Entanglement Measures, Twist Fields, and Partition Functions in Quantum Field Theory

Quantum Mechanics is the theory that describes physical phenomena at atomic scales. It defines a set of mathematical objects which characterize a physical system and specifies which mathematical operations on those objects need to be performed in order to extract information about the system. In quantum mechanics we often speak about "the state of a system" meaning its properties. Mathematically, a state is a vector with certain special properties. Similarly, an observable in quantum mechanics is any property that we can measure. Mathematically, observables are represented by matrices. The beauty of the theory is that once we have vectors and matrices, we can use standard techniques to perform computations (even if these computations can become extremely involved for complex physical systems).

At the heart of this research project lies a particular feature of quantum mechanics: it allows for the states of two different quantum systems to be entangled. This means that under certain circumstances it is possible to prepare say, two electrons in a state such that if we can measure a property of electron 1 we will automatically know the value of the same property for electron 2 without needing to perform a second, independent measurement. Entanglement is a genuine quantum phenomenon. It has no counterpart in classical mechanics (e.g. the sort of physics that describes planetary motion) and it has attracted much attention among scientists as it demonstrates in a striking way the "weird" quantum behaviour of nature at microscopic scales.

Following on from quantum mechanics, one of the greatest advancements in Physics in the 20th century has been the formulation of theories which can describe the physics of many body quantum systems. This is in essence the generalisation of quantum mechanics to the situation where we have hundreds (potentially infinitely many) elementary particles in interaction. Such highly complex systems are best described by a continuum version of quantum mechanics which also incorporates the principles of general relativity. These theories are known as quantum field theories (QFTs) and they have proven incredibly successful in describing the results of many experiments such as those performed at CERN. In this setting the state of the systems is described by a vector in a Hilbert space and the values of measurable quantities are related to expectation values of local operators acting on that space.

In this project we want to investigate the mathematical properties of various functions which given a quantum state of a many-body system, give us information about the amount of entanglement that can be stored in such a state. The functions in question are known as the entanglement entropy (EE) and the logarithmic negativity (LN) and they have been previously studied for particular kinds of quantum theories and also in the context of theoretical quantum computation and information theory. Most of the results hitherto known apply to an important subset of QFTs which are known as conformal field theories (CFTs) or critical QFTs. CFTs have many special features and many applications including to the description of emergent behaviours in many-body systems. Many-body critical systems display correlations at all length scales, meaning that small local changes to one part of the system quickly propagate to the whole system. In contrast, another family of QFTs are massive or gapped models where the correlation length is finite. Such models describe universal features of many-body systems near but not at criticality and have been less studied from the viewpoint of entanglement. Our project will contribute to filling this gap by computing measures of entanglement in massive QFTs and generalising these to systems in higher space dimensions. Along the way a new mathematical framework will be developed which is based on the use of a particular family of local fields and their correlation functions.

Knowledge: Given the theoretical nature of our research its greatest impact will be of the type broadly described as "knowledge" under EPSRC guidelines. The research will certainly lead to scientific advance as it will allow to tackle problems which are currently unresolved and for whose solutions known techniques need to be refined (e.g. computation of generic matrix elements of local fields) and some new techniques need to be devised (e.g. definition of twist fields in higher dimensions). Given the fundamental role twist fields play in the formulation of many key measures of entanglement, we envisage that the techniques we plan to develop will have a lasting impact in the scientific community as the study of entanglement moves towards new measures and more complex quantum field theories (QFTs) such as near-critical and/or higher dimensional ones. When we refer to the scientific community above we mean principally those researchers working most closely in the kind of problems we are interested in, that is, experts in low dimensional QFTs such as the integrable models and conformal field theory (CFT) communities. However, we also hope that other research communities may benefit from our research in this area. A community where this is likely to happen is that of researchers currently exploiting CFT techniques and the AdS/CFT correspondence to test expressions of the entanglement entropy (EE) in anti-de-Sitter space. At present, the research is based on employing the principle of holography to link the EE of AdS3 to the EE of two-dimensional CFT. Research in that field is now also moving towards establishing a similar duality for other measures of entanglement, such as the logarithmic negativity (LN). A problem naturally addressed in that community is the study of entanglement measures in higher dimensions. However, there are only few known results on the CFT side in higher dimensions. New results based on the use of line twist fields will help bridge that gap and could open up a new area of research in the AdS/CFT community. In recent years, research in String Theory has uncovered a deep connection between many aspects of the AdS/CFT correspondence and integrable models so that the research we plan to undertake may come to play unexpected roles in this area. For example, recent work by B. Basso, A. Sever and P. Vieira, Phys. Rev. Lett. 113, 261604 (2014) which studies particular limits of scattering amplitudes in N=4 super Yang-Mills theory (the paradigm CFT in the AdS/CFT correspondence) has identified a key type of operator as a special kind of branch point twist field. The impact above will be facilitated by the successful dissemination of our results. This will occur through publication of our results in international journals, presentation of our results at international meetings and discussions occurring during scientific visits to key players.

People: The project will provide expert training in a highly specialised area of mathematical physics to a postdoctoral researcher. Some of the work will be of a numerical nature and therefore it will give the researcher the opportunity to develop/improve upon their programming skills. Thus the project will lead to new skills being acquired and provide a set of highly specialised skills which will improve the researcher's future academic prospects if they are to continue research in this area and work prospect if they go on to join industry. The project will contribute to training the next generation of researchers in the field.

Society & Economy: We do not envisage any clear, immediate social or economic impact from this research.