We discuss formation of cosmic strings associated with a spontaneously broken approximate !Latex$U(1)$ symmetry by performing classical field-theoretical simulation. An original !Latex$U(1)$ symmetry is explicitly broken down to its subgroup !Latex$Z_{N}$. We evaluate critical values of ratios of an effect of explicit breaking to that of the spontaneous breaking to form topological defects for !Latex$N=1$ and !Latex$N=2$, respectively. For !Latex$N=1$, a cosmic string attached to a single domain wall can be formed when the amount of the explicit breaking is three order magnitude smaller than that of the spontaneous breaking. For !Latex$N=2$, no matter how large the explicit breaking is, domain walls are inevitably formed as long as cosmic temperature is high enough to restore the !Latex$Z_{2}$ symmetry. In this case, we also find that when the amount of the explicit breaking is smaller than that of the spontaneous breaking, cosmic strings are also inevitably formed.

We construct a Grassmann tensor network representing the partition function of 1+1D SU(2) lattice gauge theory coupled with staggered fermions. At finite couplings, a random sampling is applied to discretize the group integration. The initial bond dimension turns out to be 16K where K is the number of SU(2) matrices sampled for each link variable. We introduce an efficient initial tensor compression scheme to reduce the size of initial tensors. Then, Grassmann bond-weighted tensor renormalization group approach is adopted to investigate a phase diagram in the $(m, \mu)$ plane with the quark mass $m$ and chemical potential $\mu$. The free energy density, number density, and diquark condensate at different gauge couplings are computed as a function of the chemical potential. We discuss the efficiency of random sampling method, our initial tensor compression scheme, and the future application toward the corresponding higher-dimensional models.

We construct TBA equations for D-type Argyres-Douglas theories with a SU(2) flavor symmetry in the Nekrosov-Shatashvili limit based on their spectral networks. We show that the solutions of TBA equations are related to the exact quantum periods of the corresponding quantum Seiberg-Witten curves including a centrifugal term. We study the variety of TBA systems across the Coulomb branch moduli space and find they correspond to the Dynkin diagrams of Dn Lie algebras in the minimal chamber and reproduce Zamolodchikov's TBA equations for reflectionless D scattering theories in the maximal chamber.