Source-linked AI summary
Backflow Transformations via Neural Networks for Quantum Many-Body Wave-Functions
Di Luo, Bryan K. Clark
TL;DR
Accurate ground-state wave functions remain challenging in quantum many-body systems. The paper develops neural-network backflow and evaluates it across Hubbard and Kagome-lattice systems, reporting improved energies and reduced relative errors.
Problem
Ground-state wave functions remain challenging to obtain accurately, including for the Kagome-lattice Heisenberg model.
Method
Neural-network backflow uses feed-forward network transformations with forward and backward propagation to modify a mean-field wave function.
Results
NNB improves energies and relative errors across tested systems, including approximately 50% relative-error improvement on the Kagome lattice.
Takeaways & Limitations
NNB provides a systematically configurable wave-function approach that improves agreement with exact or approximate reference results in the tested systems.
Abstract
from arXiv · showhide
Obtaining an accurate ground state wave function is one of the great challenges in the quantum many-body problem. In this paper, we propose a new class of wave functions, neural network backflow (NNB). The backflow approach, pioneered originally by Feynman, adds correlation to a mean-field ground state by transforming the single-particle orbitals in a configuration-dependent way. NNB uses a feed-forward neural network to find the optimal transformation. NNB directly dresses a mean-field state, can be systematically improved and directly alters the sign structure of the wave-function. It generalizes the standard backflow which we show how to explicitly represent as a NNB. We benchmark the NNB on a Hubbard model at intermediate doping finding that it significantly decreases the relative error, restores the symmetry of both observables and single-particle orbitals, and decreases the double-occupancy density. Finally, we illustrate interesting patterns in the weights and bias of the optimized neural network.
Supplementary Materials
The supplementary materials describe a direct standard-backflow representation within NNB and compare ΨSN, ΨSN-b, and ΨPN using energy-error, variance-extrapolation, and doublon-density plots.
- Figure 6 compares percentage relative energy error against 1/m for ΨSN, ΨSN-b, and ΨPN.
- Figure 7 extrapolates percentage relative energy error against variance, while Figure 8 plots doublon density against 1/m.
- Standard backflow is implemented in a Slater-determinant mean field as ΨSN-b.
- ΨSN-b can also be represented by ΨSN through an appropriate definition of aNN.
- ΨSN and ΨSN-b have similar performance for energy and doublon-density calculations.
Optimization Scheme
The optimization scheme evaluates variational-energy derivatives through Monte Carlo sampling and updates parameters using randomized gradient-sign steps, with Slater-determinant initialization available for larger systems.
- The energy is evaluated as the normalized expectation value E = ⟨ψ|Hψ⟩/⟨ψ|ψ⟩.
- Derivatives with respect to variational parameters are computed using the local energy and Monte Carlo sampling.
- Parameters are updated using the gradient sign multiplied by a random magnitude and step size.
- NNB can represent an optimized Slater determinant by setting output-layer biases to the orbitals and other parameters to zero.
- Initializing the output-layer biases with optimized orbitals and other parameters with small random values helps optimization in large systems.
Backward propagation of neural network backflow
The backward-propagation scheme applies the chain rule through the neural network and adapts the final-layer error to differentiate Slater-determinant and Bogoliubov de Gennes backflow wave functions.
- Backward propagation of neural network backflow: Backward propagation computes parameter derivatives by propagating errors through successive network layers using the chain rule.
- Backward propagation of neural network backflow: For each final-layer output, layerwise error vectors are defined and propagated backward from the final layer.
- Backward propagation of neural network backflow: The neural-network backflow mapping is treated as a special layer on top of the final network layer, requiring a different final-layer error.
- Backward propagation of neural network backflow: The method first applies this construction to Slater-determinant neural-network backflow and then addresses Bogoliubov de Gennes backflow wave functions.
- Backward propagation of neural network backflow: For Slater-determinant backflow, derivatives pass through the determinant and the neural-network transformation before standard backward propagation.
Neural Network Backflow on various systems
NNB was optimized across Hubbard and Kagome systems spanning lattice sizes, fillings, and hidden-neuron counts. It improved energies relative to mean-field references, with variance extrapolation yielding especially strong results in larger Hubbard systems.
- NNB was optimized for 4 × 4 Hubbard models at n = 0.75 and n = 1.0, larger Hubbard lattices at n = 0.875, and a 4 × 4 × 3 Kagome Heisenberg model.All Hubbard benchmarks used U/t = 8; the Kagome benchmark used J = 1.
- 4 × 4 Hubbard: 0.631% (0.233%) was the 4 × 4 Hubbard error (variance extrapolated error) at n = 0.75, while n = 1.0 gave 2.714% (1.745%).These results were compared with exact diagonalization.
- Larger Hubbard systems: 1.534 energy units separated the optimized Slater determinant from the nh = 64 NNB on the 16 × 4 Hubbard system, decreasing energy from -46.211 to -47.745.Variance extrapolation gave approximately -49, close to the twist-averaged AFQMC result of -49.088.
- Larger Hubbard systems: 0.655% was the 12 × 8 Hubbard relative error after variance extrapolation, compared with 6.3% for the optimized Slater determinant and 3.94% for nh = 32 NNB.The authors report that NNB significantly improves on standard Slater-Jastrow methodology and is competitive with other techniques after extrapolation.
- Kagome Heisenberg model: Approximately 50% improvement in relative error was obtained for the Kagome Heisenberg model as hidden neurons increased from nh = 8 to nh = 256.The associated energies ranged from −0.4311 to −0.4339 per site, compared with an exact answer of −0.4387.