Quantum states, walks, tiles, and tensor networks

Israel Klich

A major challenge of physics is the complexity of many-body systems. While true for classical systems, the difficulty is exasperated in quantum systems, due to entanglement between system components and thus the need to keep track of an exponentially large number of parameters. In particular, this complexity places a challenge to numerical methods such as quantum Monte Carlo and tensor networks. Here, exactly solvable models are of crucial importance: we use these to test numerical procedures, to develop intuition, and as a starting point for approximations.

In this talk, I will explain our current understanding of a new solvable "walk” and “tile" models. The models show that entanglement may be more acute than previously thought, in particular, it features a novel quantum phase transition between a non-entangled phase and extensively entangled “rainbow” phase in 1d. Most remarkably, the model motivated the construction of a new tensor network, providing, after many years, the first example for an exact tensor network description of a critical system. More recently these models have been extended to higher dimensions.

Finally, I will remark on open problems, and on exciting connections to other fields such as the notion of holography in field theory, and a famous problem in non-equilibrium statistical mechanics.