# Mapping a continuum XY model to a discrete one

A low energy expansion of a system I am currently investigating is described by this XY-like model:

$$H_1 = \int_0^\beta \mathrm{d} \tau \int_{\left[ 0, L \right]^2} \mathrm{d}^2 x \left( J \left( \nabla \theta \right)^2 + K \left( \partial_\tau \theta \right)^2 \right)$$

It is essentially an anisotropic XY model. It is better analyzed, as far as my question is concerned, by considering $\tau$ a space dimension, so the model is effectively 3D. If we take $\beta=L$ and $J=K$ to make it isotropic, then it undergoes a phase transition for $T=2.20$; a mean field model is presented by Kleinert ("Gauge fields in Condensed Matter", World Scientific, 1989), there the critical temperature is estimated to be $3$.

I have some Montecarlo results for a similar lattice model:

$$H_2 = - \sum_{\langle ij \rangle} J_{ij} \cos \left( \theta_i - \theta_j \right)$$

in which $J_{ij}$ can take different values if the link is on the $x-y$ plane or along the $z$ direction, I'll call them $J_{xy}$ and $J_z$; these features should account for the anisotropy of the first system, because I'd like to map it $H_2$ to $H_1$. The system described by $H_2$ lives on a cube lattice.

How do I map the first Hamiltonian into the second one? By knowing for which values of $J_{xy}$ and $J_z$ the system $H_2$ is at the critical point, I would like to know the critical values of $J$ and $K$.

I observed that a naive mapping $J_{xy} \to J$, $J_z \to K$ doesn't work, because I can always rescale $\tau$ and $(x,y)$ in the first Hamiltonian. This is mathematically equivalent, but leads to a different mapping, so it shows that this naive mapping doesn't work. I think that the reason is that by rescaling the length I am also modifying the limits of integrations, and this cannot be ignored by just "naively" imposing periodic boundary conditions: the system is anisotropic as far as the coupling is concerned, but also as far as the box it's contained is concerned.

My guess: maybe I should rescale the lengths in the first Hamiltonian so that the system lives on a cube, i.e. $\beta \to \beta' = L$ and then maybe I am allowed to map the first Hamiltonian to the first one. Also in this case the $L \to \infty$ limit seems particularly badly behaved, I can provide more details if needed.

Note that $J_{xy}$ and $J_z$ have the dimensions of energy while $J$ and $K$ have dimension of energy/(area x time). In the process of making the continuum Hamiltonian into a discrete one, you will need to choose short distance scales for both space as well as (Euclidean) time. You can choose a square lattice of length say, $a$ for space and split the time in scales of length $b$. Then, one dimensional grounds, you should expect to see the following map: $$\frac{J_{xy}}{a^2b} \rightarrow J\quad,\quad \frac{J_z}{a^2b} \rightarrow K\ .$$ These are related to the naive scaling (of space and time) dimensions of $J$ and $K$.