Emerging correlations from strong driving: a tensor network projection variational Monte Carlo approach to 2D quantum lattice systems

Much of the technology we have is based on exploiting special materials like semiconductors. The next revolution is likely to emerge from so-called quantum materials. However, while their behaviour has the potential to be extremely useful, it is also complex to understand and control. Insights gained from this research will help determine the viability of controlling quantum materials with light and the possible exploitation of dynamical non-equilibrium properties in future nano-devices.

Controlling materials with light is interesting because it is well known that driven systems can exhibit behaviour not seen when stationary. There are two simple examples of this. The first is a so-called Kapitza pendulum. This is a normal pendulum whose pivot point undergoes vertical oscillations that are rapid but small in amplitude. What is striking about this pendulum is that the inverted position, normally unstable to gravity, is dynamically stabilised by the periodic driving. The second is a ball on a rotating saddle. The ball cannot be stably positioned at the inflection point when the saddle is stationary. However, if the saddle is rotated above some threshold angular velocity then the ball can be balanced in the time-averaged bowl swept out by the saddle. The same ideas apply to many-body systems like materials and it is becoming increasingly relevant to study their behaviour.

An important class of many-body systems are those that exhibit strong correlations due to interactions between their constituents. The everyday world is full of such systems. For example traffic jams form along roads due to a combination of many vehicles and a strong repulsion between them to avoid occupying the same piece of road. However, ants marching in a line never suffer from such traffic jams despite facing very similar restrictions because they don't overtake one another. These two examples demonstrate how subtle differences in the precise microscopic nature of interactions may lead to qualitatively different macroscopic properties. Describing such correlations poses major challenges for the theoretical study of interacting systems, and no more so than for the case of quantum systems. 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 as fractional quantum Hall physics. These effects only appear at low temperatures and typically in materials with a dominant two-dimensional character.

Since quantum materials exhibit functional properties there is a major research effort to stabilise and optimise them at higher temperatures for future technological applications. A recent approach to this is to periodically drive a many-body quantum system to "dynamically stabilise" macroscopic quantum effects beyond where they occur in equilibrium. The question is made even more compelling by spectacular advances in high-field THz generation technology. This allows selective driving of low-energy excitations of real solids, like vibrations, enabling a crystal lattice to be shaken, modulated or distorted in controlled ways. This has created an exciting interface between driven systems and many-body physics engaging a large body of researchers worldwide.

A crucial issue hampering the use of periodic driving in engineering materials is heating that might wash out the desired effects. This project examines this problem within the context of one of the most important model Hamiltonians, the Hubbard model, which captures the essential physics of strong correlations. Current numerical methods struggle to give a conclusive answer to this issue. A unique feature of this project will be the development of a combined Monte Carlo and tensor network approach potentially rich enough to accurately describe the dynamical behaviour of the driven Hubbard model. The resulting high performance software will be publically available.

Next-generation technologies exploiting so-called quantum materials for ultra-fast switches, memory and processing devices may involve integration with THz opto-electronics. This opens up new vistas of opportunities to probe, drive and control the functional behaviour of interacting quantum many-body systems with tailored laser pulses. It is against this overarching backdrop that this project has two major themes for potential long-term impact. The first is through the scientific output I will produce that is aimed at answering fundamental questions about the behaviour of an archetypal model of a quantum material when it is strongly driven. The second is through the novel methodologies I will develop in this project to answer those questions.

To fully realise this promise more advanced methods of simulating the dynamical behaviour of these systems is needed to guide development and test feasibility. As such the scientific findings of this work not only answer some important questions in this direction but also lay the foundations of a powerful numerical technique that can provide these much sought-after predictive capabilities to these problems. Long lasting impact in this growing field will be fostered by giving access to these methods via an easy to use online interface, www.tntgo.org, and by making all codes freely available within the tensor network theory library (TNT) project, http://ccpforge.cse.rl.ac.uk/gf/project/tntlibrary/. This enables both theorists and experimentalists to utilise the deliverables of this project to help realise these long-term technological advances. The workshop bolt on planned will also advertise and introduce the outputs of this project to influential individuals and key beneficiaries in the community.

Beyond this direct application I further anticipate that the inclusion of advanced stochastic methods into the TNT library will have far-reaching impact for its use in the field of Big Data analysis. Owing to the generality of tensors they can serve as the foundation for the development of new algorithms widely believed to provide the much sought after extension of standard linear algebra algorithms to multidimensional data analysis. Given the huge importance of complex classical systems in natural sciences, operations research and industry as yet unforeseen applications of hybrid TNT methods could emerge in the near future. The exploitation of such applications will be of immediate relevance to many scientists working in the UK and further afield. To aid this, industrial stakeholders like MathWorks, Wolfram and NAG will be invited to the workshop bolt on making them aware of developments and fostering on going contacts during and after this project. Finally the UK academic community and beyond will benefit from the build-up of a cohort of scientists engaged in the project with skills in developing high-quality HPC software. Further work on algorithms and numerical methods underpinning this project will carry benefits for other fields of Science, Engineering and Medicine wishing to engage HPC effectively.