WKB structure in a scalar model of flat bands

Abstract

We consider a family of periodic scalar operators for which one can define flat bands in the sense of Floquet-Bloch theory. One puzzling question originating in recent physics literature is a quantisation rule for the values of parameters at which these flat bands occur. We present a general theorem about the structure of solutions to the corresponding equation and a heuristic argument explaining their WKB structure in a specific case. That structure also explains the quantisation condition - both the WKB structure and that rule are confirmed by numerical experiments. Finally, we consider a simplified model in which separation of variables allows the use of complex WKB methods.

Figures (14)

• Figure 1: The sets $\mathcal{A}$ ( red) and ${\mathcal{B} }$ ( blue) of Theorem \ref{['t:1']} for the scalar model based on the Bistritzer--MacDonald potential \ref{['eq:defU2']}. The multiple circles indicate multiplicity (in the sense of Theorem \ref{['t:1']}, $m=1,2,3$ all occur). A movie showing the evolution of the bands $k \mapsto E_j ( \alpha, k )$, $j =1, 2$ (over the real axis in $k$) as $\alpha$ varies can be see at https://math.berkeley.edu/ zworski/bands_scalar.mp4. It illustrates the behaviour of bands at real elements of $\mathcal{A} \cup \mathcal{B}$.
• Figure 2: The sets ${\mathcal{A} }$ for the scalar model with the potential $V$ given by \ref{['eq:defHpot']}. The multiple circles indicate multiplicity (in the sense of Theorem \ref{['t:1']}, $m=1,2,3$ all occur). Except for different scaling the structure of the set is the same as for $\mathcal{A}$ shown in red in Figure \ref{['f:rot']}.
• Figure 3: The bottom panels show sets of $\alpha$'s (circles; coloured depending on the momentum variable of $y$) for which $( D_x + i D_y )^2 - \alpha^2 W ( x )$, $( x, y ) \in \mathbb R^2/2 \pi \mathbb Z ^2$ has a nontrival kernel. The top panels show curves on which $\mathop{\mathrm{Im}}\nolimits( W ( \gamma ( t ) ) \gamma' ( t ) )= 0$ (Stokes lines). The first $W$ satisfies \ref{['eq:W0']} but does not have Stokes loops \ref{['eq:stol']}; on the second $W$ has Stokes loops. The quantisation rule \ref{['eq:naive']} (shown by black dots) does not apply on the top and is very accurate on the bottom. For an animated version see https://math.berkeley.edu/ zworski/stokes.mp4.
• Figure 4: The sets ${\mathcal{A} }$, of magic $\alpha$'s for the chiral model \ref{['eq:defD']} (in red) and the scalar model \ref{['eq:scalar']} (in red) with $U$ given by \ref{['eq:defU2']}. Higher multiplicities are indicated by additional circles: in the scalar model the real $\alpha$'s have multiplicity $2$. When we interpolate between the chiral model and the scalar model, the multiplicity two real $\alpha$'s split and travel in opposite directions to become magic $\alpha$'s for the chiral model: see https://math.berkeley.edu/ zworski/Spec.mp4, where the magic $\alpha$'s are plotted as a function of $t\in[0,1]$ which linearly interpolates between the chiral and scalar models.
• Figure 5: When multiplicity of the band is one, the only possible zero of $\mathbf u$ is at $0$ and that is illustrated in this figure. For the potential \ref{['eq:defU']} in the scalar model with $V ( z ) = U_{\rm{BM}} (z ) U_{\rm{BM}} ( - z )$ this occurs only for complex $\alpha$'s.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Definition 4