Representability-Aware Neural Networks for Reduced Density Matrices: Application to Fractional Chern Insulators

We develop a representability-aware and interpolable neural network (NN) framework for predicting two-particle reduced density matrices (2-RDMs). The NN incorporates a subset of representability conditions through its architecture and loss function, and can operate on different momentum meshes, enabling evaluating the representability conditions across multiple meshes, which we call interpolated representability condition. The framework can be used either to predict 2-RDMs on large momentum meshes by interpolating exact results from small meshes, or as a variational 2-RDM ansatz optimized by energy minimization on arbitrary meshes. We apply this approach to the fractional Chern insulator in the one-band projected model of twisted bilayer MoTe$_2$ at twist angle $3.89^\circ$ and hole filling $2/3$. Trained on exact-diagonalization (ED) 2-RDMs from meshes with $12$ or $18$ momentum points using six different NN architectures, the best NN is the residual multilayer perceptron, which predicts the $6\times6$ 2-RDM with $97.07\%-98.18\%$ accuracy relative to the ED 2-RDM but predicts an energy $77.353$ meV above ED ground-state energy. We then variationally optimize the NN on several meshes including $6\times6$, predicting a $6\times 6$ energy of just $0.104$ meV below ED while maintaining $98.94\%-98.96\%$ accuracy. Compared with the conventional boundary-point semidefinite programming, which gives an energy $5.560$ meV below ED with $96.40\%-98.94\%$ accuracy, the NN achieves a more accurate energy and similar accuracy while using only less than 1/20 as many parameters. Eventually, we add a symmetric mesh of $48$ momentum points to the variational optimization of the NN, and provide a prediction of the many-body ground-state energy and the many-body quantum metric on that mesh.

Machine-Learning Accelerated Calculations of Reduced Density Matrices

$n$-particle reduced density matrices ($n$-RDMs) play a central role in understanding correlated phases of matter. Yet the calculation of $n$-RDMs is often computationally inefficient for strongly-correlated states, particularly when the system sizes are large. In this work, we propose to use neural network (NN) architectures to accelerate the calculation of, and even predict, the $n$-RDMs for large-size systems. The underlying intuition is that $n$-RDMs are often smooth functions over the Brillouin zone (BZ) (certainly true for gapped states) and are thus interpolable, allowing NNs trained on small-size $n$-RDMs to predict large-size ones. Building on this intuition, we devise two NNs: (i) a self-attention NN that maps random RDMs to physical ones, and (ii) a Sinusoidal Representation Network (SIREN) that directly maps momentum-space coordinates to RDM values. We test the NNs in three 2D models: the pair-pair correlation functions of the Richardson model of superconductivity, the translationally-invariant 1-RDM in a four-band model with short-range repulsion, and the translation-breaking 1-RDM in the half-filled Hubbard model. We find that a SIREN trained on a $6\times 6$ momentum mesh can predict the $18\times 18$ pair-pair correlation function with a relative accuracy of $0.839$. The NNs trained on $6\times 6 \sim 8\times 8$ meshes can provide high-quality initial guesses for $50\times 50$ translation-invariant Hartree-Fock (HF) and $30\times 30$ fully translation-breaking-allowed HF, reducing the number of iterations required for convergence by up to $91.63\%$ and $92.78\%$, respectively, compared to random initializations. Our results illustrate the potential of using NN-based methods for interpolable $n$-RDMs, which might open a new avenue for future research on strongly correlated phases.

Probing the Critical Point (CritPt) of AI Reasoning: a Frontier Physics Research Benchmark

While large language models (LLMs) with reasoning capabilities are progressing rapidly on high-school math competitions and coding, can they reason effectively through complex, open-ended challenges found in frontier physics research? And crucially, what kinds of reasoning tasks do physicists want LLMs to assist with? To address these questions, we present the CritPt (Complex Research using Integrated Thinking - Physics Test, pronounced "critical point"), the first benchmark designed to test LLMs on unpublished, research-level reasoning tasks that broadly covers modern physics research areas, including condensed matter, quantum physics, atomic, molecular & optical physics, astrophysics, high energy physics, mathematical physics, statistical physics, nuclear physics, nonlinear dynamics, fluid dynamics and biophysics. CritPt consists of 71 composite research challenges designed to simulate full-scale research projects at the entry level, which are also decomposed to 190 simpler checkpoint tasks for more fine-grained insights. All problems are newly created by 50+ active physics researchers based on their own research. Every problem is hand-curated to admit a guess-resistant and machine-verifiable answer and is evaluated by an automated grading pipeline heavily customized for advanced physics-specific output formats. We find that while current state-of-the-art LLMs show early promise on isolated checkpoints, they remain far from being able to reliably solve full research-scale challenges: the best average accuracy among base models is only 4.0% , achieved by GPT-5 (high), moderately rising to around 10% when equipped with coding tools. Through the realistic yet standardized evaluation offered by CritPt, we highlight a large disconnect between current model capabilities and realistic physics research demands, offering a foundation to guide the development of scientifically grounded AI tools.