Fate of a Fractional Chern Insulator under Nonlocal Interactions in Synthetic Dimensions

Abstract

Synthetic dimensions provide a powerful route to engineer topological lattice models in ultracold atomic systems, but they contain intrinsic nonlocal interactions along the synthetic direction. We investigate an extended Harper-Hofstadter model subject to infinite-range column interactions that mimic this synthetic nonlocality. By tuning this interaction strength, we demonstrate an adiabatic evolution from a Laughlin-type bosonic fractional Chern insulator to a charge-ordered Tao-Thouless-like state without closing the many-body gap. Along this path, the many-body Chern number and the topological entanglement entropy remain unchanged, despite a pronounced restructuring of the entanglement spectrum and the loss of robustness against local perturbations. This adiabatic connectivity establishes a controlled bridge between topologically ordered and effect- ively one-dimensional charge-ordered regimes, opening potential new avenues for state preparation. Our results also show that conventional topological markers may fail to diagnose the breakdown of locality-protected topological order in synthetic dimensions, and identify nonlocal interactions as a powerful knob to coherently interpolate between distinct many-body regimes.

Figures (11)

• Figure 1: Schematic of the model in Eq (\ref{['Eq:Ham']}). The lattice has a physical (horizontal, $x$) and synthetic (vertical, $y$) direction. Hopping along the physical axis has amplitude $t$, while motion along the synthetic axis carries a Peierls phase $te^{i\alpha x}$. Interactions include an onsite term $U_{\mathrm{0}}$ and an infinite-range column interaction $U_{\mathrm{i}}$ that couples all particles within the same physical column.
• Figure 2: Analysis of some standard markers of TO between $U_{\mathrm{i}} = 0$ and $U_{\mathrm{i}}= 1000$. (a) the energy spectrum of an $8\times4$ lattice with $N=4$ bosons where for all values of $U_{\mathrm{i}}$ the groundstates (blue and red dashes) are degenerate and well separated from the excited states (black dashes). The insets of (a) show the winding defect fields over the boundary twist angles $\theta_x$ and $\theta_y$ at $U_{\mathrm{i}}=0$ and $U_{\mathrm{i}}=1000$ where integrating over the enclosed region (black curve) returns the Chern number, $\mathcal{C}=1$ for the whole manifold; twofold degeneracy leads to $\mathcal{C}=\frac{1}{2}$HatsugaiMBCN. (b) is the topological entanglement entropy, extracted from the linear scaling of the entanglement entropy via spatial bipartion with the bipartition perimeter as a function of $U_{\mathrm{i}}$ . The simulation utilized TTNs for a $16\times8$ lattice with $N=8$. It maintains a value of $\gamma \sim \frac{1}{2}$ which is the expected value for a Laughlin state at $\nu=\frac{1}{2}$ for all $U_{\mathrm{i}}$. Detailed calculation methods are shown in the Appendices for the Chern number $\mathcal{C}$ and the topological entanglement entropy $\gamma$.
• Figure 3: Topologically degenerate groundstates have a finite size splitting which decays exponentially in system size such that in the thermodynamic limit the groundstates are perfectly degenerate. For topological states, this behavior should be robust to local pinning noise. (a) shows the exponential scaling of $\Delta_{01}$ for the Laughlin case, $U_{\mathrm{i}}$$=0$, without (filled triangles) and with (empty triangles and stars) pinning. (b) is the strongly interacting case, $U_{\mathrm{i}}$$=300$ for two other values of pinning strength $\delta$. Without pinning, there is an exponential decay of $\Delta_{01}$, but for very weak pinning $\delta$ the splitting plateaus to a value of order $\delta$ and there is no exponential decay. Therefore, for large $U_{\mathrm{i}}$ the groundstates are not degenerate in the thermodynamic limit and are not topologically ordered.
• Figure 4: The momentum-resolved particle entanglement spectrum (PES) for a $10\times5$ lattice with $N=5$ using ED. The reduced density matrix $\rho_A$ is computed in the $N_A=3$ sector. Both panels display the momentum-resolved eigenvalues of $\mathcal{H}_A$ given by $\rho_A \equiv \mathrm{exp}(\mathcal{-H_A})$. (a) For the Laughlin state at $U_{\mathrm{i}}$$=0$, the number of low-lying PES levels matches the expected counting $\mathcal{N}=50$. (b) For strong anisotropic interactions, $U_{\mathrm{i}}$$=10$, the PES counting changes to a value consistent with CDW structure.
• Figure 5: The four point momentum operator, Eq \ref{['Eq:4PtMomentumOp']}, at $k_{y}'=0$ for a $16\times8$ lattice with $N=8$, using TTNs, for both the Laughlin case with $U_{\mathrm{i}}=0$ and the strongly interacting case, $U_{\mathrm{i}}=300$. The Laughlin case is flat and featureless in the bulk as characteristic of topological states while there is a pronounced density wave order when $U_{\mathrm{i}}=300$.