Released by
Lei Wang, Institute of Physics, Chinese Academy of Sciences
Contact email
wangleiphy@gmail.com
Method
Variational Monte Carlo / Neural Quantum States
Challenge issue
Build a symmetric neural-network ansatz for the chiral graviton: compute E(L=2) − E(L=0) at ν = 1/3
A graviton you can actually compute
The graviton — the spin-2 quantum of the gravitational field — has never been seen by any particle-physics experiment. Yet a remarkably close cousin lives inside a much humbler system: a two-dimensional sheet of electrons in a strong magnetic field, in the fractional quantum Hall (FQH) regime. There, the electrons lock into an incompressible quantum liquid (the Laughlin state at filling ν = 1/3), and its lowest-energy collective vibration in the long-wavelength limit is a spin-2 excitation with a definite chirality (helicity −2 at ν = 1/3) — a graviton-like mode that is the quantum of fluctuations of an emergent internal geometry (Liou et al. 2019). This challenge is to compute its energy from first principles with a neural-network wavefunction.
Why is there a graviton in a puddle of electrons?
A FQH liquid is not just a density of charge — it carries a hidden geometric degree of freedom, an internal "metric" that sets the shape of the correlations binding the electrons. Distorting that metric costs energy, and the elementary quantum of that distortion is a spin-2 object: exactly the quantum number of a graviton. Because the system is chiral (the magnetic field picks a direction of circulation), the mode has spin magnitude 2 and a definite handedness — a helicity of −2 at ν = 1/3 (the sign is what "chiral" means here, and it flips at other fillings such as ν = 2/3). This is emergent gravity on a tabletop: the same spin-2 mathematics as a gravitational wave, arising from nothing but electrons, Coulomb repulsion, and a magnetic field. The mode is the long-wavelength limit of the FQH magnetoroton, the well-known neutral collective excitation of these liquids.
Why the sphere, and why L = 2
Put the N electrons on the surface of a sphere threaded by magnetic flux (the Haldane sphere) and the problem becomes beautifully clean. The sphere has the full rotational symmetry SO(3) and no edges, so every state carries an exact total angular momentum L — the same quantum number that labels atomic orbitals. The ground state is a unique L = 0 singlet. The graviton is then simply the lowest L = 2 state, because L = 2 is spin-2. (Total angular momentum L is unsigned — it measures the spin-2 magnitude; the ± handedness is not in L, and is recovered separately below.) The headline number is the neutral gap
Δ = E(L = 2) − E(L = 0) (energies in e²/εℓ_B),
for ν = 1/3 (flux 2Q = 3(N−1)) under the chord-distance Coulomb interaction. And there is a built-in correctness check: a genuine spin-2 mode must be a 5-fold (2L+1) degenerate multiplet. If your method produces five degenerate L = 2 states, you have caught a graviton.
The core of the challenge: a symmetric neural-network ansatz
Getting Δ to research quality hinges on a neural-network quantum state that is both expressive enough to capture FQH correlations and respects the two symmetries that make the problem well-posed:
- Fermionic antisymmetry under particle exchange (the electrons are identical fermions).
- SO(3) rotational equivariance on the sphere. This is the load-bearing requirement: only an equivariant network keeps total angular momentum L a good quantum number. A non-equivariant ansatz silently picks a preferred axis, mixes L sectors, and the L = 2 graviton can no longer be cleanly separated from the L = 0 ground state and the L = 1 dipole mode.
A successful design lets a single trained network represent the L = 0 ground state and the L = 2 excitation together (e.g. via state-averaged / excited-state VMC, or by carrying an explicit L-tower through one symmetric map), so the gap is a difference of energies in the same variational family — where the large common correlation energy cancels and the neutral gap comes out far more accurately than either total energy.
Goal
- Construct an expressive, exchange-antisymmetric, SO(3)-equivariant NQS ansatz for N electrons in the lowest Landau level on the sphere at ν = 1/3.
- Compute the L = 0 ground-state energy and the lowest L = 2 energy; report Δ in e²/εℓ_B with statistical error bars.
- Certify spin-2: verify ⟨L²⟩ = 6 on the excited state and that the L = 2 level is a 5-fold degenerate multiplet — the signature that the mode is genuinely the spin-2, helicity −2 graviton (Liou et al. 2019).
Why this leads to research output
Symmetric neural quantum states are the most promising route past the exponential wall of exact diagonalization. The open, publishable questions are exactly the ones a good ansatz unlocks:
- Reach system sizes beyond ED, and extrapolate Δ to the thermodynamic limit.
- Resolve the chirality. L is unsigned, so the L = 2 multiplet contains both the bright (helicity +2) and dark (helicity −2) modes. Decompose it with the chiral metric operator s⁺₂ and identify the gapped helicity, matching the predicted −2 chirality of the ν = 1/3 graviton (Liou et al. 2019).
Acceptance / deliverables
- A documented symmetric NQS ansatz (antisymmetric + SO(3)-equivariant) with an equivariance check (rotate a sample, confirm the expected transformation).
- Δ = E(L=2) − E(L=0) for ν = 1/3 at one or more N, with ⟨L²⟩ = 6 and the 5-fold degeneracy demonstrated.
- A short report: ansatz design, system sizes, MC error bars, and — for the strong version — an N → ∞ extrapolation and/or the bright/dark chirality decomposition.
- Code submitted as a PR against the harness. Small-N exact diagonalization is a tractable, L-resolved cross-check for validating the ansatz.
Background & references
- S.-F. Liou, F. D. M. Haldane, K. Yang, E. H. Rezayi, Chiral Gravitons in Fractional Quantum Hall Liquids, Phys. Rev. Lett. 123, 146801 (2019), arXiv:1904.12231.
Released by
Lei Wang, Institute of Physics, Chinese Academy of Sciences
Contact email
wangleiphy@gmail.com
Method
Variational Monte Carlo / Neural Quantum States
Challenge issue
Build a symmetric neural-network ansatz for the chiral graviton: compute E(L=2) − E(L=0) at ν = 1/3
A graviton you can actually compute
The graviton — the spin-2 quantum of the gravitational field — has never been seen by any particle-physics experiment. Yet a remarkably close cousin lives inside a much humbler system: a two-dimensional sheet of electrons in a strong magnetic field, in the fractional quantum Hall (FQH) regime. There, the electrons lock into an incompressible quantum liquid (the Laughlin state at filling ν = 1/3), and its lowest-energy collective vibration in the long-wavelength limit is a spin-2 excitation with a definite chirality (helicity −2 at ν = 1/3) — a graviton-like mode that is the quantum of fluctuations of an emergent internal geometry (Liou et al. 2019). This challenge is to compute its energy from first principles with a neural-network wavefunction.
Why is there a graviton in a puddle of electrons?
A FQH liquid is not just a density of charge — it carries a hidden geometric degree of freedom, an internal "metric" that sets the shape of the correlations binding the electrons. Distorting that metric costs energy, and the elementary quantum of that distortion is a spin-2 object: exactly the quantum number of a graviton. Because the system is chiral (the magnetic field picks a direction of circulation), the mode has spin magnitude 2 and a definite handedness — a helicity of −2 at ν = 1/3 (the sign is what "chiral" means here, and it flips at other fillings such as ν = 2/3). This is emergent gravity on a tabletop: the same spin-2 mathematics as a gravitational wave, arising from nothing but electrons, Coulomb repulsion, and a magnetic field. The mode is the long-wavelength limit of the FQH magnetoroton, the well-known neutral collective excitation of these liquids.
Why the sphere, and why L = 2
Put the N electrons on the surface of a sphere threaded by magnetic flux (the Haldane sphere) and the problem becomes beautifully clean. The sphere has the full rotational symmetry SO(3) and no edges, so every state carries an exact total angular momentum L — the same quantum number that labels atomic orbitals. The ground state is a unique L = 0 singlet. The graviton is then simply the lowest L = 2 state, because L = 2 is spin-2. (Total angular momentum L is unsigned — it measures the spin-2 magnitude; the ± handedness is not in L, and is recovered separately below.) The headline number is the neutral gap
Δ = E(L = 2) − E(L = 0) (energies in e²/εℓ_B),
for ν = 1/3 (flux 2Q = 3(N−1)) under the chord-distance Coulomb interaction. And there is a built-in correctness check: a genuine spin-2 mode must be a 5-fold (2L+1) degenerate multiplet. If your method produces five degenerate L = 2 states, you have caught a graviton.
The core of the challenge: a symmetric neural-network ansatz
Getting Δ to research quality hinges on a neural-network quantum state that is both expressive enough to capture FQH correlations and respects the two symmetries that make the problem well-posed:
A successful design lets a single trained network represent the L = 0 ground state and the L = 2 excitation together (e.g. via state-averaged / excited-state VMC, or by carrying an explicit L-tower through one symmetric map), so the gap is a difference of energies in the same variational family — where the large common correlation energy cancels and the neutral gap comes out far more accurately than either total energy.
Goal
Why this leads to research output
Symmetric neural quantum states are the most promising route past the exponential wall of exact diagonalization. The open, publishable questions are exactly the ones a good ansatz unlocks:
Acceptance / deliverables
Background & references