DMRG study of the theta-dependent mass spectrum in the 2-flavor Schwinger model

We study the $\theta$-dependent mass spectrum of the massive $2$-flavor Schwinger model in the Hamiltonian formalism using the density-matrix renormalization group(DMRG). The masses of the composite particles, the pion and sigma meson, are computed by two independent methods. One is the improved one-point-function scheme, where we measure the local meson operator coupled to the boundary state and extract the mass from its exponential decay. Since the $\theta$ term causes a nontrivial operator mixing, we unravel it by diagonalizing the correlation matrix to define the meson operator. The other is the dispersion-relation scheme, a heuristic approach specific to Hamiltonian formalism. We obtain the dispersion relation directly by measuring the energy and momentum of the excited states. The sign problem is circumvented in these methods, and their results agree with each other even for large $\theta$. We reveal that the $\theta$-dependence of the pion mass at $m/g=0.1$ is consistent with the prediction by the bosonized model. We also find that the mass of the sigma meson satisfies the semi-classical formula, $M_{\sigma}/M_{\pi}=\sqrt{3}$, for almost all region of $\theta$. While the sigma meson is a stable particle thanks to this relation, the eta meson is no longer protected by the $G$-parity and becomes unstable for $\theta\neq 0$.