  We provide a method for constructing finite temperature states of
one-dimensional spin chains displaying quantum criticality. These models are
constructed using correlators of products of quantum fields and have an
analytical purification. Their properties can be investigated by Monte-Carlo
simulations, which enable us to study the low-temperature phase diagram and to
show that it displays a region of quantum criticality. The mixed states
obtained are shown to be close to the thermal state of a simple nearest
neighbour Hamiltonian.
