  For every $\alpha < 1/3$, we construct an explicit divergence-free vector
field $\mathbf{b}(t,x)$ which is periodic in space and time and belongs to
$C^0_t C^{\alpha}_x \cap C^{\alpha}_t C^0_x$ such that the corresponding scalar
advection-diffusion equation $$\partial_t \theta^\kappa + \mathbf{b} \cdot
\nabla \theta^\kappa - \kappa \Delta \theta^\kappa = 0$$ exhibits anomalous
dissipation of scalar variance for arbitrary $H^1$ initial data:
$$\limsup_{\kappa \to 0} \int_0^{1} \int_{\mathbb{T}^d} \kappa | \nabla
\theta^\kappa (t,x) |^2 \,dx\,dt >0.$$ The vector field is deterministic and
has a fractal structure, with periodic shear flows alternating in time between
different directions serving as the base fractal. These shear flows are
repeatedly inserted at infinitely many scales in suitable Lagrangian
coordinates. Using an argument based on ideas from quantitative homogenization,
the corresponding advection-diffusion equation with small $\kappa$ is
progressively renormalized, one scale at a time, starting from the (very small)
length scale determined by the molecular diffusivity up to the macroscopic
(unit) scale. At each renormalization step, the effective diffusivity is
enhanced by the influence of advection on that scale. By iterating this
procedure across many scales, the effective diffusivity on the macroscopic
scale is shown to be of order one.
