We study a sudden interaction quench in the weak-coupling regime of the quantum sine-Gordon model. The real time dynamics of the bosonic mode occupation numbers is calculated using the flow equation method. While we cannot prove results for the asymptotic long-time limit, we can establish the existence of an extended regime in time where the mode occupation numbers relax to twice their equilibrium values. This factor two indicates a non-equilibrium distribution and is a universal feature of weak interaction quenches. The weak-coupling quantum sine-Gordon model therefore turns out to be on the borderline between thermalization and non-thermalization.
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:
deltalambda, the magnitude of the quench and most importantly,
deltat, 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
deltat.\nMotivated 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.\nThis result suggests that, for
Delta>d/2, expectation values will diverge in the limit of
deltatto0, 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.\nApart 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
deltat.
We study the time evolution of a conformal field theory deformed by a relevant operator under a smooth but fast quantum quench which brings it to the conformal point. We argue that when the quench time scale
δt is small compared to the scale set by the relevant coupling, the expectation value of the quenched operator scales universally as
δg/δt2Δ−d where
δg is the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions. We present explicit results for free scalar and fermionic field theories, supported by an analytic understanding of the leading contribution for fast quenches. Results from this Letter suggest that this scaling result, first found in holography, is in fact universal to quantum quenches. Our considerations also show that this limit of fast smooth quenches is quite different from an instantaneous quench from one time-independent Hamiltonian to another, where the Schrodinger picture state at the time of the quench simply serves as an initial condition for subsequent evolution with the final Hamiltonian.
With regard to the thermalization problem in isolated quantum systems, we investigate the dynamics following a quantum quench of the sine-Gordon model (sGM) in the Luther–Emery and the semiclassical limits. We consider the quench from the gapped to the gapless phase, as well as the reverse one. By obtaining analytic expressions for the one- and two-point correlation functions of the order parameter operator at zero-temperature, the manifestations of integrability in the absence of thermalization in the sGM are studied. It is shown that correlations in the long-time regime after the quench are well described by a generalized Gibbs ensemble. We also consider the case where the system is initially in contact with a reservoir at finite temperature. The possible relevance of our results to current and future experiments with ultracold atomic systems is also considered.
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale
TK (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable
TKt. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.
We discuss the quench dynamics near a quantum critical point focusing on the sine-Gordon model as a primary example. We suggest a unified approach to sudden and slow quenches, where the tuning parameter $\ensuremath{\lambda}(t)$ changes in time as $\ensuremath{\lambda}(t)\ensuremath{\sim}\ensuremath{\upsilon}{t}^{r}$, based on the adiabatic expansion of the excitation probability in powers of $\ensuremath{\upsilon}$. We show that the universal scaling of the excitation probability can be understood through the singularity of the generalized adiabatic susceptibility ${\ensuremath{\chi}}_{2r+2}(\ensuremath{\lambda})$, which for sudden quenches
(r=0) reduces to the fidelity susceptibility. In turn this class of susceptibilities is expressed through the moments of the connected correlation function of the quench operator. We analyze the excitations created after a sudden quench of the cosine potential using a combined approach of form-factors expansion and conformal perturbation theory for the low-energy and high-energy sector, respectively. We find the general scaling laws for the probability of exciting the system, the density of excited quasiparticles, the entropy and the heat generated after the quench. In the two limits where the sine-Gordon model maps to hard-core bosons and free massive fermions we provide the exact solutions for the quench dynamics and discuss the finite temperature generalizations.