Connecting the avoided quantum critical point to the magic-angle transition in three-dimensional Weyl semimetals

We theoretically study the interplay of short-range random and quasiperiodic static potentials on the low-energy properties of three-dimensional Weyl semimetals. This setting allows us to investigate the connection between the semimetal to diffusive metal “magic-angle” phase transition due to quasiperiodicity and the rare-region-induced crossover at an avoided quantum critical point (AQCP) due to disorder. We show that in the presence of both random and quasiperiodic potentials the AQCP becomes lines of crossovers, which terminate at magic-angle critical points in the quasiperiodic, disorder-free limit. We analyze the magic-angle transition by approaching it along these lines of avoided transitions, which unveils a rich miniband structure and several AQCPs. These effects can be witnessed in cold atomic experiments through potential engineering on semimetallic band structures.

Figure 1

Phase diagram of disorder ($W_D$) and quasiperiodic ($W_Q$) potential strengths at the Weyl node energy ($E=0)$. Solid blue lines are stable Weyl semimetal phases that terminate at magic-angle transitions (the Weyl semimetal at larger $W_Q$ is an inverted semimetal phase). At any nonzero $W_D$ the model is in the diffusive metal phase; dark red marks the semimetal regime $\rho \left(E\right)\approx \rho \left(0\right)+{\rho }^{″}\left(0\right)E^2/2$, where $\rho \left(0\right)$ is nonzero but exponentially small, and light red is past the AQCP where $\rho \left(0\right)\sim \mathrm{O}\left(1\right)$. The location of the peak in ${\rho }^{″}\left(0\right)$ as a function of $W_D$ (for fixed $W_Q$) provides an estimate of the AQCP (and the quasiperiodic transition at $W_D=0$) that we label as $W_A\left(W_Q\right)$. We compare two choices of the distribution of the disorder potential $P\left[V\right]$, showing their distinction is insignificant near the magic-angle transitions and Gaussian and binary distributions; the latter has been shown to weaken the avoidance for $W_Q=0$ [36]. This analysis is done on a system size of $L=89$, KPM expansion order $N_C=2^{10}$, and 100 samples. Sufficiently close to the magic-angle transitions additional minibands appear that give rise to more structure in $W_A\left(W_Q\right)$ (see Sec. 4); that additional miniband structure is not shown here. Dashed vertical lines indicate the transitions at $W_D=0$ into or out of diffusive phases (from left to right, $W_{M,1}\approx 0.38t$, $W_{M,1}^{\prime }\approx 0.395t$, and $W_{M,2}\approx 0.6345t$).

Figure 2

Renormalized velocity and related scales: Comparison of the renormalized velocity $v\left(W_Q\right)$ with the location of the avoided transition $W_A\left(W_Q\right)$ and the dependence of the density of states $A\left(W_Q\right)$ [see Eq. (12)]. The velocity $v\left(W_Q\right)$ of the renormalized semimetal is obtained from the disorder-free limit using ${\rho }^{″}\left(0\right)\propto v^{-3}$ (for system size $L=144$ and $N_C=2^{10}$ from Ref. [44]), $A\left(W_Q\right)$ is extracted from the rare-region dependence of $ln\rho \left(0\right)\sim A\left(W_Q\right)/{\left(W_D\right)}^2$ [from the data that are converged in $L$ and $N_C$; see Eq. (12)]), and $W_A\left(W_Q\right)$ denotes the line of AQCPs determined from the maximum in ${\rho }^{″}\left(0\right)$ as a function of $W_D$ (obtained from a system size of $L=89$ and KPM expansion order $N_C=2^{10})$. Sufficiently close to each MAT, the formation of minibands (see the Appendix) enriches the picture beyond the relations implied by these data; that requires sufficiently large $L$ and/or $N_C$ to observe (see Sec. 4).

Figure 3

Evolution of the density of states: (a) and (d) $\rho \left(E\right)$ (with $L=89$ and $N_C=2^{13}$), (b) and (e) ${\rho }^{″}\left(0\right)$, and (c) and (f) $\rho \left(0\right)$ as a function of $W_D$ and $W_Q$ for various values of the KPM expansion order $N_C$, $L=55$ (open symbols), and $L=89$ (solid symbols). The values of the quasiperiodic potential are $W_Q=0.2t$ in the semimetal phase (top row) and $W_Q=0.55t$ in the inverted semimetal phase (bottom row). These data are averaged over 1000 samples for $N_C=2^9,2^{10},2^{11}$ and 5000 samples for $N_C=2^{12},2^{13}$. Straight lines in (c) and (f) are fits to the rare-region form in Eq. (12).

Figure 4

Density of states along the line of AQCPs that is defined by $W_A\left(W_Q\right)$. We show $\rho \left(E\right)$ for $L=89$ at larger $W_D=W_A\left(W_Q\right)$ in (a) (with $N_C=2^{13}$) and closer to the transition at a much lower energy scale in (b) (with $N_C=2^{14}$) as $W_A\to 0$ [recall $W_A\left(W_Q\right)$ is shown in the phase diagram in Fig. 1]. The data are shown starting from $W_Q=0.55t$ and terminating at the magic-angle transition at $W_Q^c=0.6345t$ for $W_D=0$. The finite but low-energy dependence along the line of AQCPs is consistent with the expected behavior in Eq. (14). However, as we go from (a) to (b), we can see that a consequence of the second miniband opening up for $W_Q\approx 0.62t$ realizes a dramatically renormalized ${\rho }^{″}\left(0\right)$ seen through the shape near zero energy until we hit the MAT and a finite density of states is generated ($W_D=0$).

Figure 5

Evolution of the AQCP on approach to the magic-angle transition along $W_A\left(W_Q\right)$. (a) Focusing on ${\rho }^{″}\left(0\right)$ on approach to the transition for $N_C=2^{13}$, we see that the original peak that we associate with the AQCP splits off, leaving behind a much weaker second peak at larger $W_D$ that can be associated with an approximate AQCP due to the parts of the band outside of the second miniband. (b) We study the evolution of the peaks for $W_Q=0.625t$ as a function of the expansion order, demonstrating a converged AQCP peak at this system size ($L=89$). (c) However, for $W_Q=0.63t$ as we get closer to the MAT, we are unable to converge the peak [see also Fig. 7].

Figure 6

Demonstration of AQCPs in the first and second minibands for $W_Q/t=0.63$. We show the density of states $\rho \left(E\right)$ as a function of energy $E$ for $L=89$ and $N_C=2^{13}$ at the first maximum in ${\rho }^{″}\left(0\right)$ in black, depicting the scaling $\rho \left(E\right)\sim \left|E\right|$ in the second miniband. The location of the second weaker peak in ${\rho }^{″}\left(0\right)$ is marked in red, depicting the scaling $\rho \left(E\right)\sim \left|E\right|$ in the first miniband. For reference, see Fig. 5 for the structure of ${\rho }^{″}\left(0\right)$. At each peak in ${\rho }^{″}\left(0\right)$ we find the low-energy dependence follows the AQCP form in Eq. (14), allowing us to identify the signatures of two AQCPs as a function of $W_D$ sufficiently close to the MAT.

Figure 7

Divergence of ${\rho }^{″}\left(0\right)$ on the approach to the MAT along the line of AQCPs. (a) The solid data points represent peaks in ${\rho }^{″}\left(0\right)$ that are converged in $N_C$ that follow ${\rho }^{″}\left(0\right)\sim 1/W_A^2$ (the blue solid line is a fit). The open symbols are not yet converged [although they are close to being converged as depicted in (b)], and we show their $N_C$ dependence, with a fit to the largest $N_C=2^{16}$ (gray line), yielding ${\rho }^{″}\left(0\right)\sim 1/W_A^{2.5}$. Inset: The locations of the leading AQCPs $W_A\left(W_Q\right)$ are shown as black symbols, while the location of the subdominant peak is shown as magenta circles. (b) Dependence of the peak in ${\rho }^{″}\left(0\right)$ on the KPM expansion order $N_C$ as we approach the MAT along the line of AQCPs. At the MAT ($W_D=0$) we find this diverges as ${\rho }^{″}\left(0\right)\sim {\left(N_C\right)}^{2.5}$ (the fit to the largest three expansion orders is shown as a black line). All data here are obtained for a system size $L=89$.

Figure 8

Minibands in the clean limit, $W_D=0$: For clarity, we show the density of states in the clean limit for values of $W_Q$ that are less than and greater than required to open (a) the first miniband and (b) the second miniband, both of which are shaded green, but the energy scale is much smaller in the second miniband. Both minibands depict the expected Weyl scaling $\rho \left(E\right)\sim E^2$ at sufficiently low energy with very different coefficients. The results in (a) are for $L=89$, 1000 samples, and $N_C=2^{13}$, while (b) is for $L=89$ and 5000 samples, where $W_Q=0.6t$ has $N_C=2^{13}$ but for $W_Q=0.625t$ we take $N_C=2^{14}$ in order to show hard gaps as the second miniband forms. We also point the reader to Ref. [44] for more data on the opening of the first miniband.