**Quantum field theory in low-dimensional condensed-matter systems**

### Natália Menezes Silva Da Costa (Utrecht University)

In this talk I will present two examples of how quantum field theories may be applied to describe the long wavelength regime of condensed-matter systems.

The first example [1] concerns the study of electronic interactions on the boundary of a two-dimensional time-reversal-invariant topological insulator. While the bulk of this two-dimensional material is insulating, the boundary exhibits propagating modes that may be described in terms of a one-dimensional Dirac theory. By assuming that there is an underlying electromagnetic theory mediating the e-e interaction on the edges, and by employing a dimensional reduction procedure, I will show that the effective one-dimensional theory is a non-Fermi liquid, known as the helical Luttinger liquid (HLL). This HLL resembles a theory of free bosons, however, with a parameter in its kinematics that indicates the strength of the e-e interactions. Within the quantum-field theoretical formalism, I will show that such parameter can be written in terms of the fine structure constant, which allows one not only to predict its value but also to manipulate the nature attractive/repulsive of the interaction.

The second example [2] concerns the topological response of a fermionic model defined on the Lieb lattice in presence of an electromagnetic field. The tight-binding model is built in terms of three species of spinless fermions and supports a topological Varma phase due to the spontaneous breaking of time-reversal symmetry. In the low-energy regime, the emergent effective Hamiltonian coincides with the so-called Duffin-Kemmer-Petiau (DKP) Hamiltonian, which describes relativistic pseudospin-0 quasiparticles and goes beyond the commonly studied spin-1/2 Dirac/Weyl paradigm. By considering a minimal coupling between the DKP quasiparticles and an external Abelian gauge field, I will present both the Landau-level spectrum and the emergent Chern-Simons theory. The corresponding Hall conductivity reveals an atypical quantum Hall effect, which can be simulated in an artificial Lieb lattice.

[1] N. Menezes, G. Palumbo and C. Morais Smith, Sci. Rep. 7, 14175 (2017).

[2] N. Menezes, C. Morais Smith and G. Palumbo, arXiv:1710.07916 (2017).