PhD subjects

In this  PhD project we intend to study  the phase diagram of cuprate superconductors using techniques coming from strong coupling theory. We will focus on  a recent idea for the mysterious pseudo-gap phase of  these materials, where a finite momentum  particle-particle pair  (also called Pair Density Wave) « fractionalizes » into a modulated particle-hole pair and a uniform particle-particle pair. This phenomenon of fractionalisation opens a gap in the Fermi  surface which can be connected with the pseudo-gap of cuprates. We will test this idea to experiments and produce predictions.  Meanwhile,  on a different topic we project a study of the potential superconductivity in ABC  graphite, with a focus on its topological properties.  This will be done in collaboration with Cristina Bena from IPhT. The  potential candidate will learn different techniques during her/ his PhD, with in particular writing and solving numerically strong coupling equations, diagrammatic techniques, impurity solvers. Last but not least, a strong interaction with experimentalists, with discussion of ideas and concepts  will be expected.

In this PhD project we propose a theoretical study of the superconductivity and of the topological properties observed in the vicinity of defects in graphite. This project lies in the context of the recent observation of superconductivity in Moiré bi-layer graphene, as well as in ABC rhombohedral graphene. The key element in these system seems to be the presence of flat electronic bands. Such flat bands may also arise around impurities in graphite, which could induce local superconducting islands. We propose to perform a comprehensive theoretical analysis of the electronic properties for various graphitic compounds, in particular for a tridimensional network of rhombohedral stacking impurities. Our goal is to determine the conditions under which a bulk superconductivity with a higher critical temperature could be induced in graphite. Our project would benefit from a collaboration with experimental teams at the CEA/DEN/SRMP, and College de France, that would artificially create and stabilize impurities by irradiation of graphite, as well as explore their effect on the physical properties.

This is a very interdisciplinary topic, between mathematics and theoretical physics, with a strong mathematical side. Quantum systems are often defined 'perturbatively', from a classical system, as an asymptotic expansion series, whose coefficients can be defined either from a differential equation (e.g. Schroedinger), deformations relations, combinatorial expression, recursion,... One goal will be to show, at least in examples, that all definitions lead to a common universal recursion known as 'topological recursion'. And moreover that deformation relations satisfy an integrable system. Then we shall remark that these asymptotic series are divergent, and a resummation method is needed. We shall use the 'resurgence' method. The goal of the thesis will be to study and prove, in examples or in general, some relations between topological recursion, integrable systems and resurgence.

This project takes place in the framework of statistical physics and mathematical physics. The proposed thesis consists in the study of models of interacting particles or spins on a one-dimensional lattice and placed in situations far from thermodynamic equilibrium. The thermodynamics of systems at equilibrium is a well understood subject, but systems out of equilibrium remain much more mysterious. One possible approach is the study of the evolution of a classical or quantum integrable system from an arbitrary initial state. We propose to study non-equilibrium integrable models by seeking to determine their properties at long times. In particular, we will focus on a discretization of the temporal evolution of integrable models defined on a lattice while preserving integrability. Such a discretization has proven to be powerful numerically, but little has been done analytically. More concretely, we propose in a first step to apply these ideas to a particular integrable model which is called "Toda chain".

The Higgs boson mass is one of the most important parameters in our description of Nature. Its measured value points to one of the most spectacular failures of symmetry in physics. Our theoretical estimates based on symmetry are orders of magnitude larger than experimental results. For decades we have strived to understand this apparent failure of symmetry, but the best solutions that we have come up with have not passed direct experimental tests. They have not shown up at electron-positron colliders (LEP) and hadron colliders (LHC). During the course of this project we will study a new class of explanations that constitutes a radical departure from the standard lore. The value of the Higgs boson mass is explained by early Universe events, possibly at much higher energies than those that we can probe today in the laboratory. This makes the values of the cosmological constant and of the Higgs mass deeply interconnected and points to the sky as the ultimate laboratory to understand their origin.

This subject is in the field of condensed matter theory, and concerns the physics of electrons in solids. In recent work (Phys. Rev. B 100, 081106(R) (2019)) we have provided a new and exact formalism to describe the formation of end, edge or surface states through the evolution of impurity-induced states. We have proposed a general procedure that consists of finding the impurity states via the T-matrix formalism and showing that they evolve into boundary modes when the impurity potential goes to infinity. We have applied this technique to obtain Majorana states in 1D and 2D systems, as well as topological insulator edge states, graphene edge states, graphite surface states and Fermi-arc surface states for Weyl insulators.



Here we propose to generalize this technique to other systems for which this technique would provide significant advantages: for example we plan to study the edge states of multilayer graphene with different stackings (ABA, ABC or twisted), in the normal and superconducting regimes, in particular exploring the possibility to form topological edge states. We also intend to use this technique to study Shiba chains, i.e. chains of magnetic or non-magnetic impurities located on the surface of a superconducting substrate, in the search for the formation of topological Majorana states.



The candidates should have good knowledge of advanced quantum mechanics, quantum field theory, solid state and many body physics.

Loop models are 2d statistical models which in special cases describe percolation, polymers or random surfaces. This thesis is devoted to exploring the critical limit of loop models. This limit is a family of conformal field theories (CFT), parametrized by the central charge: a complex number whose real part must be less that 13.



The conformal bootstrap method has been used for solving CFTs analytically in two dimensions (minimal models, Liouville theory) or numerically in higher dimensions (the 3d Ising model). Recently, the method has been applied to the 2d O(n) and Potts models: two particuliar classes of loop models. The results are obtained numerically, but they sometimes have simple analytic expressions. Hence the question : can loop models be solved analytically ?

This question will be addressed using a mixture of analytic and numerical techniques :

— Analytic : special functions, representation theory, enumerative geometry.

— Numerical : computing conformal blocks and correlation functions in 2d CFT, simulating loop models on a lattice.

The project may use the following tools : Latex, Git, Python, Jupyter, MediaWiki.



References :

— Introduction to 2d CFT in the bootstrap approach : arXiv :1609.09523.

— Recent results on the 2d O(n) model : arXiv :2111.01106.

In order to fund this PhD project, a possibility is to apply for a scholarship by ED PIF. It is useless to apply without our support, so please get in touch if you are interested. Timetable of ED PIF scholarhips : ED PIF website https://www.edpif.org/



Co-advisors : Jesper Jacobsen (ENS et IPhT), Sylvain Ribault (IPhT)

E-mail : jesper.jacobsen@ens.fr , sylvain.ribault@ipht.fr

Location : IPhT et/ou ENS