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Diffstat (limited to 'aps_mm_2018.html')
| -rw-r--r-- | aps_mm_2018.html | 41 |
1 files changed, 34 insertions, 7 deletions
diff --git a/aps_mm_2018.html b/aps_mm_2018.html index 8412808..b9746c1 100644 --- a/aps_mm_2018.html +++ b/aps_mm_2018.html @@ -39,12 +39,12 @@ Described by Hamiltonians $$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i)$$ -for \\(Z\\) invariant under rotations \\(R\\): \\(Z(R(s),R(t))=Z(s,t)\\). +for \\(Z\\) invariant under rotations \\(r\\): \\(Z(r(s),r(t))=Z(s,t)\\). <table style="border-collapse: collapse; table-layout: fixed; width: 80%; margin: auto;"> <thead style="border-bottom: 2px solid #000;"> <tr> - <td class="first"></td><td>\(s\)<td>\(R\)</td><td>\(Z(s_i,s_j)\)</td><td>\(H(s)\)</td> + <td class="first"></td><td>\(s\)<td>\(r\)</td><td>\(Z(s_i,s_j)\)</td><td>\(H(s)\)</td> </tr> </thead> <tbody> @@ -58,7 +58,7 @@ for \\(Z\\) invariant under rotations \\(R\\): \\(Z(R(s),R(t))=Z(s,t)\\). <td class="first">Potts model</td><td>\(\mathbb Z/q\mathbb Z\)</td><td>addition mod \(q\)</td><td>\(\delta(s_i,s_j)\)</td><td>\(\sum_iH_i\delta(i,s)\)</td> </tr> <tr> - <td class="first">Clock model</td><td>\(\mathbb Z/q\mathbb Z\)</td><td>addition mod \(q\)</td><td>\(\cos(2\pi\frac{s_i-s_j}q)\)</td><td>\(\sum_iH_i\cos(2\pi\frac{s-i}q)\)</td> + <td class="first">Clock model</td><td>ℤ/<i>q</i>ℤ</td><td>addition mod \(q\)</td><td>\(\cos(2\pi\frac{s_i-s_j}q)\)</td><td>\(\sum_iH_i\cos(2\pi\frac{s-i}q)\)</td> </tr> </tbody> </table> @@ -76,14 +76,14 @@ class: split-40 Standard approach to modelling arbitrary stat mech system: metropolis. 1. Pick random spin. - 2. Pick random rotation \\(R\\). + 2. Pick random rotation \\(r\\). 3. Compute change in energy \\(\Delta\mathcal H\\) resulting from taking \\(s\\) to \\(R(s)\\). 4. Take \\(s\\) to \\(R(s)\\) with probability \\(\max\\{1,e^{-\beta\Delta\mathcal H}\\}\\). Problem: Scales very poorly near phase transitions. -Correlation time `\(\tau\sim L^z\)` at critical point, `\(\tau\sim t^{-z/\nu}\)` +Correlation time *τ* at critical point, *t* <sup>– *z/ν*</sup> `\(\tau\sim t^{-z/\nu}\)` approaching it. `\(z\)` takes large integer values for Ising, order-`\(n\)`, Potts model critical @@ -104,11 +104,11 @@ class: split-40 1. Pick random spin, add to cluster. 2. Pick random rotation `\(R\)`. 3. For every neighboring spin, add to cluster with probability - `\(\min\{0,1-e^{-\beta(Z(R(s),t)-Z(R(s),R(t)))}\}\)`. + `\(\min\{0,1-e^{-\beta(Z(R(s),R(t))-Z(R(s),t))}\}\)`. 4. Repeat 3 for every spin added to cluster. 5. Transform entire cluster with rotation `\(R\)`. -Relies on symmetry of `\(Z\)` +Relies on symmetry of *Z* Fast near the critical point: early studies thought `\(z\)` was zero, actually 0.1–0.4. @@ -152,6 +152,33 @@ Fast near the critical point: early studies thought `\(z\)` was zero, actually  +--- + +# Why is the extended method useful? + +<img src="figs/vector-1.svg" alt="order-n" style="float: left;"/> +<img src="figs/vector-2.svg" alt="order-n" style="float: right;"/> +<span style="font-size: 40pt; overflow: hidden; height: 6.5em; text-align: center; align-items: center; display: inline-flex; position: static; float: center"> <i>R</i> <br>→</span> + +--- + +# Why is the extended method useful? + +<img src="figs/potts.svg" alt="order-n"/> + +--- + +# Correlation time scaling + +Correlation time scales consistently in the whole phase space! + +<img src="figs/autocorr-scaling-isotherm.png" style="float: left; width: 48%" /> +<img src="figs/autocorr-scaling-temp.png" style="float: right; width: 48%" /> + +--- + +# Metastable state direct measurement + </textarea> <script src="https://remarkjs.com/downloads/remark-latest.min.js"> </script> |