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

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