[Paper Review] Nonequilibrium superfield and lattice Weyl transform transport approach to quantum Hall effect

F. A. Buot|arXiv (Cornell University)|Jan 20, 2020

Quantum and electron transport phenomena15 references4 citations

TL;DR

This paper introduces a novel nonequilibrium superfield and lattice Weyl transform (SFLWT) quantum transport framework to derive the topological Chern number in the integer quantum Hall effect. By employing a first-order gradient expansion in the effective $(\vec{p},\vec{q};E,t)$-phase space, it identifies the topological invariant directly from the transport equation, yielding exact quantization of Hall conductance in units of $e^2/h$ without relying on linear response or equilibrium Kubo formalism.

ABSTRACT

We derive the topological Chern number of the integer quantum Hall effect in electrical conductivity, using Buot's superfield and lattice Weyl transform nonequilibrium quantum transport formalism. The method is naturally straightforward, appropriate for treating nonequilibrium systems acted on by external electromagnetic fields. We have identified the topological invariant in the effective (~p; ~q; E; t)-phase space via the nonequilibrium quantum transport equation, generally not to first-order in electric field but to first-order in the gradient expansion. We have also derive the Kubo current-current correlation for the Hall current as a by-product of our new transport approach. The Berry curvature related to orbital magnetic moment is also calculated.

Motivation & Objective

• To derive the topological Chern number of the integer quantum Hall effect using a nonequilibrium quantum transport formalism.
• To establish a framework that avoids reliance on linear response theory or equilibrium Kubo formalism.
• To identify the topological invariant directly in the $(\vec{p},\vec{q};E,t)$-phase space via the SFLWT-NEGF approach.
• To demonstrate the quantization of Hall conductance as an integral multiple of $e^2/h$ using real-time superfield dynamics.
• To compute the Berry curvature and orbital magnetic moment in two-dimensional systems as a byproduct of the transport formalism.

Proposed method

• Utilizes Buot’s real-time superfield and lattice Weyl transform (SFLWT) nonequilibrium Green’s function (NEGF) formalism for quantum transport.
• Applies a first-order gradient expansion in the quantum transport equation, not first-order in the electric field per se.
• Introduces the variable $\mathcal{\vec{K}} = \vec{p} + e\vec{F}t$ to describe the time-evolving phase-space coordinates under a uniform electric field.
• Derives the topological invariant as a phase-space integral involving the Hamiltonian $H^{(a)}$ and lesser Green’s function $G^{<\left(b\right)}$ in $\mathcal{\vec{K}}$-space.
• Reconstructs the Kubo current-current correlation formula for Hall conductivity as a byproduct of the transport derivation.
• Applies the method to both gapped energy-band structures and Landau-level systems by incorporating the vector potential into $\mathcal{\vec{K}}$.

Experimental results

Research questions

• RQ1 Can the topological Chern number in the integer quantum Hall effect be derived directly from a nonequilibrium quantum transport formalism?
• RQ2 How does the SFLWT-NEGF approach enable the identification of topological invariants in phase space without relying on equilibrium linear response?
• RQ3 What is the role of the $\mathcal{\vec{K}}$-space variable in capturing the dynamics of the Hall conductance under an electric field?
• RQ4 How is the Berry curvature and orbital magnetic moment naturally recovered within this transport framework?
• RQ5 Can the Kubo formula for Hall conductivity be derived as a byproduct of this new transport approach?

Key findings

• The topological invariant is identified as a phase-space integral over $\mathcal{\vec{K}}$-space, given by $\frac{1}{(2\pi\hbar)}\int\int\int d\mathcal{\vec{K}}_x d\mathcal{\vec{K}}_y dt \left[ \frac{\partial^{(a)}}{\partial\mathcal{\vec{K}}_x} \frac{\partial^{(b)}}{\partial\mathcal{\vec{K}}_y} - \frac{\partial^{(a)}}{\partial\mathcal{\vec{K}}_y} \frac{\partial^{(b)}}{\partial\mathcal{\vec{K}}_x} \right] H^{(a)}(-iG^{<\left(b\right)})$, which yields the Chern number.
• The Hall conductance is quantized exactly in units of $\frac{e^2}{h}$, corresponding to the contact conductance per spin in mesoscopic systems.
• The method achieves quantization not to first order in the electric field, but to first order in the gradient expansion, which is more general and physically consistent.
• The Kubo current-current correlation formula for Hall conductivity is derived as a byproduct, confirming consistency with established formalism.
• The Berry curvature and orbital magnetic moment are computed directly from the phase-space wavefunctions in the $\mathcal{\vec{K}}$-representation.
• The Jacobian of the transformation from $\mathcal{\vec{K}}$ to $\vec{p}$-space is unity, preserving the topological structure across the phase-space integration.