Two-dimensional excitation information by a matrix product state method on infinite helices

Understanding the excitation spectrum in two-dimensional quantum many-body systems has long been a formidable challenge. In this study, we propose an innovative approach by introducing an excitation Ansatz based on an infinite matrix product state (MPS) with a helix structure. The use of infinite uniform MPS states allows us to accurately extract key properties such as energy, degeneracy, spectrum weight, and scaling behavior of low-energy excited states simultaneously. To validate the effectiveness of our method, we apply it to the critical point of the transverse-field Ising model. The scaling exponent of the energy gap extracted aligns closely with conformal bootstrap results. Subsequently, we extend our method to the $J_1\text{−}{J}_2$ Heisenberg model on a square lattice. Our findings reveal that the degeneracy of lowest-energy excitations serves as a reliable metric for distinguishing different phases. The phase boundaries identified by our method are consistent with previous findings. The present method provides a promising avenue for studying the excitation spectrum of two-dimensional quantum many-body systems.