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# System prepended metadata

title: Adhvik project

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# Research Project - LPS

## 1. The Renormalisation Group in Statistical Physics

Phase transitions have been studied for a long time by humanity. However, there have only been a handful of model phase transitions exactly solved and described through statistical mechanics. Critical phenomena deals with the study of phase transitions, especially with the behaviour of physical quantities close to the critical point. An interesting observation made is the fact that critical exponents can be the same for two theories which differ a lot microscopically. For example, the mean field Ising model and the mean field Van der Waals gas model share the same critical exponents. This suggests the existence of **universality classes**, i.e. classes of theories with similar critical behaviour. In other words, if we "zoom out" sufficiently, macroscopic behaviours of theories from the same universality class are the same. 

Zooming out, or coarse graining our theory, is equivalent to omitting high frequency or energy modes in Fourier space. The **Renormalisation Group** (RG) is a tool which allows us to justify our description of the physical world which omit these energy scales we don't care about. It justifies two key facts:
1. We can do physics even though we don't have the microscopic details of the system. 
2. We can deduce what behaviour other theories exhibit close to their critical points, purely based on the behaviour of another theory, from the same universality class. This is because RG flows' orbits are in the same universality classes. 

### 1.1 The Landau-Ginzburg Hamiltonian 

The Landau-Ginzburg model is one of the most useful family of models used to describe systems. The free energy of this system is a functional of an order parameter $\phi_i(x)$. There are two approaches to interpreting what $\phi$ is: we could think of $\phi$ to be the coarse-grained "magnetiztion" of the system (which is more physical), or we could think of it as a purely mathematical quantity introduced to easily deal with discrete lattices (using an equivalent of a Gaussian trick). 

We typically take $\phi_i$ to be an $n$-dimensional vector valued function. The free energy is assumed to be the most general function which is local, $\phi \leftrightarrow -\phi$ symmetric, and analytic. Thus one obtains

\begin{align}
    \beta \mathcal{F}_{LG}[\phi] & = \int d^d x\left(-\frac a2 \phi_i(x)\phi_i(x) + \frac b4 (\phi_i(x)\phi_i(x))^2\right. \\ 
    & \left. + \frac K2 (\nabla \phi_i(x))^2 + \cdots - h_i\phi_i(x) \right)\\ 
    Z_{LG} & = \int \mathcal{D}\phi\, \exp \left(-\beta\mathcal{F}_{LG}[\phi]\right)
\end{align}

The constants $a,b,K,\dots$ are parameters which depend on the temperature and external parameters of the system. They are chosen based on the nature of the model we are trying to describe, to match physial results. For instance, choosing $a \propto (T - T_C) + O(T-T_C)^2$ gives a behaviour similar to that of a mean field Ising model. 

In mean field solutions of such a system, the base assumption is that the leading contribution comes from when the order parameter is constant; $\phi(x) = \phi_0$ for all $x$. Thus from a saddle point approximation, one gets 
\begin{align}
    Z_{LG} & \approx C\int d \mathbf{\phi} \exp\left[-\frac a2 \phi^2 + \frac b4 \phi^4 + \cdots - \mathbf{h}\cdot\phi\right]\\ 
    & \approx CV\psi(\mathbf{\phi}_{\text{min}})
\end{align}
where $\mathbf{\phi}_\text{min}$ is the maxima of $\psi$:
\begin{align}
    \psi(\mathbf{\phi}) & = -\frac a2 \phi^2 + \frac b4 \phi^4 + \cdots - \mathbf{h}\cdot\mathbf{\phi}
\end{align} 

If one assumes only the $\phi^2$, $\phi^4$ and magnetic field terms to exist, the familiar Ising model behaviour is recovered. For $T>T_C$, $a>0$ and so there is exactly one minimum at $\phi_{min}=0$. If $T<T_C$, $a<0$ and so the minima are $\phi_{min} = \pm \sqrt{\dfrac ab},0$, and the maximum being at 0. This leads to the familiar critical exponents as observed in the Ising model: $\phi \propto (T_C-T)^{\frac12}$, $\chi\propto |T-T_C|^{-1}$ and so on. 

### 1.2 Interactions and Perturbation Theory

The free energy is split into its Gaussian terms and the rest, which are dubbed as "interactions", denoted $\mathcal F_0$ and $U$. We find that the Gaussian Hamiltonian is exactly solvable because its Fourier transform has no mixing of different wavelengths; only the $\pm k$ modes. On the other hand, terms like $\phi^4$ do not obey this; they mix three momentum modes together. 

The Gaussian Hamiltonian can be re-written by taking a Fourier transform as follows. 
\begin{align}
    \beta \mathcal F_0[\mathbf{\phi}] & = \int d^d x\left(-\frac a2 \phi^2(x) + \frac K2 (\nabla \phi(x))^2 + \cdots \right)\\ 
    & = \int \dfrac{d^d q}{(2 \pi)^d}\left(-\frac a2 + \frac K2 q^2 + \cdots \right) \phi_i(q)\phi_i(-q)\\ 
    & =: \int \dfrac{d^d q}{(2 \pi)^d} \frac12 E(q^2) |\phi(q)|^2
\end{align}
Two point expectation values (in the unperturbed free energy) are then given by 
\begin{align}
    \langle \phi_i(q) \phi_j(q') \rangle_0 & = \frac1{Z_0}{\int \mathcal D \mathbf{\phi}\, \phi_i(q) \phi_j(q') e^{-\beta\mathcal F_0[\phi]} }
\end{align}

The computation of such a correlation function is possible by first evaluating the partition function for a non-zero magnetic field $B(x)$, and using functional derivatives of $B$ to take down components of $\phi$:
\begin{align}
    Z_0[B] & = \int \mathcal{D}\phi\exp\left[-\beta\int\dfrac{d^d q}{(2 \pi)^d} \left( \frac12 E(q^2) |\phi(q)|^2 - B_i(-q)\phi_i(q)\right)\right]\\ 
    \implies \langle \phi_i(q) \phi_j(q') \rangle_0 & = \frac{(2\pi)^{2d}}{\beta^2Z_0[0]} \left.\frac{\delta^2 Z_0[B]}{\delta B_i(-q)\delta B_j(-q')}\right|_{B=0} 
\end{align}
Upon evaluating these path integrals by first approximating them as finitely many integrals and then taking the lattice limit, one can conclude that 
\begin{align}
    Z_0[B] & = C\exp\left[\dfrac{\beta}{2}\int \dfrac{d^d q}{(2\pi)^d}\dfrac{|B(q)|^2}{E(q^2)}\right]\\
    \langle \phi_i(q) \phi_j(q') \rangle_0 & = \dfrac{(2\pi)^d\delta(q+q')\delta_{ij}}{\beta E^2(q)}
\end{align}
The constant $C$ doesn't matter since it cancels out during statistical average computations. If interactions come into the picture, expectation values can be computed in terms of the Gaussian expectation values, as follows:
\begin{align}
    \langle f[\phi] \rangle & = \dfrac{\int \mathcal{D}\phi f[\phi] \exp[-\beta \mathcal F_0 - \beta U]}{\int  \mathcal{D}\phi \exp[-\beta \mathcal F_0 - \beta U]}\\ 
    & = \dfrac{\langle f[\phi] e^{-\beta U} \rangle_0}{\langle e^{-\beta U}\rangle_0}
\end{align}
Expansion of the interaction exponential in a power series leads to a series of integrals which can be represented using Feynman diagrams.

It is convenient to note the striking similarity with quantum field theory, where instead the weight is the exponential of the action, of the form $e^{iS}$ instead:
\begin{align}
    \langle \Omega| \mathcal{T}\{A\} |\Omega\rangle & = \dfrac{\int \mathcal{D}\phi f[\phi] \exp[i S_0[\phi] + iS_1]}{\int  \mathcal{D}\phi \exp[i S_0[\phi] + iS_1]}\\ 
    & = \dfrac{\langle A e^{iS_1} \rangle_0}{\langle e^{iS_1}\rangle_0}
\end{align}
The Feynman diagram representation is also used, with propagators being interpreted as virtual particle worldlines and interaction vertices as interactions between the particles themselves. It is also interesting to note the fact that high energy physicists came up with an enitrely different yet equivalent procedure of removing infinities from loop integrals. Their approach is more focused on recovering the values of renormalised mass (which in this case is $t$), which is not needed here. 
### 1.3 RG in Momentum Space
#### 1.3.1 The Procedure
Wilsonian RG is a means of observing how the effective theory changes as the relevant energy scale changes by a scaling parameter $b>1$. Typically, such an energy scale is introduced by means of a cutoff $\Lambda$ on the momentum $q$. In other words, it is assumed that $\phi(q)=0$ for $|q|>\Lambda$. Such a constraint comes into play with the multiplication of a Heaviside function with the two-point correlation:
\begin{align}
    \langle \phi_i(q) \phi_j(q') \rangle_0 & = \dfrac{(2\pi)^d\delta(q+q')\delta_{ij}}{\beta E^2(q)} \Theta(\Lambda - |q|)
\end{align}

RG involves the following actions:
1) Split the order parameter field into low and high frequencies $\phi(q) = \phi_+(q) + \phi_-(q)$, where $\phi_-(q)$ is non-zero for $q < \Lambda/b$ and $\phi_+(q)$ is non-zero for $\Lambda/b < q < \Lambda$. 
2) Coarse grain by integrating $\phi_+$ out of the partition function to obtain an effective free energy of $\phi_-(q)$. 
3) Rescale $q$, all constants and the field $\phi_-$ by a power of $b$ to get back a theory with cutoff $\Lambda$, to be able to compare the evolution of the theory under RG. 

For example, consider the following free energy which consists of a Gaussian term plus a $\phi^4$ interaction:
\begin{align}
    \beta \mathcal{F}[\phi] & = \int \dfrac{d^d q}{(2 \pi)^d} \frac12 E(q^2) |\phi(q)|^2\\ 
    & + u\iiint \dfrac{d^d q_1d^d q_2 d^d q_3}{(2 \pi)^{3d}} \phi_i(q_1)\phi_i(q_2)\phi_j(q_3)\phi_j(-q_1-q_2-q_3)
\end{align}
The free energy is then split into three parts - one which contains purely $\phi_-$ terms denoted $\beta\mathcal{F}_0[\phi_-]$, the second with purely quadratic $\phi_+$ terms denoted $\beta\mathcal{F}_0[\phi_+]$, and the last with the remaining terms, denoted $\beta\mathcal{F}_1[\phi_-, \phi_+]$. The partition function and the effective free energy are thus written as follows:
\begin{align}
    Z & = \int  \mathcal{D}\phi_- e^{-\beta \mathcal F_0[\phi_-]} \int  \mathcal{D}\phi_+ e^{-\beta \mathcal F_0[\phi_+] - \beta \mathcal{F}_1[\phi_-, \phi_+]}\\ 
    & =: \int  \mathcal{D}\phi_- e^{-\beta \mathcal F_{\text{eff}}[\phi_-]}\\ 
    \implies \beta\mathcal{F}_{\text{eff}}[\phi_-] & = \beta\mathcal{F}_0[\phi_-] - \log \langle e^{-\beta\mathcal{F}_1[\phi_-, \phi_+]} \rangle_+
\end{align}
where $\langle\cdot\rangle_+$ indicates a statistical average over purely the high momentum modes $\phi_+$, with respect to the free energy $\beta\mathcal{F}_0[\phi_+]$. 

Expanding the logarithmic term in a power series leads to the usual cumulant expansion, which consists of connected Feynman diagrams. To write it explicitly, we have 
\begin{align}
    \log \langle e^{-\beta\mathcal{F}_1[\phi_-, \phi_+]} \rangle_+ & = -\langle \beta\mathcal{F}_1[\phi_-, \phi_+] \rangle_+\\ 
    & + \dfrac12\left(\langle \beta^2\mathcal{F}_1^2[\phi_-, \phi_+] \rangle_+ - \langle \beta\mathcal{F}_1[\phi_-, \phi_+] \rangle_+^2\right) + \cdots
\end{align}

#### 1.3.2 $\phi^4$ Theory - First Order

The interacting terms in the free energy are given by
\begin{align}
    \beta \mathcal{F}_1[\phi_-,\phi_+] & = u\int\prod\limits_i\dfrac{d^dk_i}{(2\pi)^{d}} (2\pi)^d \delta\left(\sum k_i\right)[4\phi_{-i}(k_1)\phi_{-i}(k_2)\phi_{-j}(k_3)\phi_{+j}(k_4) \\ 
    & + 4\phi_{-i}(k_1)\phi_{+i}(k_2)\phi_{+j}(k_3)\phi_{+j}(k_4) + 4\phi_{-i}(k_1)\phi_{-j}(k_2)\phi_{+i}(k_3)\phi_{+j}(k_4) \\ 
    & + 2\phi_{-i}(k_1)\phi_{-i}(k_2)\phi_{+j}(k_3)\phi_{+j}(k_4) + \phi_{+}(k_1)\phi_{+}(k_2)\phi_{+}(k_3)\phi_{+}(k_4)]\\ 
    & + \text{gaussian terms}
\end{align}

To first order in the interaction parameter $u$, the only surviving terms of $-\langle\beta\mathcal{F}_1\rangle_+$ are the $\langle \phi_+^4 \rangle_+$ and $\langle \phi_-^2\phi_+^2\rangle_+$ terms, since they are the only terms with an even power of $\phi_+$. The first kind does not matter since it is just an additive constant to the effective free energy, which leads to it getting cancelled out in all statistical average calculations by the partition function. The second kind produces a non-trivial addition to the free energy of the form:
\begin{align}
    -\langle\beta\mathcal{F}_1\rangle_+ & = -u\int\dfrac{\prod d^dk_i}{(2\pi)^{2d}}\left[\phi_{i-}(k_1)\phi_{i-}(k_2) \delta\left(\sum k_i\right) \right.\\ 
    & \left. \times \dfrac{\delta(k_3+k_4)(4+2n)}{\beta E^2(k_3)} \Theta(|k_3| - \Lambda/b) \Theta(\Lambda - |k_3|)\right]\\ 
    & = -u\int\dfrac{d^dk }{(2\pi)^{d}}\phi_{i-}(k)\phi_{i-}(k) \int\limits_{|q| = \Lambda/b}^{|q| = \Lambda}\dfrac{d^d q}{(2\pi)^d}\dfrac{(4+2n)}{\beta E^2(q)}
\end{align}
The factor of $4+2n$ comes from the different kinds of index contractions in the two terms $2\phi_{-i}\phi_{-i}\phi_{+j}\phi_{+j} + 4\phi_{-i}\phi_{-j}\phi_{+i}\phi_{+j}$. The first term gives rise to a factor $2n$ through the $\delta_{jj}$ introduced by the propagator, and the second term does not due to the appearance of $\delta_{ij}$. The Heaviside functions regulate the momentum cutoff induced by $\phi_+$, and result in the limits of the integral in the last line. 

Thus, for $E(q^2) = t + Kq^2+\cdots$, we have to first order in $b$:
\begin{align}
    \beta\mathcal{F}_{\text{eff}}[\phi_-] & = \int \dfrac{d^d q}{(2 \pi)^d} \frac12 (t'+ Kq^2+\cdots) |\phi_-(q)|^2\\ 
    & + u\iiint \dfrac{d^d q_1d^d q_2 d^d q_3}{(2 \pi)^{3d}} \phi_{-i}(q_1)\phi_{-i}(q_2)\phi_{-j}(q_3)\phi_{-j}(-q_1-q_2-q_3) 
\end{align}
where the new constant $t'$ is defined by 
\begin{align}
    t' & = t - \int\limits_{|q| = \Lambda/b}^{|q| = \Lambda}\dfrac{d^d q}{(2\pi)^d}\dfrac{4u(n+2)}{\beta(t+Kq^2 + \cdots)}
\end{align}
The modification to the coupling constants can be represented through the legs of Feynman diagrams being joined and forced into high momentum modes. At first order, the only coupling constant which does change due to this is $t$, but that is definitely not true for higher orders as shown in the next section. 

The final step is to rescale the momenta and the fields by a power of $b$ to produce a comparable theory. This is equivalent to the variable changes $q \to b^{-1}q$ and $\phi \to b\phi$, giving to first order in $u$:
\begin{align}
    t(b) & = b^2\left[t - \int\limits_{|q| = \Lambda/b}^{|q| = \Lambda}\dfrac{d^d q}{(2\pi)^d}\dfrac{4u(n+2)}{\beta(t+Kq^2 + \cdots)}\right]\\ 
    u(b) & = b^{4-d}u
\end{align}
Thus, there is no non-trivial behaviour in $u$ at this order, while $t$ does evolve, giving rise to a non-zero anomalous dimension. 

#### 1.3.3 $\phi^4$ Theory - Second Order
Rich observations can be made at second order in $u$. The calculations, however, are naturally trickier. We suppose that $E(q^2) = t + Kq^2$. The second term in the cumulant expansion is given by:
\begin{align}
    U = 2\log \langle e^{-\beta\mathcal{F}_1[\phi_-, \phi_+]} \rangle_+^{(2)} & = \langle \beta^2\mathcal{F}_1^2[\phi_-, \phi_+] \rangle_+ - \langle \beta\mathcal{F}_1[\phi_-, \phi_+] \rangle_+^2
\end{align}
and the interacting terms in the free energy are given by
\begin{align}
    \beta \mathcal{F}_1[\phi_-,\phi_+] & = u\int\prod\limits_i\dfrac{d^dk_i}{(2\pi)^{d}} (2\pi)^{2d} \delta\left(\sum k_i\right)[4\phi_{-i}(k_1)\phi_{-i}(k_2)\phi_{-j}(k_3)\phi_{+j}(k_4) \\ 
    & + 4\phi_{-i}(k_1)\phi_{+i}(k_2)\phi_{+j}(k_3)\phi_{+j}(k_4) + 4\phi_{-i}(k_1)\phi_{-j}(k_2)\phi_{+i}(k_3)\phi_{+j}(k_4) \\ 
    & + 2\phi_{-i}(k_1)\phi_{-i}(k_2)\phi_{+j}(k_3)\phi_{+j}(k_4) + \phi_{+}(k_1)\phi_{+}(k_2)\phi_{+}(k_3)\phi_{+}(k_4)]
\end{align}
Dubbing the different $\phi^4$ terms above as terms (1) to (5), it is possible to see that a combination of two (1) terms leads to a six-point interaction, with $\phi^6$ term. Thus, the space of theories with a purely $\phi^4$ interaction is not closed under RG flows. Additionally, there are corrections to the derivative term $K$ this time around, due to a factor of $q^2$ which shows up in a correction to the quadratic term. 

The terms of the second order correction are then represented by the (+)-average of products of pairs of terms in the free energy, with each term originating from a different factor of $\beta\mathcal{F}_1$. For example, one such term which originates from (1)$\times$(1) which contributes to the $\phi_-^6$ part of the Hamiltonian is given below:
\begin{align}
    U^{(2,6)} & =  u^2\int \prod\limits_{l=1}^8 \dfrac{d^d k_l}{(2\pi)^d} (2\pi)^d \delta\left(\sum\limits_{i=1}^4 k_i\right)\delta\left(\sum\limits_{i=5}^8 k_i\right)\\ 
    & \times \langle16\phi_{-i}(k_1)\phi_{-i}(k_2)\phi_{-j}(k_3)\phi_{+j}(k_4)\phi_{-m}(k_5)\phi_{-m}(k_6)\phi_{-n}(k_7)\phi_{+n}(k_8)\rangle_+
\end{align}

For this discussion, only corrections to the quadratic and $\phi^4$ terms are discussed, since they exhibit interesting behaviour by themselves. The quadratic terms which can be formed are obtained by grouping together terms which in total have two $\phi_-$ factors. Explicitly speaking, they are formed from the products (2)$\times$(2), (3)$\times$(5) and (4)$\times$(5). 

Upon computing the corrections to $t, K$ and $u$, which are non-zero in this order as opposed to the previous one, we find the usual Gaussian fixed point at $u=t=0$ and a new fixed point dubbed the Wilson-Fischer fixed point, which is non-trivial. Moreover, this exists only for $d<4$, as shown below.

[image]

It is also possible to compute the critical exponent $\eta$. It is traditionally defined to be the exponent of how much $\phi$ scales by when a coarse-graining is performed. In this case, it is defined by the change in $K$ and $t$. While performing the final rescaling, we define $K$ to be invariant under all RG transformations. Thus, the field $\phi$ is rescaled by a factor $b^{\frac{-d+2-\eta}{2}}$. 

## 2. The Dynamical Renormalisation Group
Dynamical RG is the equivalent of the usual Wilsonian RG for fields obeying a stochastic PDE. In such a setting, temperatures are replaced by noisy fields $\eta(t,\mathbf{x})$ present in the equation of motion of a dynamical order parameter $\phi(t,x)$, adding random noise to the system. These noisy fields are typically assumed to be uncorrelated with respect to space and time, and Gaussians with zero mean.
### 2.1. The Linear Theory 
It is instructive to start with the linear theory, and describe other theories as perturbations from this. The general equation of motion is then of the form 
\begin{align}
\partial_t \phi & = F_0[\phi,\dots] + \eta\\ 
F_0 & = -(i\nabla)^\alpha (a\phi - \kappa\nabla^2 \phi)
\end{align}
The parameter $\alpha$ is introduced to be as general as possible. The case $\alpha=2$ corresponds to a noisy diffusion-type equation. Upon taking a Fourier transform from $(t,\mathbf{x}) \to (\omega,\mathbf{q})$, the equation of motion becomes as follows, and is easy to solve. 
\begin{align}
    -i\omega \phi & = -q^\alpha(a+\kappa q^2)\phi + \eta\\ 
    \implies \phi(\omega,\mathbf{q}) & = G_0(\omega,\mathbf{q})\eta(\omega,\mathbf{q})\\ 
    G_0(\omega,\mathbf{q}) & = \dfrac1{-i\omega + q^\alpha(a+\kappa q^2)}
\end{align}

We choose the noise to have correlation as follows 
\begin{align}
\langle\eta(\omega,\mathbf{q})\eta(\omega',\mathbf{q}')\rangle & = 2D q^\alpha (2\pi)^{d+1} \delta^d(\mathbf{q}+\mathbf{q}')
\end{align}
This is done so so that we have the following result:
\begin{align}
    \langle\phi(\omega,\mathbf{q})\phi(\omega',\mathbf{q}')\rangle & = (2\pi)^{d+1} C_0(\omega,\mathbf{q})\delta(\omega+\omega')\delta^d(\mathbf{q}+\mathbf{q}')\\ 
    C_0(\omega,\mathbf{q}) & = \dfrac{2Dq^\alpha}{\omega^2 + q^{2\alpha}(a+\kappa q^2)^2}
\end{align}
with the scattering factor being 
\begin{align}
    S(\mathbf{q}) = \int \dfrac{d\omega}{2\pi}C_0(\omega, \mathbf{q}) = \dfrac D{a+\kappa q^2}
\end{align}
Note: the noise is adjusted such that the scattering factor matches the canonical ensemble's predictions of being a Lorentzian at its steady state. The dependence on $\alpha$ vanishes too. 
### 2.2 Perturbation Theory
Suppose the free energy has a term non-linear in $\phi$. Such an equation is not exactly solvable, and required the expertise of perturbation theory. For example, suppose $F[\phi] = -(i\nabla)^\alpha (a\phi - \kappa\nabla^2 \phi) + v_n\phi^n$, where $v_n$ is some vertex function, typically a derivative operator. Upon taking the Fourier transform, we get the formal relation:
\begin{align}
    \phi(\omega,\mathbf{q}) & = G_0(\omega,\mathbf{q})\eta(\omega,\mathbf{q}) + G_0(\omega,\mathbf{q})[v_n\phi^n](\omega,\mathbf{q})
\end{align}
where the last term is the Fourier transform of the whole $\phi$ interaction term. The problem clearly lies in the fact that the fourier transform of the non-linear term is of the form:
\begin{align}
    [v_n\phi^n](\omega,\mathbf{q}) & = \int \left[ \prod\limits_{i=1}^n \dfrac{d^{d} k_i d\Omega_i}{(2\pi)^{d+1}} \phi(\Omega_i, \mathbf{k}_i) \right] v_n(\{\Omega_j,\mathbf{k}_j\})(2\pi)^{d+1}\delta^{d+1}\left(q-\sum k_i\right)
\end{align}

The perturbative procedure involves replacing every $\phi$ in the RHS with the whole RHS in the above formal relation, to iteratively obtain a series solution of $\phi$. This can be represented using a graphical system, similar to Feynman diagrams. With each propagator $G_0$ being represented by a line, each interaction term $v_n$ represented by a vertex with $n$ points and each noise being represened by a solid dot at the end of a line, we obtain solutions to $\phi$ as a series of "trees" with $n$-point vertices. As usual, wavevectors are conserved at each vertex as imposed by the delta functions. Noise averages being taken are replaced by $C_0$, which are represented by lines with a hollow dot in the middle. Moreover, each graph comes with a multiplicity, which is the number of ways the perturbation expansion produces the same graph. While diagrams are a nice way to represent these terms in the expansion, it is always better to refer to the actual sum while making computations. 

#### 2.2.1 Many-Point Noise Correlations
We make note that the noise satisfies a Gaussian distribution in the following sense too:
\begin{align}
    P[\eta] & = A \exp\left[-\int \dfrac{d\Omega\,d^{d}q}{(2\pi)^{d+1}} \dfrac{\eta^2(\Omega,\mathbf{q})}{4Dq^\alpha} \right]\\ 
    \langle H[\eta]\rangle & = A \int \mathcal{D}\eta\; H[\eta] \exp\left[-\int \dfrac{d\Omega\,d^{d}q}{(2\pi)^{d+1}}\dfrac{\eta^2(\Omega,\mathbf{q})}{4Dq^\alpha} \right]
\end{align}
This implies that the $\eta$-correlations satisfy the Wick-relation, i.e. an $n$-point correlation of the noise fields is equal to the sum of product of correlations of sub-pairs of the collection of noises. For example:
\begin{align}
    \langle\eta_1\eta_2\eta_3\eta_4\rangle & = \langle \eta_1\eta_2\rangle\langle \eta_3\eta_4\rangle + \langle \eta_1\eta_3\rangle\langle \eta_2\eta_4\rangle + \langle \eta_1\eta_4\rangle\langle \eta_2\eta_3\rangle
\end{align}

#### 2.2.2 Equivalent Path Integral Formulation
The above reformulation may be taken further, with the following relation (atleast in the linear theory):
\begin{align}
    P[\eta] \mathcal{D}\eta & = P[\phi[\eta]]\mathcal{D}\eta\\ 
    & = A \exp\left[-\int \dfrac{d\Omega\,d^{d}q}{(2\pi)^{d+1}} \dfrac{(G_0(\Omega,\mathbf{q})\phi(\Omega,\mathbf{q}) - \hat{U}(\Omega,\mathbf{q}))^2}{4Dq^\alpha} \right]\mathcal{D}\eta
\end{align}
This is assuming the interaction $U$ preserves an invertible mapping between $\eta$ and $\phi$. Moreover, the change of measure $\mathcal{D}\eta \to \mathcal{D}\phi$ would result in an additional factor of the Jacobian. While this approach has been studied, the direct perturbative approach proves to be easier and more useful.

### 2.3 Renormalisation Group Procedure
The RG procedure is similar to that of the static case. The field is split into high and low frequencies $\phi = \phi_- + \phi_+$, with the high frequency range being $(\Lambda/b, \Lambda)$. Perturbation theory is performed with respect to the non-linear term, and one gets terms involving both high and low momentum parts of $\phi$. The noise averages over the low momentum modes are considered to be the $O(b^0)$ contributions to the solution, while the averages over high momentum modes are interpreted as corrections to the coupling constants in the stochastic PDE. 

Similar to static RG, we take $b = e^l$ for convenience. Typically, quantities are only evaluated to $O(l)$ since that is all the information we need to determine fixed points. Here onwards we relabel $l \to \delta l$ to emphasise it is small. 

#### 2.3.1 Effective Noise
$\phi(\Omega,\mathbf{q}) = G_0(\Omega,\mathbf{q})\eta_{eff}(\Omega,\mathbf{q})$, where $\eta_{eff}$ is a new noisy term with a different variance $2D' = 2(D+\delta D)$ which can be perturbatively computed using:
\begin{align}
    (2\pi)^{d+1}\delta^{d+1}(q+q')2(D+\delta D) q^\alpha & = G_0^{-1}(q)G_0^{-1}(q')\langle \phi(q)\phi(q')\rangle
\end{align}

In the latter, one assumes  $\phi(\Omega,\mathbf{q}) = G(\Omega,\mathbf{q})\eta(\Omega,\mathbf{q})$, where $G$ is the "exact" propagator. This is calculated using 
\begin{align}
    2Dq^\alpha(2\pi)^{d+1}\delta^{d+1}(q+q')G(\Omega,\mathbf{q}) & = \langle \phi(q)\eta(q')\rangle
\end{align}
### 2.4 An Example: Model B
Consider a free energy given by 
\begin{align}
    f(\phi) & = \dfrac a2 \phi^2 + \dfrac b4 \phi^4\\ 
    \mathcal{F}[\phi] & = \int d^{d}x \left(f(\phi(\mathbf{r}) + \dfrac\kappa2 |\nabla\phi|^2 \right)
\end{align}
and the order parameter satisfying the stochastic PDE
\begin{align}
    \partial_t\phi & = \nabla^2 \dfrac{\delta\mathcal{F}}{\delta\phi} + \eta
\end{align}
This equation has a conserved current in the absence of noise, namely $\nabla\dfrac{\delta\mathcal{F}}{\delta\phi}$, due to $\alpha = 2$. 

The solution to this equation is given (formally) by
\begin{align}
    \phi(q) & = G_0(q)\eta(q) + G_0(q)u\int\dfrac{d^{d+1}k_1 d^{d+1}k_2}{(2\pi)^{2(d+1)}} \phi(k_1)\phi(k_2)\phi(q-k_1-k_2)\\ 
    G_0(q) & = \dfrac1{-i\omega+aq^2+\kappa q^4}
\end{align}
Note that the vertex of such a theory has three outgoing lines, with a vertex factor $-uq^2$. 
#### 2.4.1 Propagator corrections
The first loop order propagator correction is given by the below diagram
![image](https://hackmd.io/_uploads/S1i9RXkbge.png)

According to the Feynman rules, the vertex contributes a factor of $-uq^2$, the propagators a factor of $G_0^2(q) C_0(k)$ and the multiplicity being 3, due to there being three different ways to arrive at this diagram through noise contractions. Thus, the diagram has the value:
\begin{align}
    I(\Gamma_p^{(1)}) & = -3uq^2 G_0^2(q)\int \dfrac{d^{d+1}k}{(2\pi)^{d+1}} C_0(k)\\ 
    & = -3uq^2 G_0^2(q)\int \dfrac{d^{d}k}{(2\pi)^{d}} \dfrac{D}{a + \kappa k^2}\\ 
    & = -\dfrac{3uq^2 G_0^2(q)S_d}{(2\pi)^d} \int\limits_{\Lambda/(1+\delta l)}^\Lambda \dfrac{D k^{d-1} dk}{a+\kappa k^2}\\ 
    & = -\dfrac{3uq^2 G_0^2(q)S_d D\Lambda^d}{(2\pi)^d(a+\kappa\Lambda^2)} \delta l + O(\delta l^2)\\ 
    \implies G(q) & \approx G_0(q)\left[1 - \dfrac{3uq^2 G_0^2(q)S_d D\Lambda^d}{(2\pi)^d(a+\kappa\Lambda^2)} \delta l\right]
\end{align}
The first loop correction to the vertex factor is given by the following graph. 
[INSERT GRAPH]

This can be evaluated similarly, to give:
\begin{align}
    I(\Gamma_v^{(1)}) & = -18 u q^2\int \dfrac{d^{d+1} k}{(2\pi)^{d+1}} C_0(k)G_0(q-p_1-k)\\ 
    & = -18 u q^2\int \dfrac{d^{d+1} k}{(2\pi)^{d+1}} C_0(k)G_0(q-k)\\ 
    & = -18 u q^2\int \dfrac{d^{d} k}{(2\pi)^{d}} F(q,k),
\end{align}
where 
\begin{align}
    F[q,k] & = \int \dfrac{d \Omega}{2\pi} \dfrac{2Dk^2}{\Omega^2 + ak^2 + \kappa k^4} \dfrac1{}
\end{align}