Disentangling Tensor Network States with Deep Neural Network

TL;DR

The paper introduces Neural Tensor Network States ($ u$TNS), a novel approach combining neural networks and tensor networks to efficiently represent and analyze strongly correlated quantum states, achieving state-of-the-art results.

Contribution

It presents $ u$TNS, a new variational ansatz integrating neural networks with tensor networks, enabling compact and expressive modeling of complex quantum states.

Findings

Achieved state-of-the-art energies for the $J_1$-$J_2$ Heisenberg model.

Found evidence of a gapless quantum spin-liquid ground state.

Demonstrated scalability to large system sizes (up to 20x20).

Abstract

We introduce Neural Tensor Network States ($\nu$TNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the $\nu$TNS framework, a neural network serves as a disentangler of the wave-function, transforming the physical degrees of freedom into renormalized variables with much less entanglement. The renormalized state is then efficiently encoded by a back-flow tensor network. This construction yields a compact yet highly expressive representation of strongly correlated quantum states. Using convolutional neural networks combined with matrix product states as a concrete implementation, we obtain state-of-the-art variational energies for the spin-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice at the highly frustrated point $J_2/J_1=0.5$, for systems up to $20\times 20$ with periodic boundary conditions. Finite-size scaling of spin, dimer, and plaquette correlations exhibits power-law decay without magnetic or valence-bond long-range order, consistent with a gapless quantum spin-liquid ground state at that point.This $\nu$TNS framework is flexible and naturally extensible to other neural and tensor-network structures, offering a general platform for investigating strongly correlated quantum many-body systems.