Time evolution of an infinite projected entangled pair state: An algorithm from first principles

A typical quantum state obeying the area law for entanglement on an infinite two-dimensional (2D) lattice can be represented by a tensor network Ansatz, known as an infinite projected entangled pair state (iPEPS), with a finite bond dimension D. Its real-imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension k times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension kD; hence, in principle, it can be represented accurately by a more compact Ansatz, preferably with the original D. In this work we show how the more compact iPEPS can be optimized variationally to maximize its overlap with the new iPEPS. To compute the overlap we use the corner-transfer-matrix renormalization group. By simulating sudden quench of the transverse field in the 2D quantum Ising model with the proposed algorithm, we provide a proof of principle that real-time evolution can be simulated with iPEPS. A similar proof is provided with the same model for imaginary-time evolution of purification of its thermal states.