Optimization of tensor network states by means of a minimal principle and applications to quantum systems

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The tensor network states has a hight potential of expressing or approximating a given quantum state wave function. For an efficient optimization of each local tensor, the construction of the environment around the tensor is important. In this study, we focus on the boundary condition, i.e. the smooth boundary condition, where the interaction parameter gradually decreases to zero toward the system boundary, the hyperbolic deformation, where the parameter increases to the boundary. These conditions are imposed to one dimensional free or correlated quantum systems. Under these deformation, despite of the non-uniformity in the Hamiltonian, the corresponding ground state is uniform, and the local tensors are optimized in a homogeneous manner. The fact is confirmed by means of the density matrix renormalization group, which is a representative numerical tool to optimize local tensors. Several other trials on the tensor network state are also reported.