Teorie pole fázových přechodů

• Thesis title in Czech: Teorie pole fázových přechodů
• Thesis title in English: Field theory of phase transitions
• Key words: fázové přechody druhého druhu|kritické exponenty|renormalizační grupa|konformní teorie pole|O(N) modely|beta funkce
• English key words: second order phase transitions|critical exponents|renormalization group flow|conformal field theory|O(N) models|beta functions
• Academic year of topic announcement: 2024/2025
• Thesis type: Bachelor's thesis
• Department: Institute of Particle and Nuclear Physics (32-UCJF)
• Supervisor: Mgr. Petr Vaško, Ph.D.

Guidelines

Have you ever wondered why second order phase transitions of many different systems (e.g. ferromagnet/paramagnet or water/vapor transitions) are the same and how to make order in the plethora of critical exponents (e.g. behavior of magnetization, specific heat, susceptibility, correlation length, etc. near the critical point) that experimentalists measure to describe them? The framework for modeling physical systems near the transition is Euclidean Quantum Field Theory (typically in three dimensions, since that is where a block of magnet or gas lives). Many transitions are organized based on symmetry. In one phase (called disordered) the lowest energy configuration (vacuum) does not break a symmetry, while in the other phase (ordered) vacuum breaks that symmetry (e.g. in an Ising model describing a magnet, vacuum could be either spins pointing randomly (disordered) or spins pointing in one direction (ordered), which either does not break or breaks rotational symmetry). The field in the QFT model of the system is a fluctuation of the "order parameter" (e.g. magnetization). We imagine to provide a description at high temperature (disordered phase with unbroken symmetry) and ask how does the system behave as we lower temperature. This evolution of the theory with temperature (energy) is called the Renormalization Group (RG) flow (a Nobel prize was awarded to Kenneth Wilson) and is one of the fundamental questions in the framework of Effective Field Theories. Typically, external parameters (like magnetic field) have to be tuned to reach the critical point by RG flow, which then stops there. In other words, the critical point is a fixed point of the evolution described by a Conformal Field Theory (CFT). The challenge is that most interesting critical points are captured by strongly interacting CFTs and thus their characteristics (critical exponents) are hard to compute (perturbation theory does not work well).

In this thesis the student will learn how to: (i) model a physical system by a QFT; (ii) identify the relevant parameters that govern the low temperature dynamics of the system (and have to be controlled in a lab to reach the second order phase transition); (iii) understand and use a CFT, i.e. a tool for studying physics of the critical point and computing a set of its characteristics from which all critical exponents follow (technically corrections to scaling dimensions of relevant operators by so called beta functions); (iv) deal with the strongly interacting system by a trick (continuation in the dimension of spacetime)
The universality class of phase transitions (relevant for a perhaps surprisingly large number of physical systems) of our focus will be captured by QFTs of scalar fields (so called O(N) models). The technical part of the thesis will consist of computations of beta functions (vector fields in the space of parameters of the QFT that determine their evolution under RG flow -- like velocity determines the position under time flow) for these models, in particular their leading contributions encoded in one-loop Feynman diagrams. In general, it is a demanding thesis as some (but reasonably small) parts of graduate lectures QFT 1,2 will have to be mastered by self-study.

References

Recommended:
[1] D. Tong; Statistical Field Theory; (lecture notes: http://www.damtp.cam.ac.uk/user/tong/sft.html)
Optional:
[2] J. Cardy; Scaling and Renormalization in Statistical Physics; Cambridge University Press 1996
[3] J. McGreevy; The Renormalization group; (lecture notes)
[4] D. Simmons-Duffin; Advanced Mathematical Methods: Conformal Field Theory (Chapter 1); (lecture notes)
[5] S. Pufu; Bootstrap School 2017; (video lectures)

Preliminary scope of work in English

Advanced students might be able to reach even the last chapter of Tong's lecture notes (that will be followed rather closely). It deals with the Kosterlitz-Thouless phase transition (worth a Nobel prize). At first sight it falls outside the class of symmetry governed phase transitions. However, quite recently it was understood that even this transition can be classified by symmetry. Just not an ordinary one, but a generalized (higher-form) symmetry whose charged objects are not point particles, but extended objects (strings, membranes, etc.). The topic of generalized symmetries is however outside the scope of this thesis even for advanced students and was remarked as an appetizer/motivation for further study.