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transfer_matrix [2021/04/29 20:01]
Scott Lawrence [Transfer matrix] 		transfer_matrix [2021/04/29 20:03] (current)
Scott Lawrence [Transfer matrix]
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 = \sum_{s_1,\ldots,s_N \in \{0,1\}} \langle s_1 | T | s_N \rangle \cdots \langle s_2 | T |s_1 \rangle = \sum_{s_1,\ldots,s_N \in \{0,1\}} \langle s_1 | T | s_N \rangle \cdots \langle s_2 | T |s_1 \rangle
 \end{equation} \end{equation}
-Here we have introduced $T \equiv e^{-\beta H / N}$, the transfer matrix.+Here we have introduced $T \equiv e^{-\beta H / N}$, the transfer matrix. The final expression has the form of a path integral, as $\langle s' | T | s \rangle$ can be written $e^{-J s' s}$ for some $J$ (a function of $\mu$). As a result, we have derived a Euclidean action of the form 
 +\begin{equation} 
 +S = -J \sum_{\langle i j \rangle} s_i s_j\text, 
 +\end{equation} 
 +and we see that the two-level system lets us solve the one-dimensional Ising model!
  
 This story can be told in reverse, where we start from an apparently difficult high-dimensional integral (like the Ising model in one dimension), and reduce it to a low-dimension matrix by expressing it in terms of a transfer matrix. This story can be told in reverse, where we start from an apparently difficult high-dimensional integral (like the Ising model in one dimension), and reduce it to a low-dimension matrix by expressing it in terms of a transfer matrix.
Last modified: 2021/04/29 20:01
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