<i>Ab initio</i> quantum embedding at finite temperature with density matrix embedding theory

We present a finite-temperature extension of density matrix embedding theory (FT-DMET) for realistic crystalline systems. We describe a practical framework for constructing extended bath orbitals, solving the embedding problem, and performing DMET self-consistency at finite temperature. To reduce computational cost, we introduce strategies based on mutual-information-guided bath truncation, controlled treatment of the thermal electron number without explicit optimization, and the use of low-temperature impurity solvers and one-shot FT-DMET in the low-temperature regime. We apply this approach to periodic hydrogen chains and square lattices to characterize their finite-temperature phases. We observe the Pomeranchuk-like effect in one dimension and enhanced stability of long-range order in two dimensions.