A new computational method for finite-Temperature properties of strongly correlated electrons is proposed by extending the variational Monte Carlo method originally developed for the ground state. The method is based on the path integral in the imaginary-Time formulation, starting from the infinite-Temperature state that is well approximated by a small number of certain random initial states. Lower temperatures are progressively reached by the imaginary-Time evolution. The algorithm follows the framework of the quantum transfer matrix and finite-Temperature Lanczos methods, but we extend them to treat much larger system sizes without the negative sign problem by optimizing the truncated Hilbert space on the basis of the time-dependent variational principle (TDVP). This optimization algorithm is equivalent to the stochastic reconfiguration (SR) method that has been frequently used for the ground state to optimally truncate the Hilbert space. The obtained finite-Temperature states allow an interpretation based on the thermal pure quantum (TPQ) state instead of the conventional canonical-ensemble average. Our method is tested for the one-And two-dimensional Hubbard models and its accuracy and efficiency are demonstrated.
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