Neural Gutzwiller-projected variational wave functions

Abstract

Variational wave functions have enabled exceptional scientific breakthroughs related to the understanding of novel phases of matter. Examples include the Bardeen-Cooper-Schrieffer theory of superconductivity, the description of the fractional quantum Hall effect through the Laughlin state, and Feynman's variational understanding of large-scale quantum effects in liquid Helium. More recently, Gutzwiller-projected wave functions, typically constructed from fermionic degrees of freedom, have been employed to examine quantum spin models in the presence of competing interactions, where exotic phases with no spontaneous symmetry breaking and fractional excitations may exist. In this work, we investigate the aforementioned fermionic wave functions supplemented with neural networks, specifically with the so-called restricted Boltzmann machine (RBM), to boost their accuracy and obtain reliable approximations to the ground state of generic spin models. In particular, we apply our neural augmented fermionic construction to the description of both magnetically ordered and disordered phases of increasing complexity, including cases where the ground state displays a non-trivial sign structure. Even though the RBM state is by far more effective for Néel states endowed with a particularly simple sign structure, it provides a significant improvement over the original fermionic state in highly frustrated regimes where a complex sign structure is anticipated, thus marking the path to an understanding of strongly-correlated spin models on the lattice via neural Gutzwiller-projected variational wave functions.

Fermionic neural-network states for ab-initio electronic structure

“…(4). One conceptual interest of NQS is that, because of the flexibility of the underlying non-linear parameterization, they can be adopted to study both equilibrium 24,25 and out-ofequilibrium [26][27][28][29][30][31] properties of diverse many-body quantum systems. In this work, we adopt a simple neural-network parameterization in terms of a complex-valued, shallow restricted Boltzmann machine (RBM) 10,32 .…”

Machine learning for quantum matter

“…Originally invented by Paul Smolensky [125] and popularized by Geoffrey Hinton [126], this architecture has been recently repurposed as a representation of quantum states [116,123]. In this context, the RBM has been used to approximate the ground state of prototypical systems in condensed matter physics such as the transverse field Ising and Heisenberg models in one and two dimensions [123,127,128], the Hubbard model [128], models of frustrated magnetism [127], the Bose-Hubbard model [129,130], ground states of molecules [131], to model spectral properties of many-body systems [132], as well as to study nonequilibrium properties of quantum systems [123,133]. While the simulation of real-time dynamics remains a challenge, Ref.…”

Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines

“…However, we consider here models whose ground-state wave function can be assumed to be real and nonnegative in a suitable basis; therefore, the RBM parameters can be restricted to have real parameters. Extensions to complex-valued ground-states for, e.g., fermionic systems, have recently been addressed [36,58,59]. In Ref.…”