Transport Scaling and Critical Tilt Effects in Disordered 2D Dirac Fermions

TL;DR

The paper investigates how tilt in 2D Dirac fermions, under scalar disorder, affects transport and spectral properties, revealing that tilt qualitatively reshapes conductivity scaling and level statistics. Using models with one and two tilted Dirac nodes, it employs Gaussian disorder and the Kubo formula to compute conductivities, along with density-of-states calculations and random-matrix-theory-based level statistics to classify spectral behavior. A key finding is that a single tilted Dirac node yields g(L) behavior of the form $g(L) = a_0 + a_1 \log L$ with a tilt-dependent spike in $a_1$ at the critical tilt $\lambda=1$ for transport along the tilt, while g_yy grows monotonically with tilt; spectrally the system aligns with the GUE at finite tilt. In the two-node case, transport shows a tilt-driven localization-delocalization transition along the tilt (sign change in $a_1$ for $g_{xx}$) but localization persists orthogonally, whereas spectral diagnostics remain delocalized (GOE).Overall, tilt emerges as a decisive parameter that uncovers rich, unconventional transport behavior in 2D Dirac materials and highlights a tension between real-space localization and spectral delocalization.

Abstract

Two-dimensional (2D) Dirac fermions occur ubiquitously in condensed matter systems from topological phases to quantum critical points. Since the advent of topological semimetals, where the dispersion is often tilted around the band crossing where the Dirac fermion can appear, tilt has emerged as a key handle that controls physical properties. We study how tilt affects the transport and spectral properties of tilted 2D Dirac fermions under scalar disorder. Although our spectral analyses always show conformity to appropriate Gaussian ensembles, suggestive of delocalization, the conductivity scaling $g(L)$ shows a surprising richness. For a single Dirac node, relevant for quantum Hall transitions and topological insulator surface states, we find $g(L)\sim a_1\log(L)$ with a tilt-dependent coefficient $a_1>0$. Interestingly, when the tilt and transport directions are aligned, $a_1$ and hence $g(L)$ shows a spike at the critical point between the type-I and type-II regimes of the Dirac node. For systems with two Dirac nodes with unbroken time-reversal symmetry, pertinent to quasi-2D Dirac materials, we find $g(L)\sim L^{a_1}(\log L)^{a_2}$. However, we find a surprising tension between tilt along and perpendicular to the transport directions. For the former, $a_1$ changes sign as a function of tilt, hinting at a tilt-driven localization-delocalization transition, while $a_1<0$ for all tilts in the latter case, implying localization. These localized behaviors also reveal tension with the delocalization seen in spectral properties and suggest differing localization tendencies in real and Hilbert spaces. Overall, our work identifies tilt as an essential control parameter that uncovers rich and unconventional transport physics in 2D Dirac materials.

Figures (13)

• Figure 1: Figs. \ref{['fig:singlet1']}, \ref{['fig:singletc']}, \ref{['fig:singlet2']} are showing type I, critical, and type II single Dirac node respectively.
• Figure 2: DOS of clean Dirac fermion for different tilt parameter $\lambda$. For $\lambda = 0$ the DOS is the lowest and it continues to grow as the nodes are more and more tilted until the nodes are critically tilted.
• Figure 3: Conductivity of a single Dirac node along the $x$ direction, $g_{xx}$, for different $\lambda$. For each $\lambda$, the data was obtained by averaging over around 1000 configurations. The obtained data was fitted against $g_{xx}(L) = a_0 + a_1 \log L$. $a_0$ and $a_1$ for different $\lambda$ have been plotted in Fig. \ref{['fig:abxx']}. A spike in $a_1$ and hence the conductivity has been observed for $\lambda = 1$.
• Figure 4: Conductivity of single Dirac node along $y$ direction $g_{yy}$ for different $\lambda$. For each $\lambda$, the data was obtained by averaging over around $1000$ configurations. The obtained data was fitted against $g_{yy}(L) = a_0 + a_1 \log{L}$. $a_0$ and $a_1$ for different $\lambda$ have been plotted in Fig. \ref{['fig:abyy']}. Unlike $g_{xx}$, monotonic behavior has been observed for $g_{yy}$, and $a_1$ (most important in the thermodynamic limit) continues to grow with $\lambda$
• Figure 5: Level spacing distribution of a single Dirac fermion with $\lambda = 0, 0.8$ in Fig. \ref{['fig:ls1n']} and Fig. \ref{['fig:ls1t']} respectively. The data has been fitted against $f(s) = As^{\beta}e^{-Bs^2}$, where $A, B$ are determined from normalization. For the tilted Dirac systems with single node, we obtain a fitted value of $\beta \approx 1.85$, which is close to the expected value of $\beta = 2$ for the GUE. In contrast, for the upright Dirac fermion, the fitting yields $\beta \sim 3.93$, in good agreement with $\beta = 4$ associated with the GSE.