# Nonlinear Network Description for Many-Body Quantum Systems in Continuous Space.

@article{Ruggeri2018NonlinearND,
title={Nonlinear Network Description for Many-Body Quantum Systems in Continuous Space.},
author={Michele Ruggeri and Saverio Moroni and Markus Holzmann},
journal={Physical review letters},
year={2018},
volume={120 20},
pages={
205302
}
}
• Published 6 November 2017
• Physics
• Physical review letters
We show that the recently introduced iterative backflow wave function can be interpreted as a general neural network in continuum space with nonlinear functions in the hidden units. Using this wave function in variational Monte Carlo simulations of liquid ^{4}He in two and three dimensions, we typically find a tenfold increase in accuracy over currently used wave functions. Furthermore, subsequent stages of the iteration procedure define a set of increasingly good wave functions, each with its…
20 Citations

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## References

SHOWING 1-10 OF 68 REFERENCES

### Iterative backflow renormalization procedure for many-body ground-state wave functions of strongly interacting normal Fermi liquids

• Physics
• 2015
We show how a ground-state trial wave function of a Fermi liquid can be systematically improved by introducing a sequence of renormalized coordinates through an iterative backflow transformation. We

### Solving the quantum many-body problem with artificial neural networks

• Computer Science, Physics
Science
• 2017
A variational representation of quantum states based on artificial neural networks with a variable number of hidden neurons and a reinforcement-learning scheme that is capable of both finding the ground state and describing the unitary time evolution of complex interacting quantum systems.

### Unitary Dynamics of Strongly Interacting Bose Gases with the Time-Dependent Variational Monte Carlo Method in Continuous Space

• Physics
• 2017
We introduce the time-dependent variational Monte Carlo method for continuous-space Bose gases. Our approach is based on the systematic expansion of the many-body wave function in terms of multibody

### Continuous Matrix Product States for Quantum Fields: An Energy Minimization Algorithm.

• Physics
Physical review letters
• 2017
A cMPS optimization algorithm based instead on energy minimization by gradient methods is proposed and demonstrated by applying it to the Lieb-Liniger model (an integrable model of an interacting bosonic field) directly in the thermodynamic limit.

### Many-body wavefunctions for normal liquid He3

• Physics
• 2006
We present new trial wave functions which include three-body correlations into the backflow coordinates and a four-body symmetric potential. We show that our wave functions lower the energy enough to

### Variational theory of bulk 4 He with shadow wave functions: Ground state and the phonon-maxon-roton spectrum

• Physics
• 1998
We apply an efficient optimization scheme to shadow wave functions (SWF's) for the ground state of liquid and solid ${}^{4}\mathrm{He}$. Results improve on previous variational energies in both

### Monte Carlo studies of two-dimensional phases of helium using a shadow wave function

• Physics
• 2000
The solid and liquid phases of two-dimensional ${}^{4}\mathrm{He}$ are studied in detail using a variational shadow'' wave function. This wave function provides a unified description of both

### Restricted Boltzmann machine learning for solving strongly correlated quantum systems

• Physics, Computer Science
• 2017
The combined method substantially improves the accuracy beyond that ever achieved by each method separately, in the Heisenberg as well as Hubbard models on square lattices, thus proving its power as a highly accurate quantum many-body solver.

### Exact ground state Monte Carlo method for Bosons without importance sampling.

• Physics
The Journal of chemical physics
• 2009
Zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo and a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.

### Energy Spectrum of the Excitations in Liquid Helium

• Physics
• 1956
A wave function previously used to represent an excitation (phonon or roton) in liquid helium, inserted into a variational principle for the energy, gave an energy-momentum curve having the