Research of Dr. Benjamin Doyon
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My research interest is in quantum field theory, in particular the integrable or conformal kind. I'm quite interested in looking at QFT as a powerful theory for emergent fluctuations (collective behaviours) in many-body systems. This point of view connects it to condensed matter and statistical systems, but also provides a fundamental understanding of renormalisation group and an alternative view on the fundamental particles of physics. One of my present research paths is to develop the connection between mathematical measures (conformal loop ensembles) describing emergent fluctuations for a large class of statistical models, and the algebraic construction of conformal field theory based on the usual tools and ideas of QFT.


List of publications and other works

Overview of some recent works by subject

Conformal field theory from conformal loop ensembles:

Short talk with slides

Overview of works:
  • B. D., V. Riva and J. Cardy, Identification of the stress-energy tensor through conformal restriction in SLE and related processes, Commun. Math. Phys. 268 (2006) 687-716, preprint math-ph/0511054 (30 pages). We construct the stress-energy tensor as a local random variable (or, more precisely, a limit of a random variable) in the context of Schramm-Loewner evolution at kappa = 8/3. We prove the conformal Ward identities and tranformation properties with a central charge equal to zero, based on an assumption of differentiability.
  • B. D., Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE, submitted to Comm. Math. Phys., preprint arXiv:0903.0372 (61 pages). I overview the measure theory of conformal loop ensembles developed by Sheffield and Werner, which can be seen as a generalisation of SLE where all cluster boundaries are taken into account. I also develop CLE further, by constructing probability measures on the Riemann sphere and on doubly connected domains, as well as introducing notions of continuity and support for CLE events. I prove (with mathematical rigor) conformal invariance of the constructed probability functions as well as other theorems, based on three assumptions about the CLE measure.
  • B. D., Conformal loop ensembles and the stress-energy tensor. II. Construction of the stress-energy tensor, preprint arXiv:0908.1511 (62 pages). I construct the stress-energy tensor as a local random variable (again, more precisely as a limit of a random variable) in the context of CLE.  I prove the conformal Ward identities under a single stress-energy tensor insertion, and the transformation properties with a non-zero central charge, which involves the Schwarz derivative. The way I obtain the transformation properties is fundamentally different from that used in the SLE context, because of the presence of the infinitely many small loops leading to the central charge. I also introduce the concept of relative partition function, and prove that a certain conformal derivative of it gives rise to the stress-energy tensor one-point function. In order to do all that, I introduce the notion of renormalised probability, and prove transformation theorems for it. All theorems are proven with mathematical rigor as consequences of theorems of part I, and under the additional assumption of conformal differentiability (see below).
  • B. D., Calculus on manifolds of conformal maps and CFT, preprint arXiv:1004.0138. I introduce the notion of conformal derivatives: these are particular Hadamard derivatives where we study small variations of a function upon changes of its argument arising from conformal maps near to the identity. I prove (with mathematical rigor) many properties particular to conformal derivatives. But the main theorem is essentially that the conformal Ward identities of conformal field theory can be recast into a form that involves a particular conformal derivative. More precisely, the Riemann-Hilbert problem of constructing a function with particular singularities and boundary conditions corresponding to those of the stress-energy tensor in a correlation functions on an open domain, is solved simply by applying a conformal derivative on the correlation function without the stress-energy tensor insertion. It is this form of the conformal Ward identities that occurs in the CLE context above.

Entanglement entropy in 1+1-dimensional quantum field theory:

Slides of a long talk

Overview of works:
  • J. L. Cardy, O. A. Castro Alvaredo and B. D.,  Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Stat. Phys. 130 (2007) 129-168, preprint arXiv:0706.3384 (40 pages). We introduce the concept of branch-point twist fields in a general quantum field theoretic framework, which gives rise to partition functions on multi-sheeted coverings of the Euclidean plane. These are twist fields associated to permuttation symmetries in replicated QFT models. We show that their scaling dimension is that found by Calabrese and Cardy using CFT arguments, and we give a general prescription for evaluating their two-particle form factors in integrable quantum field theory with diagonal scattering matrix. We work out the examples of the Ising and sinh-Gordon models. We use this to derive a universal formula for the correction to saturation of the entanglement entropy at large distances in one-dimensional extended quantum systems near critical points, under the assumption of diagonal integrability of the universal scaling limit. This formula does not explicitly depend on the scattering matrix, only on the particle spectrum.
  • O. A. Castro Alvaredo and B. D., Bi-partite entanglement entropy in integrable models with backscattering, J. Phys. A 41 (2008) 275203, preprint arXiv:0802.4231 (22 pages). We develop further the theory above to the calculation of form factors of branch-point twist fields and the entanglement entropy in integrable models with non-diagonal scattering. We show that the same universal formula holds for the correction to saturation, and provide an extensive analysis of the analytic continuation in the number of Riemann sheets that is needed in the process.
  • B. D., Bi-partite entanglement entropy in massive two-dimensional quantum field theory, Phys. Rev. Lett. 102 (2009) 031602, preprint arXiv:0803.1999 (4 pages). Selected for the February 2009 issue of the Virtual Journal of Quantum Information. I show that the large-distance correction to saturation of the entanglement entropy only depends on the particle spectrum in any massive 1+1-dimensional QFT model, integrable or not. I do this by showing that the only properties of form factors that are needed are in fact valid out of integrability.
  • O. A. Castro Alvaredo and B. D., Bi-partite entanglement entropy in QFT with a boundary: the Ising model, J. Stat. Phys. 134 (2009) 105-145, preprint arXiv:0810.0219 (41 pages). We develop the form-factor approach to twist-field correlation functions and entanglement entropy in the case of the Ising model with an integrable magnetic boundary. We resum the form factor expansion in order to obtain the short-distance asymptotics. We show how to obtain in a universal way the Affleck-Ludwig boundary entropy from the entanglement entropy, by comparing short- with large-distance limits, and propose its validity in more general QFT models.
  • O. A. Castro Alvaredo and B. D., Bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories, Review article to appear in the special issue ”Entanglement entropy in extended quantum systems” of J. Phys. A, preprint arXiv:0906.2946 (51 pages). We review all these works

Quantum impurities in steady states out of equilibrium:

Slides of a long talk

Overview of works:
  • B. D. and N. Andrei, Universal aspects of non-equilibrium currents in a quantum dot, Phys. Rev. B 73 (2006) 245326, preprint cond-mat/0506235 (35 pages). Selected for publication in the July 4, 2006 issue of Virtual Journal of Nanoscale Science and Technology. We study the real-time Schwinger-Keldysh formulation of the non-equilibrium steady state in the Kondo model: a state where electrons are flowing through the Kondo impurity thanks to a different of potential (voltage). We evaluate perturbatively the non-equilibrium current up to two-loop order. In this context, we prove at all orders (all loop corrections) that the large-time limit indeed converges to a steady state, and we provide arguments that this is due to the fact that the leads' electrons represented by massless free fermions in the Kondo model play the role of a proper thermal bath.
  • B. D.,  New method for studying steady states in quantum impurity problems: the interacting resonant level model, Phys. Rev. Lett. 99 (2007) 076806, preprint cond-mat/0703249 (4 pages). I introduce a new method to evaluate perturbatively currents and other averages in non-equilibrium steady states of impurity models, with the example of the interacting resonant-level model (a simpler version of the Kondo model). This method does not rely on a real-time construction of the steady state, but rather on a scattering-state formulation. It uses general properties of these scattering states along with equations of motion. I provide a complete renormalisation-group improved perturbative analysis of the current in the interacting resonant-level model, and identify the proper infrared cut-offs related to the voltage.
  • B. D., The density matrix for quantum impurities out of equilibrium, lecture notes for the Fifth Capri Spring School on Transport in Nanostructure (5 may 2009), found on school webpage: http://tfp1.physik.uni-freiburg.de/Capri09 (42 pages). I review these works, and derive new results about the relation between the real-time formulation and the scattering-state formulation of non-equilibrium steady states. I show their equivalence in cases where the large-time limit can be proven to exist (including the Kondo model and the interacting resonant-level model), and I propose a general framework for describing the ensemble of scattering states representing a non-equilibrium steady state in the context of boundary CFT.


General description of research interests

Sometimes, many basic constituents that are interacting amongst each other in simple and understood ways, such as electrons in a metal, molecules in a liquid, or buyers and sellers on the stock market, when present in large numbers, give rise to unexpected results on large scales. This is usually called "emergent behaviour", and it is very hard to predict, in general, what such behaviours can be. Physicists have studied systems with very many constituents where the interaction is simple enough that all constituents just interact with neighbours, like in many physical situations. They came up with a very powerful theory, based on physical principles, that describes the emergent behaviours that can occur in these cases. This is quantum field theory, and it describes systems near critical points, where one can see on large scales a change of the physical properties. In fact, one of the great achievements of theoretical physics of the twentieth century is the understanding that all fundamental particles that are observed in current-day experiments can be understood as "emerging" from a simpler, more symmetrical theory: this is the standard model of quantum field theory.

Physicist then have an understanding of certain emergent behaviours, but this understanding does not form yet a mathematically coherent whole. They understand them through particles and how they scatter, through energies and how it varies locally, and through local probes and how they react to local disturbances. But they often don't know how to relate these various ideas, and a description of a given model is often based on guess work and numerical evidence.

There are some models for which quantum field theory has the opportunity of being mathematically coherent: they are called integrable models. They are models with enough symmetries and invariances that exact results can be conjectured, and sometimes obtained in mathematically rigorous ways. Although they are quite special models, they can be applied to many situations of physical, and nowadays even biological interest. This is thanks to the principle of universality, that says that it does not matter too much what the interaction between the basic constituents is, one often gets the same model of quantum field theory - that is, the same emergent behaviour. Many astonishing results have been obtained already from integrable models, but the theory is wide and it is now becoming evident that it can be developed much further in different directions, with the promise of even more astonishing results.

This is what I am interested in. Using recently discovered techniques and new ideas of my own, I will try, for instance, to understand better the connection between the spins in an idealised magnet and the emergent behaviours at critical points, to understand how to calculate properties of impurities in electronic baths and see how things change when it is not at equilibrium (like when an electric current passes through a mesoscopic device), and to try to understand how different geometries and how the combination of a temperature with quantum mechanics can be studied in integrable models.