Microscopic Mechanism of Anyon Superconductivity Emerging from Fractional Chern Insulators

TL;DR

This work shows how anyon superconductivity can emerge from repulsive interactions in fractional Chern insulators by tuning toward a semion-crystal (SX) phase near a topological quantum critical point. A parton-based field theory describes the FCI–SX transition, where the charge gap closes while the spin gap stays open, creating an energetically favorable route for spin-singlet, charge-2e Cooper pairs upon doping. Through a microscopic Hubbard-Hofstadter model at ν=2/3, tensor-network simulations reveal a continuous transition from a bilayer FCI to the SX, accompanied by enhanced Cooper-pair correlations near criticality and a stable SX as a competing phase. Finite-doping analyses (via both chargon-vortex and fermionic-parton pictures) show that doping near the critical point yields conventional charge-2 superconductivity, possibly coexisting with CDW order, with edge states reflecting alternating central charges; these insights connect to experiments in twisted MoTe$_2$ and guide efforts to realize flat-band superconductivity in moiré materials using repulsive interactions.

Abstract

Fractional quantum Hall (FQH) states and superconductors typically require contrasting conditions, yet recent experiments have observed them in the same device. A natural explanation is that mobile anyons give rise to superconductivity; however, this mechanism requires binding of minimally charged anyons to establish an unusual energy hierarchy. This scenario has mostly been studied with effective theories, leaving open the question of how anyon superconductivity can arise from repulsive interactions. Here, we show that such an energy hierarchy of anyons arises naturally in fractional Chern insulators (FCIs) at fillings $ν= 2/(4p \mp 1)$ when they are driven toward a quantum phase transition into a ``semion crystal'' -- an exotic charge-density-wave (CDW) insulator with semion topological order. Near the transition, Cooper-pair correlations are enhanced, so that a conventional charge-2e superconductor appears with doping. Guided by these insights, we analyze a microscopic realization in a repulsive Hubbard-Hofstadter model. Tensor network simulations at $ν= 2/3$ reveal a robust FCI that, with increasing interactions, transitions into the semion crystal. Finding a stable semion crystal in such a minimal model highlights it as a viable state competing with conventional CDW and FQH states. In the vicinity of this transition, we find markedly enhanced Cooper pairing, consistent with our theory that the 2e/3 anyon is cheaper than a pair of isolated e/3 anyons. Doping near the transition should in general lead to doping Cooper pairs and charge-2e superconductivity, with chiral edge modes of alternating central charge $c = \pm2$, which can coexist with translation symmetry breaking. Our framework unifies recent approaches to anyon superconductivity, reconciles it with strong repulsion and provides guidance for flat band moiré materials such as recent experiments in twisted MoTe$_2$.

Figures (9)

• Figure 1: Setup and phase diagram.(a) At various fractional fillings seemingly continuous topological phase transitions arise between FQH states and semion crystals---charge density wave states which support a chiral spin liquid. The sign of the spin Hall conductivity alternates with the filling of the FQH state. (b) Upon changing the chemical potential, a conventional superconducting (SC) state of anyons with chiral edge states arises that potentially coexists with density wave order. Two cases of 2/3 (top) and 2/5 (bottom) filling are illustrated. (c) At the transition from a singlet FQH state to the semion crystal, the charge gap closes, while the spin gap remains finite across the transition. Consequently, it is favorable for doped charges to enter as charge-2 singlets of zero spin, providing a microscopic mechanism for superconductivity upon doping close to the critical point.
• Figure 2: Parton construction. Singlet FQH states can be described using fermionic spinons in IQH states at level $C_{f_\sigma}=\pm1$ and bosonic chargons in a $1/n$ Laughlin state. To describe a continuous transition to a semion crystal, the chargons undergo a crystallization transition, forming a bosonic CDW. The spinons remain in the same state across the transition, implying that the spin gap stays finite.
• Figure 3: Doping fermionic partons. Due to the projective action of translations on the partons, there are three degenerate valleys $\mathbf{K}_i$ at $\nu=2/3$. Upon doping, partons in each valley feel a flux $\Delta\Phi \propto \delta$, leading to the formation of Landau levels. (a) Valley-symmetric doping, preserving translations. (b) When translations are broken, either explicitly or spontaneously upon doping, all doped partons enter into a unique valley.
• Figure 4: Model and phase diagram.(a) Left: Bilayer Hofstadter model on a triangular lattice with $\pi/2$ flux per triangle. The layer degree of freedom serves as a pseudo-spin, which is a good quantum number because we consider vanishing inter-layer hopping. Right: Flat Chern bands of the model for $t'=t/4$. The cut through the Brillouin zone is shown as an inset. (b) Phase diagram for $\nu=2/3$ as a function of the repulsive on-site interaction $U$ and the nearest-neighbor interaction $V$, obtained from Tensor Network simulations on an infinite cylinder with $L_y=3$ and $t'=t/4$, bond dimension $\chi=1024$. For small $V$, we find a stable $(\bar{1}\bar{1}2)$ FQH phase. Increasing $V$ leads to the formation of a CDW, whose spin order depends on $U$. For large $U$, the spins order to form a $120$ degree state. For smaller $U$, the spin-sector forms a CSL on top of the CDW, i.e., the semion crystal.
• Figure 5: Entanglement spectrum of the $(\bar{1}\bar{1}2)$-state. The two chiral edge modes of the $(\bar{1}\bar{1}2)$-state have opposite chirality. Top (bottom) panel shows the charge $Q = 0$$(1)$ sector. The counting of the lowest states agrees well with the edge theory. Here, we used $U=1.2t$, $V=U/2$, $\chi=10000$, $L_y=6$.