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I've read in a few papers things that use the fact that the $2D$ Ising model can be interpreted as a $1+1$ quantum spin model. I haven't been able to find this description and would like to read about it. Could anyone recommend a book with this description?

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Your question, "Ising model as quantum model?", really opens a Pandora's box. Anyway, yes, in general, an $N\text{-dimensional}$ quantum (Ising) model can be mapped to an $N+1$ classical (Ising) model.

For a general start, maybe you could refer this book:

You will find your questions addressed in the first chapter.

Also, and this is a bit of a spoiler alert if you want to figure it out by yourself, but the extra dimension in the classical case is time. It is best understood using the path integral formalism.

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