Efficiency of the minimally entangled typical thermal state algorithm in quantum critical chains

Minimally entangled typical thermal states (METTS) are pure states that decompose the Gibbs state. By randomly sampling these states, one can compute finite temperature properties of quantum many-body systems. It is empirically known that METTS has only a small amount of entanglement and can be efficiently represented using a matrix product state. However, there has been a lack of analysis of the computational cost of the METTS algorithm, and its superiority over other simulation methods has not been clarified. In particular, in the case of quantum critical chains, the entanglement entropy of the ground state diverges, and thus, estimates of the computational cost in terms of the entropy break down. We study the computational efficiency of the METTS algorithm in quantum critical chains. We show that the entanglement entropy of generic METTS obeys the area law and grows logarithmically slowly with the inverse temperature. Furthermore, we find that it exhibits a universal behavior characterized only by the central charge of the conformal field theory that describes the critical point. Based on these results, we argue that the computational cost of constructing METTS is parametrically smaller than that of the purification of the Gibbs state in the low-temperature limit. Finally, we demonstrate that the METTS algorithm provides a significant speedup compared to the purification method when analyzing low-temperature thermal equilibrium states in quantum critical chains.