Quantum Hall to Chiral Spin Liquid transition in a Triangular Lattice Hofstadter-Hubbard Model

Abstract

We investigate the weak interaction integer quantum Hall (IQH) phase, the intermediate interaction phase identified as a chiral spin liquid (CSL) and the transition between them in the triangular lattice Hofstadter-Hubbard model at a density of one electron per site in an orbital magnetic field corresponding to one-quarter flux per plaquette. Our primary tool is the finite system density matrix renormalization group (DMRG) method with both interaction-strength scan and fixed interaction techniques for cylinders of circumference 3, 5, and 7 and lengths up to 240. For the IQH phase, we use single particle exact diagonalization to clarify finite size effects, including an excess charge on the edges of our cylinders, and the limitations of entanglement spectra degeneracies on small circumference cylinders. For both phases, we use DMRG to study the entanglement spectra, the entanglement entropy, and the effect of flux insertion on charge and spin pumping, all of which show key differences between the two phases. To study the transition, we use interaction-strength scans extending between the two phases, and apply a scaling data collapse of a bond-dimerization order parameter to extract critical exponents. We also extract critical behavior from the divergence of correlation lengths on the IQH side, measuring decay away from edges of both the dimerization order parameter and transverse edge currents. The critical behavior and exponents are consistent with an Ising transition in 1+1 dimensions. Finally, we obtain excited states in various quantum number sectors finding that the gap to a charge neutral momentum $π$ excitation corresponding to fluctuations of the dimerization order parameter closes in the vicinity of the critical point but gaps to other excitations remain large.

Figures (19)

• Figure 1: (a) Triangular lattice on a YC$3$ cylinder, with periodic boundary conditions in the $y$‑direction. Peierls hopping phases, where nonzero, are shown. A standard cylinder starts on a "lower" $y$-column without the extra $\pi$ phases. (b) Schematic phase diagram of the model \ref{['eq:HHmodel']} at half-filling, showing the transition point $U_c$ between the integer quantum Hall (IQH) and chiral spin liquid (CSL) phases. The width of the lines are proportional to the nearest-neighbor spin correlations $\langle {\bf S}_i \cdot {\bf S}_j\rangle$ on the YC$5$ cylinders. In the CSL, a uniform offset of $–0.149$ has been subtracted from the correlations to highlight the order parameter; solid red (dashed blue) bonds indicate correlations below (above) this offset.
• Figure 2: Single particle energy levels for the model at $U=0$. (a) $64\times31$ open cylinder, showing bulk states (black circles) and left (red triangles) and right (green triangles) chiral edge bands. (b-d) $64\times5$ cylinders, with three values of an external flux along the axis of the cylinder. For $\Phi_{\rm ext}=\pi/2$, every energy level is doubly degenerate.
• Figure 3: Entanglement spectrum for different particle number sectors at $U=0$ for a $32\times15$ spinless fermion system divided between columns 16 and 17. The legend shows the number of particles on the left half of the system.
• Figure 4: Entanglement spectrum versus transverse momentum $k_y$ for a YC5 cylinder of length $L_x=80$ at bond dimension $\chi=2000$. Panels (a,b) show $U=6$; panels (c,d) show $U=15$. For each $U$ we compare adjacent bipartitions taken immediately to the right of columns 20 and 21 (even/odd cuts). At $U=6$ the cuts immediately to the right of columns 20 (panel a) and 21 (panel b) are essentially identical up to a relabeling of particle number. At $U=15$ we see that the cut immediately to the right of column 20 (panel c) has a non-degenerate entanglement spectrum ground state and an approximately three-fold degenerate next lowest state, differing markedly from the cut immediately to the right of column 21 (panel d) which has a two fold degenerate entanglement ground state and a two-fold degenerate next lowest state. Marker color encodes the particle number $N$ on the left of the cut and marker shape encodes $S_z$ of the left Schmidt sector (see legends). Only the lowest 50 entanglement levels are shown.
• Figure 5: Extrapolated entanglement entropy $S$ on YC3 cylinders versus $U$. We compare two extrapolation schemes, retaining the largest 5 (solid lines, filled markers) and the largest 10 (dashed lines, open markers) bond dimensions. Cylinder lengths $L_x$ are shown in the upper left legend, and the cut is always in the middle. At larger $U$, the curves split into two branches depending on the cut parity. Inset: zoom of the transition region.