Theory of Network Contractor Dynamics for Exploring Thermodynamic Properties of Two-dimensional Quantum Lattice Models

Based on the tensor network state representation, we develop a nonlinear dynamic theory coined as network contractor dynamics (NCD) to explore the thermodynamic properties of two-dimensional quantum lattice models. By invoking the rank-$1$ decomposition in the multi-linear algebra, the NCD scheme makes the contraction of the tensor network of the partition function be realized through a contraction of a local tensor cluster with vectors on its boundary. An imaginary-time-sweep algorithm for implementation of the NCD method is proposed for practical numerical simulations. We benchmark the NCD scheme on the square Ising model, which shows a great accuracy. Besides, the results on the spin-1/2 Heisenberg antiferromagnet on honeycomb lattice are disclosed in good agreement with the quantum Monte Carlo calculations. The quasi-entanglement entropy $S$, Lyapunov exponent $I^{lya}$ and loop character $I^{loop}$ are introduced within the dynamic scheme, which are found to display the ``nonlocality" near the critical point, and can be applied to determine the thermodynamic phase transitions of both classical and quantum systems.