Direct Observation of the Berezinskii-Kosterlitz-Thouless Renormalization Group Flow at a Quantum Phase Transition

The Berezinskii-Kosterlitz-Thouless (BKT) transition describes the proliferation of topological defects in two dimensions and is relevant in wide range of classical and quantum systems, including the (1+1)D Bose-Hubbard model, where it is driven by quantum fluctuations. While the critical behavior of the BKT transition is well established, direct observation of its renormalization group (RG) flow in quantum models has remained elusive due to the complexity of the finite-size scaling near the transition. In this work, we numerically study the BKT transition in the Bose-Hubbard model using density matrix renormalization group (DMRG) with periodic boundary conditions. We use bipartite particle fluctuations to accurately pinpoint the critical interaction strength in the thermodynamic limit and demonstrate the BKT RG flow in the critical regime. The results highlight the utility of low energy effective field theories to describe microscopic models near quantum phase transitions and the importance of boundary conditions in capturing the correct scaling behavior near the critical point.