Tensor networks are a recently developed formulation for quantum systems which enables major advances in both the conceptual understanding and the simulation of these systems. The first successful tensor network and associated algorithm, the density matrix renormalization group (DMRG) was invented by group leader Steven White in 1992. DMRG has become very widely used to study the simplified model Hamiltonians used to describe the high-temperature superconductors, quantum spin liquids and other strongly correlated systems. Group member Garnet Chan has pioneered the use of these methods in quantum chemistry to understand strongly correlated molecules. However, only recently was it realized that DMRG was just the first of a broad family of tensor networks, each with different strengths and that tensor networks are deeply connected to the field of quantum information, which studies the ways to possibly build a quantum computer and to program it. Since this connection was first made by group member Guifre Vidal around the year 2000, these fields have become intertwined, with important ramifications from string theory to quantum chemistry. Indeed, tensor networks have broadened from being an extremely useful computational algorithm to one of the hottest fields in theoretical physics.

The tensor network group aims to push the development of tensor networks and DMRG to the next level. One key focus is in developing the networks and algorithms to study two dimensional systems in the most efficient way. The most famous model Hamiltonian, the Hubbard model, is used to describe high-temperature superconducting cuprates. This model cannot be solved exactly, and solving it approximately has been the focus of hundreds of physicists in the last 20 years, with only partial success. Using both novel tensor networks optimized for 2D, and more traditional DMRG approaches, and comparing with the results from the other groups in the collaboration, we plan to advance the Hubbard simulations to the point where the phase diagram of the model is reliably determined. These techniques will also be applied to related models, such as the frustrated Heisenberg model, which are used to describe quantum spin liquids. The eventual goal is to be able to solve reliably any simple quantum lattice model in 2D.

As the simulation techniques have improved, it has also become clear that the models themselves are oversimplified and need to be improved. The existing models may capture the essence of the problem, but they certainly leave out all the chemical details that would distinguish, say, a hypothetical room temperature superconductor from one which must be cooled using liquid nitrogen. Improving these models is a challenging problem, for which progress in the past has been slow. However, tensor network techniques, along with the canonical transformation approach developed by White and Chan for quantum chemistry, offer a new approach to improving the models. The group will use these techniques, along with the 2D tensor network methods, to study the copper-oxygen plane common to all cuprates, in chemically realistic detail.

Tensor networks algorithms require sophisticated software libraries, which are only now being developed. Two of the leading tensor libraries were developed by group members. The group will work to improve the efficiency, user-friendliness and interoperability of these libraries, and work to make them widely available to other groups using tensor networks.