Topological Order in Deep State

Abstract

Topologically ordered states are among the most interesting quantum phases of matter that host emergent quasi-particles having fractional charge and obeying fractional quantum statistics. Theoretical study of such states is however challenging owing to their strong-coupling nature that prevents conventional mean-field treatment. Here, we demonstrate that an attention-based deep neural network provides an expressive variational wavefunction that discovers fractional Chern insulator ground states purely through energy minimization without prior knowledge and achieves remarkable accuracy. We introduce an efficient method to extract ground state topological degeneracy -- a hallmark of topological order -- from a single optimized real-space wavefunction in translation-invariant systems by decomposing it into different many-body momentum sectors. Our results establish neural network variational Monte Carlo as a versatile tool for discovering strongly correlated topological phases.

Figures (7)

• Figure 1: (a) Schematic of the self-attention–based neural network architecture. Particle positions ${{\boldsymbol{r}}_i}$ are mapped to high-dimensional vectors ${{\boldsymbol{h}}_i^0}$, which are processed by $L$ layers of self-attention and perceptron transformations supplementary. The final output is projected onto $N_{\rm det}$ determinants to form the variational ansatz \ref{['eq:ansatz']}. The parameters ${\theta}$ are optimized via Monte Carlo sampling of the variational energy $L(\theta) = \langle \Psi_{\theta}|H|\Psi_{\theta}\rangle / \langle \Psi_{\theta}|\Psi_{\theta} \rangle$. (b) The optimized wavefunction is decomposed to different momentum sectors at the end of optimization.
• Figure 2: Band structure of the model defined in equation \ref{['eq:minimalmodel']} obtained for (a) $\lambda = -0.23$ and (b) $\lambda = -0.26$. (c) The periodic magnetic field in real space for $\lambda = -0.23$. (d) Berry curvature of the lowest band for $\lambda = -0.23$.
• Figure 3: Ground state charge density $\rho({\boldsymbol{r}})$ at $\nu = 1/3$ for (a) the FCI phase for $\lambda = -0.23$ and (b) the CDW phase for $\lambda = -0.26$. The solid black parallelogram denotes the supercell while the dashed blue parallelogram denotes ithe unit cell of the magnetic field. In (a), we use a supercell with 24 unit cells (8 particles) and in (b), we use a supercell with 27 unit cells (9 particles) (c) Structure factor $S({\boldsymbol{q}})$ as a function of $|{\boldsymbol{q}}|$ computed in the FCI phase ($\lambda = -0.23$) and the CDW phase ($\lambda = -0.26$). (d) $|{\boldsymbol{q}}|^2 A / 4 \pi S({\boldsymbol{q}})$ in the FCI phase ($\lambda = -0.23$) approaches $3$ for small $|{\boldsymbol{q}}|$, almost saturating the topological bound onishi2024topological. All calculations are done for $r_s \approx 3.43$.
• Figure 4: (a) The overlap $|\langle \Psi(\{{\boldsymbol{r}}_i\})| \Phi_{{\boldsymbol{K}}}(\{{\boldsymbol{r}}_i\})\rangle|^2$ of the optimized wavefunction $\Psi(\{{\boldsymbol{r}}_i\})$ with its projection $\Phi_{{\boldsymbol{K}}}(\{{\boldsymbol{r}}_i\})$ onto different COM momentum sectors $K$. (b) Variational energy $E_{{\boldsymbol{K}}} = \langle \Phi_{{\boldsymbol{K}}}|H|\Phi_{{\boldsymbol{K}}} \rangle / \langle \Phi_{{\boldsymbol{K}}}|\Phi_{{\boldsymbol{K}}} \rangle$ (in units of $\hbar^2/m a_0^2$) of the wavefunction in the three different momentum sectors with non-zero weight (shown in (a)).
• Figure S1: Band structure of the model \ref{['eq:minimalmodel_SM']} for (a) $\lambda = -0.2$, (b) $\lambda = -0.23$ , (c) $\lambda = -0.26$