Universal Transport at Lifshitz Metal-Insulator Transitions in Two Dimensions

We study the charge transport across a band-tuned metal-insulator transition in two dimensions. For high temperatures <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>T</a:mi></a:mrow></a:math> and chemical potentials <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>μ</c:mi></c:math> far from the transition point, conduction is ballistic and the resistance <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>R</e:mi><e:mo stretchy="false">(</e:mo><e:mi>T</e:mi><e:mo stretchy="false">)</e:mo></e:math> verifies a simple one-parameter scaling relation. Here, we explore the limits of this semiclassical behavior and study the quantum regime beyond, where scaling breaks down. We analytically evaluate the simplest Feynman diagram of the linear-response conductivity <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>σ</i:mi><i:mo>=</i:mo><i:mn>1</i:mn><i:mo>/</i:mo><i:mi>R</i:mi></i:math> of a parabolic band endowed with a finite lifetime. Our formula shows excellent agreement for experiments for a field-tuned <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mrow><k:msub><k:mrow><k:mi>MoTe</k:mi></k:mrow><k:mrow><k:mn>2</k:mn></k:mrow></k:msub></k:mrow><k:mo>/</k:mo><k:msub><k:mrow><k:mtext>WSe</k:mtext></k:mrow><k:mrow><k:mn>2</k:mn></k:mrow></k:msub></k:mrow></k:math> moiré bilayer, and can capture the quantum effects responsible for breaking the one-parameter scaling. We go on to discuss a fascinating prediction of our model: The resistance at the quantum-critical band-tuned Lifshitz point (<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>μ</m:mi><m:mo>=</m:mo><m:mi>T</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:math>) has the , <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:msub><o:mrow><o:mi>R</o:mi></o:mrow><o:mrow><o:mi>L</o:mi></o:mrow></o:msub><o:mo>=</o:mo><o:mo stretchy="false">(</o:mo><o:mn>2</o:mn><o:mi>π</o:mi><o:mi>h</o:mi><o:mo stretchy="false">)</o:mo><o:mo>/</o:mo><o:msup><o:mrow><o:mi>e</o:mi></o:mrow><o:mrow><o:mn>2</o:mn></o:mrow></o:msup></o:mrow></o:math>, per degree of freedom, in congruence with experiment. Furthermore, we investigate whether two-dimensional metal-insulator transitions driven by strong electron correlations or disorder can also be classified by their quantum-critical resistance and come up with an, in principle, complete assignment of the transition mechanism.