Disentangling Tensor Network States with Deep Neural Network

Abstract

We introduce Neural Tensor Network States ($ν$TNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the $ν$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 $ν$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.

Figures (4)

• Figure 1: Graphical illustration of the $\nu$TNS ansatz with MPS as a concrete realization of TNS. The physical configuration is first mapped into feature space by a linear embedding, yielding the initial feature vector $X^{0}$. This vector is then processed through $\ell$ neural-network layers to produce the renormalized feature vector $X^{\ell}$, which is finally contracted with a backflow MPS to generate the $\nu$MPS wave function.
• Figure 2: Convergence of the ground state energy with the optimization step for the $6\times 6$$J_1-J_2$ Heisenberg model with periodic boundary conditions at $J_2/J_1=0.5$, obtained using the pure CNN and the CNN-MPS ansatz with $D=5$ and 20. The dotted horizontal line marks the exact result schulzMagneticOrderDisorder1996. Inset: relative energy error versus $D$.
• Figure 3: Optimization of the ground-state energy for the $L=20$$J_1-J_2$ Heisenberg model with the CNN-MPS ansatz. The $C_{4v}$ lattice symmetry is enforced after the first $5{,}000$ steps. Also shown for comparison are the ViT results reported in Ref. viteritti2026approachingthermodynamiclimitneuralnetwork, where ViT-base and ViT-full refer to calculations without and with full point-group symmetry, respectively.
• Figure 4: Log-log plot of the finite-size scaling of the order parameters $m_s^2(\pi,\pi)$, $m_p^2(\pi,0)$, and $m_d^2(\pi,0)$ for the $J_1-J_2$ Heisenberg model at $J_2/J_1=0.5$, obtained using the CNN-MPS ansatz with $(h, D, \ell)=(32, 15, 20)$.