From hidden order to skyrmions: Quantum Hall states in an extended Hofstadter-Fermi-Hubbard model

TL;DR

This paper investigates a spinful Hofstadter-Hubbard lattice with nearest-neighbor repulsion to realize and identify topologically ordered fractional Chern insulators and their spin textures. Using large-scale DMRG with spin symmetries, the authors demonstrate a spin-polarized $\nu=1/3$ FCI with a many-body Chern number $\mathcal{C}_{\rm mb}=1/3$ and hidden off-diagonal long-range order in $^3$CB correlations, consistent with a lattice Laughlin state. Around integer filling $\nu=1$, they uncover quantum Hall ferromagnetism and skyrmionic excitations stabilized by $V$, with both hole- and particle-skyrmions observed depending on doping. They show that $\nu=1/3$ lacks skyrmions in the ground state, while at $\nu=1$ skyrmions arise upon doping; a comprehensive diagnostic toolbox based on local densities, correlation functions, and spin observables is provided, suitable for quantum gas microscopy. These results establish clear experimental pathways to explore FCIs with spin textures in ultracold atoms and in solid-state platforms, and they motivate further studies of spinful topological phases and their excitations.

Abstract

The interplay between topology and strong interactions gives rise to a variety of exotic quantum phases, including fractional quantum Hall (FQH) states and their lattice analogs - fractional Chern insulators (FCIs). Such topologically ordered states host fractionalized excitations and, for spinful systems, are often accompanied by ferromagnetism and skyrmions. Here, we study a Hofstadter-Hubbard model of spinful fermions on a square lattice, extended by nearest-neighbor interactions. Using large-scale density matrix renormalization group (DMRG) simulations, we demonstrate the emergence of a spin-polarized $\frac{1}{3}$-Laughlin-like FCI phase, characterized by a quantized many-body Chern number, a finite charge gap, and hidden off-diagonal long-range order. We further investigate the quantum Hall ferromagnet at $ν=1$ and its skyrmionic excitations upon doping. In particular, we find that nearest-neighbor repulsion is sufficient to stabilize both particle- and hole-skyrmions in the ground state around $ν=1$, whereas we do not find such textures around $ν=\frac{1}{3}$. The diagnostic toolbox presented in this work, based on local densities, correlation functions, and spin-resolved observables, is directly applicable in quantum gas microscopy experiments. Our results open new pathways for experimental exploration of FCIs with spin textures in both ultracold atom and electronic systems.

Figures (16)

• Figure 1: Parameter space studied in this work and schematic phase diagram. Here, $w$ is the band width of the lowest Chern band and $V_c/J$ is the critical interaction strength needed to stabilize a fractional Chern insulator (FCI) at filling factor $\nu=1/3$. "FM" denotes ferromagnetic phases. White areas indicate metallic states with short-range antiferromagnetic correlations. The dashed gray line indicates an incompressible unpolarized state at $\nu=2/3$, the nature of which is not discussed in detail here. The insets sketch the qualitative spin-spin correlations $\braket{\hat{\vec{S}}_{x,y}\cdot\hat{\vec{S}}_{x_0,y}}$ on a cylinder and composite boson ($^m$CB) correlations $\rho^{(m)}$ on a square as defined in Eq. \ref{['eq:hodlro']} for characteristic phases. Note that for $x=x_0$ we expect $\braket{\hat{\vec{S}}_{x,y}\cdot\hat{\vec{S}}_{x,y}} = s(s+1) = 3/4$ based on the spin-1/2 of the constituent particles.
• Figure 2: Magnetization texture $\braket{\hat{\vec{S}}_{\mathbf{i}}}$ of (a) a classical skyrmion spin texture and (b) a ferromagnet. In the skyrmion, local spin alignment evolves into spin reversal at larger distances. In contrast, in the ferromagnet the spin alignment persists at all distances.
• Figure 3: (a) Ground state density profile and (b) Středa response for $\nu\approx 1/3 \ (N=6),\ V/J=1$ (top) and $\nu\approx 1 \ (N=20),\ V/J=0$ (bottom). In both cases, we find an extended bulk region with a many-body Chern number consistent with continuum predictions in the thermodynamic limit. (c) The extracted many-body Chern number for varying values of $V/J$ shows the transition towards an FCI above $V_c/J \approx 0.5$, whereas the CI persists at all interaction strengths.
• Figure 4: Two-particle density correlations as function of the Euclidean distance in units of the magnetic length $\ell_B = a/\sqrt{2\pi \alpha}$. The reference site is fixed at the center of the square lattice. Error bars correspond to a standard deviation of the average over equidistant lattice points. In the case of nearest-neighbor interactions ($V/J=1$, green circles), we observe the characteristic drop of the correlation function at short distances in qualitative agreement with the expected correlation hole for the Laughlin state (black diamonds). In contrast, the pure Hofstadter-Hubbard model ($V/J=0$, blue squares) does not exhibit the characteristic polynomial drop.
• Figure 5: (Hidden) correlations for different filling factors and interaction strengths for the square lattice of size $L=10$ as a function of the Euclidean distance. The error of the sampling (standard deviation $\sigma_{\rho^{(m)}(r)}/\sqrt{N_{\mathrm{snaps}}}$) is smaller then the marker size for all data points and thus not explicitly shown. The dotted lines in panels (b-d) indicate the algebraic fit of the data and the insets show the effective degrees of freedom displaying quasi long-range order in the different phases. While we observe the emergence of algebraically decaying correlations in the $^1$CB or $^3$CB in most cases, we find metallic correlations independent of the basis choice for $\nu\approx1/3$ ($N=6$) at $V/J=0$. These results are consistent with the expected behavior in Tab. \ref{['tab:CorrelationsDecay']}.