$c=-2$ conformal field theory in quadratic band touching

TL;DR

This paper demonstrates that a quadratic band touching fermionic system in (d+1) dimensions exhibits conformal invariance and is described by the $d$-dimensional symplectic fermion theory, revealing topological and logarithmic conformal features.

Contribution

It establishes a precise correspondence between quadratic band touching models and symplectic fermion conformal field theories, including explicit operator mappings and topological implications.

Findings

Spatial conformal invariance in quadratic band touching models

Exact correlation functions via symplectic fermion theory

Identification of anyonic excitations and topological degeneracy

Abstract

Quadratic band touching in fermionic systems defines a universality class distinct from that of linear Dirac points, yet its characterization as a quantum critical point remains incomplete. In this work, I show that a $(d+1)$-dimensional free-fermion model with quadratic band touching exhibits spatial conformal invariance, and that its equal-time ground-state correlation functions are exactly captured by the $d$-dimensional symplectic fermion theory. I establish this correspondence by constructing explicit mappings between physical fermionic operators and the fields of the symplectic fermion theory. I further explore the implications of this correspondence in two spatial dimensions, where the symplectic fermion theory is a logarithmic conformal field theory with central charge $c=-2$. In the corresponding $(2+1)$-dimensional systems, I identify anyonic excitations originating from the underlying symplectic fermion theory, even though the Hamiltonian is gapless. Transporting these excitations along non-contractible loops generates transitions among topologically degenerate ground states, in close analogy with those in topologically ordered phases. Moreover, the action of a $2\pi$ rotation on these excitations is represented by a Jordan block, reflecting the logarithmic character of the associated conformal field theory.