Abstract
It has been shown that local four-fermion interactions on the edges of two-dimensional time-reversal-invariant topological insulators give rise to a new non-Fermi-liquid phase, called helical Luttinger liquid (HLL). Here, we provide a first-principle derivation of this HLL based on the gauge-theory approach. We start by considering massless Dirac fermions confined on the one-dimensional boundary of the topological insulator and interacting through a three-dimensional quantum dynamical electromagnetic field. Within these assumptions, through a dimensional-reduction procedure, we derive the effective 1 + 1-dimensional interacting fermionic theory and reveal its underlying gauge theory. In the low-energy regime, the gauge theory that describes the edge states is given by a conformal quantum electrodynamics (CQED), which can be mapped exactly into a HLL with a Luttinger parameter and a renormalized Fermi velocity that depend on the value of the fine-structure constant α.
Highlights
Topological insulators represent a large family of materials characterized by gapped bulks and metallic edge states
By using a Hubbard-Stratonovich transformation, we determine the effective 1 + 1-dimensional gauge theory that mediates the fermionic interaction, which is given by the sum of a conformal quantum electrodynamics (CQED)[14,15] plus the 1 + 1-dimensional massless QED, known as the Schwinger model[16,17]
It preserves the dimensionality of both, the electric charge and the gauge field of the 3 + 1-dimensional QED from which the CQED will be derived by using a dimensional reduction procedure
Summary
We start by considering two-dimensional time-reversal invariant topological insulators in class AII2. The dynamics of the edge modes can be described by a 1 + 1-dimensional massless Dirac theory with a two-component Dirac spinor ψ = (ψR, ψL)T, where ψR and ψL are the right-handed spin-up and left-handed spin-down chiral modes, respectively It was theoretically proposed in refs[12,13] and experimentally confirmed in ref.[23] that these topological insulators can support HLLs on the boundary due to the presence of unavoidable electron-electron interactions. These non-Fermi liquid phases fully preserve the time-reversal symmetry and are formally described by the free Dirac theory plus suitable four-fermion interactions.
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