Tensor Network Theory for strongly correlated quantum systems

Physical systems that display strong correlations as a result of interactions between their constituents are present everywhere around us in our daily lives. For example traffic jams form on our roads every day in the morning due to strong interactions between cars that do not allow two of them to occupy the same piece of road. However, ants marching in a line never form such traffic jams despite facing very similar restrictions of not being allowed to sit on top of each other. These two examples demonstrate how subtle differences in the precise microscopic nature of interactions may lead to qualitatively different macroscopically observed properties and this poses major challenges for their theoretical study. In the quantum case strong interactions lead to some of the least well understood phenomena of condensed matter like high-Tc superconductivity, frustration, and topological phases such fractional quantum Hall physics which only appear in materials with a dominant two dimensional character.
An amazing feature of these systems is that one can readily write down simple looking models which are believed to capture the main physics on a macroscopic level. However, because of the strong interactions even these simple models turn out to be very hard to solve. Despite almost four decades of research in this area a detailed theoretical understanding of macroscopic properties in thermodynamic equilibrium emerging from strong interactions is still lacking for many of these seemingly simple models. In addition recent experimental progress now allows for the dynamical study of driven strongly correlated quantum systems far away from equilibrium and this poses new opportunities for applications in quantum enhanced devices as well as new challenges for theoretical physics research.
In this project we will develop high performance software which will enable tackling basic questions about models for strongly correlated systems in the quantum and also in the classical case. The underlying so-called tensor network algorithms have been developed over the past two decades but it is only now that a unified framework for these algorithms is known. This justifies the development of high performance computer software which will encompass existing and well-tested algorithms but is also sufficiently versatile to form the basis for future developments in this field of research. Indeed, the software developed in this project will be available to researchers throughout the UK and form the backbone of numerical studies based on tensor network algorithms for the next decade and possibly beyond.
Developing this powerful new tool for enhancing simulation methods will enable such things as the optimisation of quantum enhanced effects in promising new generations of technology. Improved numerical algorithms will enable sensors as well as energy transfer and storage devices to utilise physically enhanced processes at the scale where dynamical quantum effects are crucial. In particular the software will be required to study strongly correlated models without being hindered by boundary effects or minus-sign problems inherent in some other methods. The insights gained from this research could lead to novel superconducting materials or the exploitation of dynamical non-equilibrium properties in applications of nano-materials. Furthermore these may also be applicable to everyday classical strongly interacting systems like e.g. the formation of traffic jams or the dynamics of queues forming at box offices or in order books at the stock exchange.

The proposed work will change the accessibility and applicability of many well established algorithms for studying strongly correlated physical systems and open the way to the development and application of novel tensor network algorithms. This will benefit not only researchers in the fields of condensed matter physics and cold atom physics, but also those who have no previous experience with TNT. Additionally, it will benefit researchers in the field of numerical analysis by providing optimised general functionality for manipulating tensors.

The impact to these different groups of beneficiaries will be achieved through access to the three tiers of the software library:

(1) The tier I software library will provide basic routines for tensor manipulation that can be used by developers of tensor network algorithms and researchers, which we would usually expect to be theorists, who require non-standard usage of well-known tensor network algorithms. These routines will also be useful for commercial and academic users who wish to work with tensor manipulation in any general context.

(2) The tier II software library will provide the building blocks for tensor network algorithms using the most common network types. It will allow customised, yet highly optimised, algorithms to be written by the user with the minimum of effort, allowing simplified testing of new tensor network algorithms by researchers.

(3) The tier III software library will contain ready-to-use implementations of the most common and already widely used tensor network algorithms. It will make these algorithms accessible to a large group of users, and they will be able to use software of unprecedented power which will significantly enlarge the class of physical models that can be studied with them. This tier will contain algorithms that can be directly applied to standard strongly correlated quantum models and also routines for application to classical stochastic systems.