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International Symposium on Lattice Field Theory (LATTICE)

Bosonization of Majorana modes in arbitrary geometries

Arkadiusz Bochniak, Błażej Ruba

Published May 16, 2022• 0 citations
Publisher

Abstract

We briefly summarize the main results from [25] which extend the ones from [17,18]. The bosonization method based on the properties of Clifford algebras is presented, together with the illustrating examples: system of Majorana fermions on honeycomb lattice and its relation to Kitaev's model, and the Hubbard model on rectangular lattices. The role of bundaries together with the presence of Majorana edge modes are discussed.

DOI
10.22323/1.396.0123

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We briefly summarize the main results from [25] which extend the ones from [17,18]. The bosonization method based on the properties of Clifford algebras is presented, together with the illustrating examples: system of Majorana fermions on honeycomb lattice and its relation to Kitaev's model, and the Hubbard model on rectangular lattices. The role of bundaries together with the presence of Majorana edge modes are discussed.
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.
We propose an approach to detect the peculiarity of Majorana fermions at the edges of Kitaev magnets. As is well known, a pair of Majorana edge modes is realized when a single complex fermion splits into real and imaginary parts which are, respectively, localized at the left and right edges of a sample magnet. Reflecting both of this peculiarity of the Majorana fermions and the ground-state degeneracy caused by the existence of the Majorana edge zero modes, the spins at the edges of the sample magnet are expected to behave as a peculiar ``free'' spin which exhibits a unidirectional magnetization without any transverse magnetization when applied a sufficiently weak external magnetic field. For the Kitaev honeycomb model, we obtain the expression of the Majorana edge magnetization by relying on standard techniques to diagonalize a free fermion Hamiltonian. The magnetization profile thus obtained indeed shows the expected behavior. We also elucidate the relation between the Majorana edge flat band and the bulk winding number from a weak topological point of view.
Dimer models have long been a fruitful playground for understanding topological physics. Here, we introduce a class, termed Majorana-dimer models, wherein bosonic dimers are decorated with pairs of Majorana modes. We find that the simplest examples of such systems realize an intriguing, intrinsically fermionic phase of matter that can be viewed as the product of a chiral Ising theory, which hosts deconfined non-Abelian quasiparticles, and a topological ${p}_{x}\ensuremath{-}i{p}_{y}$ superconductor. While the bulk anyons are described by a single copy of the Ising theory, the edge remains fully gapped. Consequently, this phase can arise in exactly solvable, frustration-free models. We describe two parent Hamiltonians: one generalizes the well-known dimer model on the triangular lattice, while the other is most naturally understood as a model of decorated fluctuating loops on a honeycomb lattice. Using modular transformations, we show that the ground-state manifold of the latter model unambiguously exhibits all properties of the $\text{Ising}\ifmmode\times\else\texttimes\fi{}({p}_{x}\ensuremath{-}i{p}_{y})$ theory. We also discuss generalizations with more than one Majorana mode per site, which realize phases related to Kitaev's 16-fold way in a similar fashion.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension and transport in clean quantum wires and in the presence of isolated impurities.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension, and transport in clean quantum wires and in the presence of isolated impurities.