  We present a lattice calculation of the $K\to\pi\pi$ matrix elements and
amplitudes with both the $\Delta I = 3/2$ and 1/2 channels and $\varepsilon'$,
the measure of direct $CP$ violation. We use periodic boundary conditions
(PBC), where the correct kinematics of $K\to\pi\pi$ can be achieved via an
excited two-pion final state. To overcome the difficulty associated with the
extraction of excited states, our previous work \cite{Bai:2015nea,RBC:2020kdj}
successfully employed G-parity boundary conditions, where pions are forced to
have non-zero momentum enabling the $I=0$ two-pion ground state to express the
on-shell kinematics of the $K\to\pi\pi$ decay. Here instead we overcome the
problem using the variational method which allows us to resolve the two-pion
spectrum and matrix elements up to the relevant energy where the decay
amplitude is on-shell.
  In this paper we report an exploratory calculation of $K\to\pi\pi$ decay
amplitudes and $\varepsilon'$ using PBC on a coarser lattice size of
$24^3\times64$ with inverse lattice spacing $a^{-1}=1.023$ GeV and the physical
pion and kaon masses. The results are promising enough to motivate us to
continue our measurements on finer lattice ensembles in order to improve the
precision in the near future.
