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

TL;DR

This work shows that quadratic band touching in $(d+1)$-D free-fermion systems is a spatial conformal critical point described by the $d$-D symplectic fermion CFT with central charge $c=-2$ in 2D. It provides an explicit operator mapping between physical fermions and SF fields, and demonstrates that equal-time ground-state correlators reproduce SF correlators via a Grassmann path integral, establishing a quantum-classical correspondence akin to Rokhsar–Kivelson constructions but for fermions. In two spatial dimensions, the SF CFT yields a rich set of anyonic-like excitations (Dirichlet, Neumann, composite) whose string operators generate topological ground-state degeneracy on manifolds of nontrivial topology, and whose spin content exhibits non-diagonalizable (Jordan block) structure under rotations. The results bridge non-relativistic quantum criticality with logarithmic CFTs, showing that gapless QBT systems can host topological features typically associated with gapped topological phases, and open pathways for investigating entanglement, twists, and interactions within the SF framework.

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 $2π$ rotation on these excitations is represented by a Jordan block, reflecting the logarithmic character of the associated conformal field theory.

Figures (6)

• Figure 1: A square lattice (left) and a checkerboard lattice (right). Edges of the square lattice in the staggered orientation correspond to the vertices of the checkerboard lattice. The hopping coefficients for the Hamiltonian \ref{['discrete Hamiltonian']} are $-t_-$ along the horizontal and vertical full lines, $t_+$ along the horizontal and vertical dashed lines, and $-t = -(t_++t_-)$ along the diagonal lines.
• Figure 2: Realizations of the Dirichlet ($\theta$) excitation (left), Neumann ($\phi^*$) excitation (center), and the composite ($\phi^*\theta$) excitations (right) in a square lattice. The Dirichlet excitations are placed on vertices, and the Neumann excitations are placed on faces.
• Figure 3: Non-contractible string operators in the presence of Dirichlet and Neumann boundaries.
• Figure 4: Illustration of a Dirichlet boundary (left) and a Neumann boundary (right) on a square lattice.
• Figure 5: The $2\pi$ rotation of the Dirichlet excitation $\theta(\bm{x})$ is obtained by turning the integration contour around the point $\bm{x}$.