Quantum Phase Diagram Of The Two-Dimensional Transverse-Field Ising Model: Unconstrained Tree Tensor Network And Mapping Analysis

We investigate the ground-state phase diagram of the frustrated transverse-field Ising (TFI) model on the checkerboard lattice (CL), which consists of Neel, collinear, quantum paramagnet, and plaquette-valence bond solid (-VBS) phases. We implement a numerical simulation that is based on the recently developed unconstrained tree tensor network ansatz, which systematically improves the accuracy over the conventional methods as it exploits the internal gauge selections. At the highly frustrated region (J(2) = J(1)), we observe a second-order phase transition from the plaquette-VBS state to the paramagnet phase at the critical magnetic-field Gamma(c)= 0.28 with the associated critical exponents v = 1 and gamma similar or equal to 0.4, which are obtained within the finite-size scaling analysis on different lattice sizes N = 4 x 4, 6 x 6, 8 x 8. The stability of the plaquette-VBS phase at low magnetic fields is examined by a spin-spin correlation function, which verifies the presence of plaquette-VBS at J(2) = J(1) and rules out the existence of a Neel phase. In addition, our numerical results suggest that the transition from the Neel (for J(2) < J(1)) to the plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of the TFI on the CL to be the same model on the J(1)- J(2) square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point J(2) = J(1) on the CL is mapped to the emergent string-VBS phase on the SL at J(2) =0.5J(1).