Fractional Quantum Hall Wedding Cakes

Abstract

This work investigates the coexistence of distinct topologically ordered phases within a single setup. We demonstrate this concept through tensor network simulations of the Hofstadter-Bose-Hubbard model under a spatially modulated chemical potential. Focusing on cylindrical geometries, we realize regions exhibiting the Laughlin-1/2 phase and its particle-hole conjugate, and confirm their topological character via the local Středa's response and Laughlin's flux insertion protocol. Our approach offers a new pathway for experimentally and numerically charting entire phase diagrams within a single system, possibly eliminating the need for independent parameter scans.

Figures (5)

• Figure 1: (a) Sketch of a wedding cake-like structure with FCI regions in a system with a trapping potential. (b) Illustration of the Hofstadter-Bose-Hubbard model on a cylinder and, (c) a $L_x\times L_y$ square lattice with MPS snaking.
• Figure 2: (a) The variation of the density $\rho(x)$ as the chemical potential $\mu(x)$ changes in a local region, embedded within a uniform FCI state. The dashed line represents the density $\rho_0$. $\textbf{(b)}$ The slope of the density as a function of the variation of the chemical potential in the center of the local region (at $x=0$), obtained from varying the profile $\mu(x)$. The solid line serves as a guide for the eye.
• Figure 3: (a) The chemical potential $\mu_0(x)$, and (b) the density profile in a $90\times7$ cylinder with $\alpha=2/7$. The dashed lines represent the tripartition we make for the charge pumping experiment. (c) The entanglement entropy $S(x)$. The dashed line is the value of the entanglement entropy for a Laughlin-1/2 state in a uniform MPS simulation.
• Figure 4: (a) The adiabatic change of the chemical potential $\mu_{\beta}(x)$ for different values of $\beta$. (b) The density profiles in a $90\times7$ cylinder with $\alpha=2/7$ for different values of $\beta$. The dashed lines show $\rho=1/7,1/2,6/7$ as a guide for the eye.
• Figure 5: (a) The many-body Chern number obtained along the cylinder using Středa's response. The blue dots are the numbers obtained from our simulations. The dashed lines show the $\mathcal{C}=0,\pm 1/2$ values as a guide for the eye. We show the error bars from the fit in the FCI regions. The shaded regions correspond to the compressible phases where Středa's response is not meaningful. The inset shows the evolution of the density as a function of $\tilde{\alpha}=\alpha+\delta\alpha$ in the bulk of the Laughlin-1/2 region at $x=23$. The solid line is the linear fit we use to extract the Chern number. (b) The charge evolution as a function of the inserted flux $\phi_y$ for the left of the $\nu=1/2$ region, $\Delta \mathcal{Q}_L(\phi_y)$, and (c) the right of the $\nu^*=1/2$ region, $\Delta \mathcal{Q}_R(\phi_y)$. For each step of the adiabatic insertion, we perform DMRG ground state searches with a maximal bond dimension of $D_{\mathrm{max}}=200$.