  We present quantum Monte Carlo simulations for the chiral Heisenberg
Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with $N=4$
Dirac spinor components subject to a repulsive, local four fermion interaction
in 2+1$d$. Here we employ a two dimensional lattice Hamiltonian with a single,
spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum
relation for all finite size lattice momenta in the absence of interactions.
This allows us to significantly reduce finite size corrections compared to the
widely studied honeycomb and $\pi$-flux lattices. A Hubbard term dynamically
generates a mass beyond a critical coupling of ${U_c = 6.76(1)}$ as the system
acquires antiferromagnetic order and SU(2) spin rotational symmetry is
spontaneously broken. At the quantum phase transition we extract a
self-consistent set of critical exponents ${\nu = 0.98(1)}$, ${\eta_{\phi} =
0.53(1)}$, ${\eta_{\psi} = 0.18(1)}$, ${\beta = 0.75(1)}$. We provide evidence
for the continuous degradation of the quasi-particle weight of the fermionic
excitations as the critical point is approached from the semimetallic phase.
Finally we study the effective "speed of light" of the low-energy relativistic
description, which depends on the interaction $U$, but is expected to be
regular across the quantum phase transition. We illustrate that the strongly
coupled bosonic and fermionic excitations share a common velocity at the
critical point.
