Entanglement scaling and criticality of infinite-size quantum many-body systems in continuous space addressed by a tensor network approach

Simulating strongly-correlated quantum systems in continuous space belongs to the most challenging and long-concerned issues in quantum physics. This work investigates the quantum entanglement and criticality of the ground-state wave-functions of infinitely-many coupled quantum oscillators (iCQOs). The essential task involves solving a set of partial differential equations (Schrödinger equations in the canonical quantization picture) with infinitely-many variables, which currently lacks valid methods. By extending the imaginary-time evolution algorithm with translationally-invariant functional tensor network, we simulate the ground state of iCQOs with the presence of two- and three-body couplings. We determine the range of coupling strengths where there exists a real ground-state energy (dubbed as physical region). With two-body couplings, we reveal the logarithmic scaling law of entanglement entropy (EE) and the polynomial scaling law of correlation length against the virtual bond dimension χ at the dividing point of physical and non-physical regions. These two scaling behaviors are signatures of criticality, according to the previous results in quantum lattice models, but were not reported in continuous-space quantum systems. The scaling coefficients result in a central charge c=1, indicating the presence of free boson conformal field theory (CFT). We further show that the presence of three-body couplings, for which there are no analytical or numerical results, breaks down the CFT description at the dividing point. Our work reveals the scaling behaviors of EE in continuous-space quantum many-body systems. These results provide strong numerical evidence supporting the efficiency of TN in representing continuous-space quantum wave-functions in the thermodynamic limit and offer an efficient approach to studying entanglement properties and criticality in continuous space.