Quantum Field Theory, Conformal Field Theory and Integrable Systems

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Quantum field theory is the pillar of modern quantum theory and of high energy physics, and of part of statistical mechanics. Conformal field theories (CFT's , i.e. theories invariant under Virasoro symmetry - local angle-preserving transformations - and of possibly higher extended symmetries) are of special interest. After more than 40 years, they still undergo tremendous developments, e.g. the conformal bootstrap program and Liouville theory, non-unitary CFT's, lattice regularisationsof CFT's, with applications to condensed matter and statistical physics. A vast and partly overlapping field is the study of integrable systems, classical and quantum systems with an infinite number of conserved quantities (like energy and momentum), but non-local, associated with infinite dimensional symmetry algebras. They allow to study non-perturbatively physical systems with strong statistical and quantum fluctuations. Born in the 1930's with Bethe's ansatz, integrable systems are studied at IPhT since the 1960's until now, in many different contexts: mathematics and combinatorics (cluster algebras, plane tilings), quantum spin chains, integrable gauge theories, out-of-equilibrium physics and integrable probabilities, etc. Again, these themes overlap and interact with many others studied at IPhT.

The renormalization-group flow equation describes the evolution of the effective action of a regularized quantum field theory in terms of its infrared cutoff. The renormalized quantum field theory is recovered by taking the limit that removes the cutoff. The renormalization- group flow equation can be expanded perturbatively in the Planck constant, and solved recursively. This allows us to rigorously prove properties of renormalized correlators at all orders of perturbation theory, without computing Feynman diagrams.

We have demonstrated this technique in the case of a simple pure gauge theory. In this case we have proved that the cutoff-removing limit exists, and that the renormalized theory has a Becchi–Rouet–Stora–Tyutin symmetry.

The conformal bootstrap methods consists in classifying, defining and solving conformal field theories using only symmetry and consistency assumptions. This method can lead to exact results in two-dimensional theories, thanks to their infinite-dimensional symmetry algebras. In particular, a discrete family called minimal models has been solved in the 1980's.

We have extended this method to continuous families of theories. This is motivated by models that have a continuous parameter (the central charge), such as the 2d Potts model. Having a continuous parameters shifts the emphasis from the algebraic analysis of the structure of representations, to the analytic properties of correlators.

We have thus built and solved continuous families of both diagonal and non-diagonal theories . The family of diagonal theories is called Liouville theory with a central charge less than one. We have found an interpretation of this theory in terms of microscopic loop models, which paves the way to applications to statistical physics .

Conformal field theories that are relevant to various condensed-matter systems in 1+1 dimensions can be not only non-unitary, but also logarithmic. In order to disentangle the complicated structures of their spectrums, we have introduced lattice regularizations of these theories. They exhibit symmetry algebras such as the Temperley–Lieb algebra, and can be analyzed using known mathematical techniques. The next step of the program is to recover the properties of the continuum theories from their lattice regularizations. Out of the building blocks of correlators in the conformal bootstrap approach, we have already recovered structure constants, and we believe that we can tackle conformal blocks too.

Non-unitary quantum field theories in 1+1 dimensions appear naturally in the description of a variety of gapless systems of statistical mechanics, such as percolation and polymers, or quantum critical points separating the plateaux in 2+1-dimensional topological insulators. The loss of unitarity can be traced back to either weak non-locality, or averaging over disorder.

In order to study renormalization group flows in the absence of unitarity, we have shown that it was fruitful to study the behaviour of entanglement entropy, rather than trying to generalize Zamolodchikov’s C-theorem. We have also shown that continuum limits of non-unitary spin chains can be conformal field theories with continuous spectrums, and demonstrated this in the case of a well-known model for tricritical polymers .

Cluster algebras, invented as a pure combinatorial structure, found many applications in mathematics (representation theory,..) and statistical physics. In a nutshell, a cluster algebra is a multi-time dynamical system describing the evolution of variables via mutations. We explored the link between the cluster algebra structure of the solutions to the abstract T-system satisfied by transfer matrices of integrable quantum spin chains and the statistical mechanics of dimer models on certain graphs called networks. In particular, partition functions on fixed domains correspond to mutation invariants. A quantized cluster algebra structure was used to compute graded tensor product multiplicities of representations occurring in inhomogeneous quantum spin chains, generalizing the difference Toda equation.

Dimer models also have formulations as tilings models. On nice domains, with suitable boundary conditions, such tilings exhibit an arctic curve phenomenon where the system displays a sharp separation between frozen phases near the boundary and fluid phases away from the boundary, not unlike a freezing pond. A new method coined "tangent method" allows to compute efficiently the arctic curve using only a boundary one-point function of the model. We applied it to the study of rhombus tilings with special boundary conditions and found a direct transformation from the boundary shape to the arctic curve. A related topic is the study of Schur processes, which encompass several statistical models such as plane partitions, domino tilings of the Aztec diamond and last-passage percolation. We showed that a Schur process can be generally realized as a dimer model on a certain plane bipartite graph called Rail Yard Graph. We fully characterized the properties of this dimer model through the computation of the inverse Kasteleyn matrix using the free-fermion formalism.

The Bethe ansatz is a standard method for addressing integrable models, but it does not always apply. Then we can fall back to Baxter’s TQ relation: a difference equation for two commuting operators that depend on a spectral parameter. In particular, this relation leads to a non-linear equation from which the model’s spectrum can be deduced.

We have identified the two relevant operators in a number of interesting models, including the the Ruisjenaars–Toda chain and the Asymmetric Simple Exclusion Process. In the latter case, this has allowed us to rederive the fluctuations of the current.

We have focussed on the planar limit of a maximally supersymmetric four-dimensional gauge theory, which on the one hand can be viewed as a toy model for quantum chromodynamics, and on the other hand is dual to string theory in Anti-de Sitter space.

While the spectral problem in this theory is in principle solved, computing correlators is a challenge that goes well beyond standard integrability techniques such as the Bethe Ansatz. We have been developing other methods, using integrable spin chains, separation of variables, or infinite-dimensional symmetry algebras. In particular, we have shown that three-point functions have a Yangian symmetry in the weak coupling limit. We have been particularly interested in the limit of heavy fields, where the model is dual to classical strings. We have computed some three-point functions in this limit, and found agreement with string theory.

We maintain contacts with other researchers interested in this thematics of the Mathematical Physics group and of the Statistical Physics group: François DAVID, Bertrand DUPLANTIER, Gregory KORCHEMSKY, Kirone MALLICK, etc.

* Associate researcher

Thimothy BUDD |

Linxiao Chen |

Postdoctoral positions are available each year in the Fall. Check this page or contact any staff member of the group.

Each member of the group can be contacted via email at *name.surname@ipht.fr*.

The full postal adress of IPhT is: Institut de Physique Théorique, CEA/Saclay, Bat 774 Orme des Merisiers, 91191 Gif-sur-Yvette Cedex, France.

Here are directions to the IPhT.

Last update : 01/17 2019 (860)