fixed Tc in the figures and fixed a minor mistake in the elastic susceptibility caluclation

9 files changed, 23 insertions, 6 deletions

diff --git a/cusp.pdf b/cusp.pdf
index 337b8a9..6499a0d 100644
--- a/cusp.pdf
+++ b/cusp.pdf
diff --git a/hidden_order.bib b/hidden_order.bib
index 7a24ef4..9d289d5 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -1,4 +1,21 @@
 
+@article{el-showk_solving_2014,
+  title = {Solving the 3d {{Ising Model}} with the {{Conformal Bootstrap II}}. {$\mathsl{c}$}-{{Minimization}} and {{Preise Critial Exponents}}},
+  volume = {157},
+  issn = {0022-4715, 1572-9613},
+  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},
+  journal = {Journal of Statistical Physics},
+  doi = {10.1007/s10955-014-1042-7},
+  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},
+  keywords = {_tablet},
+  pages = {869-914},
+  file = {/home/pants/.zotero/data/storage/XB5EWQ28/El-Showk et al. - 2014 - Solving the 3d Ising Model with the Conformal Boot.pdf}
+}
+
 @book{landau_theory_1995,
   series = {Landau and {{Lifshitz Course}} of {{Theoretical Physics}}},
   title = {Theory of {{Elasticity}}},
diff --git a/main.tex b/main.tex
index c92400b..4264884 100644
--- a/main.tex
+++ b/main.tex
@@ -181,7 +181,7 @@ which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cann
     \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
     &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')}
     =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\
-    &=-\frac2b\chi^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta(x-x')
+    &=-\frac2b\chi_{ij}^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta_{ij}\delta(x-x')
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
@@ -191,25 +191,25 @@ It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the suscept
   \begin{aligned}
     \chi_{\X ij}^{-1}(x,x')
     &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\
-    &=\lambda_\X\delta(x-x')+
+    &=\lambda_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\
     &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')}
     +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ 
-    &=\lambda_\X\delta(x-x')+
+    &=\lambda_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ 
-    &=\lambda_\X\delta(x-x')+
+    &=\lambda_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} 
-    =\lambda_\X\delta(x-x')+
+    =\lambda_\X\delta_{ij}\delta(x-x')+
     \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')},
   \end{aligned}
 \end{equation}
 \end{widetext}
 whose Fourier transform follows from \eqref{eq:inv.func} as
 \begin{equation}
-  \chi_{\X ij}(q)=\frac1{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
+  \chi_{\X ij}(q)=\frac{\delta_{ij}}{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
   \label{eq:elastic.susceptibility}
 \end{equation}
 At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde r_c|^\gamma$ for $\gamma=1$.
diff --git a/stiff_a11.pdf b/stiff_a11.pdf
index 89891b7..2867849 100644
--- a/stiff_a11.pdf
+++ b/stiff_a11.pdf
diff --git a/stiff_a12.pdf b/stiff_a12.pdf
index 404e775..d3fdd9d 100644
--- a/stiff_a12.pdf
+++ b/stiff_a12.pdf
diff --git a/stiff_a22.pdf b/stiff_a22.pdf
index 92a4a99..34261a7 100644
--- a/stiff_a22.pdf
+++ b/stiff_a22.pdf
diff --git a/stiff_b1.pdf b/stiff_b1.pdf
index e2bbeb9..ceb6cfb 100644
--- a/stiff_b1.pdf
+++ b/stiff_b1.pdf
diff --git a/stiff_b2.pdf b/stiff_b2.pdf
index 93bb660..28c18d0 100644
--- a/stiff_b2.pdf
+++ b/stiff_b2.pdf
diff --git a/stiff_e.pdf b/stiff_e.pdf
index 3aa2ef1..0a7f703 100644
--- a/stiff_e.pdf
+++ b/stiff_e.pdf