Universal transport at Lifshitz metal-insulator transitions in two dimensions

TL;DR

The paper develops a minimal 2D Lifshitz-type framework for metal-insulator transitions using a parabolic band with a finite lifetime $\Gamma$, enabling an exact Kubo-conductivity expression $\sigma(\mu,T)$ that spans ballistic to diffusive regimes. It demonstrates that high-temperature data in MoTe$_2$/WSe$_2$ moiré bilayers follow one-parameter scaling, while quantum corrections near the Lifshitz point break this scaling and yield a finite residual conductance; notably, at the quantum critical point $\mu=T=0$ the resistance is universal, $R_L=\frac{2\pi h}{e^2}$ per degree of freedom. The authors further propose an $R_c$ criterion to distinguish transition mechanisms, predicting $R_c>R_L$ signals interaction-dominated MITs, with $R_c\le R_L$ offering only inconclusive guidance. Overall, the work provides a unified description of quantum-critical transport in 2D MITs and suggests a practical diagnostic using critical resistance ranges, with relevance for Moiré materials and other 2D systems.

Abstract

We study the charge transport across a band-tuned metal-insulator transition in two dimensions. For high temperatures $T$ and chemical potentials $μ$ far from the transition point, conduction is ballistic and the resistance $R(T)$ verifies a simple one-parameter scaling relation. Here, we explore the limits of this semi-classical behaviour and study the quantum regime beyond, where scaling breaks down. We derive an analytical formula for the simplest Feynman diagram of the linear-response conductivity $σ=1/R$ of a parabolic band endowed with a finite lifetime. Our formula shows excellent agreement for experiments for a field-tuned MoTe$_2$/WSe$_2$ 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 ($μ=T=0$) has the universal value, $R_L=(2 πh)/e^2$, per degree of freedom and this value is found to be compatible 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 find that $R_L$ may be useful in identifying predominantly interaction driven transitions.

Figures (8)

• Figure 1: Metal-insulator transition in a MoTe$_2$/WSe$_2$ moiré bilayer. Tuning an applied electric displacement field from $D=$0.544 V/nm (red dots) to $D=$0.427 V/nm (purple dots), the MoTe$_2$/WSe$_2$ bilayer near full filling ($f=2$) undergoes a metal-insulator transition. The inset shows the data from the experiment by Li et al.MoTe2_data provided in Ref. Disorder_Dom_Crit, which is then scaled to give the data points in the main figure. In the high-temperature scaling regime, resistance curves $R(T)$ collapse onto a metallic or insulating branch (black dashed lines $\mathcal{F}_+$ and $\mathcal{F}_-$) when dividing resistances $R(T)$ by the critical resistance $R_c(T)$ (shown in black in the inset, corresponding to $D=$0.437 V/nm) and scaling temperature with a field-dependent $T_0$ (see Suppl. Materials). Nearing the quantum critical point, $R_c(T=0)$, the experimental data strongly deviates from the semi-classical prediction ($\mathcal{F}_+$ and $\mathcal{F}_-$) that verifies one-parameter scaling, while agreeing excellently with our quantum theory (red solid lines) for all temperatures.
• Figure 1: Qualitative fit to MoTe$_2$/WSe$\boldsymbol{_2}$ experiment. Resistance curves, $R(T)$, for varying chemical potential $\mu$, with red curves in the metallic and purple curves in the insulating regime. The black lines indicate the 'critical' resistances at $\mu=0$. (a) and (b) are theoretical curves. In (a) a constant scattering rate $\Gamma=\Gamma_0=0.3$meV is used and (b) with an additional linear term, $\Gamma=\Gamma_0+bT$ with $b=0.005$meV/K is used. (c) experimental resistances for the ($f=2$) MoTe$_2$/WSe$_2$ bilayer undergoing a MIT (data from Ref. Disorder_Dom_Crit of experiment carried out in Ref. MoTe2_data), each curve corresponding to a different applied electric displacement field $D$.
• Figure 2: Quantifying transport in the quantum regime. (a) The relative deviation of the quantum resistance $R(\mu,T)$ of Eq. (\ref{['cond']}), from the semi-classical limit, $\left|1-{R^{sc}(\mu,T)}/{R(\mu,T)}\right|$, for a constant scattering rate $\Gamma$ is shown. At low temperatures close to the transition we see a breakdown of semi-classical behaviour in the metallic phase, corresponding, broadly, to the Mott-Ioffe-Regel limit. The semi-classical approximation is accurate deep in the metal, but, also, for arbitrary chemical potential, when temperature is high enough. (b) Semi-classically the resistance in the insulating phase at $T=0$K is predicted to diverge, however in the quantum theory with the inclusion of finite lifetime effects, a finite value is seen for all values of $\mu$. (c) The residual resistance is shown for an increasing value of the scattering rate $\Gamma$. At $\mu=0$ all the curves intersect, indicating a universal value for the resistance at the Lifshitz quantum critical point.
• Figure 2: Deviations from the quantum theory for larger gapped insulating statesat low temperatures. The scaled resistance curves for an applied electric displacement field $D=$0.399V/nm-0.435V/nm are shown, corresponding to all insulating curves measured in the experiment. As in the main text the theoretical curves use a scattering rate, $\Gamma= \Gamma_0+bT$ with $\Gamma_0=0.3$ meV and $b=0.005$ meV/K, that is independent of the chemical potential. It can be seen that for the four most insulating curves at low temperatures there is a deviation of the scaled data from our theory, being more resistive than predicted, however the theory still performs well close to the transition. We note that relaxing the restriction of a $\mu$ independent $\Gamma$, may help to improve the agreement between theory and experiment.
• Figure 3: Comparison of experimental critical resistance ranges with $\boldsymbol{R_L}$. Extracted critical resistance ranges from several experiments (per degree of freedom of each system), plotted against the carrier density $n_c$ at which the system was critical. The systems are labelled by the predominant mechanism which drives the MIT. In the intermediate regime where this assignment is unclear, systems are labelled "disorder + interactions". The Lifshitz resistance $R_L$, the predicted quantum critical resistance for a band-tuned transition and proposed as an upper bound for a predominantly disorder-driven transition sets the scale of the resistance and is shown as a horizontal grey dashed line. Dotted vertical lines indicate that the range extends beyond the limits of the graph. A sensible critical resistance range for data point (c), the half-filled ($f=1$) MoTe$_2$/WSe$_2$ bilayer could not be estimated using our standard method for the data available (hence the dashed line). However the transition takes place near the point indicated. Experimental references of systems shown: (a) PhysRevB.99.081106, (b) Falson2022, (c,m) MoTe2_data, (d) PhysRevLett.80.1288, (e) COLERIDGE2000268, (f) PhysRevB.72.081313, (g) PhysRevB.57.R15068, (h) Pack2024, (i) Kravchenko_MOSFET, (j) PhysRevLett.87.266402, (k) Osofsky2016.