Dirac’s method for constraints: an application to quantum wires

Schmeltzer, D.

doi:10.1088/0953-8984/23/15/155601

Cited by 9 publications

(7 citation statements)

References 42 publications

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“…In consequence, the commutators

[p_{i}, p_{j}]

turn out to satisfy the following relation with

f,_{k} \equiv \partial f / \partial x_{k}

false[p_{i}, p_{j} false] = \frac{i ℏ}{2} ((n_{j} n_{i, l} - n_{i} n_{j, l}) p_{l} + p_{l} (n_{j} n_{i, l} - n_{i} n_{j, l})),

and the Hamiltonian operator turns out to be,

H = \frac{p^{2}}{2 μ} - \frac{ℏ^{2}}{8 μ} M^{2} + V_{G} .

The second and key finding of this Letter is: With the quantum condition being imposed, the curvature‐induced potential

V_{G}

proves to be the geometric potential what has been expected for more than three and a half decades. Being

K \equiv \nabla_{N} n : \nabla_{N} n = {(n_{i, j})}^{2}

in fact the trace of square of the extrinsic curvature tensor, the geometric potential is

V_{G} = - \frac{ℏ^{2}}{4 μ} K + \frac{ℏ^{2}}{8 μ} M^{2}

…”

Section: Dirac Brackets and Quantization Conditionsmentioning

confidence: 99%

“…This CPF has a distinct feature for no presence of any ambiguity. It is thus a powerful tool to examine various curvature‐induced consequences in two‐dimensional curved surfaces or curved wires . Experimental confirmations of the geometric potential include: an optical realization in 2010 and an observation of its effects in an uneven periodic caged peanut‐shaped nanostructure in 2012 .…”

Section: Introductionmentioning

confidence: 99%

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Geometric Potential and Dirac Quantization

Lian

Liu

2018

Annalen der Physik

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A fundamental problem regarding the Dirac quantization of a free particle on an N − 1 (N ≥ 2) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature-induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin-layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator-ordering-free section is identified and is then found sufficient to lead to the expected form of geometric potential.

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“…In consequence, the commutators

[p_{i}, p_{j}]

turn out to satisfy the following relation with

f,_{k} \equiv \partial f / \partial x_{k}

false[p_{i}, p_{j} false] = \frac{i ℏ}{2} ((n_{j} n_{i, l} - n_{i} n_{j, l}) p_{l} + p_{l} (n_{j} n_{i, l} - n_{i} n_{j, l})),

and the Hamiltonian operator turns out to be,

H = \frac{p^{2}}{2 μ} - \frac{ℏ^{2}}{8 μ} M^{2} + V_{G} .

The second and key finding of this Letter is: With the quantum condition being imposed, the curvature‐induced potential

V_{G}

proves to be the geometric potential what has been expected for more than three and a half decades. Being

K \equiv \nabla_{N} n : \nabla_{N} n = {(n_{i, j})}^{2}

in fact the trace of square of the extrinsic curvature tensor, the geometric potential is

V_{G} = - \frac{ℏ^{2}}{4 μ} K + \frac{ℏ^{2}}{8 μ} M^{2}

…”

Section: Dirac Brackets and Quantization Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Geometric Potential and Dirac Quantization

Lian

Liu

2018

Annalen der Physik

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“…In the second line of Eq. (21) we have used the constraint relation which emerges from the strong spin-orbit interaction y = kx kso a, This result is interpreted as a second class constrained [20,21] The one dimensional effective model given in Eq. (21) with the potential g 2 x 2 allows to introduce a space dependent Fermi momentum, k F (x) = k so 2µ…”

Section: Using the Relation Imposed By The Open Boundary Conditions Wmentioning

confidence: 99%

Fractional Topological Insulators—A Bosonization Approach

Schmeltzer¹

2016

JMP

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A metallic disk with strong spin orbit interaction is investigated . The finite disk geometry introduces a confining potential. Due to the strong spin-orbit interaction and confining potential the metal disk is described by an effective one dimensional with a harmonic potential. The harmonic potential gives rise to classical turning points. As a result open boundary conditions must be used. We Bosonize the model and obtain chiral Bosons for each spin on the edge of the disk. When the filling fraction is reduced to ν = k F kso = 1 3 the electron-electron interactions are studied using the Jordan Wigner phase for composite fermions which gives rise to a Luttinger liquid. When the metallic disk is in the proximity with a superconductor a Fractional Topological Insulators is obtained.An experimental realization is proposed. We show that by tunning the chemical potential we control the classical turning points for which a Fractional Topological Insulator is realized.

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“…(18), we will use the algebra of the zero modes [22][23][24] where the charge and the current are given byQ Using the algebra of the zero modes, we compute to lowest order the energy of the ground state as a function of the coupling constant λ ≡ 2ĝ 2 ( πv F l ring ) −1 = 2g 2 k F l ring πv F < 1, the electronic bandwidth and the external fluxes. We find for the ground-state energy…”

Section: Persistent Currentsmentioning

confidence: 99%

Chiralpx+ipysuperconducting nanowire coupled to two metallic rings pierced by a flux

Schmeltzer¹,

Saxena²

2012

Phys. Rev. B

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We consider a p-wave superconducting nanowire weakly coupled to two metallic rings. At the two interfaces between the nanowire and the metallic rings the pairing order parameter vanishes, as a result two zero-mode Majorana fermions appear. The two metallic rings are pierced by external magnetic fluxes. By determining the correlation between the persistent currents in the two rings, we identify the special features of Majorana fermions, such as nonlocality, which can be experimentally observed in this setup.

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Dirac’s method for constraints: an application to quantum wires

Cited by 9 publications

References 42 publications

Geometric Potential and Dirac Quantization

Geometric Potential and Dirac Quantization

Fractional Topological Insulators—A Bosonization Approach

Chiralpx+ipysuperconducting nanowire coupled to two metallic rings pierced by a flux

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