The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in (2+1) dimensions is known to be $O\left(2N\right)$, 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, $USp\left(2N\right)$. All fermion bilinears belong to one of the three small irreducible representations of this group. The interacting theory that respects the $USp\left(2N\right)$ 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 $USp\left(N\right)\times USp\left(N\right)$. The symmetry in 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 $O\left(2N\right)\bigcap USp\left(2N\right)=U\left(N\right)$. Some consequences of the symplectic symmetry and its spontaneous breaking are briefly discussed.

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• Dirac semimetal
• Quantum field theory
• Continuous symmetries in condensed matter