Isometric Tensor Networks Compute Excited States in Two-Dimensions with Block-isoPEPS

Understanding the excited states of complex materials presents a significant challenge in modern physics, yet these states dictate many crucial material properties. Alec Dektor, Runze Chi, and Roel Van Beeumen, alongside Chao Yang from Lawrence Berkeley National Laboratory and the California Institute of Technology, now offer a new approach to calculating these excited states in two-dimensional systems. Their work introduces a novel method based on isometric tensor networks, extending techniques previously successful in one-dimensional materials to a more complex, two-dimensional realm. This advancement provides a scalable framework for studying excitations, promising to unlock deeper insights into the behaviour of materials beyond the limitations of existing computational methods and paving the way for the design of materials with tailored properties.

Projected Entangled Pair States and isoPEPS Methods

Scientists explore methods for determining the excited states of quantum many-body systems, focusing on Projected Entangled Pair States (PEPS) and Isometric PEPS (isoPEPS). PEPS represents a way to describe the ground state of quantum systems in two dimensions, extending the concept of Matrix Product States, while isoPEPS builds upon this by enforcing orthogonality between network components, simplifying calculations and improving efficiency. Researchers employ a tangent space method, which finds excited states by exploring variations around a known ground state, streamlined by the inherent orthogonality of isoPEPS. The team demonstrates that isoPEPS offers significant advantages over standard PEPS in calculating excited states, avoiding spurious solutions and reducing computational demands due to its stable, orthogonal basis. This work provides a detailed comparison of these techniques, outlining the mathematical framework and computational costs involved.

Block-isoPEPS for Two-Dimensional Quantum Excited States

Scientists developed a new computational method for determining the excited states of complex quantum systems interacting on two-dimensional lattices. This work pioneers a subspace iteration method built upon a block isometric projected entangled pair state, or block-isoPEPS, ansatz, extending the block matrix product state framework from one to two dimensions for a more scalable approach. The core of this method involves constructing a tensor network with specific connectivity designed to capture entanglement in two dimensions. Researchers engineered the block-isoPEPS ansatz to incorporate isometric constraints on the tensors within the network, enabling exact block orthogonalization and controlled local truncation via singular value decompositions, significantly improving computational efficiency. The team implemented a procedure where tensors are systematically updated within the subspace iteration, refining the approximation of excited states, and demonstrated the efficacy of this approach by computing excitations of the two-dimensional transverse-field Ising and Heisenberg models.

Block-isoPEPS Computes Two-Dimensional Quantum Excitations

Scientists have developed a new block-isoPEPS tensor network ansatz for computing multiple eigenpairs, ground and excited states, of quantum many-body systems in two dimensions. This work generalizes the state-averaged matrix product state (MPS) framework to extend its capabilities to two-dimensional geometries, building upon isometric constraints within the tensor network for efficient computation by simplifying contractions of large network regions. This approach retains desirable properties for handling nearest-neighbor interactions in two dimensions, offering a scalable framework for studying quantum systems beyond one dimension. The team demonstrated the method by computing excitations of the two-dimensional transverse-field Ising and Heisenberg models, highlighting attractive features like exact block orthogonalization and controlled local truncation achieved through singular value decompositions, contributing to computational efficiency. The computational cost of many algorithms based on this approach scales favorably, offering a significant improvement over generic PEPS algorithms, and provides a pathway to compute several algebraically smallest eigenpairs, addressing a gap in existing isoPEPS algorithms focused on finding only the ground state.

Two-Dimensional Excited States via Tensor Networks

This work presents a new method for calculating the low-lying excited states of quantum systems in two dimensions, extending techniques previously used for one-dimensional systems. Researchers developed a block isometric tensor network approach, building upon existing methods like matrix product states, to represent the complex interactions within these systems, utilizing a novel tensor network structure for accurate calculations and efficient evaluation of key properties. Demonstrating the method’s effectiveness, the team successfully computed excited states for the transverse-field Ising and Heisenberg models, comparing results with established techniques, and indicating that this block isometric tensor network scales favorably with system size, accurately capturing low-energy excited states across various parameter settings. Importantly, the method proves capable of analyzing systems where conventional approaches, based on the ground state, fall short. Future research will explore extending this approach to three dimensions and improving computational efficiency through parallel processing and GPU acceleration, promising to broaden the applicability and power of this new technique for studying complex quantum systems.