models:spin_models:ising_model
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models:spin_models:ising_model [2018/05/05 10:04] jakobadmin [Intuitive] |
models:spin_models:ising_model [2018/12/28 11:52] (current) jakobadmin [Why is it interesting?] |
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| + | <WRAP lag>$$ \mathcal H=-J\sum_{\langle ij\rangle}S_iS_j$$</WRAP> | ||
| ====== Ising Model ====== | ====== Ising Model ====== | ||
| Line 9: | Line 10: | ||
| <tabbox Concrete> | <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. | ||
| + | |||
| + | 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 $k_B$). | ||
| + | |||
| + | There is a second-order phase transition at temperature $T_c$. This phase transition is characterised by | ||
| + | a high temperature phase with an average magnetization zero (disordered phase) | ||
| + | and a low temperature phase with a non-zero average magnetization (ordered phase). | ||
| + | |||
| + | 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]]. | ||
| + | |||
| + | ---- | ||
| * 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 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. | ||
| Line 16: | Line 38: | ||
| ---- | ---- | ||
| - | 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 defined completely general. | + | 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. |
| ---- | ---- | ||
| - | [{{ :models:image_20171011_163300.png?nolink |Source: Where do quantum field theories come from? by McGreevy}}] | + | **Recommended Resources** |
| + | * A Python simulation of the 2D Ising model, can be downloaded [[https://github.com/Damicristi/Ising-Model-in-2D|here]]. | ||
| - | [{{ :models:image_20171011_150737.png?nolink |Source: Where do quantum field theories come from? by McGreevy}}] | ||
| - | <tabbox Abstract> | ||
| - | <note tip> | + | <tabbox Abstract> |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
| <tabbox Why is it interesting?> | <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. | 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}}] | ||
| + | |||
| + | <-- | ||
| + | |||
| + | -->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> | </tabbox> | ||
models/spin_models/ising_model.1525507453.txt.gz · Last modified: 2018/05/05 08:04 (external edit)
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