## Topological Ladder Models

What physics results if a prototypical topological model - the Su-Schrieffer-Heeger (SSH) chain - is doubled?

The SSH chain is a one-dimensional tight-binding model that can host localized bound states at its ends. It is celebrated as the simplest model having topological properties captured by invariants calculated from its band-structure. Collaborators and I studied two coupled SSH chains i.e. the SSH ladder.

The SSH ladder has a complex phase diagram determined by inter-chain and intra-chain couplings. We found three distinct phases: a topological phase hosting localized zero energy modes, a topologically trivial phase having no edge modes and a phase akin to a weak topological insulator where edge modes are not robust. The topological phase of the SSH ladder is analogous to the Kitaev chain, which is known to support localized Majorana fermion end modes. Bound states of the SSH ladder having the same spatial wavefunction proles as these Majorana end modes are Dirac fermions or bosons. The SSH ladder is consequently more suited for experimental observation than the Kitaev chain.

For quasiperiodic variations of the inter-chain coupling, the SSH ladder topological phase diagram reproduces Hofstadter's butterfly pattern. This system is thus a candidate for experimental observation of the famous fractal. Possible experimental setups for realizing the SSH ladder in its Kitaev chain-like phase include mechanical meta-material systems and ultracold atomic systems. These approaches could also be used to experimentally study the Hofstadter butterfly in the future.

Related publications:

K. Qian, D. J. Apigo, K. Padavic et al, Investigations of Topological Edge Modes in Majorana Wires, in preparation