Visualization of Entanglement Geometry by Structural Optimization of Tree Tensor Network

TL;DR

This work tackles visualizing entanglement geometry in quantum many-body states by optimizing the structure of tree tensor networks (TTNs) under a least-entanglement-entropy principle. The authors apply the TTN structural optimization to the Rainbow chain, a model whose ground state consists of distant spin-singlet pairs, and demonstrate that the optimized TTN accurately reproduces the singlet-pair distribution and associated entanglement patterns. The key contribution is showing that the TTN not only reduces truncation error but also serves as a diagnostic tool for entanglement geometry, including long-range singlet structures, with potential extensions to random-singlet-like states. They also discuss limitations arising from tiny excitation gaps and propose improvements via higher-precision numerics or tensor-network renormalization-group approaches, along with prospects for a revised high-rank tensor representation algorithm.

Abstract

In tensor-network analysis of quantum many-body systems, it is of crucial importance to employ a tensor network with a spatial structure suitable for representing the state of interest. In the previous work [Hikihara et al., Phys. Rev. Research 5, 013031 (2023)], we proposed a structural optimization algorithm for tree-tensor networks. In this paper, we apply the algorithm to the Rainbow-chain model, which has a product state of singlet pairs between spins separated by various distances as an approximate ground state. We then demonstrate that the algorithm can successfully visualize the spatial pattern of spin-singlet pairs in the ground state.

Figures (5)

• Figure 1: Example of tree-tensor network. The open circles denote tensors and the grey circles denote bare spins. The red lines represent auxiliary bonds connecting tensors, while the block lines represent physical bonds connecting a tensor and a bare spin. If one cuts the auxiliary bond indicated by a blue cross, the system is separated into two subsystems, A and B.
• Figure 2: (a) The part to be updated in the TTN. (b) The ground-state vector $\Psi$. (c) Three candidates of local network structure to be adopted after the singular-value decomposition of $\Psi$.
• Figure 3: (a) Exchange constants in the Rainbow chain with $N=10$. (b) Singlet-pair distribution in the ground state of the Rainbow chain. Solid lines represent the spin-singlet pairs.
• Figure 4: Matrix-product network used as an initial network in the calculation. The open circles denote the tensors and the grey circles denote the bare spins. The black lines represent physical bonds directly connected to the bare spins. The red lines represent auxiliary bonds.
• Figure 5: Optimal TTN obtained by the structural optimization algorithm for the Rainbow chain with $\alpha=0.1$ and $N=10$. The open circles denote the tensors and the grey circles denote the bare spins. In (a), the spins are arranged by the site number. In (b), the spins are arranged so that two adjacent spins in the panel form the singlet pairs. The numbers presented at the thin red lines represent the EE on the corresponding bond. The thick black lines represent the physical bonds with EE of $\ln 2$.