Why, in one dimension, does the topological entropy scale with the size of system as $S \sim \ln L$, while in a 2D system it scales with $S \sim L$? Why does dimensionality play such an important role here? I mean, is there any simple but straightforward idea to understand these results?
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As pointed out by Prof. Wen, the question is not "correct" in itself. Below are some facts:
(1) For both gapped and gapless 1D systems, there is no topological entanglement entropy. For 1D gapped system, the entanglement entropy is $S\sim L^0$. For 1D critical system (CFT), the entanglement entorpy is $S\sim\log L$.
(2) For 2D gapped system, the entanglement entropy is $S\sim \alpha L$, i.e., boundary law behavior. In particular, for 2D topological ordered system, there is a subleading universal piece $S\sim \alpha L-\gamma_{top}$, where $\gamma_{top}$ is called the topological entanglement entropy.
(3) For $d$ dimensional gapped system, the leading term is always the boundary law term, i.e., $S\sim L^{d-1}$.
(4) For $d$ dimensional gapless system, the result is very interesting. For example, for free fermions with a finite fermi surface, it is found that $S\sim L^{d-1}\log L$. Intuitively, one can consider the fermi surface in terms of a collection of 1D CFT. There is a good reference on this issue: http://arxiv.org/abs/0908.1724.
IsingX
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In a 1D system, all you can do is vary the size of a subset, which only ever gives $\propto L$ possibilities. Then the entropy takes the logarithm, and we have $\propto \ln L$.
In higher dimensions however, you can also vary the shape, and that is combinatorially much more powerful: you have exponentially many possibilities (think of how you can thread a wire through a grid). The logarithm is then merely able to resolve that into a linear relationship, so it's $S \propto L$ here.
leftaroundabout
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