Chapter 2 Lagrangian
The lattice Lagrangian is
L0=∑x[−2κ0∑μRe{ϕ†0(x)ϕ0(x+μ)}],L1=∑x[−2κ1∑μRe{ϕ†1(x)ϕ1(x+μ)}],LI=gL∑xRe{ϕ†1(x)ϕ30(x)}.
The fields has the constrain ϕ†iϕi=1 (i=0,1) so ϕi=eiθi.
We also have m20=1κ0−8,,φ0=√2κ0ϕ0m21=1κ1−8,,φ1=√2κ1ϕ1, and g=gL4√κ0κ3/21
In alternative we also try
- to simulate a point split version of LI
LpsI=gL∑xRe{ϕ†1(x)ϕ0(x)(18ˆ3∑μ=±−3ϕ0(x+μ))2}
- derivative couplint
LderivI=gL∑xRe{ϕ†1(x)ϕ0(x)(∂μϕ0(x))2} where we discretised with the symmetric O(a2) finite difference ∂μϕ→12(ϕ(x+μ)−ϕ(x−μ)) and (∂μϕ)2→143∑μ=0((ϕ(x+μ)(ϕ(x+μ)−2ϕ(x+μ)(ϕ(x−μ)+ϕ(x−μ)(ϕ(x−μ))