
Research of Dr.
Benjamin Doyon
home page
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
manybody 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
stressenergy tensor through conformal
restriction in SLE and related processes, Commun. Math. Phys.
268
(2006) 687716, preprint mathph/0511054
(30 pages). We construct the
stressenergy tensor as a local random variable (or, more precisely, a
limit of a random variable) in the context of SchrammLoewner 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 stressenergy 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 stressenergy tensor. II. Construction of the
stressenergy tensor, preprint arXiv:0908.1511
(62 pages). I construct the
stressenergy 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 stressenergy tensor
insertion, and the transformation properties with a nonzero 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 stressenergy tensor onepoint
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 RiemannHilbert
problem
of constructing a function with particular singularities and boundary
conditions corresponding to those of the stressenergy tensor in a
correlation functions on an open domain, is solved simply by applying a
conformal derivative on the correlation function without the
stressenergy tensor insertion. It is this form of the conformal Ward
identities that occurs in the CLE context above.
Entanglement
entropy in
1+1dimensional quantum field theory:
Slides of a long talk
Overview of works:
 J. L. Cardy, O. A. Castro
Alvaredo and B. D., Form
factors of branchpoint twist fields in
quantum integrable models and entanglement entropy, J. Stat.
Phys. 130
(2007) 129168, preprint arXiv:0706.3384
(40 pages). We introduce the concept
of branchpoint twist fields in a general quantum field theoretic
framework, which gives rise to partition functions on multisheeted
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
twoparticle form factors in integrable quantum field theory with
diagonal scattering matrix. We work out the examples of the Ising and
sinhGordon models. We use this to derive a universal formula for the
correction to saturation of the entanglement entropy at large distances
in onedimensional 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., Bipartite
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 branchpoint twist fields and the
entanglement entropy in integrable models with nondiagonal 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., Bipartite
entanglement entropy in massive twodimensional 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
largedistance correction to saturation of the entanglement entropy
only depends on the particle spectrum in any massive 1+1dimensional
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., Bipartite
entanglement entropy in QFT with a boundary: the
Ising model, J. Stat. Phys. 134 (2009) 105145, preprint arXiv:0810.0219 (41 pages). We develop the formfactor approach to
twistfield 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 shortdistance
asymptotics. We show how to obtain in a universal way the
AffleckLudwig boundary entropy from the entanglement entropy, by
comparing short with largedistance limits, and propose its validity
in more general QFT models.
 O. A. Castro Alvaredo and
B. D., Bipartite
entanglement entropy in massive 1+1dimensional
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
nonequilibrium currents
in a quantum dot, Phys. Rev. B 73 (2006) 245326, preprint condmat/0506235
(35 pages). Selected for publication in the July 4, 2006 issue of
Virtual Journal
of Nanoscale Science and Technology. We
study the realtime SchwingerKeldysh formulation of the
nonequilibrium 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 nonequilibrium
current up to twoloop order. In this context, we prove at all orders
(all loop corrections) that the largetime 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 condmat/0703249
(4 pages). I introduce a new method
to evaluate perturbatively currents and other averages in
nonequilibrium steady states of impurity models, with the example of
the interacting resonantlevel model (a simpler version of the Kondo
model). This method does not rely on a realtime construction of the
steady state, but rather on a scatteringstate formulation. It uses
general properties of these scattering states along with equations of
motion. I provide a complete renormalisationgroup improved
perturbative analysis of the current in the interacting resonantlevel
model, and identify the proper infrared cutoffs 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.unifreiburg.de/Capri09
(42 pages). I review these works,
and derive new results about the relation between the realtime
formulation and the scatteringstate formulation of nonequilibrium
steady states. I show their equivalence in cases where the largetime
limit can be proven to exist (including the Kondo model and the
interacting resonantlevel model), and I propose a general framework
for describing the ensemble of scattering states representing a
nonequilibrium 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 currentday 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.


