Tensor network methods for quantum system simulations
Understanding strongly correlated quantum many-body systems is necessary to study high-temperature superconductivity or topological phase transitions and spin liquids. For this purpose, efficient numerical approaches are highly desired. The development of such new approaches is, however, challenging. The dimension of the Hilbert space of the quantum many-body systems increases exponentially with the number of particles. This limits not only the applicability of diagonalization methods but also quantum Monte Carlo methods. One of the most powerful numeric approaches that have emerged in the last decade was explored within the EU-funded SQSNP (Simulating quantum systems numerically and physically) project. Techniques based on tensor networks offer an accurate representation of the entanglement structure of quantum many-body states. Scientists improved the efficiency of existing simulation techniques based on tensor networks. The major breakthrough of SQSNP was the successful application of the tensor network formalism to quantum Monte Carlo and series expansion approaches. Specifically, an unbiased Monte Carlo technique was introduced based on sampling all possible renormalizations of tensor networks. By simultaneously sampling multiple degrees of freedom associated with each bond of the tensor network, scientists achieved extremely low levels of statistical fluctuations. The new technique, called tensor network Monte Carlo, is quite general and can be combined with a wide variety of tensor renormalization techniques. It naturally applies to quantum systems and could be used to approach other problems in physics and beyond. The formalism developed also offered insights into how to apply tensor networks in different contexts. For instance, series expansions are often used to describe interacting systems, but scientists understood, for the first time, how to fully utilise and exploit this idea in the tensor network formalism. Lastly, scientists used tensor network solvers to simplify conventional series expansion. They worked on ways to represent the exact state of the systems as tensor network and numerical approximations to the series expansion. This line of research is expected to be further explored after the end of the SQSNP project.
Keywords
Tensor network, quantum many-body system, Monte Carlo methods, SQSNP, series expansion
