Neural Transformer Backflow for Solving Momentum-Resolved Ground States of Strongly Correlated Materials

Abstract

Strongly correlated materials host a rich variety of exotic quantum phases but remain challenging to solve due to strong interactions. We introduce the Neural Transformer Backflow (NTB) framework, a powerful neural-network ansatz formulated within a multi-band projection formalism. NTB is mean-field transcendental, parameter-efficient and fermionic intrinsic, exhibiting superior performance compared with existing neural ansatzes. By naturally enforcing momentum conservation, NTB enables direct computation of momentum-resolved many-body ground states, providing detailed access to degeneracies and energy gaps. It achieves high accuracy on small systems and scales efficiently to larger sizes and higher-band truncations far beyond the reach of exact diagonalization. We demonstrate the power of NTB in capturing diverse correlated phases in twisted MoTe$_2$, including charge density waves, fractional Chern insulators, and anomalous Hall Fermi liquids, within a unified framework. This approach offers a generic, scalable route towards understanding and discovering quantum phases in strongly correlated materials.

Figures (6)

• Figure 1: Schematic of the architecture and applications of the Neural Transformer Backflow (NTB).
• Figure 2: (a). Energy spectrum of momentum resolved ground states across different momentum sectors. 3-bands ED results are plotted against NTB results with 3-bands and 5-bands. (b). Optimization curve of $\Gamma$-point energy with $N_b = 3$ for different NN architectures. The gray dotted line is reference energy obtained from HF method. All networks are trained with $N_{\rm batch} = 10^{4}$. For (a) and (b), the parameters used are: $N_{\rm site} = 9$, $\nu = 2/3$, $\epsilon/\epsilon_0 = 10$, $\theta = 2.7^{\circ}$.
• Figure 3: $\Gamma$-point energy across different choices of $\theta$. 1-band ED results are plotted against 2-bands NTB results. The parameters used are: $N_{\rm site} = 25$, $\nu = 3/5$, $\epsilon/\epsilon_0 = 5$.
• Figure 4: Energy spectrum of momentum resolved ground states and the corresponding structure factor for filling factors $\nu=1/3$ and $\nu=2/3$. (a), (c): Structure factor calculated with $\Gamma$-point ground state. (b), (e): Energy spectrum of momentum resolved ground states at high symmetry points. The parameters used are: $N_{\rm site} = 27$, $N_b = 2$, $\epsilon/\epsilon_0 = 10$, $\theta = 2.7^{\circ}$.
• Figure 5: Various of observables for filling factors $\nu= 1/4$ and $\nu= 3/4$. (a), (d): Structure factor at $\Gamma$-point. (b), (e): Momentum distribution function at $\Gamma$-point. (c), (f): Energy spectrum of momentum resolved ground states at high symmetry points. The parameters used are: $N_{\rm site} = 36$, $N_b = 1$, $\epsilon/\epsilon_0 = 10$, $\theta = 2.7^{\circ}$.