#### Degree

Doctor of Philosophy

#### Program

Applied Mathematics

Alex Buchel

#### Abstract

In this Thesis, the problem of a quantum quench in quantum field theories is analyzed. This involves studying the real time evolution of observables in a theory that undergoes a change in one of its couplings. These quenches are then characterized by two parameters: $\delta \lambda$, the magnitude of the quench and most importantly, $\delta t$, the quench duration. In contrast to previous studies of abrupt quenches in the condensed matter theory community, we will be interested in smooth quenches with a finite $\delta t$.

Motivated by existing results in holographic theories, we studied the problem of a fast smooth quench in free field theories by quenching the mass of a free scalar and a free Dirac fermion. For specific mass profiles, exact analytic answers were found. We found that expectation values for the quenched operators obey universal scaling properties. We provide both numerical and analytic evidence for this scaling to hold in free field theories and argue that the same scaling properties should hold independently of the theory, the coupling or the quench profile. In fact, we show that the renormalized expectation value of an operator ${\cal{O}}_\Delta$ of dimension $\Delta$ should scale as $\vev{{\cal{O}}_\Delta}_{ren} \sim \delta \lambda/(\delta t)^{2\Delta -d}$, when the quench rate is fast compared to the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions.

This result suggests that, for $\Delta>d/2$, expectation values will diverge in the limit of $\delta t \to 0$, which seem to contradict previous studies of abrupt quenches. In this Thesis, we carefully analzye the relation between the two approaches and establish restrictions to the kind of objects that can be studied using the abrupt approximation.

Apart from the fast smooth quench, another regime of interest is the slow quench through a critical point. In free field theories we found that adiabatic behaviour breaks down when the system is close enough to the critical point and renormalized expectation values scale different, following expectations from the Kibble-Zurek argument. Given any finite fixed time, we are able to show how expectation values scale at any value of the quench duration $\delta t$.

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