[Paper Review] Quantum Hall effect in curved space realized in strained graphene

Glenn Wagner, Fernando de Juan+1 more|arXiv (Cornell University)|Nov 17, 2020

Graphene research and applications44 references4 citations

TL;DR

This paper demonstrates the realization of the quantum Hall effect in curved space using strained graphene, where strain-induced pseudo-gauge fields and curvature map electron dynamics to curved-space Dirac fermions. By analytically deriving a low-energy Hamiltonian up to second-order strain and confirming with tight-binding simulations, the study achieves excellent agreement with theoretical predictions of Landau level quantization in curved space, paving the way for experimental observation.

ABSTRACT

The quantum Hall effect in curved space has been the subject of many theoretical investigations in the past, but devising a physical system to observe this effect is hard. Many works have indicated that electronic excitations in strained graphene realize Dirac fermions in curved space in the presence of a background pseudo-gauge field, providing an ideal playground for this. However, the absence of a direct matching between a numerical, strained tight-binding calculation of an observable and the corresponding curved space prediction has hindered realistic predictions. In this work, we provide this matching by deriving the low-energy Hamiltonian from the tight-binding model analytically to second order in the strain and mapping it to the curved-space Dirac equation. Using a strain profile that produces a constant pseudo-magnetic field and a constant curvature, we compute the Landau level spectrum with real-space numerical tight-binding calculations and find excellent agreement with the prediction of the quantum Hall effect in curved space. We conclude discussing experimental schemes for measuring this effect.

Motivation & Objective

• To establish a direct link between strained graphene's electronic behavior and the quantum Hall effect in curved space.
• To resolve the long-standing gap between theoretical predictions of curved-space quantum Hall effects and realistic numerical simulations.
• To derive a low-energy effective Hamiltonian for strained graphene up to second order in strain, mapping it to the curved-space Dirac equation.
• To validate the theoretical framework using real-space tight-binding calculations under a strain profile generating constant pseudo-magnetic field and curvature.
• To propose feasible experimental schemes for detecting the quantum Hall effect in curved space using strained graphene.

Proposed method

• Analytically derive the low-energy Hamiltonian from a strained tight-binding model up to second order in strain.
• Map the resulting Hamiltonian to the Dirac equation in curved space, incorporating pseudo-gauge fields and intrinsic curvature.
• Design a strain profile that generates a constant pseudo-magnetic field and constant curvature, mimicking curved-space geometry.
• Perform real-space numerical tight-binding calculations to compute the Landau level spectrum under this strain profile.
• Compare the simulated Landau level spectrum with predictions from the curved-space quantum Hall effect theory.
• Propose experimental setups involving suspended graphene with controlled strain gradients to detect the predicted quantum Hall states.

Experimental results

Research questions

• RQ1 Can strained graphene realize the quantum Hall effect in curved space as predicted by effective field theory?
• RQ2 To what extent does the low-energy Hamiltonian derived from a tight-binding model in strained graphene match the curved-space Dirac equation?
• RQ3 Does a strain profile producing constant pseudo-magnetic field and constant curvature yield Landau level spectra consistent with curved-space quantum Hall theory?
• RQ4 What is the quantitative agreement between numerical tight-binding simulations and theoretical predictions in curved space?
• RQ5 What experimental configurations can be used to detect the quantum Hall effect in curved space using strained graphene?

Key findings

• The low-energy effective Hamiltonian derived from the strained tight-binding model up to second order in strain matches the form of the curved-space Dirac equation with high fidelity.
• Numerical tight-binding simulations of the Landau level spectrum under a constant pseudo-magnetic field and curvature show excellent agreement with theoretical predictions for the quantum Hall effect in curved space.
• The Landau level spacing and level degeneracy in the simulation align quantitatively with the expected behavior in curved space, confirming the emergence of curved-space quantum Hall physics.
• The study establishes a direct and quantitative bridge between microscopic tight-binding models and effective curved-space field theories in the context of strained graphene.
• The authors identify specific strain profiles and experimental geometries that could enable the detection of the quantum Hall effect in curved space using existing nanofabrication techniques.