Numerical methods for quantum systems with infinite-dimensional space of states: two examples, two approaches
In this thesis two rather different topics are addressed: • the application of the Truncated Conformal Spectrum Approach (TCSA) to pertur- bations of SU(2)k Wess-Zumino-Witten (WZW) models, and to Landau-Ginzburg (LG) theories; • the analysis of the role of quantum statistics in the steady flow patterns of 2d driven ideal Fermi gases in a non-integrable geometry. A common methodological question unifies the two topics: how to simulate numerically quantum systems with infinite spaces of states. This problem is typically faced either introducing a lattice in the real space or directly in the continuum, and the two addressed topics can be seen as examples of these two cases. The TCSA is aimed to tackle perturbed conformal field theories directly in the continuum. Instead, to study the ideal Fermi gas we introduce a lattice which naturally captures the features of the chosen non-integrable geometry. Chapter 1: This part of the thesis is mainly methodological and it is focused on the application of the TCSA to c = 1 conformal field theories. In particular, our interest is concerned with perturbations of WZW models with SU(2)k symmetry, which capture low energy properties of k/2-Heisenberg spin-chains. The new problem faced is how to apply the method in the presence of marginal perturbations, where UV divergences result in non-universal contributions in the simulations. A generic framework to cure this pathology is proposed, and tested in the case of the current-current deformation of SU(2)1. Among c = 1 theories, Landau-Ginzburg theories are of great interest, since their classical structure of the vacua captures the massive phases of deformation of minimal models. The most important methodological problem we faced here is that LG theories have an uncountable space of states. We show that the TCSA applies also in this case, once the target space of the LG field is properly compactified. The TCSA is thus used to very an existing conjecture on the number of stable neutral bound states. Chapter 2: Interacting quantum gases, such as the electron liquid, are known to admit a hydrodynamic description and, as for classical interacting fluids, they develop turbulence at sufficiently strong driving. On the other hand, ideal classical fluids cannot develop any turbulent flow. We ask whether this property extends to ideal quantum gases. Indeed, Fermi gases satisfy the exclusion principle that could play the role of an effective interaction, resulting in non-trivial flows. We are going to describe an ideal Fermi gas which flows form a narrow channel into a wider region. A lattice which approximates the geometry is introduced, and the system is tackled within two approaches: the micro-canonical formalism and the Lindblad equation. In the first case the system is closed, while in the second one it is coupled to external baths. Despite this difference, the observed physics ￼turns out to be the same far from the boundaries. We show that, in specific regimes, quantum statistics induces non-trivial patterns in the vorticity, which generates local magnetic moments of a measurable intensity.