Exactly solvable models in low-dimensional many-body physics

Resumen

Abstract The thesis is divided into two parts related to different subjects. The first part is devoted to the study of a model describing fermion pairing with p-wave symmetry. It is a variant of the BCS model for superconductivity in which the Cooper pairs form a triplet spin state, as opposed to the singlet state of the BCS case. The model has attracted a lot of attention since its non-trivial topological features ¿and the possibility of non-abelian braiding of quasiparticles in the model¿ were unveiled by Read and Green (2000) in mean field approximation. We present the exact solution in two dimensions, which is accessible via the Quantum Inverse Scattering Method (QISM). The model exhibits a critical line in which a topological phase transition takes place. The exact solution reveals novel features of the model, such as the existence of another critical boundary that limits phases with different topological properties. These may be characterized in terms of a topological order parameter whose behavior depends on the reality/rationality of the considered point in the phase diagram. The second part of the thesis is concerned with entanglement in many-body critical systems in one dimension. R ¿nyi and von Neumann entropies quantifying the amount of entanglement in ground states of critical spin chains are known to satisfy a universal law which is given by the Conformal Field Theory (CFT) describing their scaling regime. This law can be generalized to excitations described by primary fields in CFT, as was done by Alcaraz et al. (2011). The R ¿nyi entropy of excitations corresponding to a given primary field turns out to be related to the correlators of this field in the cylinder. In the second part of the thesis we derive such a generalization, along with illustrations and numerical tests in finite-size models belonging to two different universality classes. We also discuss unsolved and controversial problems in this context.