Global Team Advances Quantum Physics with Generating Function for Projected Entangled Pair States

A team of international researchers has developed the Generating Function for Projected Entangled Pair States, a concept in quantum physics that addresses a common bottleneck in modern applications of projected entangled pair states. The function is used to compute low-energy excitations of a two-dimensional quantum many-body system efficiently. The team’s results for the spin-1/2 transverse-field Ising model and Heisenberg model on the square lattice show good agreement with known results. The study also explores the dynamical properties of the spin-1/2 J1-J2 model on the same lattice. The research offers potential for significant breakthroughs in understanding quantum systems.

What is the Generating Function for Projected Entangled Pair States?

The Generating Function for Projected Entangled Pair States is a concept in quantum physics that has been developed by a team of researchers from various universities across the globe. The team includes WeiLin Tu from Keio University, Laurens Vanderstraeten from Université Libre de Bruxelles, Norbert Schuch from University of Vienna, HyunYong Lee from Korea University, Naoki Kawashima from The University of Tokyo, and JiYao Chen from Sun Yatsen University.

The researchers have extended the generating function approach for tensor network diagrammatic summation, a scheme previously proposed in the context of matrix product states. This approach is used to solve a common bottleneck in modern applications of projected entangled pair states, particularly in computing low-energy excitations of a two-dimensional quantum many-body system. The excited state can be computed efficiently in the generating function formalism, which can further be used in evaluating the dynamical structure factor of the system.

The team’s benchmark results for the spin-1/2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. They then study the spin-1/2 J1-J2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. The study concludes with a discussion on generalizations to multi-particle excitations.

What is the Importance of Strongly Correlated Quantum Systems?

Strongly correlated quantum systems hold a central position in condensed-matter physics, often triggering exotic behavior at low temperatures. For low-dimensional systems where quantum effects are pronounced, the entanglement-based tensor network method is now widely recognized as an ideal tool for both analytical and numerical studies of these systems.

Beyond its immense success in exploring ground-state properties due to its structure that adheres to the area law, tensor network methods also enable the study of low-energy properties above the ground state, including dynamical correlations and entanglement dynamics. These properties are often directly measurable in spectroscopic experiments within condensed-matter physics, such as inelastic neutron-scattering experiments in quantum magnets, while out-of-equilibrium properties are accessible in quantum simulators.

However, in systems extending beyond one spatial dimension, efforts to fully understand low-energy excitations in quantum many-body systems through tensor networks are still in their early stages. Consequently, there is a high demand for an efficient and accurate method to compute low-lying excitations and related dynamical correlation functions in two-dimensional systems.

What are Projected Entangled-Pair States (PEPS)?

Over the past 20 years, projected entangled-pair states (PEPS) have become one of the cornerstones in the study of 2D quantum many-body systems. Along with a deeper understanding of its mathematical structure and improved numerical recipes, PEPS has been demonstrated to capture the ground states of many classes of 2D phases of matter.

These include non-chiral topological states, ordered quantum magnets, chiral quantum spin liquids, and various phases in fermionic systems. Beyond ground-state properties, the PEPS toolbox has been expanded to study excited states, time evolution, and finite-temperature properties. Although PEPS-based ground-state exploration has reached a level of maturity, research into excited states continues to be a vibrant area of development.

How are Excitations Studied in Quantum Systems?

One physically motivated way of studying excitations is to use the tensor-network generalization of the Feynman-Bijl ansatz or the single-mode approximation, which is a variational ansatz for an excited state in the tangent space of the ground-state tensor manifold. In this construction, the ground state is perturbed locally by introducing an impurity tensor to represent a local quasi-particle.

By making a momentum superposition of such a local perturbation, one obtains a natural representation of a low-energy excited state. This approach has been successfully applied to build excitations on top of ground states of matrix product state (MPS) form in one-dimensional and quasi-1D systems, where contractions can be performed exactly and efficiently.

What are the Challenges in Studying Excitations in 2D Quantum Systems?

For a 2D quantum many-body system in the thermodynamic limit, the construction of excitations leads to the sum of infinitely many copies of PEPS with infinite size, so that the computation of expectation values or optimization of the variational energy is not straightforward. This can still be achieved through summing a geometric series of channel operators built from the boundary-MPS approach.

However, this process is not straightforward and presents a significant challenge in the study of excitations in 2D quantum systems. Despite these challenges, the research into this area continues to be a vibrant field of study, with the potential for significant breakthroughs in our understanding of quantum systems.