Continuous transition and gapless roton inside fractional quantum anomalous Hall states

TL;DR

The paper addresses how a symmetric FQAH state can undergo a continuous transition to an FQAH+CDW state with spontaneous translation symmetry breaking, while preserving the same Hall conductance, and eventually transition to a topologically trivial PSM under stronger CDW order. It reveals that this evolution is driven by the softening of the magnetoroton mode at finite momentum, causing a neutral gap closure while the charge gap remains open, a scenario reminiscent of Hall crystals and FQH nematics but realized in a lattice FQAH system. The authors establish the transition as consistent with the Ising universality class through iDMRG analyses and support it with ED studies across multiple parameter paths, including $V_3$-driven and $\lambda$-driven trajectories. The findings offer a generic mechanism for interaction-induced translation symmetry breaking inside topological orders, with potential experimental relevance for moiré materials and cold-atom platforms.

Abstract

Collective excitations play a vital role in understanding the exotic phases of matter and phase transitions in quantum many-body systems. For the first time, we numerically (via exact diagonalization and density matrix renormalization group) report the microscopic realization of a transition from a translationally invariant fractional quantum anomalous Hall (FQAH) state to the same FQAH state with spontaneously broken translation symmetry, by softening the magnetoroton mode (intrinsic collective excitations in such systems) through isotropic interactions in a topological flat-band model. At the critical point, the gap of collective neutral excitations closes at finite momentum, while the charge gap remains robust. This mechanism echoes with the integer quantum Hall crystals and fractional quantum Hall nematics in Landau levels, but exhibits unique features. Further through criticality analysis, we identify that this non-trivial transition is consistent with the Ising universality class. Such spontaneous translation symmetry breaking inside the topological ordered FQAH state could serve as a generic scheme in various systems, with experimental implications to the quantum moiré materials and the cold-atom systems.

Figures (8)

• Figure 1: Generic phase diagram. By tuning interaction from a symmetric FQAH state with quantized $\sigma_{xy}=\nu\frac{e^2}{h}$, the system undergoes a two-step transition to a coexisting FQAH+CDW state with the same Hall conductance and moderate CDW order and a topologically trivial CDW state with stronger charge order. The focus is the general mechanism of the first transition. The increasing interaction softens the magnetoroton mode, and at the quantum critical point, the neutral gap continuously closes at finite momentum, which leads to the spontaneously (discrete) translation symmetry breaking, while the charge gap remains open at the quantum critical point as well as the two topological states.
• Figure 2: Model realization. (a) The two-band model on checkerboard lattice with $N=N_y\times N_x\times2=3\times2\times2$ sites as an example. A(B) sublattices and different hoppings are denoted by different colors and the arrows represent the directions of the NN loop current. (b) Phase diagram at $\nu=2/3$ filling of the flat Chern band with fixed $V_1=1.1,$$V_2=1$ and changing $V_3$. The ED spectra from $3\times4\times2$ tori of each phase are shown. At small $V_3$, the ground state is a symmetric FQAH state with 3-fold degeneracy and Hall conductivity $\sigma_{xy}=\frac{2}{3}\frac{e^2}{h}$. At intermediate $V_3$, the FQAHS state has the same Hall conductivity as the FQAH state and the co-existing CDW order at $(\pi,0)$. Due to the spontaneously translation symmetry breaking at $V_3=0.61$, the ground-state degeneracy of the FQAHS state is further doubled to 6-fold. Further increasing $V_3$, the CDW order becomes stronger and the topological transition leads to the gapless and topologically trivial PSM state.
• Figure 3: Continuous FQAH-FQAHS transition. iDMRG results at different bond dimensions of a $N_y=3$ cylinder with infinite length (a) Bipartite entanglement entropy and (b) correlation length as functions of $V_3$, which both diverge at the critical point. The inset of panel (a) shows the fitting of central charge at critical point $V_3^c=0.61$ according to $S_\mathrm{E}=\frac{c}{6}\log(\xi)$ and the fitted result from the data points of largest bond dimensions is $c\sim0.6$ (we simulated up to $D=4000$ for this part). The power-law divergence of correlation length approaching the critical point at the (c) FQAH and (d) FQAHS sides are shown. When the bond dimension increases, the scalings of correlation length are both approaching $|V_3-V_3^c|^{-\nu}$ with $\nu=1$ of the 2D Ising universality class. (e) Smectic CDW order parameter and (f) structure factor at $(\pi,0)$ as functions of $V_3$. The inset of panel (e) in the double-logarithm scale shows the fitting of the order parameter according to $\delta_\mathrm{smectic}\sim (V_3-V_3^c)^\beta$ and the fitted exponent is $\beta\sim0.14$. $S(\pi,0)$ that measures the smectic fluctuations also tends to diverge at the critical point and quickly drops after the establishment of smectic order.
• Figure 4: Gapless roton with a large charge gap at critical point. The (a) single-particle Green's function and (b) density (neutral) correlation functions at different $V_3$ versus the distance are shown in the semi-log scale and they share the same labels. In panel (a), the exponential decaying behavior of fermion correlations remains across the FQAH-FQAHS transition ($\xi_\mathrm{sp}\sim0.69$), indicating the robust charge gap. In panel (b), it is clear that the correlation length of density correlations increases fast when approaching the critical point. (c) The density correlations at the critical point $V_3^c=0.61$ in the log-log scale, where the algebraic decaying is shown. With the increasing bond dimension, the scaling behavior is approaching $\sim d_{i,j}^{-\eta}$ with $\eta=0.25$ for the 2D Ising transition.
• Figure S 1: Energy spectra with fixed $V_1=1.1$ and $V_2=1$. These energy levels are obtained from ED simulations of $3\times4\times2$ tori. In panel (a), we plot the energy levels from all sectors while we plot only the $(0,\frac{2\pi}{3})$ and $(\pi,\frac{2\pi}{3})$ sectors in panel (b).The colors of different momentum sectors are defined in Fig. \ref{['fig_figS2']}(a). The red dashed lines are the phase boundaries as in the Fig. 2 (b) of the main text. The FQAH state has a 3-fold ground-state degeneracy and the FQAHS state has a 6-fold degeneracy, while the PSM state is gapless. We have labelled the number of states near the ground states of the FQAH and FQAHS phases. We also add two small red triangles in panel (a) to represent the neutral gap of the FQAH and FQAHS phases, respectively. The supporting spectra of each phase under twisted boundary conditions are also shown in Fig. 2 (b) of the main text. The first phase boundary is from the iDMRG results in the main text, while the second phase boundary is determined from the level crossing at $V_3\sim2.84$.