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The Hall Plateau Transition and non-unitary Quantum Field Theory

Periodic Reporting for period 3 - NuQFT (The Hall Plateau Transition and non-unitary Quantum Field Theory)

Reporting period: 2018-10-01 to 2020-03-31

"Quantum Field Theory (QFT) is one of the most powerful tools invented by physicists in the last 50 years. It provides, in essence, a way to think of - and, in many cases, to calculate properties of - interacting and or disordered physical systems in the ""low-energy"" regimes which are the most useful for experiments and practical applications.

The success of quantum field theory in particle physics has been well publicized in particular thanks to the successful CERN experiments that confirmed the existence of the Higgs boson. Quantum field theory is however also widely used in condensed matter or statistical physics. In this case, it can often happen that the formalism, well developed in the context of particle physics, has to be extended beyond a certain comfort zone. In particular, phase transitions occurring in a variety of disordered non-interacting electronic systems such as integer quantum Hall devices can only be tackled using quantum field theories that are not unitary. This technical term means that, after properly taking into account the disorder, one ends up having to deal with some formal processes for which probabilities are, in fact, negative! However unpleasant, the study of these processes still seems the best way to make progress on fundamental questions of current interest. Chief among these are the properties of the localization/delocalization transitions in topological insulators, which offer a wealth of potential technological applications via, in particular, the prospects of quantum computation.

The goal of this project is to build the technical apparatus necessary to handle complications arising from the loss of unitarity, and apply this apparatus to a better understanding of several types of physical systems, including the aforementioned phase transitions."
The loss of unitarity has more consequences than the apparition of negative probabilities. In particular, the action of symmetries can become considerably more complicated than in the usual case of, say, the rotation symmetry in ordinary quantum mechanics, which leads to the important role of the quantum number of spin. In the non unitary case, many more quantum numbers appear, while, to use a simple analogy, states may involve an indecomposable mixture of several values of the spin. An important part of the work done so far has thus been rather mathematical in nature, involving a lot of representation theory. The main result has been an understanding of how the scale and conformal invariance symmetries - expected to be present in the field theory describing quantum critical points - act when unitarity is lost. An example of the resulting organization of the Hilbert space is a simple model is provided in the figure.

This crucial result has then been put to use in the construction of correlation functions for, in particular, the plateau transition in the spin quantum Hall effect (class C), with the hope to be able to connect with experimental results in the near future.

Meanwhile, another consequence of the loss of unitarity is the occurence of field theories with continuous spectra of critical exponents. This is in contrast with ordinary critical points - such as the one occurring in simple ferromagnets - for which only a discrete handful of exponents are known to exist. Another important part of the work done so far has been devoted to the understanding of mathematical and physical properties of theories with such continuous spectra, which involve as a result unusually large corrections to scaling. This has led to unexpected relationships with developments in string theory, as well as applications to problems of a more geometrical nature, such as the study of dilute polymers.
"The main progress beyond the state of the art is the systematic development of a ""bottom-up"" approach, where the properties of quantum field theories are analyzed and understood via the thorough analysis of lattice models (regularizations) reproducing the properties of these theories at large distance. This has rendered possible the use of numerical techniques such as the Density Matrix Renormalization Group, analytical techniques such as the Bethe-ansatz, or mathematical techniques such as representation theory of finite dimensional associative algebras. Our approach is in sharp contrast with the ""top-down"" one used so far, where one attempts to derive properties of the systems of interest essentially using symmetry considerations, the construction of effective actions (roughly analogous to the Landau-Ginzburg actions), and ""bootstrap"" arguments. The bottom-up approach, while involving considerably more work, produces results in a systematic way. We're well on our way to solve the simplest examples of fully interacting non-unitary critical quantum field theories, and to solve or gain considerably more understanding of the integer quantum Hall critical points and its cousins."
Typical organization of eigenenergy states when the symmetry is not unitary.