Transverse Field in an Ising Model which exhibits a Kasteleyn transition

My Laidlaw project has been extremely rewarding in more ways than one: It is the first time I have performed original research in the context of theoretical physics; the opportunity to meet peers at the same level and support each other and; the wealth of knowledge provided to us in the numerous events which are held.

My research is a continuation of Doctor Hooley’s paper [1] which considered a particular magnetic model and showed that a special phase transition occurred.

The model is a two-dimensional Ising system with anti-ferromagnetic bonds horizontally and ferromagnetic bonds vertically and diagonally. An Ising model is a classical system which considers particles to have half-integer spin and couplings between nearest-neighbours. One can think of the particles as tiny bar magnets, with a north and south pole. Ferromagnetic couplings prefer to align spins, while anti-ferromagnetic couplings prefer to anti-align spins.  In this system, all the bonds cannot be satisfied simultaneously, hence the system is “magnetically frustrated”. There are six configurations in the elementary plaquette (a square with a spin particle at each corner) which correspond to the minimal achievable energy – the system is degenerate. The degeneracy is interesting because when the system is cooled to zero temperature, there is not a unique ground state and hence it retains a non-zero entropy density, which seems to violate the third law of thermodynamics.

Five elementary plaquettes and the couplings.

Any magnetic field in the z-direction magnetises all the spins upwards fully, as “all-up” is one of the degenerate ground states. Reiterating, the longitudinal magnetic field eliminates the degeneracy. Hooley showed that there is a phase transition line between the interplay of this magnetic field and the temperature. If the temperature increases beyond this line, the system changes into one with lines of down spins extending from the top end of the system to the bottom. These lines do not cross and never make loops, characterising a “Kasteleyn transition”.

My work this summer has been trying to explore this phase space when an additional x-direction magnetic field is applied. We expect to see the Kasteleyn transition being characterised by a choice of states that the system picks when the fields are varied.

The beginning third of my internship was background reading which was critical before I could proceed. These weeks allowed me to go over the various parts of statistical and quantum mechanics that I needed, and let me explore the more recent developments in this area. Although I felt particularly thrown in the deep end, it was paramount to my understanding. Reading all the time instead of solving problems can be frustrating but one of the great rewards of this was the night and day difference in my understanding of Hooley’s paper before- and after the reading period.

The rest of my internship has been spent working on the problem. To explore the effect of this second “transverse” field we approached the problems via various methods. One method is quantum mechanical perturbation theory, which is essentially the Taylor series applied to QM. Here, we consider one part of our Hamiltonian to have known solutions, and the other part to be a small change which gets weaker with the increasing power of some parameter. Perturbation theory was applied numerous times, with different parts of the Hamiltonian assumed to be the “original”.

The second main technique used is the quantum mechanical variational method, which tells us that the expectation value of a Hamiltonian with respect to any state is greater than or equal to the true ground state energy. This simple rule is useful as it allows us to guess a state, work out its expectation value, change the value of a parameter and then if the new expectation value is smaller then our guess gets better and better. Both methods allow us to find out which states are chosen when different strengths of longitudinal and transverse are applied.

I am incredibly grateful to be a part of the Laidlaw summer program and am thankful to Lord Laidlaw and the university. I would also like to thank CAPOD for the events and the support which they provide the students. Thank you to my peers who have kept me supported with good ideas and motivation. Last but certainly not least, a massive thank you to my supervisor, Doctor Chris Hooley who has led the direction of the research, and for his endless support and ideas.

My setup in the physics building.

[1]  C.A. Hooley, S.A. Grigera arXiv:1607.04657