  We construct an explicit family of arithmetic Teichm\"uller curves
$\mathcal{C}_{2k}$, $k\in\mathbb{N}$, supporting
$\textrm{SL}(2,\mathbb{R})$-invariant probabilities $\mu_{2k}$ such that the
associated $\textrm{SL}(2,\mathbb{R})$-representation on $L^2(\mathcal{C}_{2k},
\mu_{2k})$ has complementary series for every $k\geq 3$. Actually, the size of
the spectral gap along this family goes to zero. In particular, the
Teichm\"uller geodesic flow restricted to these explicit arithmetic
Teichm\"uller curves $\mathcal{C}_{2k}$ has arbitrarily slow rate of
exponential mixing.
