  A search for the non-resonant production of Higgs boson pairs in the
$HH\rightarrow b\bar{b}\tau^+\tau^-$ channel is performed using 140 fb$^{-1}$
of proton-proton collisions at a centre-of-mass energy of $13$ TeV recorded by
the ATLAS detector at the CERN Large Hadron Collider. The analysis strategy is
optimised to probe anomalous values of the Higgs boson self-coupling modifier
$\kappa_\lambda$ and of the quartic $HHVV$ ($V = W,Z$) coupling modifier
$\kappa_{2V}$. No significant excess above the expected background from
Standard Model processes is observed. An observed (expected) upper limit
$\mu_{HH}<5.9$ $(3.3)$ is set at 95% confidence-level on the Higgs boson pair
production cross-section normalised to its Standard Model prediction. The
coupling modifiers are constrained to an observed (expected) 95% confidence
interval of $-3.1 < \kappa_\lambda < 9.0$ ($-2.5 < \kappa_\lambda < 9.3$) and
$-0.5 < \kappa_{2V} < 2.7$ ($-0.2 < \kappa_{2V} < 2.4$), assuming all other
Higgs boson couplings are fixed to the Standard Model prediction. The results
are also interpreted in the context of effective field theories via constraints
on anomalous Higgs boson couplings and Higgs boson pair production
cross-sections assuming different kinematic benchmark scenarios.
