  The phase diagram of a material is of central importance to describe the
properties and behaviour of a condensed matter system. We prove that the
general task of determining the quantum phase diagram of a many-body
Hamiltonian is uncomputable, by explicitly constructing a one-parameter family
of Hamiltonians for which this is the case. This work builds off recent results
from Cubitt et al. and Bausch et al., proving undecidability of the spectral
gap problem. However, in all previous constructions, the Hamiltonian was
necessarily a discontinuous function of its parameters, making it difficult to
derive rigorous implications for phase diagrams or related condensed matter
questions. Our main technical contribution is to prove undecidability of the
spectral gap for a continuous, single-parameter family of translationally
invariant, nearest-neighbour spin-lattice Hamiltonians on a 2D square lattice:
$H(\varphi)$ where $\varphi\in \mathbb R$. As well as implying uncomputablity
of phase diagrams, our result also proves that undecidability can hold for a
set of positive measure of a Hamiltonian's parameter space, whereas previous
results only implied undecidability on a zero measure set.
