Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states

First-quantized deep neural network techniques are developed for analyzing strongly coupled fermionic systems on the lattice. Using a Slater-Jastrow inspired ansatz which exploits deep residual networks with convolutional residual blocks, we approximately determine the ground state of spinless fermions on a square lattice with nearest-neighbor interactions. The flexibility of the neural-network ansatz results in a high level of accuracy when compared to exact diagonalization results on small systems, both for energy and correlation functions. On large systems, we obtain accurate estimates of the boundaries between metallic and charge ordered phases as a function of the interaction strength and the particle density.

[1]  D. Ceperley,et al.  Monte Carlo simulation of a many-fermion study , 1977 .

[2]  R. Sugar,et al.  Monte Carlo study of a two-dimensional spin-polarized fermion lattice gas , 1984 .

[3]  Toussaint,et al.  Two-dimensional spin-polarized fermion lattice gases. , 1985, Physical review. B, Condensed matter.

[4]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[5]  M. Fannes,et al.  Finitely correlated states on quantum spin chains , 1992 .

[6]  White,et al.  Density-matrix algorithms for quantum renormalization groups. , 1993, Physical review. B, Condensed matter.

[7]  Shun-ichi Amari,et al.  Natural Gradient Works Efficiently in Learning , 1998, Neural Computation.

[8]  Ross H. McKenzie,et al.  Charge ordering and antiferromagnetic exchange in layered molecular crystals of the theta type , 2001 .

[9]  Thierry Giamarchi,et al.  Quantum physics in one dimension , 2004 .

[10]  Randall G. Hulet,et al.  Pairing and Phase Separation in a Polarized Fermi Gas , 2006, Science.

[11]  Frank Verstraete,et al.  Matrix product state representations , 2006, Quantum Inf. Comput..

[12]  D. Rocca,et al.  Weak binding between two aromatic rings: feeling the van der Waals attraction by quantum Monte Carlo methods. , 2007, The Journal of chemical physics.

[13]  M. Hastings,et al.  An area law for one-dimensional quantum systems , 2007, 0705.2024.

[14]  Adriana Moreo,et al.  Cold attractive spin polarized Fermi lattice gases and the doped positive U Hubbard model. , 2007, Physical review letters.

[15]  I. I. Mazin,et al.  Fermi surface nesting and the origin of charge density waves in metals , 2007, 0708.1744.

[16]  W. R. Czart,et al.  Charge Ordering and Phase Separations in the Spinless Fermion Model, with Repulsive Intersite Interaction , 2008 .

[17]  Edwin Langmann,et al.  Partially Gapped Fermions in 2D , 2009, 0907.1277.

[18]  Bela Bauer,et al.  Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states , 2009, 0912.0646.

[19]  R. T. Clay,et al.  Monte Carlo simulations of two-dimensional fermion systems with string-bond states , 2014 .

[20]  Matthias Troyer,et al.  Fermionic quantum critical point of spinless fermions on a honeycomb lattice , 2014, 1407.0029.

[21]  Ying-Jer Kao,et al.  Variational Monte Carlo simulations using tensor-product projected states , 2014, 1407.4107.

[22]  M. Bukov,et al.  QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains , 2016, 1610.03042.

[23]  F. Becca.  Quantum Monte Carlo Approaches for Correlated Systems , 2017 .

[24]  Andrew S. Darmawan,et al.  Restricted Boltzmann machine learning for solving strongly correlated quantum systems , 2017, 1709.06475.

[25]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[26]  Kenny Choo,et al.  Two-dimensional frustrated J1−J2 model studied with neural network quantum states , 2019, Physical Review B.

[27]  J. Carrasquilla,et al.  Neural Gutzwiller-projected variational wave functions , 2019, Physical Review B.

[28]  David Pfau,et al.  Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks , 2019, Physical Review Research.

[29]  M. Bukov,et al.  QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. Part II: bosons, fermions and higher spins , 2018, SciPost Physics.

[30]  Satoshi Morita,et al.  mVMC - Open-source software for many-variable variational Monte Carlo method , 2017, Comput. Phys. Commun..

[31]  Naftali Tishby,et al.  Machine learning and the physical sciences , 2019, Reviews of Modern Physics.

[32]  Bryan K. Clark,et al.  Backflow Transformations via Neural Networks for Quantum Many-Body Wave Functions. , 2018, Physical review letters.

[33]  F. Verstraete,et al.  Restricted Boltzmann Machines for Quantum States with Non-Abelian or Anyonic Symmetries. , 2019, Physical review letters.

[34]  G. Carleo,et al.  Fermionic neural-network states for ab-initio electronic structure , 2019, Nature Communications.

[35]  J. Stokes,et al.  Quantum Natural Gradient , 2019, Quantum.

[36]  F. Noé,et al.  Deep-neural-network solution of the electronic Schrödinger equation , 2019, Nature Chemistry.

[37]  Nikita Astrakhantsev,et al.  Generalization properties of neural network approximations to frustrated magnet ground states , 2020, Nature Communications.