  Competing and intertwined orders including inhomogeneous patterns of spin and
charge are observed in many correlated electron materials, such as
high-temperature superconductors. Introducing a new development of the
constrained-path auxiliary-field quantum Monte Carlo (AFQMC) method, we study
the interplay between thermal and quantum fluctuations in the two-dimensional
Hubbard model. We obtain an accurate and systematic characterization of the
evolution of the spin and charge correlations as a function of temperature $T$
and how it connects to the ground state, at three representative doping levels
$\delta = 1/5$, $1/8$, and $1/10$. We find increasing short-range commensurate
antiferromagnetic correlations as $T$ is lowered. As the correlation length
grows sufficiently large, a modulated spin-density-wave (SDW) appears. At
$\delta = 1/5$, the SDW saturates and remains short-ranged as $T \rightarrow
0$. In contrast, at $\delta = 1/8$ and $1/10$ this evolves into a ground-state
stripe phase. We study the relation between spin and charge orders and find
that formation of charge order appears to be driven by that of the spin order.
We identify a finite-temperature phase transition below which charge ordering
sets in and discuss the implications of our results for the nature of this
transition.
