Studies on impurities moving in Tomonaga-Luttinger Liquids

In this thesis, we have explored various aspects of the dynamics of an impurity moving in more than one 1D bath. A recurring theme has been the investigation of the orthogonality catastrophe (OC) that follows the injection of the impurity in the system. This phenomenon has been studied by calculating the Green’s function of the impurity, which also described the time evolution and spectral features of the latter. The distinctive signature of the OC is that the Green’s function shows a power-law tail at long times. This function has been calculated using suitable perturbative expansions in the impurity-bath coupling, namely the Linked Cluster Expansion (LCE) and a time-dependent perturbation theory around a nontrivial dynamics. All of our results are nonperturbative in the inter-bath hopping. In the two-bath scenario we have performed a detailed asymptotic expansion of the LCE Green's function at long times, which turned out to be very accurate in comparison with its numerical evaluation. The expansion has allowed us to obtain the renormalisation of the dispersion of the impurity bands, as well as the exponent of the power-law decay and the lifetime of the odd mode. One of our main results is that the OC, leading to the breakdown of the quasiparticle picture, survives the inclusion of a second 1D bath and dominates the long-time behaviour of all the components of the Green's function. In particular, the exponent characterising the long-time behaviour of the Green's function is given by half of the average of the exponents of the individual baths and, notably, is the same for the intra-bath Green's functions and for that connecting the two baths. In the case of two asymmetric baths, the Green's function is nonuniversal, acquiring a high-frequency component at short times and exhibiting persistent oscillations at longer times. In real experiments, the temperature of the baths is always finite, so we have extended the LCE treatment to this scenario. The effect of the temperature is to suppress the Green's function, limiting the possibility to observe its coherent oscillations and the power-law tail. At sufficiently long times, we found analytically that this suppression is exponential in time, with a different decay constant for the even and odd modes. For low temperatures, the two decay constants are approximately coinciding, and are proportional to the temperature. Using a perturbation theory in the inter-band part of the interaction we have been able to reproduce the LCE Green’s function and the OC with a method which also allows us to access the evolution of the whole impurity-bath system. The advantage of this approach is that it provides an analytic expression for the time evolution of the state of the whole impurity-baths system. This has allowed us to compute the time evolution of observables beyond the impurity Green's function, including the often-neglected properties of the baths. Moreover, the numerical effort required by the approach is sufficiently low so that we have been able to treat the case in which the impurity is initialised in a wave packet of an (almost) arbitrary shape. On the impurity side, we have analysed the time evolution of its population within each bath, observing how the persistent oscillations of the free impurity are damped and slowed down by the interaction with the baths. The impurity momentum is subjected to damping, as well. We have observed that momentum is transferred to the baths in two steps: a short transient, connected to the bath relaxation and independent of the inter-bath hopping, and a much slower decay caused by the emission of phonons during the deexcitation of the odd mode. Finally, we have also examined the time evolution of the probability density of finding the impurity in a given position and bath, for various Gaussian wave packets. We have studied the time evolution of observables describing the dynamics of the baths, which are rarely discussed in the literature on mobile impurities, but are nonetheless accessible to experiments. We have looked at the number of excited phonons, which shows a slow logarithmic divergence in time, related to the OC, superimposed with the faster growth caused by the emission of phonons from the odd mode decay. The particle density of the baths shows a semiclassical behaviour, intuitively similar to that of a pond in which a stone has been thrown. After the impurity has been injected into one of the baths, a localised density depletion forms and follows its motion. At the same time, two wave fronts are generated and propagate away. Moreover, each time the impurity oscillates between the baths, a new pair of ripples is emitted. The emission of ripples can be suppressed by employing wider wave packets. We have also found that the bath momentum density displays a behaviour analogous to that of the density. When the initial impurity wave packet has a markedly non-Gaussian shape, we find complex interference phenomena, both in the ripples and within the central trough. We examined the inter-bath, equal-times connected density and momentum density correlations, which revealed a rich structure in real space. This structure is best understood by taking “slices” of the correlation function along the relative and centre-of-mass coordinate, which show both the light-cone propagation of correlations and the motion of the impurity. We have made the first steps towards a many-baths system, in which the impurity moves in a 1D or 2D lattice of 1D baths. We have obtained the time evolution of the state of the system with the perturbative technique developed before. The impurity Green's function has revealed qualitative differences from the two-baths setup. First of all, the Green's function shows a complex short-time behaviour, caused by interference effects between the various paths of propagation within the lattice of baths. More importantly, we have found that each band of the noninteracting impurity is characterised by its own OC exponent, which is proportional to the degeneracy of the band. We have then speculated that it may be possible to tune the OC exponent either by properly designing the lattice of baths, or by changing the degeneracies by means of external magnetic fields. We also shown that for generic lattices of baths, the OC exponents vanish in the limit of an infinite number of lattice sites. Lastly, we showed that in the case of two identical baths there exists a unitary transformation that diagonalises the impurity degrees of freedom, thus achieving a complete decoupling of the latter from the baths. On the basis of this transformation, we have sketched two variational approximations for the ground-state and dynamics of the system.