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Matrix product states satisfy the entanglement area law, which should be a property of gapped states.

But usually, MPS work well in 1D quantum phase transition problems.

As far as I know, entanglement at critical point should satisfy the log-divergence.

So why do MPS work well at a critical point?

And I also hope to know the reason why MPS fail in 2D?

Thanks

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In order to capture a system with a entanglement which scales like $S\sim\log(L)$, the bond dimension has to (roughly) grow like $D\sim e^S\sim \mathrm{poly(L)}$. Thus, the computational resources required will still scale polynomially with the system size, even for critical systems.

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