diff options
Diffstat (limited to 'aps_mm_2018.html')
| -rw-r--r-- | aps_mm_2018.html | 155 |
1 files changed, 109 insertions, 46 deletions
diff --git a/aps_mm_2018.html b/aps_mm_2018.html index b9746c1..a3f6d91 100644 --- a/aps_mm_2018.html +++ b/aps_mm_2018.html @@ -19,7 +19,9 @@ <link rel="stylesheet" type="text/css" href="fonts/Concrete/cmun-concrete.css"> </head> <body> - <textarea id="source"> + + <textarea id="source"> + class: center, middle @@ -35,30 +37,30 @@ class: center, middle # Spin systems -Described by Hamiltonians +Described by Hamiltonians whose degrees of freedom are spins `\(s\in X\)` $$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i)$$ -for \\(Z\\) invariant under rotations \\(r\\): \\(Z(r(s),r(t))=Z(s,t)\\). +for \\(Z\\) invariant under the action of rotations \\(r\in R\\), or \\(Z(r\cdot s,r\cdot t)=Z(s,t)\\). -<table style="border-collapse: collapse; table-layout: fixed; width: 80%; margin: auto;"> +<table style="border-collapse: collapse; width: 100%; 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><i>X</i><td><i>R</i></td><td>Group Action</td><td>\(Z(s,t)\)</td><td>\(H(s)\)</td> </tr> </thead> <tbody> <tr> - <td class="first">Ising model</td><td>\(\{-1,1\}\)</td><td>\(s\mapsto-s\)</td><td>\(s_is_j\)</td><td>\(Hs\)</td> + <td class="first">Ising model</td><td>\(\{-1,1\}\)</td><td>ℤ/2ℤ</td><td>\(0\cdot s\mapsto s\)<br>\(1\cdot s\mapsto-s\)</td><td>\(s\times t\)</td><td>\(Hs\)</td> </tr> <tr> - <td class="first">Order-\(n\) model</td><td>\(S^n\)</td><td>\(\mathop{\mathrm{SO}}(n)\) (rotation)</td><td>\(s_i\cdot s_j\)</td><td>\(H\cdot s\)</td> + <td class="first">Order-\(n\) model</td><td>\(S^n\)</td><td>\(\mathop{\mathrm{O}}(n)\)</td><td>\(O\cdot s\mapsto Os\)</td><td>\(s^{\mathrm T}t\)</td><td>\(H\cdot s\)</td> </tr> <tr> - <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> + <td class="first">Potts model</td><td>ℤ/<i>q</i>ℤ</td><td>ℤ/<i>q</i>ℤ</td><td>\(m\cdot s\mapsto{(s+m)}\pmod q\)</td><td>\(\delta(s,t)\)</td><td>\(\sum_mH_m\delta(m,s)\)</td> </tr> <tr> - <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> + <td class="first">Clock model</td><td>ℤ/<i>q</i>ℤ</td><td>ℤ/<i>q</i>ℤ</td><td>\(m\cdot s\mapsto{(s+m)}\pmod q\)</td><td>\(\cos(2\pi\frac{s-t}q)\)</td><td>\(\sum_mH_m\cos(2\pi\frac{s-m}q)\)</td> </tr> </tbody> </table> @@ -76,17 +78,17 @@ class: split-40 Standard approach to modelling arbitrary stat mech system: metropolis. 1. Pick random spin. - 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}\\}\\). + 2. Pick random rotation \\(r\in R\\). + 3. Compute change in energy \\(\Delta\mathcal H\\) resulting from taking \\(s\\) to \\(r\cdot s\\). + 4. Take \\(s\\) to \\(r\cdot s\\) with probability \\(\max\\{1,e^{-\beta\Delta\mathcal H}\\}\\). Problem: Scales very poorly near phase transitions. -Correlation time *τ* at critical point, *t* <sup>– *z/ν*</sup> `\(\tau\sim t^{-z/\nu}\)` +Correlation time `\(\tau\sim L^z\)` at critical point, `\(\tau\sim t^{-z/\nu}\)` approaching it. - `\(z\)` takes large integer values for Ising, order-`\(n\)`, Potts model critical + `\(z\)` takes large integer values for Ising, order-`\(n\)`, Potts model critical regions. ] .column[ @@ -100,18 +102,19 @@ class: split-40 # Wolff: wow, what a solution .column[ +Non-local updates! - 1. Pick random spin, add to cluster. - 2. Pick random rotation `\(R\)`. - 3. For every neighboring spin, add to cluster with probability - `\(\min\{0,1-e^{-\beta(Z(R(s),R(t))-Z(R(s),t))}\}\)`. - 4. Repeat 3 for every spin added to cluster. - 5. Transform entire cluster with rotation `\(R\)`. + 1. Pick random spin `\(s\in X\)`, add to cluster. + 2. Pick random rotation `\(r\in R\)`. + 3. For every neighboring spin `\(t\)`, add to cluster with probability + `\(\min\{0,1-e^{-\beta(Z(r\cdot s,r\cdot t)-Z(r\cdot s,t))}\}\)`. + 4. Repeat 3 for every new spin added to cluster. + 5. Transform entire cluster by applying the action of rotation `\(r\)`. -Relies on symmetry of *Z* +Relies on symmetry of *Z* under group of rotations. -Fast near the critical point: early studies thought `\(z\)` was zero, actually - 0.1–0.4. +Fast near the critical point: early studies thought `\(z\)` was zero for 2D Ising, actually + 0.1–0.4, below one for models listed here. ] @@ -121,36 +124,50 @@ Fast near the critical point: early studies thought `\(z\)` was zero, actually --- -# We want to apply an external field, though +  + +--- + +  + +--- + +class: split-40 + +.column[ - The external field `\(H\)` is not invariant under global rotations! +# Applying an *arbitrary* field - Let's make it that way: introduce an extra spin `\(s_0\)`, let `\(R_s\)` be the rotation that takes `\(s\)` to the - identity + Introduce an extra “spin” `\(r_0\in R\)` whose possible configurations come from the set of *rotations*, not the set of spin states. + + Mark this spin a neighbor of every spin on the lattice. + + New effective coupling defined by `\[ - \tilde Z(s_i,s_j)= + \tilde Z(s,t)= \begin{cases} - Z(s_i,s_j) & \text{if $i,j\neq0$}\\ - H(R_{s_0}s_i) & \text{if $j=0$}\\ - H(R_{s_0}s_j) & \text{if $i=0$} + Z(s,t) & \text{if $s,t\in X$}\\ + H(t^{-1}\cdot s) & \text{if $t\in R$}\\ + H(s^{-1}\cdot t) & \text{if $s\in R$} \end{cases} \]` + + Preform ordinary Wolff algorithm on the extended phase space. + Exact correspondence between expectation values of operators in old and new - models: if `\(A(s)\)` is an observable on old model, `\(\tilde - A(s_0,s)=A(R_{s_0}s)\)` has the property + models: if `\(A(\{s\})\)` is an observable on old model in field, `\(\tilde + A(r_0,\{s\})=A(\{r_0^{-1}\cdot s\})\)` has the property `\[ \langle\tilde - A\rangle=\mathop{\mathrm{Tr}}\nolimits_s\mathop{\mathrm{Tr}}\nolimits_{s_0}\tilde - A(s_0,s)=\mathop{\mathrm{Tr}}\nolimits_sA(s)=\langle A\rangle + A\rangle=\mathop{\mathrm{Tr}}\nolimits_{r_0\in R}\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}\tilde + A(r_0,\{s\})=\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}A(\{s\})=\langle A\rangle \]` ---- - -  - ---- + ] -  + .column[ +<video width="320" height="640" style="float:right;"><source src="figs/wolff_ising_field.webm" type="video/webm"></video> +] --- @@ -158,13 +175,32 @@ Fast near the critical point: early studies thought `\(z\)` was zero, actually <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> +<span style="font-size: 30pt; overflow: hidden; height: 7em; text-align: center; align-items: center; display: inline-flex; position: static; float: center"> `\[\xrightarrow{r\in O(2)}\]`</span> --- -# Why is the extended method useful? +class: split-40 + +.column[ + +# Example: Clock Potts + +Consider the clock Potts model with field +`\[ +H(m)=0.01\times\begin{cases} +0 & \text{if $m=0$ (black)}\\ +\cos(2\pi/6) & \text{if $m=1$ (grey}\\ +-\cos(2\pi/6) & \text{if $m=2$ (white)} +\end{cases} +\]` -<img src="figs/potts.svg" alt="order-n"/> +<img src="figs/potts.svg" style="width:300px;"/> + +No rotation `\(r\in\mathbb Z/3\mathbb Z\)` that takes field to the identity! +] + .column[ +<video width="320" height="640" style="float:right;"><source src="figs/wolff_potts_field.webm" type="video/webm"></video> +] --- @@ -172,14 +208,41 @@ Fast near the critical point: early studies thought `\(z\)` was zero, actually Correlation time scales consistently in the whole phase space! +Pictured: scaling collapses of 2D Ising model correlation time. + <img src="figs/autocorr-scaling-isotherm.png" style="float: left; width: 48%" /> <img src="figs/autocorr-scaling-temp.png" style="float: right; width: 48%" /> --- +class: split-50 +# Applications & future work +.column[ #### Direct measurements in metastable states + +Rapid sampling of states near the critical point of models in a small field occasionally yields microstates against the direction of the field. + +<img src="figs/metastable-scaling.png" style="width: 80%; float: left;" /> +] + +.column[ + +#### Symmetry-breaking perturbations of XY model + +For various values of `\(p\)`, external fields of the form + +`\[ +H_p(s)=\cos(p\theta(s)) +\]` + +have various effects on the criticality of the XY model. +] + +--- + +# Questions? -# Metastable state direct measurement +<video><source src="figs/wolff_xy_field.webm" type="video/webm"></video> - </textarea> + </textarea> <script src="https://remarkjs.com/downloads/remark-latest.min.js"> </script> <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS_HTML&delayStartupUntil=configured" type="text/javascript"></script> |