Chapter 2 Lagrangian

The lattice Lagrangian is

L0=x[2κ0μRe{ϕ0(x)ϕ0(x+μ)}],L1=x[2κ1μRe{ϕ1(x)ϕ1(x+μ)}],LI=gLxRe{ϕ1(x)ϕ30(x)}.

The fields has the constrain ϕiϕi=1 (i=0,1) so ϕi=eiθi.

We also have m20=1κ08,,φ0=2κ0ϕ0m21=1κ18,,φ1=2κ1ϕ1, and g=gL4κ0κ3/21

In alternative we also try

  • to simulate a point split version of LI

LpsI=gLxRe{ϕ1(x)ϕ0(x)(18ˆ3μ=±3ϕ0(x+μ))2}

  • derivative couplint

LderivI=gLxRe{ϕ1(x)ϕ0(x)(μϕ0(x))2} where we discretised with the symmetric O(a2) finite difference μϕ12(ϕ(x+μ)ϕ(xμ)) and (μϕ)2143μ=0((ϕ(x+μ)(ϕ(x+μ)2ϕ(x+μ)(ϕ(xμ)+ϕ(xμ)(ϕ(xμ))