  We establish the universal torus low-energy spectra at the free Dirac fixed
point and at the strongly coupled chiral Ising fixed point and their subtle
crossover behaviour in the Gross-Neuveu-Yukawa field theory with
${n_\text{D}=4}$ component Dirac spinors in $D=(2+1)$ dimensions. These fixed
points and the field theories are directly relevant for the long-wavelength
physics of certain interacting Dirac systems, such as repulsive spinless
fermions on the honeycomb lattice or $\pi$-flux square lattice. The torus
energy spectrum has been shown previously to serve as a characteristic
fingerprint of relativistic fixed points and is a powerful tool to discriminate
quantum critical behaviour in numerical simulations. Here, we use a combination
of exact diagonalization and quantum Monte Carlo simulations of strongly
interacting fermionic lattice models, to compute the critical torus energy
spectrum on finite-size clusters with periodic boundaries and extrapolate them
to the thermodynamic limit. Additionally, we compute the torus energy spectrum
analytically using the perturbative expansion in ${\epsilon = 4 - D}$, which is
in good agreement with the numerical results, thereby validating the presence
of the chiral Ising fixed point in the lattice models at hand. We show that the
strong interaction between the spinor field and the scalar order-parameter
field strongly influences the critical torus energy spectrum and we observe
prominent multiplicity features related to an emergent symmetry predicted from
the quantum field theory. Building on these results we are able to address the
subtle crossover physics of the low-energy spectrum flowing from the chiral
Ising fixed point to the Dirac fixed point, and analyze earlier flawed attempts
to extract Fermi velocity renormalizations from the low-energy spectrum.
