Phases of itinerant anyons in Laughlin's quantum Hall states on a lattice

Abstract

We study phases of itinerant anyons when hole-doping Laughlin-like states in fractional Chern insulators (FCIs). In light of the recent observation of time-reversal-broken superconductivity near FCIs in van der Waals materials, a theoretical understanding of doped fractional quantum Hall states on a lattice has been developed by Shi and Senthil [Phys. Rev. X 15, 031069], reviving old ideas about "anyon superconductivity". We test these ideas analytically within an effective parton mean-field theory and numerically with variational Monte Carlo, pointing out that the predicted state depends on whether the Laughlin order at $ν=1/m$ is described by a U(1), or an SU(m) Chern-Simons field, the latter implying a symmetry between the m parton species. Our results demonstrate that the interplay between band Berry curvature and effective anyon dispersion has crucial implications for which anyonic phase is realized. In the experimentally relevant scenario of hole-doping the $ν=1/3$ fermionic FCI, our results uncover a mechanism for the formation of an anyon superconducting state of half-integer central charge in the case when the energetically cheapest excitations are the fundamental 1/3 charge anyons, bypassing the need for these anyons to pair into charge-2/3 composites, which has generally been assumed in similar anyon superconductivity constructions.

Figures (10)

• Figure 1: The formation of mini Landau Levels (mLLs) when a small magnetic field is added to an ideal Chern band. Depending on the direction of the field, the first mLL forms either exactly at the top of the band or an energy $\tilde{\omega}_c=|b|/m^*$ away. This is in contrast to time-reversal invariant bands, where the first mLL's forms at energy $\tilde{\omega}_c/2$ away from the extremum regardless of the sign of $b$. In $\mathcal{C}=1$ bands, the density of states per unit cell is $(1+b/2\pi)$, unlike the $b$-independent value in the presence of time-reversal symmetry. At $\nu=\frac{1}{m}$ with $b=0$, each parton's dispersion has $m$ degenerate maxima. When $b\neq0$, this degeneracy can be broken by inter-peak tunnelling processes, but the scale of this effect is $\sim \exp(-\frac{2\pi}{m^2}\frac{1}{b})$, making it negligible at small $b$.
• Figure 2: The mean-field parton energy levels before and after introducing a flux $b$. Initially, we have a flat $\mathcal{C}=1$ band (blue) of width $w\ll\Delta$ with $\Delta$ the parton band gap (which is at the scale of the physical many-body gap). Upon introducing a small field $b$, the flat-band splits into mLL's, spaced by an energy $\sim |b|$. The band now has $N_xN_y(1+\frac{b}{2\pi})$ states. Given a fixed doping, for sufficiently negative $b$, some partons must jump the gap $\sim\Delta$, which is large. The lower band will generally not be filled, and if the holes (red crosses) are at mLL filling $\tilde{\nu}$, the remaining partons fill a $\mathcal{C}=1+\tilde{\nu}$ band with a gap $\sim |b|$ and an enlarged unit cell of area $\sim 2\pi/|b|$.
• Figure 3: The parton mean-field energy of a doped $\nu=\frac{1}{2}-\delta$ FCI as a function of the flux $b_1$ of the internal gauge field, showing degenerate minima whenever the partons are gapped at the mean-field level, at $b_1=2\pi\delta /(1+2n)$ for all $n\in\mathbb Z_{\geq0}$.
• Figure 4: Parton mean-field energy of a doped $\nu=\frac{1}{3}-\delta$ FCI. The centre (orange dot) indicates a state with $b_1=b_2=b_3=0$ where all partons are equivalent and see no magnetic field, while the corner states (green diamonds) are cases where $b_1/2=-b_2=-b_3=2\pi\delta$ and only one of the partons holds quasiholes. A two-dimensional continuum of possible flux arrangements spanned by $b_1,b_2,b_3$ with the constraint $b_1+b_2+b_3=0$ lies between these extremes. The filling fractions are related to the fields by $\tilde{\nu}_p=-1-\frac{2\pi\delta}{b_p}$ and evolve non-linearly across the diagram, diverging at the origin.
• Figure 5: Bosons at $\nu=\frac{1}{2}-\delta$ on a system of $N_s=48$ unit cells ($6\times8$ grid) for $\delta\in [0.02,0.08]$. The nature of the ground state depends on the sign of $\eta=\eta_\text{geo.}+\eta_\text{int.}$ (as interactions are hard-core only, $\eta_\text{int.}=0$). When $\eta<0$, the lowest energy state has $b_1=2\pi\delta$ and forms an anyon superconducting state with $\text{U}(1)$ gauge invariance, as predicted by senthil_doping_2024. When $\eta>0$ however, $b_1=0$ is favoured for all $\delta$, leading to a gapless parton mean-field with an $\text{SU}(2)$ gauge invariance which is discussed further in Sect. \ref{['sect:undistr']}.