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--- transfer_matrix [2021/04/29 19:57]
Scott Lawrence
+++ transfer_matrix [2021/04/29 20:03] (current)
Scott Lawrence [Transfer matrix]
@@ Line 9: / Line 9: @@
 We can derive a (Euclidean) path integral for this Hamiltonian by starting from the thermal partition function and inserting many copies of the identity operator $\left|0\right>\left<0\right| + \left|1\right>\left<1\right|$.
 \begin{equation}
-Z(\beta) = \Tr
+Z(\beta) = \operatorname{Tr} e^{-\beta H}
+= \operatorname{Tr} \left(e^{-\beta H / N}\right)^N
+= \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}
+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.

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