Exactly soluble models with an odd number of Majorana fermions

Fa Yueh Wu

Tom Banks

We express Kitaev's exact solution of spin models on the honeycomb lattice as a special case of a higher dimensional duality between staggered Majorana fermions and Pauli spins. General models with bilinear nearest neighbor couplings of spins and external magnetic fields are soluble in this way. Each model gives rise to an infinite number of single particle fermion Schrodinger equations, depending on the choice of discrete gauge field flux.

Marcel Franz, Ching‐Kai Chiu, Dmitry I. Pikulin

Holger Frahm, Markus P. Pfannmüller, Alexei M. Tsvelik

An exactly soluble model describing a spin $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ antiferromagnetic chain doped with mobile $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ carriers is constructed. In its continuum limit the undoped state is described by three gapless Majorana fermions composing the $\mathrm{SU}(2)$ triplet. Doping adds a scalar charge field and a singlet Majorana fermion with different velocity. We argue that this mode survives when the Haldane gap is added.

Basil A. Orfanopoulos

Cécile Monthus

Abstract For random interacting Majorana models where the only symmetries are the parity P and the time-reversal-symmetry T , various approaches are compared to construct exact even and odd normalized zero modes Γ in finite size, i.e. Hermitian operators that commute with the Hamiltonian, that square to the identity, and that commute (even) or anticommute (odd) with the parity P . Even normalized zero-modes are well known under the name of ‘pseudo-spins’ in the field of many-body-localization or more precisely ‘local integrals of motion’ (LIOMs) in the many-body-localized-phase where the pseudo-spins happens to be spatially localized. Odd normalized zero-modes are popular under the name of ‘Majorana zero modes’ or ‘strong zero modes’. Explicit examples for small systems are described in detail. Applications to real-space renormalization procedures based on blocks containing an odd number of Majorana fermions are also discussed.