### Thermal Pure Quantum State

In textbooks of statistical physics, it is shown that finite-temperature properties (for example, internal energies or specific heat) can be calculated by the ensemble averages, which take the averages of physical properties with respect to the Boltzmann weight. To take the ensemble average numerically, it is necessary to perform the full diagonalization. The numerical cost of the full diagonalization is very high and it is almost impossible to calculate the finite-temperature properties for realistic systems.

However, recent progress of the quantum statistical physics shows that the finite-temperature properties can be calculated by the expectation values of the single wave function in principle. In the pioneering work by Imada and Takahashi, it was pointed out that finite-temperature properties can be calculated by the expectation values of the wave functions. After that, several people rediscovered or proved that ensemble average can be replaced by the expectation value of the wave function [J.Jaklic and P.Prelovsek PRB 1994, A. Hams and H. de Raedt PRE 2000, S.Lloyd 1988].

In the paper of 2012, Sugiura and Shimizu prove that the finite-temperature properties can be exactly calculated by the single wave functions in the thermodynamic limit and show the simple way to construct the single wave function. They call the wave function **“Thermal Pure Quantum (TPQ) State”**. The TPQ method is implemented in HΦ and it is easy to perform the finite-temperature calculation for a wide range of the quantum lattice model such as the Hubbard model and the Heisenberg model. Recent applications of the TPQ state using HΦ to the frustrated Hubbard model is found in the paper.