Alternating and Gaussian fermionic Isometric Tensor Network States

TL;DR

This paper introduces an improved alternating isometric tensor network state (isoTNS) for 2D quantum systems and a Gaussian fermionic extension, demonstrating superior entanglement mediation and ground state approximation capabilities.

Contribution

The paper presents an alternating isoTNS variant and a Gaussian fermionic tensor network framework, enhancing the representation of quantum many-body states and their entanglement structure.

Findings

Alternating isoTNS better captures ground states than original isoTNS.

Alternating isoTNS mediates entanglement more efficiently.

Numerical results show improved energy and bond-dimension scaling for fermionic models.

Abstract

Isometric tensor networks in two dimensions enable efficient and accurate study of quantum many-body states, yet the effect of the isometric restriction on the represented quantum states is not fully understood. We address this question in two main contributions. First, we introduce an improved variant of isometric network states (isoTNS) in two dimensions, where the isometric arrows on the columns of the network alternate between pointing upward and downward, hence the name alternating isometric tensor network states. Second, we introduce a numerical tool -- isometric Gaussian fermionic TNS (isoGfTNS) -- that incorporates isometric constraints into the framework of Gaussian fermionic tensor network states. We demonstrate in numerous ways that alternating isoTNS represent many-body ground states of two-dimensional quantum systems significantly better than the original isoTNS. First, we show that the entanglement in an isoTNS is mediated along the isometric arrows and that alternating isoTNS mediate entanglement more efficiently than conventional isoTNS. Second, alternating isoTNS correspond to a deeper, thus more representative, sequential circuit construction of depth $O(L_x \cdot L_y)$ compared to the original isoTNS of depth $O(L_x + L_y)$. Third, using the Gaussian framework and gradient-based energy minimization, we provide numerical evidences of better bond-dimension scaling and variational energy of alternating isoGfTNS for ground states of various free fermionic models, including the Fermi surface, the band insulator, and the $p_x + ip_y$ mean-field superconductor. Finally, we find improved performance of alternating isoTNS as compared to the original isoTNS for the ground state energy of the (interacting) transverse field Ising model.