Continuum Field Theory for the Ising Model My problem is to take the $d$-dimensional Ising Hamiltonian,
$$H = -\sum_{i,j}\sigma_i J_{i,j} \sigma_j - \sum_{i} \tilde{h}_i \sigma_i$$
where $J_{ij}$ is a matrix describing the couplings between sites $i$ and $j$. Applying a Hubbard-Stratonovich transformation, rewrite the partition function as
$$Z = N_0 \int d^N \psi \exp\left\{-\left[\frac{1}{4}\sum_{i,j} \psi_i K_{ij} \psi_j - \sum_{i} \ln[\cosh(h_i+\psi_i)]\right]\right\}$$
where $N_0$ is an overall normalization constant, $K_{ij} = (\beta J_{ij})^{-1}$, and $h_i = \beta\tilde{h}_i$.
This much is relatively straightforward. We write the field as $\psi_i = \phi_i - h_i$, and we can show that $\left<\phi_i\right> \propto J_{ij} \left<\sigma_j\right>$, i.e. it can be interpreted as a "mean field" at site $i$ due to the interaction with all other sites. 
Next we assume that the variation in the field is small, $\left|\phi_i\right|<<1$, we set $h_i = 0$, and expand $\ln \cosh(x) \approx \frac{1}{2}x^2 - \frac{1}{12}x^4$ to get
$$Z \approx N_0\int d^N\psi \exp\left\{-\left[\frac{1}{4}\sum_{i,j}\phi_i K_{ij} \phi_j - \sum_i \left[\frac{\phi_i^2}{2} - \frac{\phi_i^4}{12}\right]\right]\right\}$$
Now we take the continuum limit, in units where the lattice spacing is unity, labeling each site by its position $\mathbf{r}$, which gives
$$Z\rightarrow \mathcal{N} \int \mathcal{D}\phi\, \exp\left\{-\frac{1}{2}\left[\frac{1}{2}\int d\mathbf{r}\,d\mathbf{r}'\,\phi(\mathbf{r}) K(\mathbf{r}-\mathbf{r}') \phi(\mathbf{r}') - \int d\mathbf{r}\,\left[\phi(\mathbf{r})^2 - \frac{\phi(\mathbf{r})^4}{6}\right]\right]\right\}$$
This is where I am not sure how to proceed. I am told to expand $\phi(\mathbf{r}')$ as a small variation from the value at $\mathbf{r}$, i.e.
$$\phi(\mathbf{r}') \approx \phi(\mathbf{r}) + (x_\mu'-x_\mu)\partial_\mu \phi(\mathbf{r}) + \frac{1}{2}(x_\mu' - x_\mu)(x_{\nu}'-x_\nu)\partial_\mu \partial_\nu \phi(\mathbf{r}) + \cdots$$
and introduce the Fourier transform $\tilde{K}(\mathbf{q}) = \int d\mathbf{r} K(\mathbf{r}) e^{-i\mathbf{q}\cdot\mathbf{r}}$
and write the continuum action as
$$S = \int d^d\mathbf{r} \left[c_1 \left(\partial \phi\right)^2 + c_2 \phi^2 + c_4 \phi^4\right]$$
and find the coefficients in terms of $\tilde{K}(0)$ and $\tilde{K}''(0)$. 
I believe that I can argue that $K$ is only a function of $\left|\mathbf{r}-\mathbf{r}'\right|$, in which case $K(\mathbf{r}-\mathbf{r}')(x_\mu'-x_\mu)$ is odd about the point $\mathbf{r}$, and so integrating over $d\mathbf{r}'$ (treating $\mathbf{r}$ as constant) will kill any term except those which depend on the square of the difference, leaving me with
$$\int d\mathbf{r}\,d\mathbf{r}'\,\phi(\mathbf{r}) K(\mathbf{r}-\mathbf{r}') \phi(\mathbf{r}') = \int d\mathbf{r}\,d\mathbf{r}' K(\mathbf{r}-\mathbf{r}')\left(\phi(\mathbf{r})^2 + \frac{1}{2}(x_\mu'-x_\mu)^2\phi(\mathbf{r})\partial_\mu^2 \phi(\mathbf{r})\right) $$
The first term I can deal with, but it's the second term that I don't know how to deal with.
 A: You can write the (scaled) interaction part of the action as:
$$S_I \equiv \int_{\mathbb R^d}d^d \mathbf r \ \phi(\mathbf r)\int_{\mathbb R^d}d^d \mathbf r' \ K(\mathbf r-\mathbf r') \ \phi(\mathbf r')$$
Let's take the inner integral over $\mathbf r'$ first (I will call it $\mathcal I$ to make things easier). Expanding $\phi(\mathbf r')$ around $\mathbf r$ gives :
$$\mathcal I \equiv\int_{\mathbb R^d}d^d \mathbf r' \ K(\mathbf r-\mathbf r') \ \phi(\mathbf r') \approx \int_{\mathbb R^d}d^d \mathbf r' \ K(\mathbf r-\mathbf r') \ \bigg(\phi(\mathbf r)+ \sum_{i=1}^d (x_i'-x_i)\partial_i \phi(\mathbf r) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac 12\sum_{i=1}^d\sum_{j=1}^d (x_i'-x_i)(x_j'-x_j)\partial_i \partial_j \phi(\mathbf r) \bigg)$$
Now take the integral inside to get:
$$\mathcal I\approx \phi(\mathbf r) \int_{\mathbb R^d}d^d \mathbf r' \ K(\mathbf r-\mathbf r') \ + \sum_{i=1}^d \partial_i \phi(\mathbf r)\int_{\mathbb R^d}d^d \mathbf r' \ (x_i'-x_i) K(\mathbf r-\mathbf r') +\frac 12\sum_{i=1}^d\sum_{j=1}^d  \partial_i \partial_j \phi(\mathbf r)  \times\int_{\mathbb R^d}d^d \mathbf r'(x_i'-x_i)(x_j'-x_j)K(\mathbf r-\mathbf r') \bigg)$$
Now assuming that the coupling is homogenous, $K(\mathbf r-\mathbf r')\equiv K(\mathbf r'-\mathbf r)$. With that in mind, and also changing variables $\mathbf R \equiv \mathbf r'-\mathbf r$, we get:
$$\mathcal I \approx \phi(\mathbf r) \int_{\mathbb R^d}d^d \mathbf R \ K(\mathbf R) \ + \sum_{i=1}^d \partial_i \phi(\mathbf r)\int_{\mathbb R^d}d^d \mathbf R \ R_i K(\mathbf R) +\frac 12\sum_{i=1}^d\sum_{j=1}^d  \partial_i \partial_j \phi(\mathbf r) \ \ \ \ \times\int_{\mathbb R^d}d^d \mathbf R \ R_iR_jK(\mathbf R) \bigg)$$
You can relate each of the integrals over $\mathbf R$ to the Fourier transform of $K(\mathbf R)$ defined as $\tilde K(\mathbf q) \equiv \int_{\mathbb R^d} d^d \mathbf R \ K(\mathbf R)\exp(-i \mathbf{q} . \mathbf R)$: 
- First integral:
$$\int_{\mathbb R^d}d^d \mathbf R \ K(\mathbf R) = \int_{\mathbb R^d}d^d \mathbf R \ K(\mathbf R) \ e^{-i \mathbf q. \mathbf R} |_{\mathbf q =0} = \tilde K(\mathbf 0)$$
- Second integral:
$$\int_{\mathbb R^d}d^d \mathbf R \ R_i K(\mathbf R) =0$$
Because of the integrand being odd as you mentioned. 
- Third integral: 
For this one we first note that as you mentioned the integral is zero for all different $i,j$. For $i=j$, first note that:
$$\frac{\partial^2}{\partial q_i^2} \int_{\mathbb R^d}d^d \mathbf R \ K(\mathbf R) \ e^{-i \mathbf q. \mathbf R} =  \int_{\mathbb R^d}d^d \mathbf R \ (-i)(-i) R_i R_i K(\mathbf R) \ e^{-i \mathbf q. \mathbf R} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ = - \int_{\mathbb R^d}d^d \mathbf R \ R_i^2 K(\mathbf R) \ e^{-i \mathbf q. \mathbf R}  $$
Which implies:
$$\int_{\mathbb R^d}d^d \mathbf R \ R_i^2 K(\mathbf R)=-\frac{\partial^2}{\partial q_i^2} \int_{\mathbb R^d}d^d \mathbf R \ K(\mathbf R) \ e^{-i \mathbf q. \mathbf R}|_{\mathbf q=0}=-\frac{\partial^2}{\partial q_i^2}\tilde K(\mathbf q) |_{\mathbf q=0} $$
Now if you assume that the coupling is also isotropic, i.e. $\exists \mathcal K : K(\mathbf R) \equiv \mathcal K(|\mathbf R|)$, the Fourier transform of $K$ will become a single variable function, meaning that the third integral is just $-\tilde K''(0)$. 
In summary, $\mathcal I$ is:
$$\mathcal I  \approx \phi(\mathbf r) \tilde K(0) \ - \frac 12\sum_{i=1}^d  \partial_i^2  \phi(\mathbf r) \tilde K''(0)$$
Thus, the interaction term in the action is:
$$S_I = \int_{\mathbb R^d}d^d \mathbf r \ \phi(\mathbf r) \mathcal I = \int_{\mathbb R^d}d^d \mathbf r \ \phi(\mathbf r)\bigg(\phi(\mathbf r) \tilde K(0) \ - \frac 12\sum_{i=1}^d\partial_i^2 \phi(\mathbf r) \tilde K''(0)\bigg)$$
$$=\tilde K(0)\int_{\mathbb R^d}d^d \mathbf r \ \phi^2(\mathbf r)  - \frac {\tilde K''(0)}2\sum_{i=1}^d \int_{\mathbb R^d}d^d \mathbf r \ \phi(\mathbf r) \   \partial_i^2 \phi(\mathbf r) $$
Integrating by parts in the second term results in (boundary terms vanish because $\phi(\mathbf r) \to 0$ as $|\mathbf r| \to \infty$ so that the integrals converge):
$$S_I =\tilde K(0)\int_{\mathbb R^d}d^d \mathbf r \ \phi^2(\mathbf r)  + \frac {\tilde K''(0)}2\sum_{i=1}^d\int_{\mathbb R^d}d^d \mathbf r \ \partial_i \phi(\mathbf r) \   \partial_i \phi(\mathbf r)$$
$$=\tilde K(0)\int_{\mathbb R^d}d^d \mathbf r \ \phi^2(\mathbf r)  + \frac {\tilde K''(0)}2 \int_{\mathbb R^d}d^d \mathbf r \  \sum_{i=1}^d (\partial_i \phi(\mathbf r))^2$$
$$=\int_{\mathbb R^d}d^d \mathbf r \ \bigg( \tilde K(0) \phi^2(\mathbf r)  + \frac {\tilde K''(0)}2  \ \big( \partial \phi(\mathbf r) \big)^2 \bigg)$$
Plugging this in the full action finally gives:
$$S[\phi] =\int_{\mathbb R^d}d^d \mathbf r \ \bigg(  \frac {\tilde K''(0)}8  \ \big( \partial \phi(\mathbf r) \big)^2 + \left(\frac {\tilde K(0)}{4}- \frac 12\right) \phi^2(\mathbf r) + \frac 1{12} \phi^4(\mathbf r)  \bigg) $$
Notice that the coefficient of the quadratic term can change sign with temperature (through $\tilde K$), which is a sign of a phase transition.
