Topology and Quantum Phases of Low Dimensional Fermionic Systems

Abstract

In this thesis, we study quantum phase transitions and topological phases in low dimensional fermionic systems. In the first part, we study quantum phase transitions and the nature of currents in one-dimensional systems, using eld theoretic techniques like bosonization and renormalization group. This involves the study of currents in Luttinger liquids, and the fate of a persistent current in a 1D system. In the second part of the thesis, we study the different types of Majorana edge modes in a 1D p-wave topological superconductor. Further we extend our analysis to the e ect of an additional s-wave pairing and a Zeeman field on the topological properties, and present a detailed phase diagram and symmetry classification for each of the cases. In the third part, we concentrate on the topological phases in two-dimensional systems. More specifically, we study the experimental realization of SU(3) topological phases in optical lattice experiments, which is characterized by the presence of gapless edge modes at the boundaries of the system. We discuss the specific characteristics required by a such a three component Hamiltonian to have a non-zero Chern number, and discuss a schematic lattice model for a possible experimental realization. The thesis is divided into three chapters, as discussed below: In the first chapter, we study the effect of a boost (Fermi sea displaced by a finite momentum) on one dimensional systems of lattice fermions with short-ranged interactions. In the absence of a boost such systems with attractive interactions possess algebraic superconducting order. Motivated by physics in higher dimensions, one might naively expect a boost to weaken and ultimately destroy superconductivity. However, we show that for one dimensional systems the e ect of the boost can be to strengthen the algebraic superconducting order by making correlation functions fall o more slowly with distance. This phenomenon can manifest in interesting ways, for example, a boost can produce a Luther-Emery phase in a system with both charge and spin gaps by engendering the destruction of the former. In the second chapter, we study the type of Majorana modes and the topological phases that can appear in a one-dimensional spinless p-wave superconductor. We have considered two types of p-wave pairing, 4"" = 4## and 4"" = 4##., and show that in both cases two types of Majorana bound states (MBS) with different spatial dependence emerge at the edges: one purely decaying and one damped oscillatory. Even in the presence of a Zeeman term B, this nature of the MBS persists in each case, where the value of chemical potential and magnetic field B decides which type will appear. We present a corresponding phase diagram, indicating the number and type of MBS in the -B space. Further, we identify the possible symmetry classes for the two cases (based on the ten-fold classification), and also in the presence of perturbations like a s-wave pairing and various terms involving magnetic field. It is seen that in the presence of a s-wave perturbation, the MBS will now have only one particular nature, the damped oscillating behaviour, unlike that for the unperturbed p-wave case. In the third chapter, we study SU(3) topological phases in two dimension. It is shown by Barnett et.al that N copies of the Hofstadter model with 2N Abelian ux per plaquette is equivalent to an N-component atom coupled to a homogeneous non-Abelian SU(N) gauge field in a square lattice. Such models have non-zero Chern number and for N = 3, can be written in terms of the SU(3) generators. In our work, we uncover two salient ingredients required to express a general three-component lattice Hamiltonian in a SU(3) format with non-trivial topological invariant. We nd that all three components must be coupled via a gauge eld, with opposite Bloch phase (in momentum space, if the NN hopping between two components is teik, then for the other two components, this should be te ik) between any two components, and there must be band inversion between all three components in a given eigenstate. For spinless particles, we show that such states can be obtained in a tripartite lattice with three inequivalent lattice sites, in which the Bloch phase associated with the nearest neighbor hopping acts as k-space gauge eld. The second criterion is the hopping amplitude t should have an opposite sign in the diagonal element for one of the two components, which can be introduced via a constant phase ei along the direction of hopping. The third and a more crucial criterion is that there must also be an odd-parity Zeeman-like term (as k ! k, the term changes sign), i.e. sin(k) z term, where z is the third Pauli matrix defined with any two components of the three component basis. In the presence of a constant vector potential, the kinetic energy of the electron gets modified when the vector potential causes a flux to be enclosed. This can generate the desired odd parity Zeeman term, via a site-selective polarization of the vector potential. This can be achieved in principle by suitable modifications of techniques used in Sisyphus cooling, and with a suitable arrangement of polarizer plates, etc. The topological phase is a firmed by edge state calculation, obeying the bulk-boundary correspondence.