Leg-Tied Tensor Network States: Entanglement Beyond Virtual Bonds

Standard Matrix Product States (MPS) are highly effective for one-dimensional quantum systems but struggle to represent long-range correlations in two- and three-dimensional lattice models. The authors seek to overcome this limitation by developing a more expressive tensor network ansatz that can explicitly encode long-range dependencies without requiring the prohibitively large virtual bond dimensions that would otherwise be necessary in a standard MPS.

To address this, the authors introduce Leg-Tied Tensor Network States (LETTA). LETTA retains the linear virtual backbone of an MPS, which enables efficient, deterministic variational optimization using density matrix renormalization group (DMRG)-like algorithms. However, it extends this structure by adding explicit physical ties—shared physical legs—between local tensors. These ties allow the state to encode long-range correlations directly. The authors develop a deterministic variational optimization algorithm for LETTA, including gauge-fixing and symmetry-preserving techniques, and test the approach on the 2D frustrated J1–J2 Heisenberg model and the 3D transverse-field Ising model.

LETTA demonstrates superior performance compared to standard MPS across the tested models. For the 6x6 J1–J2 Heisenberg model, LETTA with a bond dimension of D=4 outperforms an MPS with D=32, using only 7% of the variational parameters. Similarly, in 3D transverse-field Ising clusters, LETTA consistently achieves lower ground-state energies than MPS at the same bond dimension. The authors show that LETTA is particularly effective in regimes where entanglement entropy rises rapidly, as the physical ties provide additional channels for cross-cut entanglement that standard virtual bonds cannot capture efficiently.

This work provides a powerful, compact alternative to traditional tensor network methods for higher-dimensional quantum systems. By explicitly incorporating long-range correlations into the tensor network structure, LETTA offers a path toward high-accuracy simulations of strongly correlated quantum matter, including potential applications in quantum chemistry and disordered systems, where traditional MPS approaches often fail to reach the necessary precision.