Quantum phase transitions, spontaneous symmetry breaking, mean field theory

Quantum Condensed Matter Physics: Lecture 9
Theoretical physicist Dr Andrew Mitchell presents an advanced undergraduate / introductory Master's level lecture course on Quantum Condensed Matter Physics at University College Dublin. This is a complete and self-contained set of lectures, in which the theory is built up from scratch, and requires only a knowledge of basic quantum mechanics.
In this lecture I discuss phase transitions in quantum many body systems, and focus in particular on the phenomenon of spontaneous symmetry breaking in spin systems. We will take as a concrete example the onset of ferromagnetism in the Heisenberg model on cooling through the Curie temperature, and see how the system spontaneously develops magnetization and becomes polarized. The low-temperature phase is ordered and of a lower symmetry than that of the high-temperature disordered phase. We solve the model and analyze the phase transition using a simple but powerful approximation: mean field theory.
Navigate through the lectures of this course in order using the playlist:
• Quantum condensed matt...
Recommended course textbook: "Many-body quantum theory in condensed matter physics" by Bruss and Flensberg

[@loveof567code.exfitruso8]

Brilliant.

[@GeoffryGifari]

wait... even as we raise the temperature from the initial one, we still compare the symmetry of the *ground state* at the two temperatures? wouldn't high temperature -> more excitations?

[@insightfool]

Great explanation at 22:00 ! Thanks.

[@giljorge7479]

You meant 12:00 probably

[@GeoffryGifari]

If spontaneous symmetry breaking followed from fragmentation of phase space due to large energy barrier between degenerate region of phase space, can we know if such barriers exist (and where they are in phase space) from the mathematical form of the Hamiltonian?

[@DevashishTiwari-cc7dv]

yes one can always know whether such barrier do exist or not. The answer is number of possible vacuum for the system, if there are two vacuum of the system, or more, this means, there either can be tunneling possible or maybe constrained because of the time. The very simple answer can be given from field thweory, when there are multiple vacuua exist, this means, you need to expand around one of the minima, and consider those quantum fluctuations, once considered, one can check that symmetry which initially you had has been lost now. This gives yiu the answer that there is some barrier. for sure.

[@GeoffryGifari]

when we cool down a ferromagnetic metal from high temperature to below its critical temperature *without* external magnetic field, can we predict the direction of magnetization? at first glance it should be random.... but maybe amplification of small spin fluctuation might help us

[@GeoffryGifari]

we know that the phase transition from gas to liquid is followed by the reduction of entropy, but what is the difference in symmetry between gas and liquid? i don't think its as clear cut when compared to crystalline solid

[@DevashishTiwari-cc7dv]

the difference in symmetry is, the gas and liquid, are almost equally symmetric, because gases have both the continuous translation and rotation symmetry. the only difference is in there density, which is quite high in the liquid but lesser in the gas. This low density symmetry is broken during the phase transition of liquid-gas phase.

[@taibilimunduan]

symmetry breaking:; "Let´s get DOWN to work"

[@arkopro30]

Mean field approximation for anti ferromagnetic Heisenberg model is not successful!