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Study of the Two-Dimensional Frustrated J1-J2 Model with Neural Network Quantum States

Kenny Choo, Titus Neupert, Giuseppe Carleo

arXiv:1903.06713v1cond-mat.str-elcond-mat.dis-nnquant-ph

TL;DR

The paper asks whether neural-network quantum states can address the unresolved two-dimensional frustrated J1-J2 model, where existing methods give conflicting conclusions. It uses a fully convolutional neural network as a variational ansatz and finds competitive, often improved energies and properties, with weaker performance near maximal frustration. The study also identifies substantial room for improvement because its networks are much smaller than modern deep-learning models.

  • Problem

    The J1-J2 model is an unsolved frustrated magnet whose possible spin-liquid phase near maximal frustration has drawn conflicting computational conclusions.

  • Method

    The paper represents the wavefunction with a fully convolutional neural network and optimizes the variational state using stochastic reconfiguration.

  • Results

    CNN predictions for ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods, except in a small maximally frustrated region.

  • Takeaways & Limitations

    The results provide numerical evidence that neural-network quantum states are competitive variational ansätze for challenging frustrated magnets.

  • Takeaways & Limitations

    Performance is comparatively weaker near maximal frustration, where finite-size effects and imposed sign structures make accurate results difficult.

Abstract

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The use of artificial neural networks to represent quantum wave-functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parameterizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ansatz, we study the frustrated spin-1/2 J1-J2 Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ansatz exhibits comparatively good but not best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.

I. INTRODUCTION

The paper tests neural-network quantum states on the unresolved, frustrated two-dimensional J1-J2 model, where established methods have produced conflicting conclusions. A convolutional neural network is used as a variational ansatz and benchmarked against exact and state-of-the-art many-body calculations.

  • I. INTRODUCTION: The J1-J2 model is an open challenge because competing computational methods have produced conflicting conclusions about a possible spin liquid near maximal frustration.The relevant regime is around J2/J1 ≈ 0.5.
  • I. INTRODUCTION: The study uses a feedforward convolutional neural network as a variational wavefunction for the two-dimensional frustrated magnet.The network architecture has six convolutional layers followed by an output layer that sums the preceding values.
  • I. INTRODUCTION: CNN variational energies improve upon other techniques at several phase-diagram points, while remaining competitive but not cutting-edge in a small maximally frustrated window.This identifies both strong performance across much of the parameter space and a localized weakness.

II. MODEL

The paper studies the antiferromagnetic spin-1/2 J1-J2 Heisenberg model on a periodic square lattice, focusing on its highly frustrated intermediate regime. The ground-state character there remains unresolved, so the model serves as a demanding benchmark for variational methods.

  • II. MODEL: The spin-1/2 J1-J2 Heisenberg model contains nearest- and next-nearest-neighbor interactions on a square lattice with periodic boundary conditions.Both couplings are antiferromagnetic, and J1 is fixed to 1.
  • II. MODEL: The analysis is restricted to the zero-magnetization sector because total magnetization is conserved and the ground state is expected there.This restriction applies to the spin configurations used in the variational analysis.
  • II. MODEL: At J2/J1 ≈ 0.5, frustration is maximal and candidate ground states include plaquette valence-bond, columnar valence-bond, and gapless spin-liquid phases.The correct ground state in this intermediate regime remains unknown.
  • II. MODEL: The authors make no claim about the frustrated regime’s physical phase and instead present the neural-network construction as a variational ansatz.They compare its variational energies with DMRG and projected-fermion VMC.

III. NEURAL NETWORK QUANTUM STATES

Neural network quantum states represent complex wavefunction amplitudes with neural networks; this work uses a fully convolutional architecture. Its spatially local, translationally invariant structure keeps the parameter count fixed across system sizes.

  • III. NEURAL NETWORK QUANTUM STATES: Neural network quantum states interpret a neural network’s output as the complex amplitude Ψ(σ) of each spin configuration.The paper uses a feedforward CNN rather than the earlier RBM ansatz.
  • III. NEURAL NETWORK QUANTUM STATES: The network uses complex-valued weights, biases, and nonlinearities, including complex ReLU and logarithmic hyperbolic cosine activations.The first layer uses glncosh(z) = log[cosh(z)], while later layers use complex ReLU.
  • III. NEURAL NETWORK QUANTUM STATES: The CNN uses spatially local convolutional layers with multiple channels and filters to transform the spin-configuration input.The final layer sums outputs across channels, implementing average pooling.
  • III. NEURAL NETWORK QUANTUM STATES: The full CNN contains 3838 complex-valued parameters independent of system size.This parameter sharing follows from the convolutional architecture.
  • III. NEURAL NETWORK QUANTUM STATES: The architecture represents an explicitly translationally invariant, zero-momentum function over Hilbert-space configurations.The CNN maps computational-basis configurations σ to complex values.
  • III. NEURAL NETWORK QUANTUM STATES: The same network structure and parameter count is used for every system size studied.The fully convolutional design also allows parameters learned on smaller systems to initialize larger-system optimization.

Sign structure of the ground state

The variational ansatz incorporates sign conventions known exactly in the unfrustrated limits, then optimizes the resulting complex CNN state. Choosing the subset defining the sign convention introduces bias and can make optimization difficult when the convention is inappropriate.

  • Sign structure of the ground state: In the limits J1 = 0 or J2 = 0, the ground-state wavefunction follows a simple sign rule with nonnegative amplitude Ψ(σ).The sign is determined by the parity of up spins on a selected subset A.
  • Sign structure of the ground state: For J2 = 0, subset A is one bipartite component of the square lattice, whereas for J1 = 0 it can be every other row or column.These choices yield the corresponding limiting sign conventions.
  • Sign structure of the ground state: The known sign conventions can be encoded in variational parameters, but the authors initialize the ansatz with a limiting convention before optimization.Both limiting conventions are tested and the one with lower variational energy is selected.
  • Sign structure of the ground state: The chosen subset A biases the ansatz, and optimization becomes extremely challenging when the imposed sign structure is inappropriate.The complex CNN can in principle change the sign structure, but the initial convention still affects optimization.

Enforcing C4 Symmetry

The CNN is translationally invariant but not explicitly C4 symmetric, so the wavefunction is symmetrized to enforce the lattice's fourfold rotational symmetry. This ensures correlation functions have the correct spatial symmetry, particularly in the large-J2 striped phase.

  • Enforcing C4 Symmetry: The CNN is explicitly translationally invariant but does not automatically transform within a C4 irreducible representation.The square lattice's fourfold rotation group is Abelian, with one-dimensional irreducible representations.
  • Enforcing C4 Symmetry: The wavefunction is symmetrized using the generator of the C4 group and an associated character.
  • Enforcing C4 Symmetry: C4 symmetrization ensures that correlation functions have the correct spatial symmetry.This is especially important in the striped-order phase at large J2, where the initial sign structure is not rotationally invariant.

IV. VARIATIONAL MONTE CARLO OPTIMISATION

The variational energy is estimated with Monte Carlo sampling, while stochastic reconfiguration updates neural-network parameters through an imaginary-time-evolution procedure. Regularization addresses ill conditioning in the resulting linear system.

  • IV. VARIATIONAL MONTE CARLO OPTIMISATION: Stochastic reconfiguration optimizes the variational parameters and can be interpreted as imaginary time evolution.
  • IV. VARIATIONAL MONTE CARLO OPTIMISATION: Small parameter changes are represented through logarithmic derivatives of the wavefunction before applying the stochastic-reconfiguration update.
  • IV. VARIATIONAL MONTE CARLO OPTIMISATION: The updated parameters minimize the distance to the imaginary-time-evolved wavefunction under the Fubini–Study metric.
  • IV. VARIATIONAL MONTE CARLO OPTIMISATION: The stochastic-reconfiguration linear system has complexity O(Nw^2), compared with O(Nw) for stochastic gradient descent.The iterative conjugate-gradients algorithm is used, and stochastic reconfiguration is reported to perform better for small to midsized networks.
  • IV. VARIATIONAL MONTE CARLO OPTIMISATION: Expectation values are estimated by Monte Carlo samples drawn from a probability distribution proportional to |Ψ(σ)|^2.The Metropolis algorithm generates samples, sparse Hamiltonian matrix elements make averages efficient, and identity regularization alleviates ill conditioning.

V. RESULTS AND DISCUSSION

The study applies the CNN variational ansatz with stochastic reconfiguration to 6 × 6 and 10 × 10 square lattices with periodic boundary conditions.

  • V. RESULTS AND DISCUSSION: The reported calculations use the variational ansatz together with stochastic reconfiguration on 6 × 6 and 10 × 10 square lattices.Both systems have periodic boundary conditions.

A. Comparison with ED

On the 6 × 6 lattice, the CNN reproduces exact energies and spin properties across most of the parameter range, with its largest errors near maximal frustration. The total-spin behavior indicates substantial overlap with the singlet sector.

  • A. Comparison with ED: Relative energy errors are of order 10^-3 or lower across most of parameter space, but the largest error occurs at J2 = 0.55 near maximal frustration.The authors associate this difficulty with strong violation of the prior sign structure, which biases the simulation.
  • A. Comparison with ED: The 10 × 10 comparison includes CNN energies, DMRG results, a Gutzwiller-projected fermionic ansatz, and exact Green's-function quantum Monte Carlo at J2 = 0.
  • A. Comparison with ED: The spin-spin structure factors agree accurately with exact values except in the maximally frustrated transition region.For J2 ≲ 0.5 the relevant order is Néel with q = (π,π), while for J2 ≳ 0.5 it is stripe order with q = (π,0) or (0,π).
  • A. Comparison with ED: The total-spin spike coincides with the relative-error spike, while the expectation value remains much less than the next allowed eigenvalue, 2.This supports good overlap between the variational wavefunction and the singlet sector.

B. Benchmarking with state of the art methods

On the 10×10 cluster, the CNN ansatz produces variational energies competitive with or better than established methods across most of the parameter space, with weaker performance near maximal frustration.

  • The benchmark compares CNN energies with density matrix renormalization group and Gutzwiller-projected mean-field fermionic variational results.
  • The CNN ansatz yields variational energies very close to or better than existing state-of-the-art results.The authors identify residual sign-structure optimization and symmetry violations as possible sources of systematic error near the least favorable point.
  • At J2/J1 = 0.55, performance is non-optimal, coinciding with the region where prior sign structures are most strongly violated.
  • On a 10×10 cluster with J2 = 0.5, the paper reports energy per site −0.4952, compared with −0.4736 from a previous CNN ansatz.

C. Discussion

The discussion presents the neural-network states as competitive for frustrated magnets while identifying finite-size effects, model scale, and sign-structure learning as unresolved constraints.

  • The results provide numerical evidence that neural-network quantum states are competitive variational ansätze for challenging frustrated-magnet problems.
  • Because the system is gapless, finite-size effects are large and accurate extrapolations to the thermodynamic limit are necessary.
  • A more computationally demanding simulation campaign is required for a firm finite-size extrapolation of the reported magnetic correlations.
  • The networks are much smaller in depth and trainable-parameter count than state-of-the-art models used in modern deep learning applications.
  • Learning the correct sign structure challenges the stochastic-reconfiguration optimization used with the adopted ansatz.

APPENDIX: VARIATIONAL ENERGIES

The appendix presents exact variational energies from this work and a comparison table spanning CNN, DMRG, fermionic VMC, and exact diagonalization benchmarks.

  • The appendix reports exact values, including error bars, for the variational energies obtained in this work.
  • Table I compares the CNN with Gutzwiller-projected mean-field fermionic VMC and DMRG results on the 10×10 case.
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