Chapter 2 Lagrangian

The lattice Lagrangian is

\[ {\cal L}_0= \sum_{x}\left[ -2\kappa_0\sum_\mu \mbox{Re}\left\{ \phi_0^\dagger(x) \phi_0({x+\mu})\right\} \right]\,,\\ {\cal L}_1= \sum_{x}\left[ -2\kappa_1\sum_\mu \mbox{Re}\left\{ \phi_1^\dagger(x) \phi_1({x+\mu})\right\} \right]\,,\\ {\cal L}_{I}= g_L \sum_{x}\mbox{Re}\left\{ \phi_1^\dagger(x) \phi_0^3(x)\right\} \,. \]

The fields has the constrain \(\phi^\dagger_i\phi_i=1\) (i=0,1) so \(\phi_i=e^{i\theta_i}\).

We also have \[\begin{align} & m_0^2=\frac{1}{\kappa_0}-8\,,\quad \,,\quad \varphi_0=\sqrt{2\kappa_0}\phi_0\\ & m_1^2=\frac{1}{\kappa_1}-8\,,\quad \,,\quad \varphi_1=\sqrt{2\kappa_1}\phi_1 \,, \end{align}\] and \[\begin{align} g=\frac{g_L}{4\sqrt{\kappa_0}\kappa_1^{3/2}} \end{align}\]

In alternative we also try

• to simulate a point split version of \({\cal L}_I\)

\[ {\cal L}_{I}^{ps}= g_L \sum_{x}\mbox{Re}\left\{ \phi_1^\dagger(x) \phi_0(x) \left(\frac{1}{8}\sum_{\mu=\pm-3}^{\hat3} \phi_0(x+\mu)\right)^2\right\} \]

• derivative couplint

\[ {\cal L}_{I}^{deriv}=g_L \sum_{x}\mbox{Re}\left\{ \phi_1^\dagger(x) \phi_0(x) \left(\partial_\mu \phi_0(x)\right)^2\right\} \] where we discretised with the symmetric \(O(a^2)\) finite difference \(\partial_\mu \phi\to \frac{1}{2}(\phi(x+\mu)-\phi(x-\mu))\) and \[ \left(\partial_\mu \phi\right)^2\to \frac{1}{4}\sum_{\mu=0}^3 \left((\phi(x+\mu)(\phi(x+\mu)-2\phi(x+\mu)(\phi(x-\mu)+\phi(x-\mu)(\phi(x-\mu) \right) \]