /4) sinh(̺/a) 2 where a is the curvature radius and ̺ is the geodesic distance from a fixed center. Thus the potential is rotationally symmetric and unbounded likewise as in the Euclidean case. The eigenvalue equation leads to the differential equation of spheroidal functions. We provide a basic numerical analysis of eigenvalues and eigenfunctions in the case when the value of the angular momentum, m, equals 0.

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Effect of surface curvature on magnetic moment and persistent currents in two-dimensional quantum rings and dots

Cited by 50 publications

References 39 publications

Aharonov-Bohm effect on the Poincaré disk

Aharonov-Bohm effect on the Poincaré disk

Geometry-driven shift in the Tomonaga-Luttinger exponent of deformed cylinders

On the harmonic oscillator on the Lobachevsky plane

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