  We explore the relation between the entanglement of a pure state and its
energy variance for a local one dimensional Hamiltonian, as the system size
increases. In particular, we introduce a construction which creates a matrix
product state of arbitrarily small energy variance $\delta^2$ for $N$ spins,
with bond dimension scaling as $\sqrt{N} D_0^{1/\delta}$, where $D_0>1$ is a
constant. This implies that a polynomially increasing bond dimension is enough
to construct states with energy variance that vanishes with the inverse of the
logarithm of the system size. We run numerical simulations to probe the
construction on two different models, and compare the local reduced density
matrices of the resulting states to the corresponding thermal equilibrium. Our
results suggest that the spatially homogeneous states with logarithmically
decreasing variance, which can be constructed efficiently, do converge to the
thermal equilibrium in the thermodynamic limit, while the same is not true if
the variance remains constant.
