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--- 김민재:스터디:임계현상:3장._가우스_근사 [2017/10/05 16:24] – minjae
+++ 김민재:스터디:임계현상:3장._가우스_근사 [2023/09/05 15:46] (current) – external edit 127.0.0.1
@@ Line 33: / Line 33: @@
 $$ \langle\boldsymbol \sigma_{i\boldsymbol k}\rangle=0,$$
 $$ G(k)=\langle|\boldsymbol\sigma_{i\boldsymbol k}|\rangle^{2}=\frac{1}{2}(a_{2}+ck)^{-1},$$
-$$FL^{d}=a_{0}L^{d}-\frac{1}{2}T\sum_{k<\Lambda}n\ln\left[\frac{\pi}{(a_{2}+ck^{2})\right]\quad(F\equiv\frac{f}{L^{d}})$$
+$$FL^d=a_0L^d-\frac{1}{2}T\sum_{k<\Lambda}n\ln\left[\frac{\pi}{(a_2+ck^2)}\right]\quad\left(F\equiv\frac{f}{L^d}\right)$$
 이 된다. 앞서, $a_{2}=a_{2}^{\prime}(T-T_{c})$라 두었기 때문에
 $$\lim_{T\rightarrow T_{c}}G(k)\propto k^{-2},$$
@@ Line 44: / Line 44: @@
 \begin{equation}\notag
 \begin{split}
-C&\equiv -T\left(\frac{\partial^{2}F}{\partial T^{2}}\right) \\
+C&\equiv -T\left(\frac{\partial^2F}{\partial T^2}\right) \\
-&=\frac{a_{2}^{\prime}^{2}}{2}T^{2}n(2\pi)^{-d}\int d^{d}k(a_{2}+ck^{2})^{-2}+l.s
+&=\frac{{a_2}^{\prime~2}}{2}T^2n(2\pi)^{-d}\int d^dk(a_2+ck^2)^{-2}+l.s
 \end{split}
 \end{equation}
@@ Line 56: / Line 56: @@
 \begin{equation}\notag
 \begin{split}
-C&=n\left[\frac{a_{2}^{\prime}^{2}}{2}T^{2}(2\pi)^{-d}\int d^{d}k(a_{2}+ck^{2})^{-2}\right]+l.s \\
+C&=n\left[\frac{{a_2}^{\prime~2}}{2}T^2(2\pi)^{-d}\int d^dk(a_2+ck^2)^{-2}\right]+l.s \\
-&=n\left[\frac{a_{2}^{\prime}^{2}}{2}T^{2}(2\pi)^{-d}\int d^{d}\left(\frac{k^{\prime}}{\xi}\right)(c\xi^{-2}+ck^{\prime}^{2}\xi^{-2})^{2}\right]+l.s \\
+&=n\left[\frac{{a_2}^{\prime~2}}{2}T^{2}(2\pi)^{-d}\int d^{d}\left(\frac{k^\prime}{\xi}\right)(c\xi^{-2}+ck^{\prime}^{2}\xi^{-2})^{2}\right]+l.s \\
-&=n\left[\frac{1}{2}(Ta_{2}^{\prime})^{2}(2\pi)^{-d}c^{-2}\int d^{d}k^{\prime}(1+k^{\prime}^{2})^{-2}\right]\xi^{4-d}+l.s \\
+&=n\left[\frac{1}{2}(T{a_2}^\prime)^2(2\pi)^{-d}c^{-2}\int d^dk^\prime(1+k^{\prime~2})^{-2}\right]\xi^{4-d}+l.s \\
-&\equiv C_{0}\xi^{4-d}+l.s
+&\equiv C_0\xi^{4-d}+l.s
 \end{split}
 \end{equation}