[Paper Review] Noncommutative quantum Hall effect in graphene

Willien O. Santos, G. M. A. Almeida+1 more|arXiv (Cornell University)|Jan 10, 2018

Graphene research and applications3 citations

TL;DR

This paper investigates the noncommutative quantum Hall effect in single-layer graphene by introducing momentum noncommutativity via a parameter η. Using the Streda formula and experimental Hall resistivity data, it derives a correction to the Hall conductivity and finds an upper bound of √η ≤ 2.5 eV/c, providing a phenomenologically accessible constraint on noncommutative momentum effects in condensed matter systems.

ABSTRACT

We discuss the quantum Hall effect on a single-layer graphene in the framework of noncommutative (NC) phase space. We find it induces a shift in the Hall resistivity. Furthermore, comparison with experimental data reveals an upper bound on the magnitude of the momentum NC parameter $η$ in about $\sqrtη\leq 2.5 \, \mathrm{eV}/c$.

Motivation & Objective

• To investigate the impact of momentum noncommutativity on the quantum Hall effect in graphene.
• To derive a correction to the Hall conductivity and resistivity due to noncommutative momentum space.
• To estimate an upper bound on the noncommutative momentum parameter η using experimental data.
• To extend previous studies on noncommutative graphene by focusing on the quantum Hall effect.

Proposed method

• Formulating the noncommutative Dirac Hamiltonian for graphene using noncanonical commutation relations [p̂i, p̂j] = iηij.
• Implementing the noncommutative gauge transformation via coordinate shifts to incorporate a magnetic field.
• Defining noncommutative ladder operators and Landau level quantization in the presence of momentum noncommutativity.
• Applying the Streda formula to relate Hall conductivity to electron density and magnetic field, incorporating noncommutative corrections.
• Using experimental Hall resistivity data from Ref. [35] with δρH ≤ 15×10⁻⁶ to constrain the noncommutative parameter η.
• Deriving an analytical expression for η in terms of the relative resistivity shift δρ̂H to extract an upper bound.

Experimental results

Research questions

• RQ1 How does momentum noncommutativity modify the quantum Hall effect in graphene?
• RQ2 What is the form of the corrected Hall conductivity and resistivity in a noncommutative momentum space framework?
• RQ3 Can experimental measurements of the Hall resistivity in graphene be used to constrain the noncommutative momentum parameter η?
• RQ4 How does the noncommutative correction compare to standard quantum Hall behavior in graphene?

Key findings

• The noncommutative momentum space induces a shift in the Hall resistivity, modifying the standard quantum Hall effect in graphene.
• The corrected Hall conductivity is given by ŝσH = ge e²/hc (1 - ηc/eBℏ)(1 - ηc/2eBℏ)⁻², showing a nontrivial dependence on η.
• Using experimental data with δρH ≤ 15×10⁻⁶ at B ~ 10⁴ G, the paper derives an upper bound √η ≤ 2.5 eV/c.
• This bound is of the same order of magnitude as previous estimates in other systems, but higher than those from hydrogen atom and Zeeman effect studies.
• The result provides a phenomenologically accessible constraint on noncommutative momentum effects in a real material system.