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arXiv:2605.18625

Emergent Thiemann coherent states in the near-kernel sector of quantum reduced loop gravity

Ilkka Mäkinen, Hanno Sahlmann, Waleed Sherif

Published May 18, 2026• 0 citations
arXivHTMLPDF

Abstract

We study the near-kernel sector of the Hamiltonian constraint operator in the one-vertex model of quantum reduced loop gravity using variational Monte Carlo methods with neural quantum states. The analysis is based on the symmetric Hamiltonian containing both Euclidean and Lorentzian contributions, and on the variational minimization of the positive quadratic operator $\hat{\mathcal Q}=\hat C \hat C^\dagger$ in truncated Hilbert spaces with spin cutoff up to $j_{\mathrm{max}}=1001$. The resulting near-kernel states are found to organize into three qualitatively distinct classes. At low cutoffs, we find solutions that do not factorize across the three edge degrees of freedom. At larger cutoffs, we find two different factorized branches, both described to very high accuracy by products of one-edge wavefunctions but localized in different spin regimes. One of these branches is matched with near-unit fidelity by reduced Thiemann coherent states, providing evidence for an emergent semiclassical organization of the near-kernel sector. The other is likewise strongly factorized, but its one-edge factors are not well described by the same coherent-state family.

DOI
10.48550/arxiv.2605.18625

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Neural network quantum states (NQS) have been widely applied to spin-1/2 systems, where they have proven to be highly effective. The application to systems with larger on-site dimension, such as spin-1 or bosonic systems, has been explored less and predominantly using spin-1/2 Restricted Boltzmann Machines (RBMs) with a one-hot/unary encoding. Here, we propose a more direct generalization of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labeling freedom for the visible variables and gauge equivalence to the tensor network formulation. To test this new approach, we present variational Monte Carlo (VMC) calculations for the spin-1 anti-ferromagnetic Heisenberg (AFH) model and benchmark it against the one-hot/unary encoded RBM demonstrating that it achieves the same accuracy with substantially fewer variational parameters. Furthermore, we investigate how the hidden unit complexity of NQS depend on the local single-spin basis used. Exploiting the tensor network version of our RBM we construct an analytic NQS representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state in the xyz spin-1 basis using only M=2N hidden units, compared to M∼O(N2) required in the Sz basis. Additional VMC calculations provide strong evidence that the AKLT state in fact possesses an exact compact NQS representation in the xyz basis with only M=N hidden units. These insights help to further unravel how to most effectively adapt the NQS framework for more complex quantum systems.