updated exponents, added more explanation to performance section

3 files changed, 68 insertions, 9 deletions

diff --git a/figs/fig_correlation-times.gplot b/figs/fig_correlation-times.gplot
index 88abe42..4cb6dc2 100644
--- a/figs/fig_correlation-times.gplot
+++ b/figs/fig_correlation-times.gplot
@@ -8,7 +8,7 @@ betaIsing2D = 1.0 / 8.0
 nuIsing2D = 1.0
 deltaIsing2D = 15.0
 gammaIsing2D = 7.0 / 4.0
-zIsing2D = 0.213
+zIsing2D = 0.23
 
 dataIsing3D = "data/correlation-times/ising-3d.dat"
 betaIsing3D = 0.326419
@@ -24,7 +24,7 @@ beta3Potts2D = 1.0 / 9.0
 nu3Potts2D = 5.0 / 6.0
 delta3Potts2D = 14.0
 gamma3Potts2D = 13.0 / 9.0
-z3Potts2D = 0.54
+z3Potts2D = 0.55
 
 data4Potts2D = "data/correlation-times/potts-4.dat"
 beta4Potts2D = 1.0 / 12.0
@@ -40,14 +40,14 @@ beta2Vector3D = 0.3470
 nu2Vector3D = 0.6703
 delta2Vector3D = 4.79539
 gamma2Vector3D = 1.3169
-z2Vector3D = 0.2
+z2Vector3D = 0.17
 
 data3Vector3D = "data/correlation-times/heisenberg-pert.dat"
 beta3Vector3D = 0.3662
 nu3Vector3D = 0.7073
 delta3Vector3D = 4.79465
 gamma3Vector3D = 1.3895
-z3Vector3D = 0.12
+z3Vector3D = 0.13
 
 # define colors
 
diff --git a/monte-carlo.bib b/monte-carlo.bib
index f018281..0ca69fb 100644
--- a/monte-carlo.bib
+++ b/monte-carlo.bib
@@ -15,6 +15,22 @@
 	file = {APS Snapshot:/home/pants/.zotero/data/storage/RMHXAR3A/PhysRevB.16.html:text/html;José et al. - 1977 - Renormalization, vortices, and symmetry-breaking p.pdf:/home/pants/.zotero/data/storage/M2V99JHC/José et al. - 1977 - Renormalization, vortices, and symmetry-breaking p.pdf:application/pdf}
 }
 
+@article{wu_potts_1982,
+	title = {The {Potts} model},
+	volume = {54},
+	url = {https://link.aps.org/doi/10.1103/RevModPhys.54.235},
+	doi = {10.1103/RevModPhys.54.235},
+	abstract = {This is a tutorial review on the Potts model aimed at bringing out in an organized fashion the essential and important properties of the standard Potts model. Emphasis is placed on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective. Topics reviewed include the mean-field theory, duality relations, series expansions, critical properties, experimental realizations, and the relationship of the Potts model with other lattice-statistical problems.},
+	number = {1},
+	urldate = {2018-04-04},
+	journal = {Reviews of Modern Physics},
+	author = {Wu, F. Y.},
+	month = jan,
+	year = {1982},
+	pages = {235--268},
+	file = {APS Snapshot:/home/pants/.zotero/data/storage/TJ83QK5V/RevModPhys.54.html:text/html;Wu - 1982 - The Potts model.pdf:/home/pants/.zotero/data/storage/KN9S57Z9/Wu - 1982 - The Potts model.pdf:application/pdf}
+}
+
 @article{coniglio_exact_1989,
 	title = {Exact relations between droplets and thermal fluctuations in external field},
 	volume = {22},
@@ -167,6 +183,24 @@
 	file = {Geyer - 1992 - Practical Markov Chain Monte Carlo.pdf:/home/pants/.zotero/data/storage/UAU5QJNP/Geyer - 1992 - Practical Markov Chain Monte Carlo.pdf:application/pdf}
 }
 
+@article{el-showk_solving_2014,
+	title = {Solving the 3d {Ising} {Model} with the {Conformal} {Bootstrap} {II}. \$\$\$\$-{Minimization} and {Preise} {Critial} {Exponents}},
+	volume = {157},
+	issn = {0022-4715, 1572-9613},
+	url = {https://link.springer.com/article/10.1007/s10955-014-1042-7},
+	doi = {10.1007/s10955-014-1042-7},
+	abstract = {We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge {\textbackslash}(c{\textbackslash}) in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several {\textbackslash}({\textbackslash}mathbb \{Z\}\_2{\textbackslash})-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension {\textbackslash}({\textbackslash}Delta \_{\textbackslash}sigma = 0.518154(15){\textbackslash}), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.},
+	language = {en},
+	number = {4-5},
+	urldate = {2018-04-04},
+	journal = {Journal of Statistical Physics},
+	author = {El-Showk, Sheer and Paulos, Miguel F. and Poland, David and Rychkov, Slava and Simmons-Duffin, David and Vichi, Alessandro},
+	month = dec,
+	year = {2014},
+	pages = {869--914},
+	file = {Full Text PDF:/home/pants/.zotero/data/storage/XB5EWQ28/El-Showk et al. - 2014 - Solving the 3d Ising Model with the Conformal Boot.pdf:application/pdf;Snapshot:/home/pants/.zotero/data/storage/9P7WJY86/s10955-014-1042-7.html:text/html}
+}
+
 @article{janke_nonlocal_1998,
 	title = {Nonlocal {Monte} {Carlo} algorithms for statistical physics applications},
 	volume = {47},
@@ -377,6 +411,23 @@
 	file = {APS Snapshot:/home/pants/.zotero/data/storage/Q5RL6Q8U/PhysRevE.58.html:text/html;Redner et al. - 1998 - Graphical representations and cluster algorithms f.pdf:/home/pants/.zotero/data/storage/WYEU9G6Y/Redner et al. - 1998 - Graphical representations and cluster algorithms f.pdf:application/pdf}
 }
 
+@article{guida_critical_1998,
+	title = {Critical exponents of the {N} -vector model},
+	volume = {31},
+	issn = {0305-4470},
+	url = {http://stacks.iop.org/0305-4470/31/i=40/a=006},
+	doi = {10.1088/0305-4470/31/40/006},
+	abstract = {Recently the series for two renormalization group functions (corresponding to the anomalous dimensions of the fields \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img1.gif] and \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img2.gif] ) of the three-dimensional \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img3.gif] field theory have been extended to next order (seven loops) by Murray and Nickel. We examine the influence of these additional terms on the estimates of critical exponents of the N -vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within the errors of the previous evaluation. Exponents such as \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img4.gif] (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou-Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined. The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values. Finally, because an error has been discovered in the last order of the published \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img5.gif] expansions (order \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img6.gif] ), we have also re-analysed the determination of exponents from the \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion. The conclusion is that the general agreement between \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion and three-dimensional series has improved with respect to Le Guillou-Zinn-Justin.},
+	language = {en},
+	number = {40},
+	urldate = {2018-04-18},
+	journal = {Journal of Physics A: Mathematical and General},
+	author = {Guida, R. and Zinn-Justin, J.},
+	year = {1998},
+	pages = {8103},
+	file = {IOP Full Text PDF:/home/pants/.zotero/data/storage/K468APXL/Guida and Zinn-Justin - 1998 - Critical exponents of the N -vector model.pdf:application/pdf}
+}
+
 @article{stauffer_scaling_1979,
 	title = {Scaling theory of percolation clusters},
 	volume = {54},
diff --git a/monte-carlo.tex b/monte-carlo.tex
index cf45bcd..fe6227d 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -547,7 +547,13 @@ large-field regime where the crossover happens the correlation length is on
 the scale of the lattice spacing and better algorithms exist, like
 Bortz--Kalos--Lebowitz for the Ising model \cite{bortz_new_1975}. Also plotted
 are lines proportional to $h^{-z\nu/\beta\delta}$, which match the behavior of
-the correlation times in the intermediate scaling region.
+the correlation times in the intermediate scaling region. Values of the
+critical exponents for the models were taken from the literature
+\cite{wu_potts_1982, el-showk_solving_2014, guida_critical_1998} with the
+exception of $z$ for the energy in the Wolff algorithm, which was determined
+for each model by making a power law fit to the constant low field behavior.
+These exponents are imprecise and are provided with only qualitative
+uncertainty.
 
 \begin{figure*}
   \include{fig_correlation-times}
@@ -557,7 +563,10 @@ the correlation times in the intermediate scaling region.
     various models of Table~\ref{table:models}. Critical exponents are
     model-dependent. Colored lines and points depict values as measured by the
     extended algorithm. Solid black lines show a plot proportional to
-    $h^{-z\nu/\beta\delta}$ for each model.
+    $h^{-z\nu/\beta\delta}$ for each model. The dynamic exponents $z$ are
+    roughly measured as \twodee Ising: 0.23(2), \threedee Ising: 0.28(2),
+    \twodee 3-State Potts: 0.55(1), \twodee 4-State Potts: 0.94(5),
+    \threedee O(2): 0.17(2), \threedee O(3): 0.13(2).
   }
   \label{fig:correlation_time-collapse}
 \end{figure*}
@@ -604,9 +613,8 @@ large argument. We further conjecture that this scaling behavior should hold
 for other models whose critical points correspond with the percolation
 transition of Wolff clusters.  This behavior is supported by our numeric work
 along the critical isotherm for various Ising, Potts, and $\mathrm O(n)$
-models, shown in Fig.~\ref{fig:cluster_scaling}. Fields for the Potts and
-$\mathrm O(n)$ models take the form $B(s)=(h/\beta)\sum_m\cos(2\pi(s-m)/q)$
-and $B(s)=(h/\beta)[1,0,\ldots,0]s$ respectively. As can be seen, the average
+models, shown in Fig.~\ref{fig:cluster_scaling}. Fields are the canonical ones
+referenced in Table~\ref{table:models}. As can be seen, the average
 cluster size collapses for each model according to the scaling hypothesis, and
 the large-field behavior likewise scales as we expect from the na\"ive Ising
 conjecture.