Spin and Charge Orders in the Doped Two-Dimensional Hubbard Model at Finite Temperature

Competing orders, including inhomogeneous spin and charge orders, are observed in many correlated electron materials, including the high-temperature superconductors. The two-dimensional Hubbard model provides a minimal paradigm for studying these orders. Using constrained-path auxiliary-field quantum Monte Carlo, We study the interplay between thermal and quantum fluctuations in this model. Reaching large supercell sizes to extract properties in the thermodynamics limit, we obtain an accurate and systematic characterization of the behaviors of the spin and charge orders as a function of temperature. In all three electron densities, we find increasing short-range antiferromagnetic correlations as temperature is lowered. As the correlation length grows sufficiently large, a modulating wave appears to produce spin-density-wave (SDW). In the case of ρ=0.9 and 0.875, this evolves smoothly into the ground-state long-range SDW order. In the case of ρ=0.8, the SDW remains short-ranged as temperature is lowered to zero. We study the interplay between spin and charge orders and find that formation of charge order appears to follow that of spin order. This leads to a very low upper bound for the transition temperature for CDW or stripe order.