  In this paper, we perform a comprehensive study of the renormalization group
(RG) method on thermal tensor networks (TTN). By Trotter-Suzuki decomposition,
one obtains the 1+1D TTN representing the partition function of 1D quantum
lattice models, and then employs efficient RG contractions to obtain the
thermodynamic properties with high precision. The linearized tensor
renormalization group (LTRG) method, which can be used to contract TTN in an
efficient and accurate way, is briefly reviewed. In addition, the single-layer
LTRG can be generalized to a bilayer form, dubbed as LTRG++, in both finite-
and infinite-size systems, with accuracies significantly improved. We provide
the details of LTRG++ in finite-size system, comparing its accuracy with
single-layer algorithm, and elaborate the infinite-size LTRG++ in the context
of fermion chain model. We show that the LTRG++ algorithm in infinite-size
system is in essence equivalent to transfer-matrix renormalization group
method, while expressed in a tensor network language. LTRG++ is then applied to
study an extended Hubbard model, where the phase separation phenomenon,
groundstate phase diagram, as well as quantum criticality-enhanced
magnetocaloric effects under external fields, are investigated.
