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| -rw-r--r-- | monte-carlo.bib | 32 | ||||
| -rw-r--r-- | monte-carlo.tex | 96 |
2 files changed, 116 insertions, 12 deletions
diff --git a/monte-carlo.bib b/monte-carlo.bib index 4d8669e..70f3f49 100644 --- a/monte-carlo.bib +++ b/monte-carlo.bib @@ -8,7 +8,7 @@ number = {3}, urldate = {2018-04-04}, journal = {Physical Review B}, - author = {Jose, Jorge V. and Kadanoff, Leo P. and Kirkpatrick, Scott and Nelson, David R.}, + author = {José, Jorge V. and Kadanoff, Leo P. and Kirkpatrick, Scott and Nelson, David R.}, month = aug, year = {1977}, pages = {1217--1241}, @@ -420,7 +420,7 @@ number = {22}, urldate = {2018-04-24}, journal = {Physical Review B}, - author = {Carmona, Jose Manuel and Pelissetto, Andrea and Vicari, Ettore}, + author = {Manuel Carmona, José and Pelissetto, Andrea and Vicari, Ettore}, month = jun, year = {2000}, pages = {15136--15151}, @@ -507,8 +507,9 @@ author = {Dotsenko, Vl. S. and Selke, W. and Talapov, A. L.}, month = jan, year = {1991}, + keywords = {monte-carlo, rfim, cluster-algorithm}, pages = {278--281}, - file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/39ZPGDAE/Dotsenko et al. - 1991 - Cluster Monte Carlo algorithms for random Ising mo.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/8YGYCZZ8/037843719190045E.html:text/html} + file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/ZULKCCN9/Dotsenko et al. - 1991 - Cluster Monte Carlo algorithms for random Ising mo.pdf:application/pdf;ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/39ZPGDAE/Dotsenko et al. - 1991 - Cluster Monte Carlo algorithms for random Ising mo.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/7Z9PAX48/037843719190045E.html:text/html;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/8YGYCZZ8/037843719190045E.html:text/html} } @incollection{rieger_monte_1995, @@ -545,11 +546,20 @@ random field Ising model and finally of quantum spin glasses.}, file = {APS Snapshot:/home/pants/.zotero/data/storage/GD9PHBAV/RevModPhys.51.html:text/html;Mermin - 1979 - The topological theory of defects in ordered media.pdf:/home/pants/.zotero/data/storage/ZJE9JPN6/Mermin - 1979 - The topological theory of defects in ordered media.pdf:application/pdf} } -@misc{bierbaum_ising.js_nodate, - title = {ising.js}, - url = {https://mattbierbaum.github.io/ising.js/}, - urldate = {2018-05-15}, - author = {Bierbaum, Matthew K.}, - note = {Source: https://github.com/mattbierbaum/ising.js}, - file = {ising.js:/home/pants/.zotero/data/storage/XR534SY3/ising.html:text/html} -} +@article{ossola_dynamic_2004, + title = {Dynamic critical behavior of the {Swendsen}–{Wang} algorithm for the three-dimensional {Ising} model}, + volume = {691}, + issn = {0550-3213}, + url = {http://www.sciencedirect.com/science/article/pii/S0550321304003098}, + doi = {10.1016/j.nuclphysb.2004.04.026}, + abstract = {We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen–Wang algorithm for the three-dimensional Ising model at the critical point. For the dynamic critical exponents associated to the integrated autocorrelation times of the “energy-like” observables, we find zint,N=zint,E=zint,E′=0.459±0.005±0.025, where the first error bar represents statistical error (68\% confidence interval) and the second error bar represents possible systematic error due to corrections to scaling (68\% subjective confidence interval). For the “susceptibility-like” observables, we find zint,M2=zint,S2=0.443±0.005±0.030. For the dynamic critical exponent associated to the exponential autocorrelation time, we find zexp≈0.481. Our data are consistent with the Coddington–Baillie conjecture zSW=β/ν≈0.5183, especially if it is interpreted as referring to zexp.}, + number = {3}, + urldate = {2018-09-19}, + journal = {Nuclear Physics B}, + author = {Ossola, Giovanni and Sokal, Alan D.}, + month = jul, + year = {2004}, + keywords = {Autocorrelation time, Cluster algorithm, Dynamic critical exponent, Ising model, Monte Carlo, Potts model, Swendsen–Wang algorithm}, + pages = {259--291}, + file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/MKA8WYZZ/Ossola and Sokal - 2004 - Dynamic critical behavior of the Swendsen–Wang alg.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/YHGX7CDT/S0550321304003098.html:text/html} +}
\ No newline at end of file diff --git a/monte-carlo.tex b/monte-carlo.tex index e258818..370ac0f 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -369,7 +369,31 @@ interest include $(n+1)$-dimensional spherical harmonics \cite{jose_renormalization_1977} and cubic fields \cite{bruce_coupled_1975,blankschtein_fluctuation-induced_1982}, which can be applied with the new method. The method is -quickly generalized to spins whose symmetry groups other compact Lie groups +quickly generalized to spins whose symmetry groups other compact Lie groups. + +At low temperature or high field, selecting reflections uniformly becomes +inefficient because the excitations of the model are spin waves, in which the +magnetization only differs by a small amount between neighboring spins. Under +these conditions, most choices of reflection plane will cause a change in +energy so great that the whole system is always flipped, resulting in many +highly correlated and inefficiently generated samples. To ameliorate this, one +can draw reflections from a distribution that depends on how the first spin +flip is transformed. We implement this in the following way. Say that the seed +of the cluster is $s$. Generate a vector $t$ taken uniformly from the space of +unit vectors orthogonal to $s$. Let the plane of reflection that whose normal +is $n=s+\zeta t$, where $\zeta$ is drawn from a normal distribution of mean +zero and variance $\sigma$. It follows that the tangent of the angle between +$s$ and the plane of reflection is also distributed normally with zero mean +and variance $\sigma$. Since the distribution of reflection planes only +depends on the angle between $s$ and the plane and that angle is invariant +under the reflection, this choice preserves detailed balance. The choice of +$\sigma$ can be inspired by mean field theory. At high field or low +temperature, spins are likely to both align with the field and each other and +the model is asymptotically equal to a simple Gaussian one, with in the limit +of large $L$ the expected square angle between neighbors being +\[ + \avg{\theta^2}\simeq\frac{(n-1)T}{D+H/2} +\] \subsection{The Potts model} In the $q$-state Potts model spins are described by elements of $\{1,\ldots,q\}$. Its symmetry group is the symmetric group @@ -536,6 +560,75 @@ perturbations on spin models can be tested numerically \cite{jose_renormalization_1977, blankschtein_fluctuation-induced_1982, bruce_coupled_1975, manuel_carmona_$n$-component_2000}. +<<<<<<< HEAD +\appendix + +\section{$\mathrm O(n)$ model at high field} + + +\[ + \H=-\sum_r\sum_{i=1}^D\sum_{j=1}^ns_r^js_{r+e_i}^j + -\sum_r\sum_{j=1}^nH^js_r^j +\] +under the constraint +\[ + 1=\sum_{j=1}^ns_r^js_r^j +\] +Let $s=m+t$ for $|m|=1$ (usually $m=H/|H|$). Suppose without loss of +generality that $m=e_1$. +\[ + 1=|s|^2=1+2m\cdot t+|t|^2 +\] +whence $m\cdot t=-\frac12|t|^2$. Then +\begin{align} + s_1\cdot s_2 + &=1+m\cdot(t_1+t_2)+t_1\cdot t_2\\ + &=1-\frac12(|t_1|^2+|t_2|^2)+t_1\cdot t_2 +\end{align} +and +\[ + H\cdot s=|H|(1+m\cdot t)=|H|(1-\frac12|t|^2) +\] +For small perturbations, there are only $n-1$ degrees of freedom. We must have +(for $t$ in the same hemisphere as $m$) +\[ + t_\parallel=\sqrt{1-|t_\perp|^2}-1 +\] +\[ + t_1\cdot t_2=t_{1\perp}\cdot t_{2\perp}+O(t^4) +\] +Since there are $2D$ nearest neighbor bonds involving each spin, +\[ + \H + \simeq\H_0 + -\sum_{\langle ij\rangle}t_{i\perp}\cdot t_{j\perp} + +(D+|H|/2)\sum_i|t_{i\perp}|^2 +\] +Taking a discrete Fourier transform on the lattice, we find +\[ + \H + \simeq\H_0 + -\sum_k|\tilde t_{k\perp}|^2(D+|H|/2-\sum_{i=1}^D\cos(2\pi k_i/L)) +\] +It follows from equipartition (and the fact that $t_{k\perp}$ is an $n$-1 +component complex number) that +\[ + \avg{|\tilde t_{k\perp}|^2}=\frac + {n-1}2T\bigg(D+\frac{|H|}2-\sum_{i=1}^D\cos(2\pi k_i/L)\bigg)^{-1} +\] +whence +\begin{align} + \avg{\theta^2} + &=\avg{\cos^{-1}s_i\cdot s_j} + \simeq2(1-\avg{s_i\cdot s_j})\\ + &=2(\avg{|t|^2}-\avg{t_i\cdot t_j}) + \simeq2(\avg{|t_\perp|^2}-\avg{t_{i\perp}\cdot t_{j\perp}})\\ + &=2\sum_k(1-\cos(2\pi k_1/L))\avg{|\tilde t_{k\perp}|^2}\\ + &=\frac{(n-1)T}{L^D}\sum_k\frac{1-\cos(2\pi + k_1/L)}{D+|H|/2-\sum_{i=1}^D\cos(2\pi k_i/L)}\\ +\end{align} + +\section{Calculating autocorrelation time} \begin{figure*} \include{fig_correlation-times} \end{figure*} @@ -546,5 +639,6 @@ bruce_coupled_1975, manuel_carmona_$n$-component_2000}. \bibliography{monte-carlo} + \end{document} |