Bosonization of Majorana modes and edge states

Abstract

We present a bosonization procedure which replaces fermions with generalized spin variables subject to local constraints. It requires that the number of Majorana modes per lattice site matches the coordination number modulo two. If this condition is not obeyed, then bosonization introduces additional fermionic excitations not present in the original model. In the case of one Majorana mode per site on a honeycomb lattice, we recover a sector of Kitaev's model. We discuss also decagonal and rectangular geometries and present bosonization of the Hubbard model. For geometries with a boundary we find that certain fermionic edge modes naturally emerge. They are of different nature than edge modes encountered in topological phases of matter. Euclidean representation for the unconstrained version of a spin system of the type arising in our construction is derived and briefly studied by computing some exact averages for small volumes.

Higher-dimensional Jordan-Wigner Transformation and Auxiliary Majorana Fermions

“…Note Added-While finishing this work, a related paper appeared [36]. Part of the present work overlaps with Ref.…”

Symmetric Jordan-Wigner transformation in higher dimensions

“…It is instructive to compute how the partons transform under Ξ, which can be conveniently done by first decomposing the complex partons into Majorana fermions, and then interpreting the action of e −iπ φ13 r as a root of parity (Appendix B). The upshot is Ξ : ĉr → − ĥr ; ĥr → ĉr ; ûr → − dr ; dr → ûr , (80) which takes the famous "Dirac form" as is proposed in the half-filled lowest Landau level context [57]. In addition, one can compute Ξ2 explicitly as…”

“…Furthermore, before restricting to the physical Γ − subspace we have Ξ2 = r Γr , which is the simply the total parton parity as is anticipated from the Dirac form in Eq. (80).…”

Exactly Solvable 1D Quantum Models with Gamma Matrices

“…1 See [35] and references therein for a recent exposition on the general conditions when such rewriting is possible and obeys the anticommutation relation…”

“…As mentioned before, the quantum phase transitions can be diagnosed using the gap closing condition given in (37). There may exist a number of conditions over the values of the system parameters involved in (35) for which this condition can be satisfied, and each of these conditions will, in principle, provide a quantum phase transition occurring in the g-XYG models for d = 2. For the purpose of demonstration, we consider the simplest critical point of the g-XYG model, which is the analogue of order-disorder transition in the transverse-field Ising model, which takes place at the vanishing momentum, i.e k = 0.…”