Tensor renormalization group study of (3+1)-dimensional $Z_2$ gauge-Higgs model at finite density

We investigate the critical endpoints of the (3+1)-dimensional $Z_2$ gauge-Higgs model at finite density together with the (2+1)-dimensional one at zero density as a benchmark using the tensor renormalization group method. We focus on the phase transition between the Higgs phase and the confinement phase at finite chemical potential along the critical end line. In the (2+1)-dimensional model, the resulting endpoint is consistent with a recent numerical estimate by the Monte Carlo simulation. In the (3+1)-dimensional case, however, the location of the critical endpoint shows disagreement with the known estimates by the mean-field approximation and the Monte Carlo studies. This is the first application of the tensor renormalization group method to a four-dimensional lattice gauge theory and a key stepping stone toward the future investigation of the phase structure of the finite density QCD.