# Difference between multi-tape Turing machine and single tape machine

A beginner's question about "fine-grained" computational power.
Let $$M_k$$ be a $$k$$-tapes turing machine, and let $$M$$ be a single tape turing machine. We know that $$M_k$$ and $$M$$ both have the same "computable power". In addition, one can simulate $$M_k$$ on $$M$$ in a way that every computation which takes $$O(t(n))$$ on $$M_k$$ will take $$O(t(n) \log(t(n))$$ on $$M$$.
Here is my question:
Is there a language $$L$$ such that $$L$$ can be decided in $$O(n)$$ time in $$k$$-tape Turing machine (for fixed $$k$$, say 2), but can't be decided in $$O(n)$$ time in a single tape machine? (every single tape machine which decides $$L$$ needs $$\Omega(n \log n)$$ time).
In addition, are there any examples of two computational models (classical, not the quantum model) with the same computable power, but with fine-grained differences in their running time? (I guess that major changes in running time would contradict the extended Church-Turing thesis, which is less likely).