Tensor networks (or TNs) have played a central role in furthering our understanding of classical and quantum many-body systems. TNs not only provide a practical tool for quantum simulations but form a key building block of various fields in theoretical physics. Given the importance of theoretical backgrounds of various TNs, a team of researchers from Japan set out to provide a unified description of the developments in TNs from the statistical mechanics perspective.

They began by looking at the famous 2D Ising model. They showed that using a vertex model representation with 4-leg tensors and a row-to-row transfer matrix T leads very naturally to a variational state in the matrix product state (or MPS) form, which reflects the transfer matrix structure of the square lattice. Further consideration of the variation of the largest eigenvalue of T showed the equivalence of this variation with the variational principle of the MPS for the 1D quantum system. Notably, this consideration also led to the corner transfer matrix (or CTM).

Next, the team highlighted that a systematic formulation of recursive relations for CTMs leads to a real space renormalization group algorithm known as corner transfer matrix renormalization group. In this, the variational principle for the partition function is reformulated for the CTM. The researchers then pointed out the difficulties of recasting this CTM formulation to the 1D quantum system and how they can be bypassed in the formulation of infinite time-evolved block decimation and density matrix renormalization group.

They also highlighted the higher dimension generalization of MPS, such as tensor product states or projected entangled pair states. They then moved onto tensor renormalization groups or TRGs, focusing on their fixed point structures. Lastly, they discussed how tensor network renormalization can help overcome the difficulties associated with TRGs for critical systems, and the multi-scale entanglement renormalization ansatz.

Overall, an understanding of the fundamental background of TN approaches could lead to the development of reliable TN simulations and accelerate the development of quantum technologies, such as quantum computers.