Perfect Born Sampling of Symmetric Thermal Tensor Network for Quantum Lattice Models
Accurate calculations of quantum lattice models at low temperatures constitute a major challenge in many-body physics. Stochastic sampling of tensor-network states offers a promising route to tackle this problem; however, existing schemes have long faced a fundamental dilemma---sampling efficiency and symmetry acceleration \textit{cannot} be achieved simultaneously. Here we propose a perfect Born sampling approach for thermal tensor networks, which performs importance sampling directly from the purified density matrix via the Born rule and incorporates Abelian and non-Abelian symmetries by sampling symmetry quantum numbers. We benchmark the method, realized as both stochastic matrix product states (stoMPS) and stochastic projected entangled pair states (stoPEPS), on large-scale quantum lattice models. Using stoMPS, we accurately simulate the square-lattice Hubbard model on cylinders up to width $W=10$, and study the triangular-lattice Hubbard model down to $T/t = 1/64$, revealing scalar chiral order at half filling and kinetic ferromagnetism upon electron doping. We further extend the stoMPS method to compute finite-temperature quantum dynamics, as demonstrated by the optical conductivity of the Hubbard model, and generalize it to stoPEPS, as showcased on the $20\times20$ square-lattice quantum Ising model at its quantum critical point. Our method combines high sampling efficiency with full symmetry acceleration, and can be used as a state-of-the-art framework for studying both equilibrium and dynamical properties down to ultralow temperatures.