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--- models:spin_models:ising_model [2018/02/23 17:35]
jakobadmin
+++ models:spin_models:ising_model [2018/12/28 11:52] (current)
jakobadmin [Why is it interesting?]
@@ Line 1: / Line 1: @@
+<WRAP lag>$$ \mathcal H=-J\sum_{\langle ij\rangle}S_iS_j$$</WRAP>
 ====== Ising Model ======
-<tabbox Why is it interesting?>
-The Ising model is the simplest model with a critical point and thus ideal to understand notions like [[advanced_notions:critical_exponent|critical exponents]] and the [[advanced_tools:renormalization_group|renormalization group]] in a simplified setup.
+<tabbox Intuitive>
+{{ :models:isingmodel.png?nolink&200|}}
+The Ising model consists of a linear chain (or lattice) where at each node we have an either an arrow that points upwards or an arrow that points downwards. The force between two neighboring lattice nodes depends on whether they point in the same direction or not.
+<tabbox Concrete>
+The Hamiltonian of the Ising model is:
+\begin{equation}
+ \mathcal H=-J\sum_{\langle ij\rangle}S_iS_j\,,\label{HIsing}
+\end{equation}
+where the summation runs only over nearest-neighbor pairs $\langle ij\rangle$ on
+the lattice. $J$ is the interaction strength.
-<tabbox Layman>
+The [[theories:statistical_mechanics|statistical properties]] of the system are obtained from the corresponding **partition function**
+\begin{equation}
+ Z=\sum_{\mathcal C} e^{-\beta  E({\mathcal C})}\,,
+\end{equation}
+where the summation runs over all possible configurations $\mathcal C$ and $E({\mathcal C})$ denotes the energy of a given configuration. Moreover, $\beta\equiv 1/(k_B T)$ is the inverse temperature
+(temperature $T$ and Boltzmann constant $k_B$).
-<note tip>
+There is a second-order phase transition at temperature $T_c$. This phase transition is characterised by
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
+a high temperature phase with an average magnetization zero (disordered phase)
-</note>
+and a low temperature phase with a non-zero average magnetization (ordered phase).
-<tabbox Student>
-The best introduction can be found in Critical point phenomena: universal physics at large length scales by Bruce, A.; Wallace, D. published in the book The new physics, edited by P. Davies
+The Ising model is exactly solvable in one and two dimensions. In four-dimension, we can calculate the critical properties using the [[advanced_tools:renormalization_group|renormalization group]].
-It was originally devised as a simplified model of a ferromagnet, but is applicable also in a much broader sense. There is no need to refer to any ferromagnetic setup, but instead the Ising model can be defined completely general.
+----
-<tabbox Researcher>
-<note tip>
+  * The best introduction can be found in Critical point phenomena: universal physics at large length scales by Bruce, A.; Wallace, D. published in the book "The New Physics", edited by P. Davies.
-The motto in this section is: //the higher the level of abstraction, the better//.
+  * Another great introduction can be found [[ http://www.nptel.ac.in/courses/115103028/download/ch4_asm_MODELS_nptel.doc.pdf|here]].
-</note>
+  * See also [[https://www.maa.org/sites/default/files/pdf/upload_library/22/Hasse/00029890.di991727.99p0087h.pdf|An introduction to the Ising model]] by Barry A. Cipra, which is a good introduction for beginners.
---> Common Question 1#
+----
+The Ising model was originally devised as a simplified model of a ferromagnet, but is applicable also in a much broader sense. There is no need to refer to any ferromagnetic setup, but instead the Ising model can be used in a generalized sense.
-<--
---> Common Question 2#
+----
+**Recommended Resources**
-<--
-<tabbox Examples>
---> Example1#
+  * A Python simulation of the 2D Ising model, can be downloaded [[https://github.com/Damicristi/Ising-Model-in-2D|here]].
+<tabbox Abstract>
+<tabbox Why is it interesting?>
+The Ising model is the simplest model with a critical point and thus ideal to understand notions like [[advanced_notions:critical_exponent|critical exponents]] and the [[advanced_tools:renormalization_group|renormalization group]] in a simplified setup.
+<blockquote>The Ising Model has been called the Drosophila of condensed matter physics: a simple and well understood case on which the phenomena associated with phase transitions can be studied. <cite>http://philsci-archive.pitt.edu/8339/1/Is_more_different_2006.pdf</cite></blockquote>
+<tabbox FAQ>
+-->Where does the Ising model come from?#
+[{{ :models:image_20171011_163300.png?nolink |Source: Where do quantum field theories come from? by McGreevy}}]
 <--
---> Example2:#
+-->In what forms does the Ising model appear?#
+[{{ :models:image_20171011_150737.png?nolink |Source: Where do quantum field theories come from? by McGreevy}}]
+<tabbox Abstract>
 <--
 <tabbox History>
+  * A good summary of the history of the Ising model can be found in  [[http://personal.rhul.ac.uk/uhap/027/ph4211/PH4211_files/brush67.pdf|History of the lenz-ising model]] by Stephen G. Brush.
+  * A more personal account is [[https://arxiv.org/abs/1706.01764|The Fate of Ernst Ising and the Fate of his Model]] by Thomas Ising et. al.
 </tabbox>

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