# The quantum field description of the vortex in the Superconductor-Insulator phase transition

I was reading professor Naoto Nagaosa's book on Quantum Field Theory in Condensed Matter Physics about the problem on universal conductivity. On page 156, he showed a method to obtain the QFT description of the vortices. But I have some confusions on his approach.

In (5.3.50), he defined the partition function of a specific loop:

$\begin{gathered} \hspace{5cm} Z_{loop}=\int_{0}^{\infty}dt\ P(t)\ e^{-At} \end{gathered}$

here $A$ is the action ( $\textit{constant}$ ) along one unit segment, and $P(t)$ is the number of possible configurations of a closed loop consisting of $t$ steps. Later on, in order to get $P(t)$, he introduced a probability distribution $p(x,x',t)$ for the probability of reaching point $x$ after $t$ steps having started at $x'$, and he said the $p(x,x',t)$ obeys the diffusion equation

$\begin{gathered} \hspace{5cm} \frac{\partial p(x,x',t)}{\partial t}=\frac{1}{2D}\nabla_x^2 p(x,x',t) \end{gathered}$

I don't really understand why the $p(x,x',t)$ here has to satisfy this equation ( $\textbf{first doubt}$ ), however, this equation reminds me of the form of Schrodinger equation, so I guess this can be formulated in the way like:

\begin{aligned} \hspace{5cm} p(x,x',t)&=\langle x|e^{-t \hat{H}}|x'\rangle \\ \hat{H} &= \frac{1}{2D}\hat{P}^2 \\ \langle x| \hat{P}| \psi \rangle &= \frac{1}{i}\partial_x \langle x| \psi\rangle \end{aligned}

from which I can easily get the path integral representation of the $p(x,x',t)$ in (5.3.52). But then I started to feel confused about the meaning of the action $A$, because now it becomes more clear that the action of a vortex along a path should be governed by the "Hamiltonian" defined above, why should it just be a constant in the first equation? Also, later on, it seems that he assumed each path with $t$ steps is equally weighted, because he defined the number of paths $\Gamma(x,x',t)$ going from $x'$ to $x$ in $t$ steps acording to: \begin{aligned} \hspace{5cm} \Gamma(x,x',t) = p(x,x',t)\ (2D)^t \end{aligned}

I hope someone who understands the approach here can give me an explanation of the logic here, and why should $p(x,x',t)$ satisfy the second equation.

• Perhaps you can write to Nagaosa directly for a better understanding. I would naively say he tries the simplest known model for transport of probability distribution, that is, a diffusion equation. If the results he finds nicely fit some observations, then he's happy with his model. If $p$ is the probability distribution of the vortex center, then a diffusion equation can be justified by hand, simply from observation that the vortices diffuse. A more refined model could be the use of complicated potential to relate the sticking of vortices. Microscopic justification might be really difficult. – FraSchelle Apr 24 '17 at 8:20