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We have proposed a new framework to solve various quantum many-body problems named a matrix product renormalization group (MPRG) [1]. MPRG solves the sign problem of conventional Monte Carlo methods and can be regarded as generalization of a density matrix renormalization group (DMRG) in one dimension. Compared with DMRG, MPRG is directly applicable to infinite systems, higher-dimensional systems, finite-temperature systems, and even to open quantum systems. In particular, a nonvariational variant of MPRG can be used to simulate non-Hermitian models like the Yang-Lee model with a Yang-Lee edge singularity.A variational variant of MPRG has a further application to many Hermitian systems. By utilizing a continuous projected entangled pair state (cPEPS), we can even solve two-dimensional systems at finite temperature. As for the accuracy, cPEPS outperforms PEPS with about a one-digit-higher precision when the same bond dimension is used. The finite-temperature observables like a specific heat are also calculated and compared with a quantum Mote Carlo simulation. Due to the absence of a sign problem, a Trotter error, or a finite-size effect, the observables can easily be extrapolated to the thermodynamic limit only by the bond dimension scaling.

[1] Masahiko G. Yamada et al., arXiv:2212.13267 (2022).

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