Symplectic symmetry of quadratic-band-touching Hamiltonians in two dimensions
Igor F. Herbut, Samson C. H. Ling
TLDR
This paper reveals that 2D quadratic-band-touching Hamiltonians possess an internal USp(2N) symplectic symmetry, analogous to Dirac systems.
Key contributions
- Identifies USp(2N) as the internal symmetry for 2D quadratic-band-touching Hamiltonians.
- Classifies fermionic bilinears into small irreducible representations of this USp(2N) group.
- Constructs an interacting theory with USp(2N) symmetry, allowing two independent interaction terms.
- Shows lattice symmetries like honeycomb are U(N), the overlap of orthogonal and symplectic groups.
Why it matters
This work establishes a fundamental internal symmetry for quadratic-band-touching systems, analogous to Dirac Hamiltonians. Understanding this USp(2N) symmetry is crucial for exploring their interacting phases and potential topological phenomena. It provides a new framework for theoretical condensed matter physics.
Original Abstract
The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in $2+1$ dimensions is known to be $O(2N)$, where $N$ is the number of two-component Dirac fermions. Here we point out that there exists an analogous internal symmetry of the single-particle quadratic-band-touching Hamiltonian in two spatial dimensions, and it is the unitary symplectic group, $USp(2N)$. All fermionic bilinears belong to one of the three small irreducible representations of this group. The interacting theory that respects the $USp(2N)$ symmetry and the spatial rotations is constructed and found to allow two independent interaction terms. When these interactions are infrared-relevant the symplectic symmetry either remains preserved or becomes spontaneously broken to $USp(N) \times USp(N)$. The symmetry in the lattices such as honeycomb to infinite order in the dispersion's expansion in powers of local momentum is given by the overlap of the symplectic and the orthogonal groups. We show that this overlap is $O(2N) \bigcap USp(2N) = U(N)$.
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