February this year was a happy period. Chinese new year came on 12 Feb., so we have many small dinner parties in the dormitory. However, physics learning has never stopped. I finally find time to finish the last condensed matter course in PSI program: PSI18-19 Condensed Matter Review lectured by Professor Alioscia Hamma. Other two condensed matter courses are reviewed in my previous posts, one covering entanglement in many-body systems and tensor network states, the other covering magnetism. Notice the first one has a more convenient online course form.

I started Prof. Hamma’s course on 10 Feb and finished all its lecture videos yesterday. This condensed matter review course focuses on 1) the notion of quantum phase transition, and 2) how equilibrium in a quantum many-body system is reached. Both are topics I’m very interested in.

## Quantum Phase Transition

Phase transition means there are something singular happening in the system. For a quantum system, it usually happens when *the gap between the ground state energy and the first excited state energy with zero momentum closes*. To be honest, I haven’t fully understand why this statement should be true, but let’s treat it as a working definition for now.

The simplest example to demonstrate this is the 1D quantum Ising model. The two phases separated by a phase transition point are paramagnetic phase and symmetry-breaking phase. One confusing point is that why nature will choose a symmetry-breaking bases $|\Uparrow \rangle, |\Downarrow \rangle$ instead of forming the symmetry superposition $|\Uparrow \rangle \pm |\Downarrow \rangle$. Prof. Hamma gives a very physical argument by thinking about the stability of these states when exposed to local perturbations. The lifetime of the symmetry-breaking state will be very long, and thus, meta-stable. I wonder whether anyone has made such arguments precise.

The second kind of phase transition is more peculiar. No symmetry-breaking here, but instead, you can form the arbitrary superposition in the ground state degenerate subspace, without ruining the meta-stability of the state! We say such state has topological order. Prof. Hamma uses a $Z_2$ lattice gauge theory, toric code, to demonstrate this kind of phase. This part is very interesting. The excitation of such a model will make boson, fermion and semion emergent.

## Light-cone emerges from locality

This part is even more exciting! Prof. Hamma shows that a fuzzy light-cone can emerge from a *non-relativistic* quantum many-body system with only local interactions. This intriguing feature was first point out by Elliott Lieb and Derek W. Robinson, and is quantitatively encoded in an inequality called Lieb-Robinson bound.

This fact indicates many interesting things can emerge from many-body interactions. It also has many practical implications. For example, in a short review paper, it is mentioned that this light-cone-like propagation of information indicates that although the area law of the many-body state holds during the time evolution, the pre-factor of this area law will grow exponentially.

## Statistical mechanics and irreversibility emerges from closed quantum many-body systems

From lecture 12 to lecture 14, Prof. Hamma tells the class how the equilibrium is reached for a closed quantum many-body system. Strictly speaking, a close quantum system should not equilibrate simply due to the unitarity of the evolution. I think the situation is quite similar in our old classical regime, where we have Liouville’s theorem. Although Kardar discusses how statistical mechanics arises from classical mechanics in his 8.333 statistical mechanics course, I’m still now sure how it works.

Here, Prof. Hamma does a similar thing in quantum regime. The key idea here to solve the above paradox is to examine the time average and show the variant is very small, so the average value is typical, and can be seen as steady state. Many interesting theorems, Poincare recurrence theorem, for example, appear during the discussion.

I want to study more about the dynamics of many-body systems and how many-body (both classical and quantum) systems get to equilibrium later. A simple question like “why hot things cool, cold things warm” really provoke many interesting discoveries.