Gradient-based search of quantum phases: discovering unconventional fractional Chern insulators

Abstract

The discovery and understanding of new quantum phases has time and again transformed both fundamental physics and technology, yet progress often relies on slow, intuition-based theoretical considerations or experimental serendipity. Here, we introduce a general gradient-based framework for targeted phase discovery. We define a differentiable function, dubbed "target-phase loss function", which encodes fingerprints of a quantum state, thereby recasting phase search as a tractable optimization problem in Hamiltonian space. The method is broadly applicable to a wide range of symmetry-broken and topological orders and can be interfaced with most many-body numerical solvers. As a demonstration, we apply it to spinless fermions on the kagome lattice using exact diagonalization and discover two distinctive fractional Chern insulators (FCIs): (i) at filling $ν= 1/3$, a "non-ideal" Abelian FCI whose band geometry lies far beyond the Landau-level mimicry paradigm and all recent generalizations; and (ii) at $ν= 1/2$, a non-Abelian FCI stabilized purely by finite-range two-body interactions. These results provide the first explicit realization of such types of FCIs and establish a versatile paradigm for systematic quantum-phase discovery.

Figures (6)

• Figure 1: (a) Diagram illustrating the general procedure for targeting phases of interest. A quantum ground state (e.g., a topologically ordered state, density wave, superconductor, or Fermi liquid) is characterized by a collection of sharp features (boxed), some of which can be leveraged to define a target-phase loss function $\mathcal{L}$. Optimizing $\mathcal{L}$ over a family of Hamiltonians leads to the identification of microscopic models whose ground states realize the phase of interest. (b) Schematic workflow for gradient-based search for a target phase (see text).
• Figure 2: Generic spectra of a many-body Hamiltonian at two distinct parameter points (a) $\mathbf{p}_1$ and (b) $\mathbf{p}_2$ and a given flux $\Phi$, with energy $E$ as a function of symmetry sectors $\mathcal{K}$. The target and complement manifolds are shown in orange and blue, respectively. The per-flux loss $\ell(\mathbf{p}_1, \Phi)$ is negative, indicating that the system is potentially in the target phase with many-body gap $-\ell(\mathbf{p}_1, \Phi)$, whereas $\ell(\mathbf{p}_2, \Phi) > 0$ captures a spectral distance to the target phase in parameter space.
• Figure 3: ED evidence for a non-ideal FCI, located at $(\lambda_1, t_2, \lambda_2, V_2) = (-0.169, -0.992, -0.157, 0.25)$. (a) The kagome lattice model with first- and second-neighbor complex hoppings, as well as first- and second-neighbor density-density interactions. (b) Single-particle Berry curvature $\mathcal{F}(\mathbf{k})$ (top) and momentum-resolved trace violation $T(\mathbf{k})$ (bottom) distributions in the Brillouin zone. High-symmetry momenta $\Gamma = (0, 0), K = (2\pi/3, 2\pi/\sqrt{3}), K' = (4\pi/3, 0)$ are labeled. Inset: zoomed in Berry curvature distribution near the $\Gamma$ point, showing that it vanishes at $\Gamma$, consistent with a quadratic band touching with small gap. (c) Zero-flux ED spectrum for a $6x6$ cluster as a function of center-of-mass momentum at filling $\nu=1/3$. (d) ED spectrum for a $6x5$ cluster as a function of magnetic flux $\Phi_1$ along reciprocal lattice vector $\mathbf{b}_1$. (e) Zero-flux gap $\Delta_0$ as a function of inverse system size $1/N$. All clusters with $N < 24$ have vanishing gap. (f) Particle entanglement spectrum on a $6x5$ cluster for subspace particle number $N_A = 4$. The number of states below the dashed line is 9975, consistent with a Laughlin ground state.
• Figure 4: (a) Phase diagram as a function of $(\lambda_1, t_2)$ at fixed $(\lambda_2, V_2) =(-0.157, 0.25)$ for a $6 \times 4$ cluster. The FCI gap $\Delta_{0, \pi} = \min(\Delta_0, \Delta_\pi)$, combining the gaps at fluxes $\Phi=\{0, \pi\}$, is shown in blue. FCI and Fermi liquid (FL) regions are highlighted, and dashed lines are guides for the eye. (b) $\Delta_{0, \pi}$ as a function of trace violation $T$ and the ratio of second-neighbor to first-neighbor interaction strengths $V_2/V_1$ along a 1D path in the FCI region found. Red star in (a) and (b) indicates the parameter point hosting the non-ideal FCI investigated here.
• Figure 5: ED evidence for a non-Abelian FCI from finite-range two-body interactions, located at $(\lambda_1, t_2, \lambda_2, V_2, V_3) = (-0.952, 0.956, -0.369, 0.781, 0.987)$. (a) The kagome lattice model with first- and second-neighbor complex hoppings, as well as first-, second- and third-neighbor density-density interactions. (b) Zero-flux ED spectrum for a $6x5$ cluster as a function of center-of-mass momentum at filling $\nu=1/2$. (c) ED spectrum for a $6x5$ cluster as a function of magnetic flux $\Phi_1$ along reciprocal lattice vector $\mathbf{b}_1$. Inset: zoomed in spectrum showing spectral flow of the ground state manifold. (d) Particle entanglement spectrum on a $6x5$ cluster for subspace particle number $N_A = 4$. The number of states below the dashed line is 25185, consistent with a MR ground state. (e) Zero-flux gap $\Delta_0$ as a function of inverse system size $1/N$. All systems with $N < 18$ have vanishing gap.