Chapter 4 E2 connected
We define the generator of the correlation function as \[ Z[J_0,J_1]=N\int D\phi_0D\phi_1 e^{-S+\int J_0\phi_0+\int J_1\phi_1} \,. \] with \(N=1/Z[0,0]\) such that \(Z[0,0]=1\). the correlation function acting with derivativesrespect to \(J\) and then setting it to zero, for example \[ \langle \phi_0(x)^2 \phi_0(y)^2\rangle= \frac{\partial^2}{\partial^2 J_0(x)}\frac{\partial^2}{\partial^2 J_0(y)}Z[J_0,J_1]\bigg|_{J_0=J_1=0}\,. \] We can define the connected correlation function deriving logarithm of \(Z\) i.e. \[ \langle \phi_0(x)^2 \phi_0(y)^2\rangle_c= \frac{\partial^2}{\partial^2 J_0(x)}\frac{\partial^2}{\partial^2 J_0(y)}\log Z[J_0,J_1]\bigg|_{J_0=J_1=0}\\ =\langle \phi_0(x)^2 \phi_0(y)^2\rangle-\langle \phi_0(x) \phi_0(y)\rangle^2 -\langle \phi_0(x)^2\rangle\langle \phi_0(y)^2\rangle \] we use the fact that \(\langle \phi (x)\rangle=0\).
Below we plot the effective mass computed from
- \(\langle \phi_0(x)^2 \phi_0(y)^2\rangle\)
- \(\langle \phi_0(x)^2 \phi_0(y)^2\rangle-\langle \phi_0(x) \phi_0(y)\rangle^2\)
- \(\langle \phi_0(x) \phi_0(y)\rangle^2\)
In all cases we remove the constant \(\langle \phi_0(x)^2\rangle\) term shifting the correlator.
4.0.0.1 ../out/G2t_T32_L24_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.000000_rep0_output
E2_0(L24T32) = 0.26571(17) \(\chi^2/dof=\) 0.45542
E2_0_con(L24T32) = 0.27326(28) \(\chi^2/dof=\) 0.17525
E1_0_sq(L24T32) = 0.25850(10) \(\chi^2/dof=\) 1.0445
E1_0(L24T32) = 0.129285(59) \(\chi^2/dof=\) 0.039038
4.0.1 FT \(\phi^2\)
Below we add to the plot the FT of the operator \(\phi(x)^2\) difined as
\[ \widetilde{\phi(t,p)^2}=\sum_{\vec x}\phi(t,\vec x)^2 e^{i\vec{p}\cdot\vec{x}} \] we call its effective mass E2_0_p0
4.0.1.1 ../../g0.025/out/G2t_T64_L28_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output
E2_0(L28T64) = 0.26027(12) \(\chi^2/dof=\) 1.1137
E2_0_con(L28T64) = 0.26479(21) \(\chi^2/dof=\) 0.45586
E1_0_sq(L28T64) = 0.255803(69) \(\chi^2/dof=\) 2.0787
E2_0_p0(L28T64) = 1.3590(23) \(\chi^2/dof=\) 824.81
E1_0(L28T64) = 0.127807(50) \(\chi^2/dof=\) 0.81852