The Ising model occupies a distinguished place in various branches of theoretical physics. In statistical physics, the classical two-dimensional Ising model is a paradigmatic model of spontaneous symmetry breaking and continuous phase transitions [1]. This model can be mapped to the one-dimensional quantum Ising spin chain which is a canonical example of quantum phase transitions that occur in the ground state of quantum many-body systems [2].  

 

The Ising spin chain is an integrable model thanks to the fact that it can be reformulated in terms of a system of free fermions using the Jordan–Wigner transformation. However, this mapping is nonlocal, and the most important observable of the model, the magnetization, can only be expressed in a complicated way in the language of free fermions. As a result, calculating correlation functions of the magnetization is a notoriously difficult task.

 

In the vicinity of continuous phase transitions and quantum critical points, the correlation length becomes much larger than the lattice constant. This on the one hand serves as an explanation for the universality of critical behavior, and on the other hand enables a continuum, field-theoretic description. Field theories therefore provide a universal, low-energy effective description of the behavior around (quantum) critical points. One of the simplest examples of this is the Ising field theory which arises in the scaling limit of the Ising spin chain. The study of this field theory can lead to the understanding not only of the Ising spin chain, but of all other models belonging to the Ising universality class.

 

The topic of the thesis is the investigation of the correlation functions of magnetization in the Ising field theory. Despite the fact that this is an area that has been researched for quite a long time, our knowledge regarding finite-temperature, two-time (dynamic) correlators is still incomplete. The student's first task is to review and understand the existing results in the literature. The starting point of the actual calculations is an infinite form factor series representation of the correlation function that can be reformulated as a Fredholm determinant [3]. The student will evaluate this Fredholm determinant numerically both in the ferromagnetic and paramagnetic phases of the model, and analyze the results as a function of spatial and temporal separations at various temperatures. Special focus will be given to the low and high temperature limits and to the asymptotic behavior at large space-time separations. This will allow the student to check the existing analytical results and conjectures in the literature. 

 

In the course of the work, the student will become familiar with the Ising model, which, due to its simplicity, is suitable for learning several important phenomena and concepts related to quantum criticality and field theories. In addition, he/she can also acquire valuable numerical skills that can be used in other context as well.

 

 

 

[1] S. Sachdev, Quantum Phase Transitions, Cambridge University Press (2011).

 

[2] G. Mussardo, Statistical Field Theory, Oxford University Press (2010).

 

[3] B. Doyon, Finite-Temperature Form Factors: a Review, SIGMA 3, 011 (2007)