  Accurate simulations of the two-dimensional (2D) Hubbard model constitute one
of the most challenging problems in condensed matter and quantum physics. Here
we develop a tangent space tensor renormalization group (tanTRG) approach for
the calculations of the 2D Hubbard model at finite temperature. An optimal
evolution of the density operator is achieved in tanTRG with a mild $O(D^3)$
complexity, where the bond dimension $D$ controls the accuracy. With the tanTRG
approach we boost the low-temperature calculations of large-scale 2D Hubbard
systems on up to a width-8 cylinder and $10\times10$ square lattice. For the
half-filled Hubbard model, the obtained results are in excellent agreement with
those of determinant quantum Monte Carlo (DQMC). Moreover, tanTRG can be used
to explore the low-temperature, finite-doping regime inaccessible for DQMC. The
calculated charge compressibility and Matsubara Green's function are found to
reflect the strange metal and pseudogap behaviors, respectively. The
superconductive pairing susceptibility is computed down to a low temperature of
approximately $1/24$ of the hopping energy, where we find $d$-wave pairing
responses are most significant near the optimal doping. Equipped with the
tangent-space technique, tanTRG constitutes a well-controlled, highly efficient
and accurate tensor network method for strongly correlated 2D lattice models at
finite temperature.
