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arXiv:cond-mat/9703080

Uncertainty Relation For Quantized Magnetic Fields and The Quantum Hall Effect

F. Ghaboussi

Published Mar 7, 1997• 0 citations
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Abstract

We discuss a model where the relation between Hall potential $A_H$ = and gate potential $A_G$, which is manifested by the stepped QHE = curve, is described by an uncertainty relation $ΔA_H . = ΔA_G =3D B(\hbar/e)$.

DOI
10.48550/arxiv.cond-mat/9703080

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We discuss a model where the relation between Hall potential AHA_HAH​ = and gate potential AGA_GAG​, which is manifested by the stepped QHE = curve, is described by an uncertainty relation ΔAH.=ΔAG=3DB(ℏ/e)ΔA_H . = ΔA_G =3D B(\hbar/e)ΔAH​.=ΔAG​=3DB(ℏ/e).
We formulate according to the quantum mechanical uncertainty relation a new quantum electrodynamical uncertainty relation ΔA˘.Δl∼ℏ/eΔ\breve{A} . Δl \sim \hbar/eΔA˘.Δl∼ℏ/e where A˘\breve{A}A˘ and Δl≥lBΔl \geq l_BΔl≥lB​ are the electromagnetic pure gauge potential, the position uncertainty and the magnetic length, respectively. Then, we show that the observed potential drops on the edge of QHE samples are varifications of this uncertainty relation, where the quantum potential drop of the relevant component of potential can be considered as its quantum uncertainty ΔA˘Δ\breve{A}ΔA˘.
A linear response expression for the Hall conductance ${\ensuremath{\sigma}}_{\mathrm{xy}}$ of interacting two-dimensional electrons in a strong magnetic field is derived. When the system is dissipationless, ${\ensuremath{\sigma}}_{\mathrm{xy}}$ can be written solely in terms of the density-density response function ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow}) of the electron liquid. Thus, a specific behavior of ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow}) is required on a Hall plateau. This behavior is ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow})\ensuremath{\propto}q2{q}^{2}q2 for small wave vector q, with a proportionality constant given by the Hall quantum number.
Rarely noted paradoxes in applications of fundamental quantum relations are pointed out, with their resolution leading to emergent non-Hermitian behaviors due to boundary terms – even for closed systems and with real potentials. The role played by these non-Hermiticities on the consistency of quantum mechanical uncertainty relations is discussed, especially in multiply-connected spaces (more generally, for any system that satisfies the Bloch theorem of Solid State Physics). These subtleties – reflections of topological quantum anomalies – follow their own patterns (for any dimensionality, for both Schrödinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities Jg. In continuous nonrelativistic models, these have the forms that had earlier been used by Chemists to describe atomic fragments of polyatomic molecules, while for Dirac/Weyl or other lattice models they have more interesting relativistic forms only recently worked out in graphene models. In spite of the deep mathematical origin as quantum anomalies examples are provided here, where such non-Hermiticities have a direct physical significance (for both conventional and topological materials). In all stationary state examples considered, these non-Hermitian boundary terms turn out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2D Landau system in an external in-plane electric field), where some particular Jg seems to be related to the well-known dissipationless edge currents. More generally, the non-Hermitian terms play a subtle role on Berry curvatures in solids and seem to be crucial for the consistent application of the so called Modern Theories of Polarization and Orbital Magnetization. It is emphasized that the above systems can be _closed_ (in multiply-connected space, so that the boundaries disappear, but the non-Hermiticity remains), a case in non-Hermitian physics that is not usually discussed in the literature; it is also stressed that a mapping between the above non-Hermiticity (for continuous systems) and the many recent available results in tight-binding solid state models (leading to the so-called exceptional points) is expected to promote enhanced understanding of quantum behavior at the most fundamental level.