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Edit: Sorry, my first answer was not the one expected. I let it below the correct answer I detail now: It's always tricky to discuss exponentials of operators. The good thing is that it's always the same trick: you use $e^{A}\approx1+A$ in both directions, valid for small $A$. To get exact results you also use that $e^{At}=\prod_{i}\left(1+A\Delta ...


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 ...

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