Quantization of Charge Transport in Gapped Ground States

In this thesis we investigate the topological structure of the set of gapped ground states of charge conserving systems. The connected components of this set are called gapped phases. The approach taken is to find topological invariants of gapped ground states taking values in a discrete set, and changing continuously under continuous deformations of the ground state. As such, the topological invariants are constant within the same gapped phase. We pursue this strategy in the context of many-body lattice systems, either spins or Fermions. The prototypical example of such a topological index is the Hall conductivity of a two-dimensional system. Much time is devoted in this thesis to providing a full understanding of the Hall conductivity in gapped ground states. In particular, we relate explicitly the Hall conductivity (as a linear response coefficient) to the adiabatic curvature of flux threading. This is an essential starting point to reveal the topological nature of the Hall conductivity. The adiabatic curvature just mentioned is the local curvature of a fibre bundle over a `flux torus'. The Hall conductivity is interpreted as the Chern number of this fibre bundle. The Chern number is the integral of the curvature over the entire flux torus, so to make this interpretation possible, we show that the curvature of the fibre bundle is asymptotically constant (in the thermodynamic limit). This construction is only possible under the assumption that the Hamiltonian remains gapped when threading fluxes, it is argued why this assumption is reasonable. We then move on to propose a general index theorem pertaining to pairs of unitary maps and a certain class of clustering states (to be thought of a gapped ground states of charge conserving Hamiltonians). The index takes values in Z/q if the clustering state is q-fold degenerate, and it is stable under perturbations of the unitary map and of the state. Physically, the index represents the amount of charge moved across a fiducial line by a `process' represented by the unitary map. The index is shown to reduce to (i) an index of projections in the case of non-interacting Fermions, (ii) quantization of the charge density of translation invariant systems (the Lieb-Schultz-Mattis theorem), and (iii) quantization of the Hall conductivity in two dimensions. This last application of the index theorem provides a short proof of quantization of the Hall conductivity that is free from the extra assumption needed in the Chern n