Imagine an isolated system of many interacting particles, which is initially prepared in some non-generic state, far from equilibrium (e.g. a finite box of atoms, where initially all atoms are squeezed into one corner). Thermodynamics and statistical physics are founded on the idea that eventually, after enough time, such a system will reach equilibrium where the atoms fill the box uniformally, and forget their initial conditions. Despite the importance of this process to the emergence of thermodynamics, the quantum mechanical description of this concept is not completely understood. I am interested in understanding the new aspects of this problem that arise from the quantum perspective. For example, how does quantum information and quantum operators propagate during such an evolution process? How is this propagation affected by disorder, conserved quantities and the nature of the evolution? When does thermalization breakdown and how? etc...

To make the discussion concrete it is useful to have a particular setup in mind. Imagine a chain of spins that are interacting with one another. To connect to the picture above, where we start from some peculiar state, we can choose the simplest initial state we can imagine, say a product state, where all the spins are pointing upwards. This state is non-generic because it is far from thermal equilibrium, for example, the entropy of this state is zero! Now let us allow this system to evolve quantum mechanically.

A quantity that is very useful to consider when looking at a closed system is entanglement entropy (or von Neumann entropy), which we denote by S. This quantity is associated with a partition of the system, say into two parts A and B, where A is everything to the left of x and B is the rest of the system, as shown in the figure below.

S(x) is quantifies how entangled is A with B. (Note that we can make more complicated partitions, e.g. multiple partitions etc., so this is just a simple and intuitive one).

For example, in the figure below the entanglement entropy S(x), divided by log 2, for a system of 459 spins-1/2 is plotted for different values of time, where the evolution with time is random. The figure is taken from a paper we wrote together with Adam Nahum, Sagar Vijay and Jeongwan Haah (note that here we used a trick to evolve such a large system of spins to such a long time - it is not possible in the general case due to the explosively large Hilbert space of a many-body system!). The color coding is such that initial-to-late times corresponds to light-to-dark blue.

Below, I also attached a video showing the entanglement growth for a shorter spin chain (100 spins) growing under similar random 2-site unitary evolution:

In the paper we showed that the randomly growing entanglement you see above, is described by the same equations that describe a classical stochastic growing surface. Thus we made a new connection between classical and quantum physics.

Going back to the general case of thermalization and starting from some non generic state, the dynamics of the entanglement entropy can be roughly divided into two regions. At short times the entanglement grows linearly in time. At longer times it reaches its maximal value. The latter saturation value is extensive in the size of the region of interest and reflects the thermodynamic limit. As a result, after a long enough period of time, the region closer to the edge saturate into a "wedge" like structure.

One of the most interesting features that came out of this paper came from looking at what is called the "entanglement velocity", denoted by vE. Basically, this is the slope of the average entanglement growth (see figure below where the entropy at the center of the chain S(L/2) over it maximal value at thermal equilibrium, Smax ~ L/2, is plotted against vE * t / Smax, for different values of length, L)

vE quantifies how fast a system thermalizes. But interestingly, it is also the speed at which entanglement propagates in the system. Moreover, one can compare this to other velocities in the system. For example, consider time evolution of a local operator in the Heisenberg picture . Because of interactions the extent of the operator grows in space. Turns out that this velocity, which is sometimes called the butterfly velocity is, in the generic case, twice the entanglement velocity vE = vB/2. This was a surprise because when quenching into an integrable model (CFT) the velocities are equal vE = vB. In general this is just another indication that the thermalziation process is complex and rich and we still have much to understand. In a later paper, with Adam Nahum and David Huse, we showed that entanglement and operators can grow subballistically, with different powers, when the system is disordered.

Looking forward, I am interested in understanding in questions along these lines in general. For example, what new can we learn by looking at the thermalziation of eigenvalues? When do quantum systems fail to thermalize? etc ...