Attention is all you need to solve chiral superconductivity
Authors
Chun-Tse Li, Tzen Ong, Max Geier, Hsin Lin, Liang Fu
Abstract
Recent advances on neural quantum states have shown that correlations between quantum particles can be efficiently captured by {\it attention} -- a foundation of modern neural architectures that enables neural networks to learn the relation between objects. In this work, we show that a general-purpose self-attention Fermi neural network is able to find chiral $p_x \pm i p_y$ superconductivity in an attractive Fermi gas by energy minimization, {\it without prior knowledge or bias towards pairing}. The superconducting state is identified from the optimized wavefunction by measuring various physical observables: the pair binding energy, the total angular momentum of the ground state, and off-diagonal long-range order in the two-body reduced density matrix. Our work paves the way for AI-driven discovery of unconventional and topological superconductivity in strongly correlated quantum materials.
Concepts
The Big Picture
Finding a new quantum phase of matter is hard when you don’t know what you’re looking for. A group of physicists at MIT, Academia Sinica, and USC just pulled it off anyway, turning up one of the most exotic states in condensed matter: chiral superconductivity.
Superconductivity is already strange enough. Below certain temperatures, electrons in some materials pair up and flow without any electrical resistance. No energy wasted, no heat generated.
Chiral superconductivity is stranger still. The paired electrons carry angular momentum, rotating around each other like partners in a waltz. The state has a built-in handedness: run time backwards, and it looks different. Physicists call this breaking time-reversal symmetry.
Traditional methods for discovering such states either fail when particles interact too strongly, or require physicists to tell the algorithm what to look for before it starts. Neither is great when the whole point is to find something unexpected.
A team led by MIT physicists showed that a single general-purpose transformer neural network, the same architecture behind large language models, can find chiral p_x ± ip_y superconductivity entirely on its own. Starting from random weights and minimizing energy, the network found the state without any hints about pairing or superconducting order.
A self-attention neural network, given no information about pairing or superconductivity, independently discovers a topologically nontrivial chiral superconducting state.
How It Works
The researchers work with spin-polarized fermions in two dimensions, subject to an attractive Gaussian potential. The physics is clean and controlled, but solving it exactly is impossible: the number of quantum states grows exponentially with particle number. Brute-force approaches are hopeless for anything beyond toy systems.
Their solution uses self-attention, the core mechanism in transformers. In a language model, attention lets the network learn which words relate to which other words. Here, it learns which particles relate to which other particles, capturing the quantum correlations that define exotic phases.
The wavefunction is built on a FermiNet-style architecture that automatically satisfies the Pauli exclusion principle. It expresses the many-body state as a sum of generalized Slater determinants, mathematical structures that enforce antisymmetry by construction.
Particle coordinates enter as sine and cosine embeddings that respect the periodic boundary conditions of the simulation box. These “particle tokens” pass through multiple layers of multi-head self-attention and MLP blocks, then get projected into many-body orbital matrices whose determinants give the final wavefunction.
Training uses variational Monte Carlo (VMC): many random particle configurations are sampled, and parameters are nudged to minimize expected energy. No pretraining, no warm-starting from BCS mean-field theory, no problem-specific modifications. The same architecture used for electron gases, atoms, and molecules is applied here unmodified.
Once converged, three independent diagnostics confirm the result is a chiral topological superconductor:
- Pair binding energy, defined as E(N) + E(N+2) − 2E(N+1), measures whether the system energetically prefers even or odd particle numbers. Conventional superconductors prefer even numbers (pairs). Topological chiral superconductors flip this: odd-N states are favored because an unpaired state sits below the Fermi level. The network reproduces exactly this reversed odd-even effect across a range of coupling strengths.
- Total angular momentum: chiral p_x ± ip_y pairing means the ground state carries net angular momentum. Measurements from the optimized wavefunction match the expected chiral quantum number.
- Off-diagonal long-range order (ODLRO): the smoking gun of superconductivity, visible in the two-body reduced density matrix. The wavefunction shows the characteristic long-range correlations of true superconducting order.
All three diagnostics agree across a broad range of interaction strengths, from weak coupling where BCS theory applies to strong coupling where it breaks down entirely. In the strong-coupling regime, the neural network goes beyond mean-field theory, picking up quantum fluctuation effects that perturbative methods miss.
Why It Matters
That reversed odd-even binding energy is the topological fingerprint. Topological superconductors are predicted to host Majorana zero modes, quasiparticles that are their own antiparticles and could be the basis for fault-tolerant quantum computing. Finding these states computationally, with no assumptions about pairing baked in, opens a route to systematic surveys of which materials might harbor them.
Previous neural quantum state studies of superconductivity required modifying the wavefunction ansatz to include pairing structure by hand. The self-attention approach needs none of that scaffolding.
The real prize is generality. High-temperature superconductors, topological insulators, quantum spin liquids: these are all governed by strongly correlated Hamiltonians that have resisted decades of effort. A solver that can work through this space unguided could reach territory that has been inaccessible.
The paper aims squarely at AI-driven discovery of unconventional superconductivity in real materials. Given what the network did here, that goal looks a good deal closer.
Starting from random weights and no pairing ansatz, a self-attention neural network discovers chiral p_x ± ip_y superconductivity on its own, showing that general-purpose neural architectures can identify topological quantum phases without human guidance.
IAIFI Research Highlights
Transformer self-attention, originally developed for language, turns out to be a natural framework for quantum correlations between electrons, connecting modern AI architecture directly to condensed matter physics.
A single general-purpose self-attention wavefunction architecture generalizes across diverse quantum phases, including topologically nontrivial superconductors, without any problem-specific modifications.
For the first time, chiral *p_x ± ip_y* superconductivity with its characteristic reversed odd-even binding energy signature has been obtained variationally from an unbiased neural network, going beyond BCS mean-field theory in the strong-coupling regime.
Future work targets strongly correlated electron materials where unconventional superconductivity remains poorly understood. The full paper is available at [arXiv:2509.03683](https://arxiv.org/abs/2509.03683).