Part 2: Simone Giombi, Igor R Klebanov and Grigory Tarnopolsky: Conformal QED_{ d }, *F*-theorem and the *ε* expansion

Since 2009, Journal of Physics A has awarded a Best Paper Prize, which serves to celebrate well written papers that make a significant contribution to their field. In 2017, the Editorial Board have awarded three prizes using the criteria of novelty, achievement, potential impact and presentation.In the second of a series of interviews, meet Simone Giombi, Igor Klebanov and Grigory Tarnopolsky, the authors of Conformal QED d , F-theorem and the *ε* expansion Simone Giombi *et al* 2016 *J. Phys. A: Math. Theor.***49** 135403.

### Meet the authors

Simone Giombi is an Assistant Professor at Department of Physics, Princeton University, where Grigory Tarnopolsky is a graduate student. Igor Klebanov is the Eugene Higgins Professor of Physics and Associate Director of the Princeton Center for Theoretical Science.

### What prompted you to pursue this field of research?

Quantum Electrodynamics (QED) is one of the original and best understood models of quantum field theory. It describes the quantum interactions of the electromagnetic field with charged matter (electrons). Its physics in three space plus one time dimensions is very well understood. In fact, QED is one of the most precisely tested theories in all of science: it is capable of reproducing experimentally measured values of quantities like the anomalous magnetic moment of the electron to a precision better than one part in a billion!The behavior of QED in two space plus one time dimensions is more subtle and not yet completely understood. This is relevant for applications to condensed matter physics, where QED in two spatial dimensions has been proposed as an effective theory for some strongly correlated systems. It is well established that, if the electromagnetic field is coupled to a large number of charged fermions of negligible mass, at low energies the model is well described by a conformal field theory – a quantum field theory with no intrinsic length scale and long range interactions. When the number of light fermions is one or two, which are the cases most relevant for condensed matter applications, the behavior of the model at low energies remains an open question. It has been proposed that below a certain critical number of fermions, the model may undergo a so-called “chiral symmetry breaking”, upon which the fermions acquire a dynamically generated mass. There is no consensus in the literature about the value of this critical number of fermion species, and understanding this better was one of the main motivations of our work.

### What is the winning paper all about?

Our approach was to combine modern tools like the F-theorem, with older ones like the Wilson-Fisher epsilon expansion and 1/N expansion, in order to obtain some constraints on the low energy behavior of QED in two plus one dimensions, in particular trying to shed light on the possible chiral symmetry breaking and the critical number of fermion species. The F-theorem is a relatively new tool in quantum field theory that can be used to constrain the behavior of an interacting system as a function of the energy scale. Essentially, it states that a quantity ‘F’, which is a measure of the number of degrees of freedom in the system, must always decrease with decreasing energy scale. In our paper we explained how to use the Wilson-Fisher epsilon expansion – a well-established approximation scheme where one uses space-time dimension as a formal continuous parameter – to compute the quantity ‘F’ for the QED model in two plus one dimensions. Then, we used our results to provide an upper bound for the critical number of fermion species below which the chiral symmetry breaking can occur.

### What do you plan to do next?

While the F-theorem is a powerful tool to study strongly interacting quantum field theories, calculating the quantity ‘F’ is typically a very hard problem. It turns out that ‘F’ is also related to the concept of “entanglement entropy”, a measure of entanglement in quantum many-body systems. The calculation of this quantity in interacting quantum field theory is in general a very challenging problem. The methods we developed in our work give a systematic way to approach this calculation and can in principle be applied not just to QED, but also to a variety of other models of interest in condensed matter physics and quantum field theory more generally. In particular, we hope to use the quantity `F’ to study the boson-fermion dualities, which have been proposed for QED in two spatial dimensions.We thank the authors and congratulate them on their award.To see the rest of the winners and the nomination process for the Journal of Physics A Best Paper Prize 2018, visit this page.

**Read More:**

Conformal QED_{ d }, *F*-theorem and the *ε* expansion

On *C _{J} *and

*C*in the Gross–Neveu and

_{T}*O*(

*N*) models

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Categories: Journal of Physics A: Mathematical and Theoretical, JPhys+