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arXiv:2404.06745 · Newton

Continuous transition and gapless roton inside fractional quantum anomalous Hall states

Hongyu Lu, Han‐Qing Wu, Bin-Bin Chen, Zi Yang Meng

Published Nov 13, 2025• 7 citations
arXivHTMLPDF

Abstract

Collective excitations are crucial for understanding exotic phases and phase transitions in quantum many-body systems. In this work, we numerically demonstrate the microscopic realization of a transition from a translationally invariant fractional quantum anomalous Hall (FQAH) state to a spontaneously broken-translation-symmetry FQAH state, driven by softening of the magnetoroton mode via isotropic interactions in a topological flat-band model. At the critical point, the neutral excitation gap closes at finite momentum, while the charge gap remains stable. This transition shares similarities with Landau-level quantum Hall crystals and nematics but exhibits unique features. Criticality analysis shows that this non-trivial transition belongs to the Ising universality class. Such spontaneous translation symmetry breaking within a topologically ordered FQAH state could serve as a universal mechanism across various systems, with potential implications for quantum moiré materials and cold-atom systems.

DOI
10.1016/j.newton.2025.100300

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Collective excitations are crucial for understanding exotic phases and phase transitions in quantum many-body systems. In this work, we numerically demonstrate the microscopic realization of a transition from a translationally invariant fractional quantum anomalous Hall (FQAH) state to a spontaneously broken-translation-symmetry FQAH state, driven by softening of the magnetoroton mode via isotropic interactions in a topological flat-band model. At the critical point, the neutral excitation gap closes at finite momentum, while the charge gap remains stable. This transition shares similarities with Landau-level quantum Hall crystals and nematics but exhibits unique features. Criticality analysis shows that this non-trivial transition belongs to the Ising universality class. Such spontaneous translation symmetry breaking within a topologically ordered FQAH state could serve as a universal mechanism across various systems, with potential implications for quantum moiré materials and cold-atom systems.
Fractional quantum anomalous Hall (FQAH) effect, a lattice analogue of fractional quantum Hall effect, offers a unique pathway toward fault-tolerant quantum computation and deep insights into the interplay of topology and strong correlations. The exploration has been successfully guided by the paradigm of ideal flat Chern bands, which mimic Landau levels in both band topology and local quantum geometry. Yet, given the boundless potential for Bloch bands in lattice systems, it remains a significant open question whether FQAH states can arise in scenarios fundamentally distinct from this paradigm. Here we turn to a class of gapless flat bands, featuring (i) ill-defined band topology, (ii) non-quantized Berry flux, (iii) divergent quantum geometry at singular band touchings, (iv) highly fluctuating and far-from-ideal quantum geometry across the Brillouin zone (BZ). Our exact diagonalization and density matrix renormalization group calculations unambiguously demonstrate FQAH phase that is virtually independent of the interaction strength, persisting from the weak-interaction to the strong-interaction limit. We find the stability of the FQAH states does not uniquely correlate with the singularity strength or the BZ-averaged quantum geometric fluctuations. Instead, the many-body topological order can adapt to the singular and fluctuating quantum geometric landscape by spontaneously developing an inhomogeneous carrier distribution, while its quenching accompanies the drop in the occupation-weighted Berry flux. Our work reveals a profound interplay between local quantum geometry and many-body correlation, and significantly expands the exploration space for FQAH effect and correlated phenomena in general.
Geometric fluctuations of the density mode in a fractional quantum Hall (FQH) state can give rise to a nematic FQH phase, a topological state with a spontaneously broken rotational symmetry. While experiments on FQH states in the second Landau level have reported signatures of putative FQH nematics in anisotropic transport, a realistic model for this state has been lacking. We show that the standard model of particles in the lowest Landau level interacting via the Coulomb potential realizes the FQH nematic transition, which is reached by a progressive reduction of the strength of the shortest-range Haldane pseudopotential. Using exact diagonalization and variational wave functions, we demonstrate that the FQH nematic transition occurs when the system's neutral gap closes in the long-wavelength limit while the charge gap remains open. We confirm the symmetry-breaking nature of the transition by demonstrating the existence of a "circular moat" potential in the manifold of states with broken rotational symmetry, while its geometric character is revealed through the strong fluctuations of the nematic susceptibility and Hall viscosity.
The fractional quantum Hall (FQH) nematic state is a conjectured state of matter in which FQH topological order coexists with spontaneously broken rotation symmetry. Recent experiments on GaAs quantum wells suggest this state, as well as the concomitant isotropic-nematic phase transition, may be realized in a two-dimensional electron gas at certain rational filling factors. Although effective field theories of the FQH nematic state have been constructed, there has been so far no explicit realization of this state as the ground state of a microscopic model of interacting electrons. The authors show via a variational approach and numerical exact diagonalization that tuning Haldane pseudopotentials in a model of interacting electrons in the lowest Landau level induces a quantum phase transition out of the isotropic Laughlin FQH liquid and into the FQH nematic state.
We study the phase diagram of the ${\ensuremath{\nu}}_{e}=7/3$ state in the N=1N=1N=1 Landau level in the presence of band mass anisotropy. Using the density matrix renormalization group on an infinite cylinder geometry, we find a continuous transition from the topologically ordered Laughlin fractional quantum Hall state to a stripe phase with a period of approximately five and a half magnetic lengths. The transition is driven by the condensation of the magnetoroton mode which becomes gapless at the critical point. We interpret the transition within the composite-boson theory as the onset of stripe order in a superfluid background, resulting from the roton mode going soft.
We study possible many body phenomena in the Quantum Anomalous Hall system of weakly interacting spinor bosons in a square lattice. There are various novel spin-bond correlated superfluids (SF) and quantum or topological phase transitions among these SF phases. One transition is a first order one driven by roton droppings ( but with non-zero gaps $ Δ_R $ ) tuned by the Zeeman field $ h $. Another is a second order bosonic Lifshitz transition with the dynamic exponents $ z_x=z_y=2 $ and an accompanying $ [C_4 \times C_4]_D $ symmetry breaking. It is driven by the softening of the superfluid Goldstone mode tuned by the ratio of spin-orbit coupled (SOC) strength over the hopping strength. The two phase boundaries meet at a topological tri-critical (TT) point which separates the $ h=0 $ line into two SF phases with $ N=2 $ and $ N=4 $ condensation momenta respectively. At the $ h=0 $ line where the system has an anti-unitary $ Z_2 $ Reflection symmetry, there are infinite number of classically degenerate family of states on both sides. We perform a systematic order from quantum disorder analysis to find the quantum ground states, also calculate the roton gaps $ Δ_R $ generated by the order from disorder mechanism on both sides of the TT point. The $ N=2 $ and $ N=4 $ SF phases have the same spin-orbital XY-AFM spin structure, respect the anti-unitary symmetry and break the $ [C_4 \times C_4]_D $ symmetry, so they be distinguished only by the different topology of the BEC condensation momenta instead of by any differences in the symmetry breaking patterns. All these novel quantum or topological phenomena can be probed in the recent experimentally realized weakly interacting Quantum Anomalous Hall (QAH) model of $ ^87 Rb $ by Wu, {\sl et.al}, Science 354, 83-88 (2016).
Abstract Interaction-driven topological phases significantly enrich the class of topological materials and thus are of great importance. Here, we study the phase diagram of interacting spinless fermions filling the two-dimensional checkerboard lattice with a quadratic band touching (QBT) point. By developing new diagnosis based on the state-of-the-art density-matrix renormalization group and exact diagonalization, we determine accurate quantum phase diagram for such a system at half-filling with three distinct phases. For weak nearest-neighboring interactions, we demonstrate the instability of the QBT towards an interaction-driven spontaneous quantum anomalous Hall (QAH) effect. For strong interactions, the system breaks the rotational symmetry realizing a nematic charge-density-wave (CDW) phase. Interestingly, for intermediate interactions we discover a symmetry-broken bond-ordered critical phase sandwiched in between the QAH and CDW phases, which splits the QBT into two Dirac points driven by interaction. Instead of the direct transition between QAH and CDW phases, our identification of an intermediate phase sheds new light on the theoretical understanding of the interaction-driven phases in QBT systems.