A new numerical approach to strongly correlated quantum physics in 2D.

Quantum physics in two dimensions is technologically relevant and fundamentally interesting; it is also difficult - available methods are generally limited to small system sizes or unrealistic approximations. The main problem is the number of degrees of freedom one must consider, which grows exponentially with system size. Methods for solving anything but the most trivial problems must select which of these degrees of freedom really matter and discard the rest.
The technique known as the 'density matrix renormalisation group' (DMRG) has revolutionised numerical studies of quantum systems by selecting the essential degrees of freedom in a remarkably efficient manner. DMRG has proven to be a highly accurate and robust tool for calculating properties of materials that are quasi-1D: systems that can be described as a one dimensional lattice or 'chain' of sites. Unfortunately this method stumbles in two spatial dimensions and above - it quickly becomes inefficient as system size grows.
A result from quantum information theory, known as the 'area law' shows the failure of conventional 2D DMRG is linked to the enhanced growth of quantum entanglement above 1D. Too much entanglement between different subregions of the system causes DMRG to grind to a halt. The area law states that entanglement scales with the boundary or 'area' between two subregions. In 1D the boundary can only be one or two points, independent of the total system size. In 2D the area will in fact scale like the perimeter of a subregion. In other words, it will increase linearly as the system gets bigger, until the DMRG approach is too inefficient to be useful.

Effective numerical techniques are vital because analytic methods based on simple approximations fail for the most interesting problems, where quantum fluctuations are dominant, while more sophisticated exact approaches available in 1D do not have analogues in higher dimensions.

By considering the anisotropic 2D case of a coupled array of exactly solvable chains we can minimise the relevant boundary area, thus minimising the entanglement problem. Combining the properties of the exactly solvable subunits with the power of DMRG, leads to an algorithm that can efficiently perform larger scale simulations than competing techniques.
Using an anisotropic representation does not prohibit us from the applying the results to isotropic systems as we are generally interested in 'universal' quantities that are independent of such microscopic details.
I intend to develop this algorithm beyond its proof-of-concept implementation into a general tool for studying the properties of two dimensional quantum systems, including their quantum information content.
In so doing, I will apply it to an important benchmark problem, relevant to the cuprate high temperature superconductors (materials that conduct electricity without resistance). I will also take advantage of the underlying 'matrix product state' structure of the technique to extend it to the emerging field of out-of-equilibrium quantum problems, where a system is 'quenched' by suddenly changing one of its properties (for example the strength of interactions).

Quantum systems that are effectively two dimensional exist in nature - either as thin films or as important subunits of otherwise 3D materials - making them experimentally and technologically relevant. However theoretically it is very difficult to understand their properties. The primary impact of this proposal is then the production of knowledge, to benefit the physics, quantum information and nanotechnology communities in understanding fundamental physical concepts. In time, our understanding of these concepts will shape the development of future materials and their applications.

The knowledge takes two forms: firstly knowledge of the properties of particular 2D quantum systems, discerned by application of the algorithm. For example, I intend to explore the nature of the mysterious 'pseudogap' phase of cuprate high temperature superconductors, by applying the algorithm to the 2D Hubbard model near half filling. This also includes more general knowledge of the behaviour of quantum entanglement measures in many-body systems.

Secondly, developing the method constitutes knowledge in itself that will allow it to be extended to new situations, such as non-equilibrium 2D systems, and provide clues to increasingly effective numerical treatments of problems in 2D and 3D. This impact also lies in producing a general algorithm, into which one can 'drop' a suitable 2D model. This will allow non-specialists to take advantage of its power in their own investigations.
The algorithmic work is expected to increase the role of the UK in the density matrix renormalisation group and matrix product state communities, within physics.

The results for specific 2D models will be of benefit to scientists (both theoretical and experimental workers) and quantum chemists in the UK and internationally. Strong efforts will be made to foster international collaboration in order to maximise the impact of the work beyond the UK.

By having an early role in this science, the UK has a better chance of participating in the development of related numerical methods and their resulting applications. Because these methods are based on both mathematical and computational techniques of some sophistication, their use by young scientists at the graduate and undergraduate level will provide training in analytical and computational skills.

The participation of Dr. Robert Konik as a visiting researcher (estimated by Dr. Konik as an effective contribution of $12500 per year by Brookhaven National Laboratory) will provide impact via new collaborative connections between UK and US research, particularly with the experimental and theory groups at his host institution. It will also allow for access to powerful computing clusters at BNL including the Bluegene supercomputing facility.