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| -rw-r--r-- | ising_scaling.tex | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index d867714..b8c4d79 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -123,7 +123,7 @@ spatial dimension in two dimensions, while its coefficient is chosen as a matter of convention, fixing the scale of $u_t$. The form $u_f$ of the free energy is known as the singular part of the free energy, since it is equivalent to subtracting an analytic function of the control variables from the free -energy, breaking it into pieces that scale homogenously and inhomogenously with +energy, breaking it into pieces that scale homogeneously and inhomogeneously with renormalization. Solving these equations for $u_f$ yields @@ -163,7 +163,7 @@ For the scale of $u_t$ and $u_h$, we adopt the same convention as used by the parameters $t$ and $h$ is system-dependent, and their form can be found for common model systems (the square- and triangular-lattice Ising models) in the literature \cite{Mangazeev_2010_Scaling, Clement_2019_Respect}. To connect the -results of thes paper with Mangazeev and Fonseca, one can write $\mathcal +results of this paper with Mangazeev and Fonseca, one can write $\mathcal F_0(\eta)=\tilde\Phi(-\eta)=\Phi(-\eta)+(\eta^2/8\pi) \log \eta^2$ and $\mathcal F_\pm(\xi)=G_{\mathrm{high}/\mathrm{low}}(\xi)$. @@ -426,7 +426,7 @@ use of the identity 0=\oint_{\mathcal C}d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \end{equation} where $\mathcal C$ is the contour in Figure \ref{fig:contour}. The integral is -zero because there are no singularites enclosed by the contour. The only +zero because there are no singularities enclosed by the contour. The only nonvanishing contributions from this contour as the radius of the semicircle is taken to infinity are along the real line and along the branch cut in the upper half plane. For the latter contributions, the real parts of the integration up |