Elastic Response and Instabilities of Anomalous Hall Crystals

Abstract

Anomalous Hall crystals (AHCs) are exotic phases of matter that simultaneously break continuous translation symmetry and exhibit the quantum anomalous Hall effect. AHCs have recently been proposed as an explanation for the observation of an integer quantum anomalous Hall phase in a multilayer graphene system. Despite intense theoretical and experimental interest, little is known about the mechanical properties of AHCs. We study the elastic properties of AHCs, first by utilizing a continuum model with a quadratic dispersion and uniform Berry curvature. We find using time-dependent Hartree-Fock that the stiffness of the AHC is an order of magnitude smaller than that of the WC, which we attribute to the finite Chern number of the AHC preventing exponential localization of the charge density. By modifying the dispersion relation to include a local minimum modeled after that of rhombohedral pentalayer graphene (R5G), we find that deformations away from the triangular lattice minimize the kinetic energy of the AHC, which overwhelms the small stiffness and generates a mechanical instability. Using a microscopic model of R5G, we observe a similar mechanical instability over an experimentally relevant parameter regime. We conclude that the topologically limited stiffness of AHCs makes them susceptible to mechanical instabilities, an important consideration when interpreting experiments in terms of AHCs.

Figures (4)

• Figure 1: Representative real space charge density modulations for (a) the WC and (b) the $\mathcal{C}=1$ AHC, arising in the ideal parent band with $V_{c}/A_{\text{u.c.}}=7.63$. Maximum charge density variation $\Delta\rho(\boldsymbol{r})\equiv\max[\rho(\boldsymbol{r})] - \min[\rho(\boldsymbol{r})]$ in the HF ground state obtained by keeping the 97 closest reciprocal lattice points and with $n_1=23$ for (c) a WC $(\mathcal{B}=0)$ and (d) AHCs with $\mathcal{B}A_{1\text{BZ}}=2\pi$, $4\pi$, and $6\pi$.
• Figure 2: (a) A depiction of the dilation deformation of the triangular lattice employed to compute the stiffness. The stiffness of (b) the WC with $\mathcal{B}=0$ and (c) the AHC with $\mathcal{B}=2\pi$, both as a function of the interaction strength, with the HF and correlation energy contributions depicted individually, in addition to the total stiffness. The dashed line corresponds to a perturbative expression for the HF ground state energy exact in the large-$V_c$ limit. The difference between the HF ground state energy of the (d) WC with $\mathcal{B}=0$ and (e) AHCs with $\mathcal{B}=2\pi,$$4\pi,$ and $6\pi$ on the square and triangular lattices, as a function of interaction strength. The dashed lines in (e) utilize the same perturbative expansion as in (c). (f) The change in the HF ground state energy as system size increases, scaling as $1/n_1^3$. The black dashed line is a guide to the eye, and $\Delta E(n_1) = E(n_1 + 1) - E(n_1).$
• Figure 3: The stiffness of the WC with $\mathcal{B}A_{\text{1BZ}}=0$ (black) and the AHC with $\mathcal{B}A_{\text{1BZ}}=2\pi$ (blue) arising in the parent band model with a Mexican hat dispersion as a function of interaction strength. The minimum of the dispersion is located at $k_0=0.6$, and both $V_c$ and $C_{66}$ are scaled by $D$. The inset shows the dispersion as a function of momentum, with the 1BZ depicted by the dashed lines.
• Figure 4: (a) Continuum R5G dispersion in a strong displacement field corresponding to an interlayer potential difference of $U_d=-36$ meV. The vertical dashed lines indicate the location of $K$ point of the 1BZ that spontaneously emerges when the triangular lattice crystal forms. (b) Dilation stiffness of the R5G AHC as a function of the inverse dielectric constant calculated with $n_1=23$, $U_d=-36$ meV and $n_{\text{bands}}=7$. Error bars are from uncertainties in evaluating the second-order derivative. (c) Convergence with respect to system size $n_1 \cross n_1$ (for $n_{\text{bands}}=7$) of the ground state energy per conduction electron on undistorted ($u_d=0$) and distorted ($u_d=-0.15$) triangular lattices for $U_d=-36$ meV and $\epsilon=8.07$. The same convergence is also shown in the presence of a moiré potential induced by a hexagonal boron nitride substrate on the bottom layer with twist angle $\theta=0.77^\circ$. (d) Convergence of the same quantity as (c) but with respect to the number of conduction bands $n_{\text{bands}}$ (for $n_1=23$).