phi4-analysis
2021-12-20
Chapter 1 Lagrangian
L=L0+L1+LIZ2L0=12∂μφ0∂μφ0+12m20φ20+λ0φ40L1=12∂μφ1∂μφ1+12m21φ21+λ1φ41LIZ2=μφ20φ21LI=gφ1φ30 On the lattice we can discretise the derivative ∂μφ(x)=1a(φ(x+μ)−φ(x)). On the lattice the above lagrangian can be written in a more convenient way for simulations L0=∑x[−2κ0∑μϕ0(x)ϕ0(x+μ)+λ0L(ϕ0(x)2−1)2+ϕ0(x)2],L1=∑x[−2κ1∑μϕ1(x)ϕ1(x+μ)+λ1L(ϕ1(x)2−1)2+ϕ1(x)2],LIZ2=μL∑xϕ0(x)2ϕ1(x)2,LI=gL∑xϕ1(x)ϕ0(x)3. With m20=1−2λ0Lκ0−8,λ0=λ0L4κ20,φ0=√2κ0ϕ0m21=1−2λ1Lκ1−8,λ1=λ1L4κ21,φ1=√2κ1ϕ1, and μ=μL4κ0κ1,g=gL4√κ0κ3/21