Can we prove the lower bound for the sorting problem just by Turing machine model? It seems that available proof of sorting is based on the assumption that the algorithm only uses comparison so we can model it by decision tree and establish the lower bound of the number of comparisons.
If you are speaking specifically of sorting lists of integers on a multitape TM, then I think the answer is no. For example, comparison-based sorts, when implemented on a TM and sorting integers of bit-length $w$, take $\Theta(n w \log n)$ time, b/c it takes $\Theta(w)$ to compare two $w$-bit integers. But radix sort takes $O(nw)$ time, which is optimal as it is linear in the input size (assuming all the integers have the same number of bits).
See also fusion trees (but I think those are in the word RAM model and I'd have to carefully think through its translation to multitape TMs...)
The paper "Sorting and Element Distinctness on One-Way Turing Machines" by Holger Petersen shows a lower bound for sorting on a Turing machine with one work tape and one-way input.